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advancement of science ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor Dr. Jose C. Principe, for his great inspiration and encouragement throughout the course of my research. Not just that. He has really become a mentor and guide during pivotal times of my life, which I would have to say regretfully that I did not take full advantage of. One could ask for no more from such an advisor. I wish to thank the members of my committee, Dr. John Harris, Dr. Jianbo Gao, and Dr. Mingzhou Ding, for their valuable time and interest in serving on my supervisory committee, as well as their comments, which helped improve the quality of this dissertation. I am grateful for Dr. Andreas Keil's expertise on psychology as well as his support, which made our collaboration fruitful. I would like to thank my friends and colleagues at the Computational NeuroEngineering Laboratory. They have made my stay in Florida during the past four years an enjoyable experience. Last but not least, I wish to thank my parents, who raised me up. Without them, all is in vain. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... LIST OF TABLES .............. ...... ........................ ............... LIST OF FIGURES ................................... .. .... ..... ................. .9 A B S T R A C T ............ ................... ............................................................ 10 CHAPTER 1 INTRODUCTION ............... ................. ........... .............................. 12 1.1 B asic C concepts of the E R P ................................... ............ .....................................12 1.1.1 G generation of the ERP .............................. ............................................. 12 1.1.2 The ERP C om ponents.................... ....... ................................. ............... 13 1.2 Estim action of the ERP....................................................................... ............... 16 2 SINGLETRIAL ERP ESTIMATION ............ ..... ........ ................... 19 2.1 SingleTrial ERP Estimation Using SingleChannel Recording ..............................19 2.1.1 Tim eInvariant D igital Filtering ............................................. ............... 20 2.1.2 TimeVarying Wiener Filtering ............................... ...............20 2.1.3 A adaptive Filtering ......... ..................................... ...... .. .. ........ .... 21 2.1.4 K alm an Filtering ......... .......... ............... ........ ...... .......... .............. 22 2.1.5 Subspace Projection and Regularization......................................................22 2.1.6 Parametric Modeling.......................................................... 23 2.1.7 Other Methods Using SingleChannel Recording ..........................................24 2.2 SingleTrial ERP Estimation Using MultiChannel Recording..............................25 2.2.1 Generative EEG M odel ................................................................................ 25 2.2.2 What Is a Spatial Filter and What Can It Do? ....................................... 27 2.3 Review of Spatiotemporal Filtering Methods.................. ............ ................28 2.3.1 Principal Component Analysis (PCA)............................................................28 2.3.2 Independent Component Analysis (ICA)........................................................ 32 2.3.3 Spatiotemporal Filtering Methods for the Classification Problem ..................36 3 NEW SPATIOTEMPORAL FILTERING METHODOLOGY: BASICS.............................40 3.1 Spatial Filter as a Noise Canceller in the Spatial Domain ...........................................40 3.2 D term inistic A approach ......... ......... ............... .. ........................... ............... 42 3.2.1 Finding Peak Latency ................................................. ............................. 42 3.2.2. Finding Scalp Topography and Peak Amplitude................... .............. 44 3.3 Stochastic A approach ............................................ ................... ........ 46 3.4 Sim ulation Study ................. .... .... .. .................................................... .. ... 48 3.4.1. Gamma Function as a Template for ERP Component ........... ...............49 3.4.2. Generation of Simulated ERP Data ....................................... ............... 49 3.4.3 Case Study I: Comparison with Other Methods .........................................50 3.4.4 Case Study II: Effects of Mismatch............... .. ..... .................... 53 4 ENHANCEMENTS TO THE BASIC METHOD.............. ... .............. .............. 59 4.1 Iteratively R efined Tem plate ............................................... ............................ 59 4.2 R egularization .............................61................................................61 4.2.1 Constrained O ptim ization ........................................................ ............... 61 4.2.2 Unconstrained Optim ization ................................................... ................. 64 4.3 Robust Estim action: the CIM M etric ......................................................... ......... 65 4.4 Bayesian Formulations of the Topography Estimation .............................................68 4.4.1 M odel 1: A dditive N oise M odel ............................................. ............... 69 4.4.2 Model 2: Normalized Additive Noise Model .......................................... 70 4.4.3 M odel 3: O original M odel ............................................... ........................ 71 4.4.4 Comparison among the Three M odels.................................. ............... 71 4.4.5 O line E stim action ................. ........... .. .. .......... .............. .............. 72 4.5 Explicit Compensation for Temporal Overlap of Components ...................................72 5 APPLICATIONS TO COGNITIVE ERP DATA ...................................... ............... 90 5.1 O ddball Target D etection.................................................. ............................... 90 5.1.1 M materials and M ethods............................................................ .....................90 5.1.2 E stim action R results ......................... .... ..................... .... .. ........... 92 5 .1.3 D iscu ssio n s ................................................................9 8 5.2 H abituation Study .......................... .......... .. ......... .............. .. 99 5.2 .1 M materials and M ethods............................................................ .....................99 5.2.2 E stim action R results ......................... .... ..................... .... .. ........... 99 6 CONCLUSIONS AND FUTURE RESEARCH ........................................................110 6 .1 C o n c lu sio n s ................................................... ................. ................ 1 10 6.2 Future Research ............................ .................. .......... .. ...... ....... 12 APPENDIX A PROOF OF VALIDITY OF THE PEAK LATENCY ESTIMATION IN (310) ................114 B GAMMA FUNCTION AS AN APPROXIMATION FOR MACROSCOPIC ELECTRIC FIELD .................. .................. .............................. ...... ... .... 115 C DERIVATION OF THE UPDATE RULE FOR THE CONSTRAINED OPTIM IZA TION PR OBLEM ............................................. ...................... ............... 116 D DERIVATION OF THE UPDATE RULE FOR THE UNCONSTRAINED OPTIM IZA TION PR OBLEM ............................................. ...................... ............... 117 E MAP SOLUTION FOR THE ADDITIVE MODEL.................................. .................118 F NORMALIZED ADDITIVE NOISE MODEL 1: MAP SOLUTION..............................120 G NORMALIZED ADDITIVE NOISE MODEL 2: MAP SOLUTION..............................122 L IST O F R E F E R E N C E S ......... .................................... ......................................................... 123 B IO G R A PH IC A L SK E T C H ......... .............................................................. ........................... 135 LIST OF TABLES Table page 31 Latency estimation: mean and standard deviation ......... ................. ...................55 32 Amplitude estimation: mean and standard deviation............................................. 55 33 Scalp topography estimation: correlation coefficient.................... ..................55 34 Effects of mismatch I: SNR = 20dB............................ .... .................................56 35 Effects of mismatch II: SNR = 10dB ............................................... ................... 56 41 Latency estimation: mean and standard deviation...... ............................75 42 Amplitude estimation: mean and standard deviation.............................. ............... 75 43 Scalp topography estimation: correlation coefficient............. ..... ..................75 44 Estimation results for the iteratively refined template method............. ................76 45 Estimation with MCC for the mismatch case at SNR = OdB.................. ............ 76 46 Estimation with MCC for the mismatch case at SNR = 20dB ......................................76 47 Amplitude estimation for three Bayesian models.................................. ............... 77 48 Scalp topography estimation for three Bayesian models............................................77 51 Correlation statistics for the 4 subjects: Scalp topography estimation .........................102 52 Regression statistics for response time and estimated amplitude.................................. 102 LIST OF FIGURES Figure page 31 Gamma functions with different shapes and scales. .................................. ............... 57 32 Waveforms of synthetic and presumed ERP component................................................58 41 Mean and standard deviation of the estimated amplitude under different SNR c o n d itio n s.......................................................................... .. 7 8 42 The waveforms of the synthetic component, presumed template and refined template under 4 SNR conditions. ........................................ ........ .. ....... ...... 79 43 Waveforms of two overlapped components used in regularization ..............................80 44 Scalp topography of two overlapped ERP components used in regularization ................81 45 Amplitude and scalp topography estimation I with regularization (constrained optim ization) under 3 SNR conditions.. ......................... ............................................82 46 Amplitude and scalp topography estimation II with regularization (constrained optim ization) under 3 SN R conditions ........................................ ......................... 84 47 Amplitude and scalp topography estimation I with regularization (unconstrained optimization) under 3 SNR conditions. ........................................ ......................... 86 48 Amplitude and scalp topography estimation II with regularization (unconstrained optim ization) under 3 SNR conditions. ........................................ ......................... 88 51 Pictures used in the experiment as stimuli................................. ...............103 52 Cost function in (3.9) versus time lag for different regularization parameters for subject #2 ........... ... ...................... ....... .......... 104 53 Estimated pdf of time lags corresponding to local minima of the cost function in (3 9) using the Parzen windowing pdf estimator with a Gaussian kernel size of 4.2.. ........105 54 Scalp topographies for the four subjects.............. ......... .. ......... ........... ............... 106 55 Scatter plot of the response time versus the estimated amplitude for each single trial for the four subjects. ............................................... ..............107 56 Estimated scalp topography for mixed and habituation phase......................................108 57 Estimated amplitude for mixed and habituation phase .................................................109 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SPATIOTEMPORAL FILTERING METHODOLOGY FOR SINGLETRIAL EVENTRELATED POTENTIAL COMPONENT ESTIMATION By Ruijiang Li December 2008 Chair: Jose Principe Major: Electrical and Computer Engineering Eventrelated potential (ERP) is an important technique for the study of human cognitive function. In analyzing ERP, the fundamental problem is to extract the waveform specifically related to the brain's response to the stimulus from electroencephalograph (EEG) measurements that also contain the spontaneous EEG, which may be contaminated by artifacts. A major difficulty for this problem is the low (typically negative) signaltonoise ratio (SNR) in EEG data. The most widely used tool analyzing ERP has been to average EEG measurements over an ensemble of trials. Ensemble averaging is optimal in the least square sense provided that the ERP is a deterministic signal. However, over four decades of research have shown that the nature of ERP is a stochastic process. In particular, the latencies and the amplitudes of the ERP components can have random variation between repetitions of the stimulus. Under these circumstances, estimation of the ERP on a singletrial basis is desirable. Traditional singletrial estimation methods only consider the time course in a single channel of the EEG. With the advent of dense electrode EEG, a number of spatiotemporal filtering methods have been proposed for the singletrial estimation of ERP using multiple channels. In this work, we introduce a new spatiotemporal filtering method for the problem of singletrial ERP component estimation. The method relies on modeling of the ERP component local descriptors (latency and amplitude) and thus is tailored to extract faint signals in EEG. The model allows for both amplitude and latency variability in the actual ERP component. The extracted ERP component is constrained through a spatial filter to have minimal distance (with respect to some metric) in the temporal domain from a template ERP component. The spatial filter may be interpreted as a noise canceller in the spatial domain. Study with simulated data shows the effectiveness of the proposed method to signal to noise ratios down to 10 dB. The method is also tested in real ERP data from cognitive experiments where the ERP are known to change, and corroborate experimentally the expected behavior. CHAPTER 1 INTRODUCTION 1.1 Basic Concepts of the ERP Eventrelated potential (ERP) is an important and wellestablished technique for neuroscientists and psychologists to study human cognitive function. In this section, we briefly review some of the basic facts and concepts related to ERP generation and analysis (Coles et al. 1995). 1.1.1 Generation of the ERP When a pair of electrodes are attached to the surface of the human scalp and connected to a differential amplifier, the output of the amplifier reveals a pattern of voltage variation over time. This voltage variation is known as the electroencephalograph (EEG). The amplitude of the normal EEG varies between approximately +100 /V and most of the EEG frequency contents range between 0.5Hz and 40Hz. Here, we do not review the recording techniques of EEG (Ruchkin 1987). If we present a stimulus to a human subject while recording the EEG, we can define a period of time (an epoch or a trial) where some of the EEG components are timelocked to the stimulus. Within this epoch, there may be voltage changes that are specifically related to the brain's response to the stimulus. These voltage changes constitute the eventrelated potential, or ERP. Although it is not completely understood how the measurements at the scalp relate to the underlying brain activity, the following points appear to be clear and are generally accepted (Scherg and Picton 1991, Wood 1987). First, ERP recorded from the scalp represents net electrical fields associated with the activity of sizeable neuron populations. These neuron populations act as current sources whose electrical fields propagate to the entire scalp through volume conduction. Second, the individual neurons that compromise such a population must be synchronously active and have a certain geometric configuration to produce measurable potentials at the scalp. In particular, these neurons must be configured in such a way (usually in a parallel orientation) that their individual fields summate to yield a dipolar field. Therefore, the ERP recorded at the scalp is selective of the totality of the brain activity. This is advantageous in that the resultant measurements would otherwise be so complex as to be difficult or impossible to analyze. On the other hand, we should also be aware that there are certainly numerous functionally important neural processes that cannot be detected by the ERP technique. 1.1.2 The ERP Components The issue of ERP components has aroused much controversy among the ERP research community, particularly the question of the definition of an ERP component. Suppose for the moment that we have obtained the ERP using some method. A simple way to define a component is to focus on some feature of the resulting waveform (for instance, a peak or trough), and this feature becomes the component of interest. Some common features include the amplitude and latency parameters of a particular peak or trough. A major problem with the simple approach mentioned above is component overlap, both spatially and temporally. Since the brain is a conducting medium, activity generated in one spatial location may be propagated through the brain tissue and appear at other locations. Thus, the waveform we observe by measuring the voltage at the scalp may well be attributed to a variety of different sources in different spatial locations of the brain. One consequence of volume conduction is that there need be no direct correspondence between the timing of the distinctive features of an ERP waveform (peaks and troughs) and the temporal characteristics of the underlying neural systems. For instance, an ERP peak with a latency of 300 ms, might reflect the activity of a single neural generator maximally active at that time, or the combined activity of two (or more) neural generators, maximally active before and after 300 ms, but with fields summating to a maximum at that time. Due to these ambiguities surrounding the interpretation of peaks and troughs in ERP waveforms, other definitions for ERP components have been proposed. Naatanen and Picton (1987) adopted what might be called the physiological approach to component definition. They proposed that a defining characteristic of an ERP component is its anatomical source within the brain. According to this definition, to measure a particular ERP component, we must have a method of identifying the contributing sources. Dochin (1979, 1981) adopted what might be called the functional approach to ERP component definition, which is concerned more with the information processing operations with which a particular component is correlated. According to this definition, it is entirely possible for a component to be identified with a particular feature of the waveform that reflects the activity of multiple generators within the brain, so long as these generators constitute a functionally homogeneous system (Coles, et al.1995). Although the above physiological and psychological approaches to component definition seem to be counteractive, for many investigators it is more appropriate to combine both approaches. A classical approach to component definition, was proposed by Dochin et al. (1978). They argued that an ERP component should be defined by a combination of its polarity, its characteristic latency, its distribution across the scalp and its sensitivity to characteristic experimental manipulations. Notice that polarity and scalp distribution imply a consistency in physiological source, while latency and sensitivity imply a consistency in psychological function. ERP components can be broadly classified into two types: exogenous and endogenous components. Characteristics of the exogenous components (amplitude, latency and scalp projection) largely depend on the physical properties of sensory stimuli, such as their modality and intensity. On the other hand, endogenous components largely depend on the nature of the subject's interaction with the stimulus. These components vary as a function of such factors as attention, task relevance and the nature of the information processing required by the stimulus. The dichotomy of the exogenousendogenous distinction turned out to be an oversimplified version of the reality. Many early 'sensory' components have been shown to be modifiable by cognitive manipulations (e.g., attention) and many of the later 'cognitive' components have been shown to be influenced by the physical attributes of stimulus (e.g., modality). In what follows, we briefly discuss one particular wellknown ERP component, the P300. For a comprehensive review on other wellknown components, we refer to Coles, et al. 1995. The P300 is probably the most important and the most studied component of the ERP. It was first described in the 1960s by Sutton et al. (1965). The P300 is evoked by a task known as the oddball paradigm. During this task a series of one type of frequent stimuli is presented to the experimental subject. A different type of nonfrequent (target) stimulus is also presented. The task of the subject is to react to the presence of target stimulus by a given motor response, typically by pressing a button, or just by mental counting to the target stimuli. Virtually any sensory modality (auditory, visual, somatosensory, olfactory) can be used to elicit the P300 response. (Polich 1999). The shape and latency of the P300 differs with each modality. This indicates that the sources generating the P300 differ and depend on the stimulus modality (Johnson 1989). There are several theories on the neural processes underlying the origin of the P300. The most cited and most criticized theory was proposed by Donchin and Coles (1988a, b). According to their theory, the P300 reflects a process of context or memory updating by which the current model of the environment is modified as a function of incoming information. Several investigators (e.g., Johnson 1986) have pointed out that the P300 does not appear to be a unitary component, and instead may represent the activity of a widely distributed system which may be more or less coupled depending on the situation. More information about the underlying neural systems is required before a consensus is attained about the functional significance of this component. For a more recent review on the research of P300, we refer to Linden, 2005. 1.2 Estimation of the ERP The fundamental problem in the analysis of ERP is to extract the signals that are brain's specific response to the stimulus from the EEG measurements that also contain 'noise'. By noise, or the background EEG, we mean the electrical activities from heart, muscles and eye movements as well as the spontaneous brain activities that are not related to brain's response to the stimulus. A major difficulty with the extraction of ERP is that, in most cases, ERP signals are small (on the order of microvolts) relative to the background EEG (on the order of tens of microvolts) in which they are embedded. For this reason, it is necessary to employ signal processing techniques to estimate the ERP signals in the presence of noise. By far the most commonly used technique has been the averaging of the EEG measurements over an ensemble of timelocked epochs. This is optimal in the mean square sense, given the assumption that the ERP is a deterministic signal timelocked to the stimulus and the additive background EEG is zeromean and uncorrelated with the ERP. However, for over four decades it has been evident that the nature of ERP is more or less random. In particular, the amplitudes and latencies of the peaks in the ERP can have random variations between repetitions of the stimuli (Brazier 1964). In addition, the variations may be trendlike and the mean of the amplitudes and the latencies can change across the trials. Under these circumstances, the information regarding to these variations in ERP is lost through averaging. Furthermore, the average waveform may not, in fact, resemble the actual ERP waveform that is recorded in an individual trial. The resulting estimates for the ERP, therefore, may not correspond to the underlying neural processes and inference about the cognitive function may be misleading. Estimation of the ERP on a singletrial basis is desired for the situations when the peak amplitude and latency of a particular component change significantly across trials. A major difficulty with singletrial ERP estimation is again the very low signaltonoiseratio (SNR) in the singletrial EEG, typically lower than 10dB. Statistically speaking, the average ERP, or the sample mean, is an example of the use of the first order statistics, where only the first order moment of the population parameters is estimated. The next obvious improvement is to use the second order statistics, i.e., covariance analysis. The most common approach is to form an estimator (filter) with which the unwanted contribution of the background EEG can be filtered out. To find such an estimator, some models or assumptions are imposed on the ERP and background EEG concerning their respective second order statistics. The estimator that satisfies the minimum mean square criterion can then be derived. The performance of the estimator then largely depends on how realistic these assumptions are. For historical reasons, these traditional singletrial estimation methods only consider the time course of a single recording channel in the EEG. In some cases, simple ERP components, e.g., the brainstem auditory evoked potentials, can be adequately examined using a single channel. However, for most ERPs, simultaneous recording from multiple electrode locations is necessary to disentangle overlapping ERP components on the basis of their topographies, to recognize the contribution of artifactual potentials to the ERP waveform, and to measure different components in the ERP that may be optimally recorded at different scalp sites (Picton et al. 2000). Today, highdensity EEG can simultaneously record scalp potentials in up to 256 electrodes. This increased number of sensors and thus increased spatial resolution has created a need for signal processing methods that can simultaneously analyze the time series of multiple channels. Recently, various methods have been proposed for singletrial analysis that linearly combine the time series in multiple channels to generate a representation of the observed data that is easier to interpret (Chapman and McCrary 1995, Makeig et al. 1996, Parra et al. 2002). This linear projection combines the information from all the available sensors into a single channel with reduced interference from other neural sources and may provide a better estimate of the underlying neural activity than the EEG measurements in a single channel. The linear projection, in this sense, may be called a 'spatial filter' and these methods can be generally called spatiotemporal filtering methods. We will review these and other singletrial estimation methods in detail in Chapter 2. CHAPTER 2 SINGLETRIAL ERP ESTIMATION In this chapter, we review the existing methods for singletrial ERP estimation. The methods are broadly categorized into two classes: those based on singlechannel EEG recording and those based on multichannel EEG recording. Methods based on singlechannel recording rely solely on the modeling of temporal characteristics of the ERP and EEG, while methods based on multichannel recording investigate both the spatial and temporal characteristics of the ERP and EEG, and are termed with the general notion of spatiotemporal filtering methods. 2.1 SingleTrial ERP Estimation Using SingleChannel Recording For the estimation of ERP using a single channel, all the available information is contained in a single time series x(t): the EEG recording at a certain electrode. We assume that the measurements consist of two parts: the signal of interest (ERP) and additive noise (background EEG), denoted by s(t) and n(t), respectively. The observation model for the EEG can be written as: x(t) = s(t) + n(t) (21) In this review, the EEG measurements for a single trial x is a finitelength vector with elements sampled from the original continuoustime waveform. When the time series in (21) are interpreted as stochastic processes, x becomes a random vector. Its length equals the number of samples in one trial. In vector form, the observation model is: x = s + n (22) Here we do not attempt to give an exhaustive review on the topic of singletrial ERP estimation with a single channel. For other reviews on the general ERP estimation problem, we refer to Aunon et al. (1981), Ruchkin (1987), McGillem and Aunon (1987), Silva (1993), Karjalainen (1997). 2.1.1 TimeInvariant Digital Filtering Digital filtering is a good place to start for time series analysis. The simplest approach is to design digital filters that have a desired frequency response. Ruchkin and Glaser (1978) used simple moving average FIR filters to estimate ERP on a single trial basis. More complicated ones may be designed to estimate some particular component such as P300 (Farwell, et al. 1993). Wiener filtering may also be used to estimate singletrial ERP, provided that the power spectra of the ERP and background EEG can be estimated appropriately (Aunon and McGillem 1975, Cerutti, et al. 1987). The major problem with linear timeinvariant filtering is the fact that the ERP is typically a transient and smooth waveform with no periodicity. The spectrum for this kind of signal is not defined properly. Consequently, the spectra of the ERP and background EEG (if they are estimated) were usually found to overlap significantly (Krieger et al. 1995, Spreckelsen and Bromm 1988, Steeger et al. 1983). Thus the application of digital filters with constant frequency response is not expected to give desirable results in most cases. The effects of digital filtering are studied in (Ruchkin and Glaser 1978, Maccabee, et al. 1983, Nishida et al. 1993). 2.1.2 TimeVarying Wiener Filtering When the optimal Wiener solution is computed for singletrial EEG data, the filtering becomes timevarying with respect to each trial. In general, these methods necessitate some analytic model for the ERP. Yu and McGillem (1983) introduced what was called the time varying minimum mean square error filter. A crucial task for their method is to obtain a good estimate for the crosscovariance between the ERP and measurements. Under the assumption that the ERP and background EEG are uncorrelated, the crosscovariance becomes the auto covariance of the ERP signal itself. The ERP is parametrically modeled by a superposition of the components with random location and amplitude. The parameters for the ERP are then calculated from the Wiener solution on a single trial basis. 2.1.3 Adaptive Filtering The use of adaptive filtering for the analysis of singletrial ERP, particularly the use of the least mean square (LMS) algorithm (Widrow 1985), was extensively studied during the 1980s (Madhavan et al. 1984, 1986; Vila et al. 1986; Thakor 1987; Doncarli 1988). For these methods, the measurement x(t) is selected as the desired signal, and several choices are proposed for the input signal. Thakor (1987) is probably the one of the most cited works among the adaptive filtering methods for ERP estimation. From the principles of adaptive noise cancellation, Thakor proposed a novel way of choosing reference and primary inputs. Two sets of singletrial measurements x& (t), x, (t) (i, j being the trial index) serve as the reference and primary inputs, respectively. The idea is to estimate the primary input with a set of delayed version of the reference input on which some form of ensemble averaging is performed. The criticism of Thakor's work is summarized in (Madhavan 1988), where the author asserted that if the ERP signal is assumed to be identical across trials, the above approach does not provide any signalto noise ratio improvement and distorts the signal at frequencies where signal and noise power spectra overlap. Madhavan (1992) proposed a modified adaptive line enhancement method. In this method the prestimulus EEG data are adaptively modeled with an autoregressive (AR) model, which is then used to filter the poststimulus EEG data. The notion of 'modified' means that a non adaptive filter is used to process the poststimulus data. 2.1.4 Kalman Filtering AlNashi (1986) adopted the Kalman filtering approach for the ERP estimation problem. It is assumed that the ERP can be modeled as a deterministic signal with additive random noise. The additive noise is assumed to be an autoregressive moving average (ARMA) process and another ARMA model is used for the background EEG. The scalar Kalman filter is then used to predict the singletrial ERP. The basic assumption for AlNashi's approach is that the difference between the singletrial ERP and the ensemble average is a stationary process. This is not consistent with (Ciganek 1969), which found that the differences are usually larger in late components than in early components. Liberati et al. (1991) model the singletrial ERP as a timevarying AR process using the ensemble average data and model the background EEG as a stationary AR process using the pre stimulus data. The AR parameters are then used to create the state and observation equations with the ERP as the unknown states. The singletrial ERP is then estimated using the Kalman filter equations (Kalman 1960). 2.1.5 Subspace Projection and Regularization The subspace projection approach starts with the linear observation model: x= s + n = HO + n (23) where, 0 contains the parameters to be estimated and the ERP signal is constrained to lie in the subspace spanned by some basis vectors, namely the columns of the matrix H. If the ERP is assumed to consist of positive and negative humps, sampled Gaussian functions may be a good choice for the basis vectors instead of a generic basis (e.g., polynomials). An alternative is to choose the eigenvectors of the EEG data autocorrelation matrix that correspond to the first few largest eigenvalues. This is motivated by the fact that the eigenvectors constitute a basis set with the minimum number of basis vectors that are required to model the ERP, assuming that the ERP spans nearly the same subspace with the EEG measurements. The least square solution with this basis set is equivalent to the principal component regression approach (Lange 1996, Karjalainen 1997). The above two basis sets may be combined into a single criterion, with the Gaussian basis vectors modeling the ERP, and the subspace spanned by the eigenvectors representing the prior information about the problem. This leads to the subspace regularization method, which is closely related to the Bayesian mean square estimation (Karjalainen et al. 1999). The ERP may also be estimated recursively using Kalman filtering (Karjalainen et al. 1996). 2.1.6 Parametric Modeling There is one type of parametric models, which uses damped sinusoids as basis function for the modeling of singletrial ERP. The model for the ERP with additive noise can be written as: p x(t)= p, sin(ot)+ n(t) (24) i=1 Estimation of the parameters A, p,, 0, is a nonlinear problem, which can be solved with an approximation method called Prony's method (Marple 1987). Its use with generalized singular value decomposition was proposed by Hansson and Cedholt (1990) and Gansler and Hansson (1991). The Prony's method was utilized by Hansson et al. (1996) for the estimation of singletrial ERP and robust performance was achieved for EEG data with SNR>10dB. A more recent improvement, called piecewise Prony's method was proposed by Garoosi and Jansen (2000) to deal with nonstationary characteristics of the sinusoids. Another well studied tool that can be used for the analysis of singletrial ERP is the wavelet transform (Daubechies 1992). Wavelets provide a tiling of timefrequency space that gives a balance between time and frequency resolution and they can represent both smooth signals and singularities. This makes them suitable models for the analysis of transient and nonstationary signals like the ERP (Thakor 1993; Schiff et al. 1994; Samar 1995; Coifman 1996; Basar et al. 1999; Effern et al. 2000; Quian and Garcia 2003). The idea is based on a technique called 'wavelet shrinkage' or 'wavelet denoising', which can automatically select an appropriate subset of basis functions and the corresponding wavelet coefficients. This relies on the property that natural signals, such as images, neural activity, can be represented by a sparse code compromising only a few large wavelet coefficients. Gaussian noise, on the other hand, compromises a full set of wave coefficients whose size depends on the noise variance. By shrinking these noise coefficients to zero using a thresholding procedure (Donoho and Johnstone 1994), one can denoise data. However, the application of the wavelet method to singletrial ERP analysis requires some form of ensemble averaging in order to derive an 'optimal' wavelet basis set that is tuned to the ERP signal. Sometimes the appropriate selection of the ERP ensemble may be a difficult task due to the effects of internal and external experimental parameters (Effern et al. 2000). 2.1.7 Other Methods Using SingleChannel Recording Some methods exist that try to explicitly estimate the latencies of the singletrial ERP components. A simple approach is to use crosscorrelation of the signal with a template waveform and find the maximum point of the correlation (Gratton et al. 1989). Pham et al. (1987) applied a maximum likelihood (ML) method to estimate the latencies of ERP assuming a constant shape and amplitude. The ML method was extended in (Jaskowski and Verleger 1999) incorporating variable amplitude into consideration. Truccolo et al. (2003) developed a Bayesian inference framework for estimation of single trial multicomponent ERP termed differentially variable component analysis(dVCA). Each component is assumed to have a trialinvariant waveform with trialdependent amplitude scaling factors and latency shifts. A Maximum a Posteriori solution of this model is implemented via an iterative algorithm from which the component's waveform, singletrial amplitude scaling factors and latency shifts are estimated. The method works well for relatively lowfrequency and large amplitude eventrelated components. 2.2 SingleTrial ERP Estimation Using MultiChannel Recording The use of multichannel recording for the estimation of singletrial ERP gave rise to a number of spatiotemporal filtering methods. These methods assume, either explicitly or implicitly, a generative EEG model, which we will introduce in the next section. We then explain what is meant by a 'spatial' filter and illustrate what it can do for us in estimating ERP on a singletrial basis. A review on existing spatiotemporal filtering methods is furnished in the following section, where we concentrate on methods that are based on wellknown statistical principles. 2.2.1 Generative EEG Model We start with the 'neural generator' assumption of EEG data, i.e., neuron populations in cortical and subcortical brain tissues act as current sources (Caspers et al. 1980, Sams 1984). Within the EEG frequency range (below 100Hz), brain tissues can be assumed to be primarily a resistive medium (Reilly 1992). Thus, according to Ohm's law, the electrical potentials collected at each sensor (channel) as a result of volume conduction, is basically a linear combination of neural current sources (and nonneural artifacts). The linear generative model for EEG data can be written in matrix form: X=AS (25) or: N X = a,s, (26) where, N is the number of current sources. where, N is the number of current sources. We denote the singletrial EEG data with a D x T matrix X, with D channels and T samples; S is a Nx T matrix with each row s, representing the time course of the current density of the ith current source; A is an unknown Dx N matrix. Strictly speaking, the number of the neural current sources N is necessarily much larger than the number of channels D. It is usually assumed that the numbers of sources and sensors are equal for the purpose of convenience. The column vector a, of the matrix A represents the projection of the ith current source to each sensor at the scalp and is called the forward model associated with the source. This scalp projection is generally unknown and depends on the location and orientation of the dipolar current source as well as the conductivity distribution of the underlying brain tissues, skull, skin and electrodes (Parra et al. 2005). Thus, if the scalp projection can be estimated, it may provide us some further evidence to the neurophysiological significance of the corresponding estimated source. An equivalent way to write the generative EEG model (25) is to use the notation of time series: x(t)= A.s(t) (27) where, x(t)= [x,(t), ., xD (t)] is a column vector representing the EEG recordings in D channels; s(t) = [s (t), ..., s (t)] is a column vector representing the time course of the current density of the sources. A is the same matrix defined as before. There is some degeneracy in the model, i.e., the scaling factor of the current source s, and its corresponding scalp projection a,. In this case, either the current sources or the scalp projections are constrained to have unit power to avoid ambiguity. We wish to point out that there is no ambiguity whatsoever if we want to extract or eliminate from the EEG data the contribution of the ith current source, i.e., X' I asT 2.2.2 What Is a Spatial Filter and What Can It Do? To illustrate our point, we begin with a simple example. Suppose we measure the EEG from 3 electrodes where only 3 sources are present. Using the time series notation (the numbers are selected for illustration purposes): x, (t)=s, (t)+ 2 s t)+ s,(t) x () = 2.1(t) + S2 (t) + s (t) (28) x (t) = s,(t)+ s (t) + 2 s(t) We would like to recover each of the three sources using the EEG measurements from all the available sensors. Sine the measurements are linearly related to the current sources, we speculate that this could be done by linearly combining all the EEG measurements through a weight vector w: D y(t) = w'x(t)= x, (t) (29) i=1 In fact, if we select the weight vector in (29) as: w' = [1,3, 1], we will get, y(t) = x, (t) + 3. x (t) x (t) = 4s, (t) (210) which is exactly the first current source with a scaling factor. The other two sources can be recovered by using the weight vectors: [3, 1, 1], [1,, 1,3] respectively. Of course the above example is simplistic, because in reality, there are certainly numerous current sources simultaneously active in the brain and the number of sensors is usually up to a few hundred. We also do not know any of the elements of the matrix A in general. However, the following point should be clear: by combining the EEG measurements from multiple channels with a simple weight vector, we are able to recover (or estimate) many sources of interest that could not be recovered using a single channel. In principle, the rejection of the interference can be perfect, as shown in (210) if the coefficients are known. The weight vector w in (29), which operates on the EEG measurements in the sensor space, is called a spatialfilter. Just like a filter operating in the time domain, a spatial filter can have either lowpass or highpass characteristics in the spatial frequency domain. For instance, a spatial filter summing the measurements from a group of neighboring sensors have a lowpass characteristic; the use of a single channel recording corresponds to a highpass spatial filter with an 'impulse' response attenuating the data from all the other channels to zero. The selection of the spatial filter w is usually based on some constraints or desired characteristics of the output y(t). Different constraints will generally lead to different methods of extracting the outputs. Loosely speaking, maximum power of the outputs leads to principal component analysis (PCA); statistical independence among the outputs leads to independent component analysis (ICA); and maximum difference between the outputs leads to linear discriminant analysis (LDA). We will review these and other spatiotemporal filtering methods in the following section. 2.3 Review of Spatiotemporal Filtering Methods In this section, we provide a review on existing spatiotemporal filtering methods for single trial ERP analysis. Particularly, we focus on two popular and well established methods, namely, principal component analysis (PCA), independent component analysis (ICA). 2.3.1 Principal Component Analysis (PCA) From the early days of cognitive ERP research, principal component analysis (PCA) was already proposed as a linear, multivariate datareduction approach (Donchin, 1966). Since then, PCA has been one of most widely used tools among psychologists for ERP analysis (Glaser and Ruchkin, 1976; Donchin and Heffley, 1978; Mocks and Verleger 1991; Chapman and McCrary 1995; van Boxtel 1998). By identifying unique variance patterns in a given set of ERP data, PCA decomposes the variance structure of the observed data into a set of latent variables that ideally correspond to the individual ERP components. In the ERP research community, these latent variables are usually called 'factors' instead of 'components' to avoid confusion with the ERP components. Among the vast literatures on PCA applied to ERP analysis, one classical method using the PCAVarimax strategy, is particularly popular and is the primary analytic tool for many ERP researchers (Gaillard and Ritter 1983). The method treats the recorded potential at a given time of the EEG epoch as variables. The domain of the observations is taken to be the Cartesian product of the recording channels, experimental conditions, participants. Suppose we have T samples in a given EEG epoch, D recording channels, C experimental conditions and P participants. The data matrix for this method has a dimension of T x (D x C x P). This particular arrangement of the data matrix leads to the socalled temporal PCA approach, which gives orthogonal factors (eigenvectors of the covariance matrix). The PCA solution is then followed by the Varimax rotation (Kaiser, 1955). The Varimax rotation is an orthogonal rotation that aims to maximize the values that are large for a factor and minimize the values that are small (by maximizing the fourth power the factor). This corresponds to the 'maximum compactness' criterion, which will make the new factors have a small number of large values and a large number of zero (or small) values. This is reasonable for ERP estimation because for the most part, ERP components appear to be monophasic and compact in time. It is easy to see that the above PCAVarimax approach is a spatiotemporal filtering method. We denote X,p as the singletrial EEG data defined in (25) from cth experimental condition and pth participant. Then the covariance matrix is: 1 C= XT X (211) where, 1 PC X =P X, (212) PCp=l c=l Formally, PCA is equivalent to the singular value decomposition (SVD) of the data matrix defined in (212), which is the average EEG data matrix for all experimental conditions and all participants. Suppose we have the SVD of X as follows: X = UVT (213) where, U, V are orthogonal matrices of dimension D x D and T x T, and contain the left singular and rightsingular vectors, respectively. Y contains the singular values of the data matrix. Equation (213) can be equivalently written as: UTX= _VT (214) The Varimax rotation procedure simply adds another D x D orthogonal matrix R multiplied on the both sides of (214): RUTX = RV'T (215) The right side of (215) is a DxT matrix, whose rows can be seen as the factors extracted by the PCAVarimax approach. We define a Dx D matrix: T W1 W= : =RUT (216) T wD_ We further denote: y1 Y= : = RV'T (217) LYD Thus, we have: Y=WX (218) or: y, =wI.X, for i= 1,...,D (219) This is the familiar form for the spatial filter defined in (29), which is now written in matrix form. Clearly, the matrix W is an orthogonal matrix. So PCA finds a number of (D) outputs that are uncorrelated with the constraint that the spatial filters are orthogonal. On the other hand, the PCAVarimax method searches for outputs that are maximally compact in time while still constraining the spatial filters to be orthogonal. In the context of the generative EEG model, PCA basically assumes that there are equal number of sources and channels. If we multiply W 1 on both sides of (218), we get: X = W 'Y (220) Thus, the rows of the output matrix Y contain the time course of the current sources, while the columns of the matrix W1 constitute the scalp projections of the corresponding sources. This means that the scalp projections for the underlying current sources are orthogonal to each other, which is a highly dubious assumption. Due to the above problem, an oblique rotation like Promax (Hendrickson and White 1964) has been proposed as a postprocessing stage after Varimax, to relax the orthogonality constraint on the scalp projections. Studies with both simulated and real dataset have shown that temporal PCA with Promax extracted markedly more accurate ERP components (Dien 1998). An alternative approach to the popular temporal PCA is the spatial PCA (Duffy et al. 1990, Donchin 1997, Spencer, et al. 1999), which treats the recorded potential at a given channel as variables. The EEG data matrix is formed with channel as one dimension, and time by experimental condition by participant as the other dimension. The same rotation procedures follow as in temporal PCA. However, spatial PCA still assumes orthogonality of the scalp projections. Two other welldocumented problems for the PCA approach are the misallocation of variance (Wood and McCarthy 1984) and the issue of latency jitter. Dien (1998) using extensive simulations, has argued that spatial PCA as a complement to temporal PCA, together with parallel analysis (Horn, 1965) to identify noise factors, and oblique rotation to allow for correlated factors, can address these and other shortcomings of PCA. More recent developments include a combined spatial and temporal PCA approach that is successfully applied to real ERP data extracting known ERP components (Spencer et al. 1999, 2001). Dien et al. (2005) have presented a standard protocol to optimize the performance of PCA when it is applied to ERP datasets, recommending the use of covariance matrix over correlation matrix, and Promax rotation over Varimax rotation, etc. 2.3.2 Independent Component Analysis (ICA) ICA was originally proposed to solve the blind source separation (BSS) problem, to recover a number of source signals after they are linearly mixed and observed in a number of sensors, while assuming as little as possible about the mixing process and the individual sources (Comon 1994). The most basic ICA model assumes linear and instantaneous mixing, which means that the source signals arrive at the sensors without time delay and are mixed in the sensors linearly with other source signals. This basic ICA model naturally fits into the generative EEG model in (25), which we repeat here, assuming that the sources and measurements are random vectors: x = As (221) In the ICA literature, the observation x is called 'mixtures', and the unknown matrix A is called 'mixing matrix'. Usually it is assumed that the number of sources is equal to the number of sensors. In this case, the mixing matrix becomes a square matrix. The key assumption used in ICA to solve the BSS problem is that the time courses of the sources are as statistically independent as possible. Statistical independence means that the joint probability density function (pdf) of the outputs factorizes. For the linear instantaneous BSS problem, the solution is in the form of a square 'demixing' matrix W, specifying spatial filters that linearly invert the mixing process. If the mixing matrix is invertible, the outputs should be identical to the original source signals, except for scaling and permutation indeterminacies (Comon 1994): y = Wx (222) There are a multitude of algorithms that have been proposed to solve the basic ICA problem, among which, Infomax (Bell and Sejnowski 1995, Lee et al. 1999), FastICA (Hyvarinen and Oja, 1997), JADE (Cardoso, 1999), SOBI (Belouchrani et al. 1997) are probably the most widely used. Some algorithms are based on the canonical informationtheoretic contrast function for ICA, i.e., mutual information, or its approximations (Infomax, FastICA, etc.); others utilize higherorder statistics of the data (e.g., forthorder cumulant) to perform source separation (JADE); still others make use of the difference in the temporal spectra of the source signals (SOBI). For a more detailed review on ICA and its applications to BSS problems, we refer to the following: Cardoso (1998); Hyvarinen et al. (2001); Roberts and Everson (2001); Cichocki and Amari (2002); James and Hesse (2004); Choi et al. (2005). Review papers comparing different ICA algorithms and their relationships are also available: Hyvarinen (1999); Lee et al. (2000). We notice that while PCA requires the spatial filters to be orthogonal, here in the case of ICA, there is no more constraint on the spatial filters (or the demixing matrix W). On the other hand, while PCA only uses secondorder statistics (the covariance matrix), to decorrelate outputs, ICA imposes a much stronger condition, i.e., independence on the outputs. The fact that ICA tries to factorize the joint pdf of the outputs implies that all the higherorder statistics (HOS) are taken into consideration by ICA. This means that for nonGaussian data, the structures contained in HOS (e.g., kurtosis), while totally ignored by PCA, may be captured by ICA. Since many natural signals are nonGaussian distributed (e.g., speech signals usually follow a Laplacian distribution), ICA may be more suitable for this and other applications than PCA. Since Makeig et al. (1996) published their seminal paper on the application of ICA to EEG data, there have been numerous studies during the last decade dedicated to this research topic (Makeig et al. 1997, 1999, 2002, 2004; Vigario et al. 1998, 2000; Jung et al. 1999, 2000, 2001; Delorme et al. 2002, 2003, 2007; Debener et al. 2005). Until now, ICA and its variants still remains a powerful tool for the analysis of EEG and ERP data. The application of ICA to the study of EEG data requires that the following conditions be satisfied: (1), statistical independence of all the underlying neural source signals; (2), their linear instantaneous mixing at the sensors; (3), the stationarity of the mixing process. Since most of the energy in EEG data lies below 100Hz, the quasistatic approximation of Maxwell equations holds. So there is (virtually) no propagation delay of the electrical potentials from the neuronal sources to the sensors through volume conduction. Thus the assumption of instantaneous mixing is valid. The linearity of the mixing follows from the Maxwell equations as well. The stationarity of the mixing process corresponds to a constant mixing matrix. For the dipole source model, this means that the dipolar neuronal sources should have fixed locations and orientations. Although there is no reason to believe that these neuronal sources are spatially fixed over time, for those that are involved in a specific information processing task and therefore are of interest to ERP researchers, they should at least have a relatively stable configuration or a stable scalp projection, which is congruent with the definition of ERP components as proposed by Fabiani et al. (1987). We have seen that conditions (2) and (3) are approximately valid for EEG data. The most debatable and perhaps perplexing condition is the first one: statistical independence of all the underlying neural source signals. The independence criterion applies solely to the amplitudes of the source signals, and does not correspond to any consideration of the morphology or physiology of the neural structures. However, the different nature of the sources originated from completely different mechanisms often yields signals that appear to be statistically independent. Particularly, analysis of the distributions of artifacts such as the cardiac cycle, ocular activity has shown the statistical independence assumption approximately holds (Vigario 2000). Although ICA continues to be a useful tool for EEG and ERP analysis, there are also some limitations to it. First, ICA can decompose up to (or at most) D sources from data recorded at D scalp electrodes (D may be ranged from several dozen to a few hundred). On one hand, the researcher has to analyze the extracted D components one by one (including the time course and scalp projection), which is laborious when is D large and the results are subject to interpretation. If he/she chooses to analyze only a part of the all the components, the subsequent analysis is correlated with the retention criterion. (Note that PCA also has this problem). On the other hand, the effective number of statistically independent signals contributing to scalp EEG is almost certainly much larger than the number of electrodes D. Using simulated EEG data, Makeig et al. (2000) has found that given a large number of sources with a limited number of available channels, ICA algorithm can accurately identify a few relatively large sources but fails to reliably extract smaller and briefly active sources. This suggests that ICA decomposition in high dimensional space is an illposed problem. Second, the assumption of statistical independence used by ICA is violated when the training dataset is too small or separate topographically distinguishable phenomena nearly always cooccur in the data (Li and Principe, 2006). In the latter case, simulations show that ICA may derive a single component accounting for the cooccurring phenomena, along with additional components accounting for their separate activities (Makeig et al. 2000). These limitations imply that the results obtained with ICA must be validated by researchers using behavioral and/or physiological evidence before their functional significance can be correctly interpreted. Current research on applications of ICA is focused on incorporating domainspecific knowledge into the ICA framework. Recently there has been work on combining ICA with the Bayesian approach (Tsai, et al. 2006) or with the regularization technique (Hesse and James, 2006). 2.3.3 Spatiotemporal Filtering Methods for the Classification Problem It is worthwhile to point out a related but different approach, which is the (supervised) singletrial EEG classification problem. It is generally less difficult than the (unsupervised) singletrial estimation problem in the sense that the availability of label information for classification facilitates learning. Many spatiotemporal filtering methods have been proposed for the singletrial EEG classification problem, which include, but not limited to, common spatial patterns (CSP) (Ramoser et al., 2000), common spatiospectral patterns (CSSP) (Lemm et al., 2005), linear discriminant analysis (Parra et al., 2002), bilinear discriminant component analysis (Dyrholm et al., 2007). The common spatial patterns method was initially proposed by Koles et al. (1990) to classify normal versus abnormal EEG (Koles et al. 1994). The method has been used for singletrial EEG classification in braincomputer interface (BCI) systems (MullerGerking et al. 1999; Ramoser et al. 2000). Given the singletrial EEG data for two different experimental conditions, the CSP method decomposes the EEG data into spatial patterns, which maximize the difference between the two conditions. The spatial filters are designed such that the variances of the outputs are optimal (in the leastsquare sense) for the discrimination of the two conditions. This is realized by simultaneously diagonalizing the two covariance matrices of the EEG associated with the two experimental conditions. The two resulting diagonal matrices (containing the eigenvalues for the two covariance matrices) add up to the identity matrix. Thus, the spatial filters that give the (n, an integer) largest variance in their outputs (associated with the largest eigenvalues) for one condition, will accordingly give the (n) smallest variance in their outputs for the other condition; and vice versa. It is along these directions that the largest differences between the two conditions lie. In (MillerGerking et al. 1999), the CSPs are called the source distribution matrix (equivalent to the mixing matrix in ICA), and the spatially filtered outputs are claimed to be the source signals, although the EEG data were temporally bandpass filtered between 830Hz prior to analysis. Para et al. (2005) showed that the simultaneous diagonalization of the covariance matrices is equivalent to the generalized eigenvalue decomposition, and according to Parra and Sajda (2003), the CSP method is in fact estimating the independent components of the temporally filtered EEG data. The original CSP method does not take into account the temporal information of the filtered EEG data. In light of this, Lemm et al. (2005) proposed an algorithm called common spatio spectral pattern (CSSP), which utilized the method of delay embedding and extended the CSP algorithm to the state space (with only one tapdelay). Dornhege et al. (2006) further improved the CSSP algorithm by optimizing an arbitrary finiteimpulse response (FIR) filter within the CSP framework. The overfitting of the spectral filter is controlled by a regularizing sparsity constraint. The CSP method and its variants all use the relevant oscillatory brain activity for EEG classification. Sometimes it is more appropriate to use coherent evoked potentials (of lowpass nature) instead. Para et al. (2002) proposed a spatiotemporal filtering method using conventional linear discrimination to compute the optimal spatial filters for singletrial detection in EEG. Specifically, the search for the optimal spatial filter given the singletrial EEG data as in (27), is based on constraining the output y(t) to be maximally discriminating between two different experimental conditions. The optimality criterion is restricted to a prespecified time interval, i.e., the time corresponding to a number of samples prior to an explicit button push. After finding the optimal spatial filter using conventional logistic regression, the output is averaged within that period of time to obtain a more robust feature. The detection performance is then evaluated using receiver operating characteristic (ROC) analysis on a singletrial basis. Unlike other conventional methods such as ICA, where the scalp projections are given directly by inverting the demixing matrix containing all the spatial filters, here since there is only one spatial filter and one output, other techniques have to be sought in order to estimate the scalp projection associated with that output. Parra et al. did this by projecting the EEG data to the discriminating output y(t) assuming that the output is uncorrelated with all other brain sources, and found that the discrimination model captured information directly related to the underlying cortical activity. The method was improved in (Luo and Sajda, 2006), where the prespecified time interval is allowed to be different and optimized for each EEG channel. This effectively defines a discrimination trajectory in the EEG sensor space. CHAPTER 3 NEW SPATIOTEMPORAL FILTERING METHODOLOGY: BASICS In this chapter, we propose a new spatiotemporal filtering method for singletrial ERP estimation. The method relies on modeling of the ERP component descriptors and thus is tailored to extract small signals in EEG. The model allows for both amplitude and latency variability in the actual ERP component. We constrain through a spatial filter w the extracted ERP component to have minimal distance (with respect to some metric) in the temporal domain from a presumed ERP component. Note that we do not constrain the entire ERPs, but instead a single ERP component. We maintain the point in the next section that the spatial filter may be interpreted as a noise canceller in the spatial domain. We then introduce two approaches for the proposed method: the deterministic approach and the stochastic approach. 3.1 Spatial Filter as a Noise Canceller in the Spatial Domain Since the method deals with one ERP component at a time, we wish to distinguish between 'signal' and 'noise' instead of using the general term 'sources'. To accommodate this distinction, we rewrite the generative EEG model in (26) as follows: N X=aST + Zb,n f (31) where, s is the time course of the ERP component to be extracted, n, denotes noise in general. The distinction between 'signal' and 'noise' is somewhat arbitrary, e.g., when P300 is the signal of interest, N100 will become noise in the model. Note that for notational convenience, we have assumed the effective number of sources to be N+1. The EEG model in (31) can in turn be rewritten as: N X = ca.s+so ,b .no, (32) i=1 where, ao, so, bo,, no, are the normalized versions (with respective to some norm, e.g., 12) of their counterparts in (31) and o, > o, ,. The scalars o,,o, may be seen respectively as the overall contributions of the signal and noise to the singletrial EEG data. In the case of independent noise, we may define the SNR for the singletrial EEG data (note that it is different from SNR in a single channel.) as: SNR = 20log J 2 (33) The vectors ao, b', represent the scalp topography of the corresponding signal and noise. For a meaningful ERP component, it must have a stable scalp topography a,. Thus, we may assume that a, is fixed for all trials. We also assume that the waveform of a particular ERP component so dimensionlesss) remains the same for all trials, although its amplitude o, may change across trials. Next, we claim the following lemma, which is basically a direct consequence of the linear generative EEG model in (31). Lemma: There exists a spatial filter w, that will completely reject the interference from the first D 1 largest noise in the output when it is applied to the singletrial EEG data, if, det[a b, ... bD1] (34) Further, the extracted ERP component will approach the actual ERP component if, , >> O (35) Such a spatial filter w can be found by taking the first row of the inverse of the matrix in (3 4). Note that (34) implies that, angle(a,b,) >,i=1,...,D1 and, angle(b,,b,) 0,1< i which means that at least the scalp topography of the source and the first D 1 noise should not be the same or very similar to each other from a computational perspective. Here, we wish to stress the point that the spatial filter specified in the above lemma may be interpreted as a noise canceller in the spatial domain. It may or may not be the optimal spatial filter for enhancing the SNR in the extracted component. In addition, the SNR enhancement due to the spatial filter increases monotonically with the number of channels (electrodes) if the EEG data were measured in those channels. This means that the more channels we use to record the EEG, the higher SNR we will get (theoretically) in the extracted component. Note that the lemma is an existence statement, it does not tell us how to find such a 'optimal' spatial filter. This will be the subject of the next two sections. 3.2 Deterministic Approach Most ERP components are monophasic waveforms with compact support in time. The morphology of the waveform can be considered relatively fixed due to the common cytoarchitecture of the neocortex and similar neuron populations, but may vary in both its peak latency and amplitude from trial to trial. Based on this, we assume that a particular ERP component can be modeled by a fixed dimensionless template (e.g. no physical unit), in the temporal domain, denoted by so(l) (where / is the unknown peak latency), multiplied by an unknown and possibly variable amplitude o across trials. We attempt to estimate the variable peak latency and amplitude on a singletrial basis. 3.2.1 Finding Peak Latency Since we do not know the peak latency in a single trial, we denote the template as so(r) with a variable time lag parameter r and slide it one lag a time to search for the peak latency. The search for the optimal filter w could be realized by minimizing some distance measure between the spatially filtered output w' X and the assumed waveform s (r) for the particular ERP component. We propose the following cost function based on secondorder statistics (SOS): min W Xso(r)' 2 (37) W 2 Note that the above optimization is with respect to w only, with r fixed. The optimal solution for w is given by: w(r)= (XXT) .X.so(r) (38) Obviously, the optimal spatial filter w depends on which ERP component is to be extracted, and also is a function of the variable time lag r. From (37) and (38), we obtain the cost solely as a function of the time lag r : J(r)= s(r.X' (XX') X I (39) The peak latency of the ERP component can be set as the time lag where the local minimum of J(r) occurs within the meaningful range of peak latencies (T, ) for that particular component (provided that its waveform is monophasic) i.e., = argmin J(r) (310) IGTs The estimated ERP component is then (this need not be normalized): y,(/)= X' .w(/) (311) It can be shown that under certain conditions, the solution in (310) is identical to the true peak latency of the ERP component (Appendix A). Exact match between the modeled and actual ERP component is not a necessary condition for the solution in (310) to be correct. For instance, it is easy to show that when both components are symmetric waveforms, then (310) also gives the correct latency. 3.2.2. Finding Scalp Topography and Peak Amplitude In the following, we make the index for trial number k explicit. Denote the estimated ERP component for k th trial by (the peak latency depends on the trial number): Yk () =kYk (4) (312) We can absorb the scalar ok into a variable scalp topography: ak = kao (313) In order to estimate the unknown scalp topography and amplitude of the ERP component, we assume that the ERP component is uncorrelated with all the noise sources. Replacing the dimensionless ERP component in (32) by its estimate in (311) and multiplying yk (l) on both sides of (32), we will get an estimate for the singletrial scalp topography (the cross terms T n,, yk k() vanish because of the uncorrelatedness assumption): a Xk'sk(k) ka =(4 (314) Yk (skT)Ysk (1k) Taking the normalized version (note that ak is in Volts) we have, ak (315) Ideally, the normalized aok should be the same as the dimensionless scalp topography ao. However, in lowSNR EEG data, the above estimation in (314) is very poor, due to the interference from background activity in the finitesample data. To estimate the scalp topography for a stable ERP component, we propose the following cost function: K min aa k 2 (316) ao k= This corresponds to a maximum likelihood (ML) estimator for a, under the assumption that each entry of the normalized singletrial scalp topography is an independent identically distributed (i.i.d.) Gaussian random variable. The optimal solution for (316) is a simple average of the estimated singletrial scalp topographies for all K trials. Taking the normalized version, the following estimate for a, is obtained: 1K K (317) no = K n'ok fiK [ ok (317) k= k=1 2 Notice that (317) is in fact a weighted average of the estimates in (314). We also point out that (317) is different from summing up directly (314) for all trials since the peak latency parameter is involved and it changes from trial to trial. In the ideal case, the two vectors a, and ak are identical except for a scaling factor, which is exactly the unknown amplitude ak associated with the ERP component in the k th trial. Replacing their respective estimates in (314), we can find 0k using again a SOS criterion: min aik ka (318) Simple calculation leads to the following estimate for the amplitude: =k oT k (319) This estimate involves information from all the available channels. In order to eliminate the indeterminacies of the linear generative EEG model, we set the peak amplitude as the maximum of the estimated ERP component in (312), i.e., k ybk (k). (All the amplitudes in the rest of the paper refer to this quantity). Accordingly, the contribution of the ERP component to the EEG data may be computed by: X, = r yg k) (320) These estimates for the scalp topography, peak latencies and amplitudes of ERP component can be used to analyze its psychological significance on a singletrial basis. Note that we do not directly compute the amplitude from the estimated component, nor do we measure it in any single channel. Instead, the amplitude is computed in (319) indirectly through an inner product of two scalp topographies, which involves information from all available channels. These estimates for the scalp topography, peak latencies and amplitudes of ERP component can then be used to analyze their psychological significance on a singletrial basis. We wish to point out that in contrast to all the spatiotemporal methods mentioned before, where only one, representative spatial filter (or matrix consisted of spatial filters), is computed given all the EEG data, here the optimal spatial filter is computed on a singletrial basis, i.e., given a singletrial EEG data matrix, we can get a spatial filter, as in (38). The reason we did it in this way is that, we believe that in theory the optimal spatial filter that is designed to extract small signals should change from trial to trial. Note in (32) the 'noise' sources are sorted in decreasing order of their power. It is likely that the noisy sources that have relatively large power change drastically across trials. In effect, this will change the configuration in (32) and accordingly, the optimal spatial filter will also change. 3.3 Stochastic Approach In the deterministic approach, the ERP component is considered to be a deterministic signal except for a random latency and amplitude across trials. It does not take into account the intrinsic error in the modeling of the ERP component itself. Here we propose a stochastic approach for the spatiotemporal filtering method. The idea is to constrain the extracted ERP component to be close to the presumed component with respect to some statistic. We still use the generative EEG model as in (31), but here both the signal and the noise are interpreted as stochastic processes. The approach utilizes the following observation model to search for the optimal spatial filter: XTw = + v (321) where, s is the actual ERP component and v is the observation noise appearing in the spatially filtered EEG data. They are both random vectors with each entry as a sample within a certain time interval from the corresponding stochastic processes. We assume that the ERP component is generated from the following additive model: s = g+u (322) where, g is a fixed signal with a certain morphology serving as the template for the ERP component, and u represents the model uncertainties of the ERP component and is assumed to be independent of the observation noise v. Given the above model in (321) and (322), the optimal spatial filter may be found by maximizing the loglikelihood of the filtered EEG data y = X'w : max L(w) = log p(y I g) (323) or, max L(w) = log p(u + v) (324) The loglikelihood function has a simple form under the assumption that u and v are zero mean Gaussian stochastic processes with covariance matrices C, CD respectively, and they are independent of each other, since u + v is nothing but another Gaussian stochastic process with covariance matrix CM + CD. In this case, maximizing the log likelihood of the (transformed) observed data given the observation model and the template of the ERP component yields the following solution for the optimal spatial filter: w, = argmin ('w g)1 (C + CD 1 (X'w g) (325) w This is a quadratic form of w, so it has a closedfrom solution: w = (X1X) .X lg (326) where, Y = CM + C (327) The estimate for the ERP component is then: Yo = X wo (328) We point out that (325) suggests that in the Gaussian assumption, observational uncertainties and model uncertainties simply combine by addition of their respective covariance matrices. We also note that if CM and CD are both chosen as identity matrix (i.e., incorporating the least amount of a priori information into the model), then the solution in (326) essentially reduces to (38) in the deterministic approach. The optimality of the spatial filter will depend on how we choose the two covariance matrices. The estimation of the scalp topography and amplitude follow the same procedures as described in (315) (320). 3.4 Simulation Study We present in this section a simulation study with synthetic data and real EEG data recorded from subjects during a passive pictureviewing experiment. The goal of the simulation study is to evaluate the latency and amplitude precision of synthetically generated transients immersed in real EEG background with different SNRs and waveform mismatching conditions. 3.4.1. Gamma Function as a Template for ERP Component Lange et al. (1996) have used a Gaussian function as the template for an ERP component. Here, we prefer the Gamma function for the shape of the synthetic ERP component because this is a very flexible function for waveform modeling and has been used extensively in neurophysiological modeling (Koch et al., 1983; Patterson et al., 1992). Freeman(1975) argued that the macroscopic EEG electrical field is created from spike trains by a nonlinear generator with a secondorder linear component with real poles. According to this model, the impulse response of the system is a monophasic waveform with a single mode, where the rising time depends on the relative magnitude of the two real poles (Appendix B). This may be approximated by a Gamma function, which is expressed by: g(t) = c tk exp(t / 0), t >0 (329) where, k > 0 is a shape parameter, 0 > 0 is a scale parameter and c is a normalizing constant. The Gamma function is a monophasic waveform with the mode at t = (k 1)0, (k > 1). It has a short rise time and a longer tail for small k, and approximates a symmetric waveform for large k. This makes it a good candidate for modeling both early and late ERP components that tend to be symmetric in the early components and have longer tails in the late components. Figure 31 shows four Gamma functions with different shape and scale parameters. 3.4.2. Generation of Simulated ERP Data EEG data were recorded from subjects during a passive pictureviewing experiment, consisting of 12 alternating phases: the habituation phase and mixed phase. Each phase has 30 trials. During the 30 trials of the habituation phase, the same picture was repeatedly presented 30 times. During the mixed phase, the 30 pictures are all different. Each trial lasts 1600 ms, and there is 600 ms prestimulus, and 1000 ms poststimulus. The scalp electrodes were placed according to the 128channel Geodesic Sensor Nets standards. All 128 channels were referred to channel Cz and were digitally sampled for analysis at 250Hz. A bandpass filter between 0.01Hz and 40Hz was applied to all channels, which were then converted to average reference. Ocular artifacts (eye movement) were corrected with EOG recordings. The scalp topography of the synthetic ERP component is chosen as the normalized P300 scalp topography from another study (Li, et al., 2006). The simulation data were created by taking the superposition of the 600ms (150 samples) prestimulus data from 120 trials as the background EEG data and the scalp projected Gamma waveform as a proxy for the ERP component. The SNR levels given the background EEG data can be easily adjusted by modifying the normalizing constant c in (17). We define the SNR given the singletrial EEG data as: SNR= 20logT (330) Note that since the 'actual' ERP component is a nonstationary signal, the magnitude o, is taken be its peak amplitude. This is different from the conventional definition of SNR. The SNR levels given the background EEG data can be easily adjusted by modifying the normalizing constant c in (329). 3.4.3 Case Study I: Comparison with Other Methods We will test the performance of the proposed spatiotemporal filtering method at varying SNR levels (from 20dB to 12dB), where we have access to the 'actual' (synthetic) ERP component. Two scenarios will be investigated: one where there is an exact match between the synthetic and the ERP component template and the other where there is a mismatch between the two components. For the case of exact match, we use the parameters k = 3, 0 = 13 for both ERP components. For the mismatch case, the synthetic ERP component remains the same, but the template has a different waveform with parameters k = 5, 0 = 6. Figure 32(a) shows the waveforms of the two components for the mismatch case. We fixed the peak latency of the synthetic component at 200ms for all the 120 trials. For Woody's filter, we selected channel Pz for analysis, and use the initial ensemble average as the template, avoiding the iterative update on the template (since the true latency is fixed, this is the bestcase scenario for Woody's filter). We search around the true latency within 100ms for maximum correlation. We estimate the peak amplitude by taking the average of the peak value and its two adjacent values (corresponding to a noncausal low pass filter with cutoff frequency of 12Hz). For spatial PCA, we select the eigenvector which has the maximal correlation with the P300 scalp topography (Note this is an ideal case for PCA, since in reality we do not know exactly the true scalp topography, nor the exact time course). The simulation results are summarized in Table 31, 32 and 33, which show the estimation mean and standard deviation for the estimated latency and the ratio between estimated and true peak amplitude, as well as the correlation coefficient between estimated and true scalp topography. Since PCA does not give an explicit estimate for latency and amplitude, we will omit the latency estimation and only compute the amplitude ratio between the estimated ERP component and the synthetic ERP component. First, we note that the singletrial estimation of the peak latency is very stable in the case of exact match. Notably, for EEG data with SNR higher than 4dB, the method estimates the latency correctly for all the trials. Second, the amplitude estimate for the exact case is also stable but is more variable than the latency estimation. The mean amplitude approaches to one and the standard deviation decreases to zero as the SNR increases. We may say that in the case of exact match between the model and the component, the estimator for the amplitude is asymptotically unbiased and asymptotically consistent with increasing SNR. The mismatch between the model and the generated component effectively introduces a bias in the estimation of the latency for realistic SNRs. From Table 31, we can see that the mean latency approaches to 188ms, yielding a bias of around 3 samples and the standard deviation is around 9ms (around 2 samples). However, the mean does converge to its true value (200ms) and the variance does approach to zero with increasing SNR, although very slowly. For instance, for SNR as high as 40dB and 60dB, the estimated latency has a meanstd statistic of 193 4ms and 200 + 0.7ms, respectively. Therefore, empirically we can see that the estimation for latency under the mismatch case is also asymptotically unbiased and asymptotically consistent with increasing SNR. The mismatch of components introduced a bias in the estimation of the amplitude for realistic SNRs, which is partly due to the difference in the waveforms of the synthetic component and template. The estimated amplitude has a statistic of 1.0458 0.0075 and 1.0015 0.0002 at a SNR of 40dB and 60dB, respectively. Thus in the case of mismatch, the estimator for the amplitude is also asymptotically unbiased and asymptotically consistent with respect to SNR, although the convergence is much slower than the exact match case. Of course, the estimation variance does increase notably as SNR decreases. But as evident from Table 42, our method at 12dB still gives a estimation variance smaller than Woody's method at OdB. Table 31 and 32 clearly show the advantage of using spatial information, in contrast to the Woody filter based on singlechannel analysis. Specifically, the estimation variance of Woody filter for both latency and amplitude are much larger for realistic (negative) SNR conditions. PCA overestimates the amplitude of the ERP component for low SNR data. In contrast to our method, PCA gives a statistically significant bias even at OdB from the baseline at 12dB (p value less than 0.0001). This means that varying SNR (below OdB) imposes a serious problem on the application of PCA in low SNR conditions. Finally, the simulated mismatch of components affects the estimation of the scalp topography negligibly for the proposed method when SNR is higher than 10dB. In fact, the estimation for scalp topography with mismatch with our method at 20 dB is comparable to PCA at 4 dB. 3.4.4 Case Study II: Effects of Mismatch The second simulation concerns the effects of the mismatch on the estimation, specifically mismatch in the spread parameter which is the most important. We use the same synthetic component as before and vary the spread parameter with K fixed. Fig. 32(b) shows the waveforms of the synthetic and 3 of the templates, including a Gaussian with a spread of 20. The results are summarized in Table 34 and 35, for SNR = 20dB and 10dB, respectively. We have included a new quantification for the amplitude estimation: coefficient of variation (CV), which is defined as the ratio of the standard deviation to the mean of a random variable. It is used as a measure of dispersion of the estimated amplitude (since its true mean is fixed at 1). From Table 34 and 35, we can see that at the same SNR level, the mean and variance of the estimated amplitude systematically change with respect to the spread parameter, i.e., larger spread parameter gives smaller amplitude. The degree of variability in the amplitude estimation (measured by coefficients of variation) for mismatch cases exceeds the exact match case by less than 10% except at 10dB for a spread parameter of 7. In some cases, CV is even smaller than the exact match case, which gives a better estimate, but only in terms of the amplitude. This is important because although the estimated amplitudes differ in the mismatch case, as long as we use the same template, these amplitudes on average will always be magnified or attenuated by a constant factor at a certain SNR level. Intuitively, this means that given a fixed template and varying SNR (>20dB), the dominant source of variability in amplitude estimation mainly comes from the estimation variance (not bias) and this variability (measured by CV) is well bounded from above for a range of spread parameters. Therefore, these amplitude estimates may still be effectively compared across experimental conditions as long as the same (meaningful) template is used and the SNR of the ERP data does not fall below 20dB. However, the mismatch clearly introduces a bias in the latency estimation, which may be as large as 50ms in absolute terms. This may or may not be significant depending on the applications. We can also see from the two tables that choosing a higher spread parameter will lead to a slightly better estimation for the scalp topography. But of course, this comes at the cost of much worse latency estimation. Table 31. Latency estimation: mean and standard deviation SNR (dB) Woody Exact match 191 60 190 + 60 190 + 61 191 + 63 195 + 60 195 + 54 199 + 45 199 + 32 200 + 15 202 10 201 +7 201 5 200 3 200 2 200 + 1 200 0 200 0 200 0 True latency: 200ms. Table 32. Amplitude estimation: mean and standard deviation SNR (dB) Woody PCA Exact match Mismatch 20 0.22 9.32 10.7 2.12 1.68 2.20 2.22 2.93 16 0.07 + 5.95 6.80 + 1.35 1.34 + 1.29 1.71 + 1.75 12 0.40 3.84 4.33 0.86 1.18 0.77 1.49 1.05 8 0.36 2.60 2.78 0.55 1.10 +0.47 1.34 0.64 4 0.53 + 1.75 1.86 + 0.36 1.06 + 0.29 1.26 + 0.40 0 0.64+ 1.14 1.35 0.23 1.04 0.18 1.23 0.25 4 0.80 0.74 1.13 0.13 1.02 0.12 1.20 +0.16 8 0.90 0.45 1.05 0.08 1.01 0.07 1.19 0.10 12 0.98 0.24 1.02 0.05 1.01 0.05 1.18 0.07 True amplitude: 1 Table 33. Scalp topography estimation: correlation coefficient SNR (dB) PCA Exact match Mismatch 20 0.568 0.829 0.759 16 0.579 0.910 0.854 12 0.601 0.959 0.926 8 0.649 0.982 0.966 4 0.759 0.993 0.986 0 0.906 0.997 0.994 4 0.979 0.999 0.998 8 0.996 1.000 0.999 12 1.000 1.000 1.000 Mismatch 190 + 16 189 + 14 188 11 188 + 10 188 + 10 187 + 10 187 + 10 188 + 10 188 9 Table 34. Effects of mismatch I: SNR = 20dB Spread Sprea Latency(ms) Amplitude parameter 7 150 20 2.32 3.0; 9 175 16 2.18 2.7L 11 191 + 12 1.93 2.5 13 202 10 1.68 2.2 15 215 11 1.43 1.8; 17 228 + 15 1.24 1.6( 19 246 + 18 1.07 + 1.41 20 (Gaussian) 210 + 22 2.14 2.89 1 ) 7 ) Coefficient of Variation 1.30 1.25 1.30 1.31 1.31 1.29 1.31 1.35 CC. of scalp topography 0.68 0.73 0.79 0.83 0.87 0.88 0.87 0.76 Table 35. Effects of mismatch II: Spread Spread Latency(ms) parameter 7 151 21 9 175 16 11 190 8 13 200 4 15 211 5 17 220 7 19 229 9 20 (Gaussian) 208 11 SNR = 10dB Amplitude 1.30 + 0.92 1.34 + 0.78 1.24 + 0.69 1.14 + 0.60 1.03 + 0.53 0.94 0.46 0.86 + 0.41 1.30 + 0.78 Coefficient of Variation 0.71 0.58 0.55 0.53 0.51 0.49 0.47 0.60 CC. of scalp topography 0.93 0.95 0.96 0.97 0.98 0.98 0.99 0.95 0.9 0.8 0.7 0.6 0.5 E 0.4 0.3 / 0.2 0.1 0  0 2 4 6 8 10 time  K=3,theta=2 K=5,theta=l K=2,theta=2 K=9,theta=1 12 14 16 18 20 Figure 31 Gamma functions with different shapes and scales. presumed ERP component synthetic ERP component ) 0.6 0.4 E 8 0.4 time (ms) K=3, theta=13 K=3, theta=7 K=3, theta=19 S Gaussian B Figure 32 Waveforms of synthetic and presumed ERP component. A) Synthetic component Gamma: K= 3, 0 = 13; presumed component: Gamma: K= 5, 0 = 6. B) Synthetic component Gamma: K= 3, 0 = 13; presumed components: Gamma: K= 3, 0 = 7, and 0 = 19, Gaussian with a spread of 20. CHAPTER 4 ENHANCEMENTS TO THE BASIC METHOD In chapter 3, we developed the basic spatiotemporal filtering method for singletrial ERP component estimation. In this chapter, we will consider some modifications to the basic method. Some serves as a heuristic postprocessing technique (Section 4.1: iteratively refined template); some aims to deal with large salient EEG artifacts (Section 4.2: robust estimation); some utilizes the apriori knowledge on the scalp topography of the ERP component (Section 4.3: regularization); some provides alternative formulations for our previous results and also derives new ones (Section 4.4: Bayesian formulations of the topography estimation); still others try to deal with the interference from other overlapping ERP components (Section 4.5: explicit compensation for temporal overlap). The details are presented below. 4.1 Iteratively Refined Template The deterministic approach of our estimation in Chapter 3 uses a fixed waveform for the template, regardless of the SNR. The stochastic approach incorporates some degree of variability in the template, but it is still implicit. We would like to explicitly utilize the posterior information from the data to update or refine our apriori assumed template. Intuitively, this should improve our estimation at least for high SNR conditions. We use the estimated scalp topography ao as a spatial filter. The output is optimal in the sense that it has the largest correlation coefficient with the actual component with the uncorrelated noise assumption. (Of course, we use the estimate as a proxy for the true topography. Note that it is different from w). The refined template is the ensemble average of spatially filtered data, with ao as the filter. The results are shown in Table 41, 42 and 43 (with one iteration of refining). We can see that for the latency estimation, the refined template method has a larger bias below 12dB than the original template, but improves quickly and approaches to the exact match case for positive SNR conditions. For the amplitude estimation, the refined template consistently beats the original template for all SNR conditions in terms of both bias and variance. It also approaches to the exact match case for positive SNR conditions. Also see Figure 41. For the scalp projection estimation, the refined template is worse below 8dB but is very close to the exact match case above 8dB. Figure 42 shows the waveforms of the synthetic component, original template and refined template at 4 SNR conditions: 20dB, 12dB, OdB and 12dB. The spatially filtered ensemble average is still quite erratic below 10dB. That possibly accounts for the worse performance of refined template for very low SNR conditions. So, there exists a critical SNR below which, using the spatially filtered ensemble average will probably worsen performance (in terms of scalp topography, but not amplitude) compared with the original template. For this particular data set, it would be safe to use a refined template as long as the SNR is above 10dB. With 1 iteration, there comes 2, 3 and infinity. The natural question then is: Will it converge? If so, what does it converge to? Theoretically, these are difficult questions. Aside from the variable latency parameter, the scalp topography is still nonlinearly related to the data. However, we can experimentally determine the limiting results. These are shown in Table 44 for 3 SNR conditions. The algorithm converges within 10 (sometimes 2) iterations. Compared with the first iteration, the estimation for latency and scalp projection barely changes, but there is a reduced bias and variance in terms of amplitude for negative SNR conditions. Of course, all these results are a function of the mismatch in the waveforms and the number of trials we use to compute the ensemble average (if we had 1000 trials instead of 120, it would be a different story). At this point, it seems that, what matters the most to deal with negative SNR is to accumulate more data, either in space or in time. 4.2 Regularization Sometimes, we have some a priori knowledge of the scalp topography of the ERP component. For instance, the P300 component usually has a large positive projection around Pz area. In these cases, we should utilize that information and incorporate it into our model. 4.2.1 Constrained Optimization Assuming that the ERP component latency has been estimated, we attempt to minimize the following cost function with respect to the amplitude a and scalp topography a (which is constrained to have a unit norm in 12): argmin XcasT +A aa,, (41) ,a F S.t., a2 =1 where, A is a regularization parameter and a, is a normalized vector representing the a priori knowledge of the scalp topography of the ERP component. F denotes the Frobenius norm of a matrix. The reason that we chose this norm will be evident later: the minimization of this norm gave the same solution as (3.14), which was derived with the uncorrelatedness assumption. s s In Appendix C we derive a fixpoint update equation: sas (42) Aa, + oXs a=   Aa0 + cXs 2 Particularly, when A = 0 (no regularization), the optimal solution becomes: Xs sTs s s (43) Xs a= When A 0o the optimal solution is: aT Xs C = s "sS (44) When A takes intermediate values, the scalp topography is a weighted average of the two extreme case solutions. This is a "real" singletrial estimation scheme in that the amplitude is computed from one singletrial data matrix. For the case of no regularization, it is different from our original formulation, where the amplitude is the inner product between Xs and the normalized average scalp topographies from all the trials. The original formulation makes the reasonable assumption that the ERP component has a fixed scalp topography and utilizes that information. We demonstrate the effectiveness of regularization to deal with the interference from other overlapping (possibly unknown, if they are all known, we can explicitly compensate for that See Section 4.5) ERP components. Specifically, we will investigate the effect of regularization on the estimation of amplitude and scalp topography under well controlled conditions. We assume that there are 2 overlapping ERP components and their latencies are fixed and known. Their waveforms are shown in Figure 43. They are both Gamma functions with the same parameters K = 3, 0 = 10, with a peak interval of 80ms. The two ERP components have a correlation coefficient of 0.36. We use templates that are exactly matched with the synthetic components. These components are projected to spontaneous EEG data to generate simulated ERP data. Their scalp topographies are shown in the Figure 44. We use that for a0 (exact a priori knowledge). We then find the optimal solution of c for a given A. The fixed point update always converges within 2 steps. We summarize the results for 3 SNR conditions (12dB, 12dB,  20dB), shown in Figure 45. For high SNR (12dB) data, regularization brings little difference. Since the overlapping ERP components have the exact opposite scalp topography, it is expected that the estimated amplitude is smaller than 1 for high SNR data. It converges to around 0.42 and 0.41 for component 1 and 2 for large A, respectively. Note the huge bias in the estimated amplitude for low SNR (especially 20dB) without regularization. But it converges to as small as 0.69 and 0.27 for component 1 and 2 for large A, respectively. This demonstrates the necessity of regularization for the constrained optimization problem. A reasonable regularization parameter for all the SNR conditions is between 104 and 105, where an unbiased estimation for amplitude could be achieved. Also notice that there is a hump for the standard deviation of the estimated amplitude. Interestingly, this is near the reflection point of the mean amplitude. In practice, this can give us some hint for finding a reasonable regularization parameter. Note that for these choices of A, the correlation coefficient of scalp topography is already very close to 1. This is an example when the overlapping components have negative correlation on the scalp. What about positive correlated components? Figure 46 shows the results for the same components, except that now they have the same scalp topography. The estimated amplitude is expected to be larger than 1 for high SNR data. It converges to 1.6 for both components for large A. Note the huge bias for low SNR (especially 20dB). It converges to 1.8 and 1.7 for component 1 and 2 for large A, respectively. This also demonstrates the benefits of regularization for the constrained optimization problem: using a sufficiently large regularization parameter in this case can reduce the bias of the amplitude estimation, while the variance are not affected very much. As expected, for large A, the correlation coefficient of scalp topography converges to 1. 4.2.2 Unconstrained Optimization Parallel with the above constrained optimization, we can also frame the problem into an unconstrained optimization one. argmin Xa.s' +A a/a 2 a, 1 (45) Note that the optimization variable a contains the amplitude parameter as well as the topography information. As shown in Appendix D, a fixedpointed update can be obtained: Xs+Aa0 a a = (46) s sAa a/a  Particularly, when A = 0 (no regularization), the optimal solution is the same as our original solution in (314). When A > oc, the optimal solution is not unique (any scaled version of a, can be a solution). The problem becomes illposed. We have obtained the regularized solution for singletrial scalp topography. The same procedures for estimating the amplitude follow: take the average of normalized singletrial scalp topography as our estimate for the overall scalp topography, then the amplitude for a particular trial is just the inner product between this vector and the corresponding scalp topography. We test the performance of regularization under the same conditions as in the constrained optimization. We find the optimal solution for a given A. Fixed point update usually converges within 10 steps. We summarize the results for 3 SNR conditions (12dB, 12dB, 20dB) shown in Figure 47 and 48. As in constrained optimization, for high SNR (12dB) data, regularization has little effect on the estimation. The amplitude converges to around the same values as before (0.4 for opposite topography and 1.6 for the same topography). The variance is not affected much for all the SNR conditions, either. As expected, the scalp topography gets monotonically better as A increases. The difference is that the estimated amplitude mean increases monotonically (except for a small interval) with increasing regularization parameter A. This translates to a larger bias (particularly for low SNR data) when the overlapping components have positively correlated topographies. The estimated amplitude becomes unstable for large A. There is no evidence at this point that regularization can benefit the estimation for general scalp topography configurations (both positive and negative topography correlations). So the unconstrained optimization formulation need not be regularized, at least for overlapping components with positively correlated topographies. This lends support to our original solution in (3.14), which is exactly the unregularized solution to the unconstrained optimization problem here. We also point out that, unlike the constrained optimization problem, here we utilize the reasonable assumption that an ERP component has a fixed scalp topography. So it does not suffer from the huge bias problem in the constrained optimization. For instance, at 20dB without regularization, the estimated mean amplitude is around 2.3 and 1.9 for the two components with positively correlated topographies, a modest increase from 1.6 at 12dB. 4.3 Robust Estimation: the CIM Metric We have seen in Section 4.2 that the estimation for singletrial scalp topography in (3.14) can be found equivalently from the minimization the following criterion: arg min Xk ak, ykT (47) where, F denotes the Frobenius norm. It is evident from (314) that the estimate for singletrial scalp topography bears a linear relationship with the EEG data. Because of the noisy nature of EEG (particularly large salient artifacts), this gives a noisy estimate for the singletrial scalp topography (with large variance) and in turn translates into a noisy estimate for the singletrial amplitude in (318). We would like to derive a robust estimator in order to reduce the effects of large EEG artifacts. We can replace the Frobenius norm in (47) with other norms (e.g., li norm) or metrics. Here we will consider a special metric: correntropy induced metric (CIM) proposed by Liu et al (2007). First we introduce what is called correntropy. Given two scalar random variables X and Y, correntropy is defined as: Vh(X,Y) = E[kh(X Y)] (48) where, k(X Y) is the Gaussian kernel (h is the kernel size), kh(X Y) exp (49) j2h 2h2) The correntropy function is a localized similarity measure in the joint probability space, which is controlled by the bandwidth parameter h (also called kernel size in kernel methods). It induces a metric (CIM) in the sample space which behaves like the 12 norm when the sample point is close to the origin (relative to the kernel size); when the sample point gets further apart from the origin, the metric is similar to the 11 norm and eventually saturates and approaches to the 10 norm (Liu, 2007). As such, CIM practically incorporates the 12 norm as a special case (if h is chosen to be sufficiently large). Minimization of CIM is equivalent to the maximum correntropy criterion (MCC). It has been shown that MCC has a close relation to Mestimation (Huber, 1981) and since correntropy is inherently insensitive to outliers, MCC is especially suitable for rejecting impulsive noise (Liu, 2007). Now, treating each entry of the matrix in (47) as a realization of a random variable, we can write our new cost function as: argmin Xk ak yk T (410) The nuisance parameter h (kernel size) should be tuned to the data (most notably to the standard deviation). Here, we use the Silverman's rule as a baseline to quantify different values of kernel sizes that we use in the simulation. It is given by h = 1.06dN 02, where N is the number of samples, and d the standard deviation of the data (Silverman, 1986). While minimization of the Forbenius norm has a closedform solution, minimization of CIM does not. So we have to search for a local minimum, using the standard gradient descent method. The convergence to a certain local minimum is guaranteed by adopting a stopping criterion that the change in the correlations of the estimate and the MSE solution between the previous and current iteration is less than 106. Table 45 and 46 summarize the estimation results for 2 SNR conditions (0dB and 20dB) for the mismatch case. We also include the MSE solution for comparison. The results are mixed: MCC gives a slightly higher variance than MSE for the estimation of amplitude, but the bias is marginally reduced; it also gives a more accurate estimate for the scalp topography. We also notice that at OdB, the results of MCC barely change from MSE. Intuitively, when the SNR becomes sufficiently large, the optimal MCC solution should converge to the MSE solution (both agree with the true values of the parameters). These differences in the results are by no means statistically significant. We venture 2 reasons why the MCC results do not change very much from MSE. First, the EEG data are already preprocessed and relatively clean. Large artifacts have already been removed. The resulting distribution is not far from Gaussian. So MSE should give a solution already close to optimal. MCC has its edge when there is large noise, especially impulse noise. Strictly speaking, it is only optimal (in the sense of maximum likelihood) for one particular type of distribution, just as MSE is strictly optimal for Gaussian distribution. It is not clear that how these two criteria compare when the data distribution changes in a neighborhood of their optimal ones. In reality (when EEG data are usually preprocessed and artifacts are removed), there is no reason to believe that MCC will outperform MSE uniformly. Another less compelling reason concerns the optimization process associated with MCC. The initial condition of MCC is set to be the MSE solution (starting a random initial condition will seldom beat MSE). When the kernel size is small, the performance surface is highly irregular, so the optimization will never go far from MSE solution (it is stuck around the local minimum near the MSE solution). When kernel size is sufficiently large, it is easy to see that MCC approaches to MSE. Only intermediate values of kernel size will produce somewhat different results from MSE. This is seen in both SNR levels, though less evident for OdB. There are two other cost functions in the estimation of amplitude that use the MSE criterion, i.e., (3.14) and (3.16), which can also be replaced by MCC. Li et al (2007) have tested its performance and the improvement was shown to be marginal. In practice, one has to weigh the small improvement in the performance of MCC against its high computational cost (and no guarantee of convergence to global optima). 4.4 Bayesian Formulations of the Topography Estimation The amplitude estimation consists of 3 steps. First we estimate the singletrial scalp topography ak (either by the uncorrelated assumption or equivalently through the minimization of the Frobenius norm of Xk akk ). Then we compute the normalized scalp topography a, as the normalized version of a weighted average of the singletrial scalp topography ak (the weights being their respective 12 norm). The third step is the minimization of the 12 norm ak oa 2, which gives the optimal singletrial amplitude as the inner product between ak and a,. We have seen that the first step gives a near optimal solution (as opposed to CIM) if the EEG data have been cleaned. We have also shown that in the third step, using other metrics (e.g., CIM) gives a marginally better results than using the 12 norm. Here we will investigate other alternatives to the estimation of the normalized scalp topography a, in the second step in (3.17). We maintain that after estimating the singletrial scalp topography in the first step, we treat ak as known and given. Then we ask the question: what is the best estimate for the normalized scalp topography a, given ak for K trials. This is a divideandconquer approach and simplifies matters. Naturally the problem is best formulated in a Bayesian framework. Of course, the formulation will depend on the model we assume for the data. Next, we will present three different models and compare their performance. 4.4.1 Model 1: Additive Noise Model ak = oa +uk (411) where, ok is the singletrial amplitude, and uk is the error (model uncertainty). This model is consistent with our linear generative EEG model. We shall assume conditional independence between a I o,, a and a, I c, ao, for any i j. Of course, we also assume o, and oa are independent for any i j, and they are all independent of the normalized scalp topography a,. We wish to maximize aposteriori probability (MAP): arg max p(a,, O... a,...aK) (412) ao,,i OK It can be shown (Appendix E) that the MAP solution occurs when ao is the normalized eigenvector of the matrix A corresponding to the largest eigenvalue, where, K K A= iAk = akk (413) k=l k=l Now we have solved the MAP problem under model 1. But, there is a weakness: the model error has a constant covariance across all trials. This is a dubious assumption. Intuitively, when the data are noisy, the variance in uk will increase accordingly. We should somehow "normalize" the data in our model. This leads to model 2. 4.4.2 Model 2: Normalized Additive Noise Model = ao +uk (414) Ok The left hand side is in fact a proxy for aok in our linear generative EEG model, except that it may not be normalized. uk is again zeromean i.i.d. Gaussian noise, but now with unit norm. The model can also be written as: ak = ok aOk + k uk. Again, it can be shown (Appendix F) that the MAP solution occurs when ao is the normalized eigenvector of the matrix B corresponding to the largest eigenvalue, where, B=: ak k (415) k=l k=l ak ak Note that the components Bk is a "normalized" version of Ak in that the trace of Bk is always 1. We also note that, we can treat ok only as a normalizing factor, not necessarily as the singletrial "amplitude". In this case, we separate the estimation of ok and a,. As before, we can compute the amplitude by: ak = aoT ak. 4.4.3 Model 3: Original Model Our last model is actually simpler. We do not include the unknown amplitude in this stage and only consider the estimation of a,. The posteriori probability is simply arg max p(ao I a,...a) . ao Given the model: k = a + uk, Appendix G shows that the MAP solution actually coincides with our original solution in (3.17). The singletrial amplitude estimation is the same as before. 4.4.4 Comparison among the Three Models We compare the amplitude and scalp topography estimation for the three models. The results are summarized in Table 47 and 48. We can see that model 3 (original solution) consistently gave the best results among the three under different SNR conditions. The topography estimation with model 1 is poor at negative SNR conditions. Model 2 is an obvious improvement. Note that its amplitude estimation has a huge bias and variance for low SNR data. This may be due to the improper prior we assign to ok (it assign large probability to large values) in the derivation of the MAP estimator. We also included the amplitude estimation with the traditional inner product. There is a significant improvement in both the bias and variance, especially in low SNR conditions. 4.4.5 Online Estimation Sometimes we wish to know the singletrial parameters after recording each trial. It is then necessary to obtain an online estimation method. We assume that the stimulus onset time is known to us. Again, the problem is best formulated in a Bayesian framework in order to utilize all the information in previous trials. In fact, it is trivial given the above analysis. Here, we adopt model 3. After the first trial, set: ai = a/ al2 1 = al 2 K At each trial, we store the running average: ck = ak/la 2 k=1 When finishing recording trial K+1, we update the topography estimate: a(K+l) K +aK+l aK+ 2 o cK+ a+/aK+1 and the amplitude for the newly recorded trial: K+1 = aK+1 K a1) 4.5 Explicit Compensation for Temporal Overlap of Components In developing our basic method in Chapter 3, we assumed that ERP components are uncorrelated with each other. In reality, this is seldom satisfied. Because ERP components have relatively stable waveforms and latency, when they overlap in time, there will generally be a nonzero correlation among them. Here we are mainly concerned with the situation where ERP components overlap in time, but latency jitter across trials is relatively small. Consider two ERP components overlapping in time (it is easy to generalize to multiple components). We assume that the time courses of the components s, and s2 are known from physiological knowledge. We also assume that the latency is given or can be estimated, e.g., from ensemble average, and it is relatively fixed. For the purpose of amplitude estimation, latency jitter is considered here as a minor issue compared with the possibly heavy overlap of components. Note that, in the case of two overlapping (correlated) components, we lose the ability to estimate the latency simply from the cost function in (39). If we assume that all other components are uncorrelated with these two components (or have negligible temporal overlap with them), then we can compensate for the correlation (due to temporal overlap) to get an unbiased estimate for both components' amplitudes. We write the linear generative EEG model in this case: N X=a s, +a2 s2 2T + bnf (416) We wish to estimate the scalp topographies for the two overlapping components. With the uncorrelated assumption, we can get two set of equations: {a, s1 s + a2 *2T *S =X X(417) 1 (417) a1 sT S2+a2 2 S 2 = Xs2 Solve for a, and a2: D2Xs1 CX.s2 al ID2 (418) DX s2 CXs, 2 DD 2 _C2 where, D s = s, s, (i = 1, 2), and C = s, .s2 = s2T S . If C = 0, we have the same solution as before: Xs a, = ,i=1,2 (419) s s, The procedures for estimating the amplitude are the same as before. The above technique assumes that accurate estimates of the waveforms of the components are available, since the crosscorrelation in (418) depends on the tails of the overlapping components. This restricts its applicability in practice. But if the researchers believe that the ERP components are heavily overlapped and are fairly certain of their waveforms, this technique should serve as a first attempt to reduce the bias in the estimation. Table 41. Latency estimation: mean and standard deviation SNR (dB) Exact match Mismatch 20 202 +10 190+ 16 16 201 7 189 14 12 201 5 188 11 8 200 3 188 10 4 200 2 188 10 0 200+ 1 187 10 4 200 0 187 10 8 200 0 188 10 12 200 +0 188 9 True latency: 200ms. Refined template 182 12 183 13 187 12 191 + 12 197 6 199 4 200 + 1 200 + 1 200 + 1 Table 42. Amplitude estimation: mean and standard deviation SNR (dB) Exact match Mismatch Refined template 20 1.68 + 2.20 2.22 + 2.93 2.17 + 2.73 16 1.34 1.29 1.71 + 1.75 1.61 + 1.76 12 1.18 + 0.77 1.49 + 1.05 1.33 1.05 8 1.10 + 0.47 1.34 0.64 1.18 0.59 4 1.06 + 0.29 1.26 + 0.40 1.11 + 0.34 0 1.04 + 0.18 1.23 + 0.25 1.06 + 0.20 4 1.02 0.12 1.20 + 0.16 1.04 0.12 8 1.01 + 0.07 1.19 + 0.10 1.02 + 0.07 12 1.01 + 0.05 1.18 + 0.07 1.01 + 0.05 True amplitude: 1 Table 43. Scalp topography estimation: correlation coefficient SNR (dB) Exact match Mismatch Refined template 20 0.829 0.759 0.644 16 0.910 0.854 0.772 12 0.959 0.926 0.891 8 0.982 0.966 0.963 4 0.993 0.986 0.987 0 0.997 0.994 0.995 4 0.999 0.998 0.998 8 1.000 0.999 0.999 12 1.000 1.000 1.000 Table 44. Estimation results for the iteratively refined template method SNR Latency estimation Amplitude estimation Scalp topography (dB) Refined Iterative Refined Iterative Refined Iterative template Refined template Refined template Refined 20 182 12 183 12 2.17 2.73 1.83 2.36 0.644 0.644 12 187 12 187 12 1.33 1.05 1.23 1.01 0.891 0.879 0 199 4 199 2 1.06 0.20 1.06 0.20 0.995 0.995 Table 45. Estimation with MCC for the mismatch case at SNR = OdB Kernel size Correlation coefficient of (multiples of h) Amp e Scalp projection 0.5 1.22 + 0.25 0.994 1 1.22 0.26 0.994 2 1.22 0.27 0.995 5 1.22 0.26 0.995 10 1.22 0.25 0.994 20 1.23 + 0.25 0.994 MSE 1.23 + 0.25 0.994 Table 46. Estimation with MCC for the mismatch case at SNR = 20dB Kernel size Correlation coefficient of Amplitude . (multiples of h) Amp e Scalp projection 0.5 2.22 + 2.95 0.758 1 2.21 + 3.05 0.754 2 2.17 3.18 0.786 5 2.18 3.14 0.789 10 2.21 3.00 0.770 20 2.22 + 2.93 0.759 MSE 2.22 + 2.94 0.759 Table 47. Amplitude estimation for three Bayesian models Model 3 SNR (dB) Model 1 Model 2 Model 2 (2) (origin (original) 20 0.38 + 1.30 12.2 + 378 1.95 3.79 2.22 2.93 12 0.86 + 0.85 9.96 60 1.42 1.34 1.49 1.05 0 1.20 + 0.26 1.66 + 0.26 1.23 + 0.25 1.23 + 0.25 12 1.18 0.07 1.21 0.07 1.18 0.07 1.18 0.07 Table 48. Scalp topography estimation for three Bayesian models SNR (dB) Model 1 Model 2 Model 3 (original) 0.188 0.558 0.971 0.998 0.642 0.841 0.994 1.000 0.759 0.926 0.994 1.000 exact match exact match mismatch ,\ mismatch refined template refined template 25 2 2 \ 15 E 2 1 5 1 05 0 20 15 10 5 0 5 10 15 20 15 10 5 0 5 10 15 SNR(dB) SNR(dB) Figure 41 Mean and standard deviation of the estimated amplitude under different SNR conditions. The refined template method approaches to the exact match case for high SNR conditions. A 5so 100 150o 5o 100 150o sample sample 10 10  synthetic  synthetic 8 \ presumed 8 / \\ presumed refined / \  refined 4 4 C 2 2 D S 0 50 100 150 0 50 100 150 sample sample Figure 42 The waveforms of the synthetic component, presumed template and refined template under 4 SNR conditions. A) 20dB. B) 12dB. C) OdB. D) 12dB. The refined template appears erratic for low SNR and approaches to the synthetic component for high SNR. 9/ 8 7 6 5 4 3 2 0 0 50 100 150 Figure 43 Waveforms of two overlapped components used in regularization. The correlation coefficient between the two waveforms is around 0.36. 0.2 0.15 0.1 0.05 O 0.05 0.1 0.15  0.2 0 20 40 60 80 100 120 140 channel number Figure 44 Scalp topography of two overlapped ERP components used in regularization.  mean  std 102 10 104 10 10o REG PARAMETER LAMDA 102 10 104 105 10 REG PARAMETER LAMDA S0999 8 0 998 S85 0 998 8 0 997 5 0 997  107 0996 5 102 0 999 o 0999 %6 o 0999 S4 0999 2 8 0999 0998 8 0 998 107 6 102 10 10 10 10 REG PARAMETER LAMDA 10 10 10 10 REG PARAMETER LAMDA A Figure 45 Amplitude and scalp topography estimation I with regularization (constrained optimization) under 3 SNR conditions. A) 12dB. B) 12dB. C) 20dB. The overlapping ERP components have the exact opposite scalp topography. So the estimated amplitude is smaller than 1 for high SNR data. It converges to 0.42 and 0.41 for component 1 and 2 for large A, respectively. Notice the huge bias for low SNR (especially 20dB) without regularization. But it converges to as small as 0.69 and 0.27 for component 1 and 2 for large A, respectively. 102 10 10 10 10 REG PARAMETER LAMDA mean / 095 09 t o 0 85 S08 o / 8 075 07J 107 065 102 10 104 105 10 107 REG PARAMETER LAMDA 0 95 S09 S0 85 1d 1 10d 10 10 o 10 REG PARAMETER LAMDA 08 1i 10 10 1 Id 17 REG PARAMETER LAMDA 10 10 10 10 10 10 REG PARAMETER LAMDA 10 10 10 10 10 10' REG PARAMETER LAMDA C Figure 45 Continued 09 08 8 8 07 06 05 04 102 10 104 105 10 10 REG PARAMETER LAMDA 1 09 08 8 8 07 06 05 05 04 10d 0d 1id 10d d d1 REG PARAMETER LAMDA  w"" $ mean std 05 REG PARAMETER LAMDA S 0999 9 S0 999 0999 8 0999 8 107 0 999 7 102 10 10 10 10 REG PARAMETER LAMDA mean std REG PARAMETER LAMDA 1 1 o S 1 L 0999 0 999 o 0999 9 0999 9 0 999 IJ 9 16i 10 10 10 10 REG PARAMETER LAMDA A Figure 46 Amplitude and scalp topography estimation II with regularization (constrained optimization) under 3 SNR conditions. A) 12dB. B) 12dB. C) 20dB. The overlapping ERP components have the same scalp topography. So the estimated amplitude is larger than 1 for high SNR data. It converges to 1.6 for both components for large A. Notice the huge bias for low SNR (especially 20dB) without regularization. It converges to 1.8 and 1.7 for component 1 and 2 for large 1, respectively. 5 102 10REGRAMETE1 10 10 10d 10 10 REG PARAMETER LAMDA 0  102 10 104 105 10 REG PARAMETER LAMDA mean std 6\ 5 4 3 2 10 10 10o 10 10 10 REG PARAMETER LAMDA mean \ 6 std 5 4 \ 3 2 10 10 104 10 106 10 REG PARAMETER LAMDA C Figure 46 Continued std 099 0 0 98 S097 0 96 S107 095 102 10 10 10 10 10 REG PARAMETER LAMDA mean / std a S0995 S099 0 985 10 1i2 10 10 10 10 10 REG PARAMETER LAMDA 0 95 o /  09 o 85 08 O 12 10i 1 0 10 1 10 REG PARAMETER LAMDA 0 99 o 098 S097 S096 8 095 0 94 093 10 10o 10 105 10 10 REG PARAMETER LAMDA I J 102 10 104 10 REG PARAMETER LAMDA 12 10f 10 10 REG PARAMETER LAMDA S0999 S0 998 S85 0 998 8 0 997 5 S0997  106 0996 5 102 0 999 8 o 0999 %6 o 0999 S4 0999 2 S0999 0998 __ 8 0 998 100 6 102 10 10 10 REG PARAMETER LAMDA 10 10 10 REG PARAMETER LAMDA A Figure 47 Amplitude and scalp topography estimation I with regularization (unconstrained optimization) under 3 SNR conditions. A) 12dB. B) 12dB. C) 20dB. The overlapping ERP components have the exact opposite scalp topography. So the estimated amplitude is smaller than 1 for high SNR data. The estimated amplitude mean generally increases with increasing A while the variance is not affected much. It becomes unstable for large A. mean std / 10 10 10L REG PARAMETER LAMDA 0 95 E0 S085 08 o 08 8 075 07 10 065 102 10 10 10 REG PARAMETER LAMDA S095 S09 o 085 Mean 3 std 2 2 15 1 0  10 103 104 10 10o REG PARAMETER LAMDA mean 6 std 12 1 1 1 1 1RE 0 ~ ~ i  REG PARAMETER LAMDA 6 std 2 10? 10" 104 105 REG PARAMETER LAMDA C Figure 47 Continued 10 10 10 REG PARAMETER LAMDA 10 10 10 REG PARAMETER LAMDA o 06 05 10 10 10 REG PARAMETER LAMDA 3 25 E S12 1i 10 10 REG PARAMETER LAMDA  mean std REG PARAMETER LAMDA o 0 999 88 0 999 0 8 106 0999 7 1C0 1 0 0999 S0 999 & 9 0999 9 0999 9 1 0 999 ,f 9 102 10 10 10 REG PARAMETER LAMDA 10R 10P 10L REG PARAMETER LAMDA A Figure 48 Amplitude and scalp topography estimation II with regularization (unconstrained optimization) under 3 SNR conditions. A) 12dB. B) 12dB. C) 20dB. The overlapping ERP components have the same scalp topography. So the estimated amplitude is larger than 1 for high SNR data. The estimated amplitude mean generally increases with increasing A while the variance is not affected much. It becomes unstable for large A. 103 10 10 REG PARAMETER LAMDA S099 0 098 o S097 o 0 96 0 95 102 10 10 10 REG PARAMETER LAMDA mean 0 102 10 104 105 10 REG PARAMETER LAMDA 10 mean 8 std 0 6 4 0  10 103 10 10 106 REG PARAMETER LAMDA 10 103 104 105 REG PARAMETER LAMDA C Figure 48 Continued 0 995 8 o 0 099 0 985 1o2 0 95  09 085 08 1o2 0 99 o 098 8 097 S096 8 095 0 94 0 93 102 10 10 10 REG PARAMETER LAMDA 10" 104 10" REG PARAMETER LAMDA R E 103 104 105 REG PARAMETER LAMDA CHAPTER 5 APPLICATIONS TO COGNITIVE ERP DATA In this chapter, we apply the spatiotemporal filtering method proposed in Chapter 3 to the singletrial ERP estimation problem in two different experiments. The first application is an oddball target detection task with different pictures as stimuli, where the difficulty of the task or saliency of the stimuli leads to decreased P300 amplitude. The second one is the habituation study where the subjects were repeatedly presented identical pictures and the amplitude of certain ERP components is expected to decrease rapidly with respect to trials. 5.1 Oddball Target Detection 5.1.1 Materials and Methods Because we were interested in singletrial, singlesubject analyses of amplitude and latency, we selected 4 participants that met a minimum signaltonoise ratios based on their averaged ERPs, from a pilot study (n=8) on implicit content processing during feature selection. They were righthanded according to the Edinburgh Handedness Questionnaire and all had normal or corrected vision. Stimuli consisted of pictures from the International Affective Picture System, depicting adventure scenes, emotionally neutral social interactions, erotica, attack scenes, and mutilations. Their color content was manipulated such that they contained only shades of green or shades of red, and for each, color brightness was systematically manipulated to yield one bright and one dim version (Figure 51). All pictures were presented for 200 ms on the center of a 21inch monitor, situated 1.5 m in front of the subjects. From this viewing distance the checkerboards subtended 4.0 deg. x 4.0 deg. of visual angle. A fixation cross was always present, even when no picture was presented on the screen. Target stimuli (p = 0.25) were defined for each experimental block (see below) by a combination of color and brightness.. All pictures were presented in randomized order, with an interstimulusinterval varying randomly between 1000 to 1500 ms in 4 blocks of 120 trials each. One block lasted 7 min. on average. At the beginning of each block subjects were instructed to attend either to the bright/dark green or red pictures and to press the space bar of the computer keyboard when they detected a target. The target color and brightness were designated in counterbalanced order. Furthermore, the responding hand was changed half way through the experiment, and the sequence of hand usage was counterbalanced across subjects. Subjects were also instructed to avoid blinks and eyemovements and to maintain gaze onto the central fixation cross. Practice trials were provided for each subject for each condition to make sure that every subject had fully understood the task. EEG was recorded continuously from 257 electrodes using an Electrical GeodesicsTM (EGI) EEG system and digitized at a rate of 250 Hz, using Cz as a recording reference. Impedances were kept below 50 kM, as recommended for the Electrical Geodesics high input impedance amplifiers. A subset of EGI net electrodes located at the outer canthi as well as above and below the right eye was used to determine horizontal and vertical Electrooculogram (EOG). All channels were preprocessed online by means of 0.1 Hz highpass and 100 Hz lowpass filtering. Epochs of 1000 ms (280 ms pre, 720 ms poststimulus) were obtained for each picture from the continuously recorded EEG, relative to picture onset. The mean voltage of a 120msec segment preceding startle probe onset was subtracted as the baseline. In a first step, data were lowpass filtered at a frequency of 40 Hz (24 dB / octave) and then submitted to the procedure proposed by (Junghofer et al., 2000), which uses statistical parameters to exclude channels and trials that are contaminated with artifacts. This procedure resulted in rejection of trials that were contaminated with artifacts (including ocular artifacts). Artifacts were also evaluated by visual inspection and respective trials were rejected. Recording artifacts were first detected using the recording reference (i.e. Cz). Subsequently, global artifacts were detected using the average reference and distinct sensors from particular trials were removed interactively, based on the distribution of their mean amplitude, standard deviation and maximum slope. Data at eliminated electrodes were replaced with a statistically weighted spherical spline interpolation from the full channel set. The mean number of approximated channels across conditions and subjects was 20. With respect to the spatial arrangement of the approximated sensors, it was ensured that the rejected sensors were not located within one region of the scalp, as this would make interpolation for this area invalid. Spherical spline interpolation was used throughout both for approximation of sensors and illustration of voltage maps (Junghofer et al., 1997). Single epochs with excessive eyemovements and blinks or more than 30 channels containing artifacts in the time interval of interest were discarded. The validity of this procedure was further tested by visually inspecting the vertical and horizontal EOG as computed from a subset of the electrodes that were part of the electrode net. Subsequently, data were arithmetically transformed to the average reference, which was used for all analyses. After artifact correction an average of 69 % of the trials were retained in the analyses. The present analysis highlighted the most reliable signal available in this featurebased target identification task, which is the P300 component in response to a target stimulus (defined by a combination of color and brightness, irrespective of picture content). Thus, all subsequent analyses focused on amplitude and latency estimates for single trials belonging to the target condition. 5.1.2 Estimation Results The present study illustrates the application of the method for a single late potential component. In reality, we do not know a priori how many ERP components there are in a single trial recording, nor do we know exactly when they occur. However, we may be able to estimate these values from singletrial EEG data in the data analysis session. This is a good time to mention one technical requirement of our latency estimation. The singletrial latency is estimated from the cost function in (3.9), which involves the inversion of the matrix XX In reality, this matrix is usually illconditioned for densearray EEG data (it will certainly be rankdeficient if there are any bad channels which were linearly interpolated from other channels.). This poses a computational problem in practice. Thus, the solution in (3.9) somehow has to be regularized. Here, we adopt a simple approach and add a regularization term AI (A > 0 ) to the matrix XX' before taking the matrix inversion operation. The regularization parameter A acted as a smoother to the cost function in (3.9). Generally, the solution is rather irregular without regularization, leading to too many local minima and spurious candidates for singletrial latencies due to large noise. With increasing A, the cost function becomes smoother. This is clearly seen in Fig. 52, which shows the cost function in (3.9) for four different A, for a particular trial from subject 2. With a smooth cost function, we can avoid the dilemma of choosing the right latency from too many candidates. Now we have to select an appropriate value (or a meaningful range) for the regularization parameter A. A good value for A is one that achieves a balance between two extremes: too few and too many local minima. The idea is this: for a particular A, we group all the candidates for singletrial latencies (time lags corresponding to local minima) together and perform ID density estimation on these candidates. We count the number of modes (peaks) from the estimated probability density function (pdf). If this number is close to the average number of candidates for each trial, then the regularization parameter A is at least internally consistent. Otherwise, it will contradict with itself and should not be used. We illustrate our point using the results from one subject. Fig. 53 shows the estimated pdf of the candidates for singletrial latency from 200ms up to 600ms after stimulus onset when the regularization parameter A equals 10 5. We used the Parzen windowing pdf estimator (Parzen, 1962) with a Gaussian kernel size of 4.2. The kernel size was selected according to Silverman's rule (Silverman, 1986), which is given by h = 1.06cN 02, where Nis the number of samples, and a is the standard deviation of the data. The number of peaks depends on the kernel size, but we found that a kernel size between 0.5h and 2h will give the same number of peaks in the estimated pdf for this data. We can see that the pdf consists of 4 modes (peaks) after 200ms of the stimulus onset. There are 418 local minima and 102 trials in total, so the average number of local minima for each trial is about 4.1 (very close to the number of peaks in estimated pdf). This indicates that =10 5 gives an internally consistent estimate for latency. We can repeat the above procedures for a wide range of regularization parameters and compute the ratio of the number of peaks in estimated pdf to the average number of local minima for each trial. For instance, the ratio was computed as around 4.75, 1.03, 0.96, 0.74 for the 4 regularization parameters in Fig. 52 respectively. Clearly, the first and last regularization parameter should not be used since they generate selfcontradictory results. It is interesting to note that for a wide range of regularization parameters (from 10 5 to 100), the results are quite similar. This can also be seen from Fig.52, where both cost functions display 4 local minima and all time lags are near to their counterparts. For practical purposes, we can select any value from this range as a regularization parameter. We were primarily interested in the P300 component, preferably the largest one. From the ensemble average, we know that the maximum ERP occurred around 380ms after stimulus onset. In Fig. 53, the estimated pdf displays a mode around 420ms. Thus, we searched around this latency and set the singletrial peak latency as the one that was closest to it. The mode of latency is 360ms, 420ms and 400ms for the other three subjects, respectively. We should point out that since there is about 1 local minimum per mode, the search need not be around the true mode for latency (we do not know this anyway). The results would be almost the same as long as the estimated mode is not skewed to its two neighboring true modes. Figure 54 shows the scalp topographies for the four subjects plotted using EEGLAB (Delorme et al., 2004). As expected for a P300 topography, it has a large positive topography around the Pz area. To evaluate the singletrial estimation of the scalp topography, we compute the correlation between the singletrial scalp topography in (3.15) and the overall normalized scalp topography (3.17). For comparison, we also compute the correlation between the single trial scalp topography in (3.15) and the scalp topography obtained from ensemble average for each subject. We name these two correlations r, and r2 respectively. Statistical inference based directly on the correlation itself is difficult since its distribution is complicated. A popular approach is to first apply the Fisher Z transformation to correlation and then do inference on the transformed variable. The Fisher Z transform is given by (Fisher, 1915): Z = 0.51n (51) (1r) Z has a simpler distribution and it converges more quickly to a normal distribution. We can calculate the mean and confidence interval of Z based on the correlation, if we assume that the estimation error in (315) is a normal distribution. The statistics of the correlation can be easily obtained from the inverse transform of (51). The results are summarized in Table 51. We can see that there is a moderate amount of correlation between the singletrial and overall scalp topography (the average mean correlation for 4 subjects is around 0.40) although the mean correlation is lower for subject 3 at around 0.20. There is a small degradation in mean correlation when the overall scalp topography is computed from the ensemble average. This is expected since the estimate in (317) is close to the ensemble averaged estimate. The correlation between these two estimates for the four subjects are: 0.80, 0.85, 0.89, 0.79 respectively. To evaluate the effectiveness of the singletrial amplitude estimation, we related our estimates to a behavioral measure of target identification: response time in target trials. Response time was selected because task difficulty was relatively low, and therefore error rate did not show pronounced variability, with only limited numbers of misses (mean of 3.9 % across 4 participants) and false alarms (mean of 1.2 % across 4 participants). Thus, response time was used as a measure of target identification, with short response times indicating facilitated discrimination and long response times indicating difficulties with identification in a given trial. Using these measures, we were interested in the relationship between P300 amplitude and response time, expecting that trials in which participants found discrimination relatively easy (short RT trials) should be associated with greater P300 amplitude, which also indicates successful encoding of the target features and preparation for responding to a target that has been identified. There seems to be little relationship between the response time and estimated singletrial peak latency. The correlation coefficients between these two for the four subjects are: 0.022,  0.248, 0.168 and 0.093 respectively. However, there were reliable negative correlations between the response time and estimated singletrial amplitude. Figure 5 shows the scatter plot of the response time versus the estimated amplitude for each single trial for the four subjects. To evaluate the statistical significance of the results, we performed linear regression on the response time and estimated singletrial amplitude for the four subjects. The results are summarized in Table 52. the negative slope parameter estimated from linear regression is statistically significant under a significance level of 0.05 for all the four subjects, which supports our hypothesis that larger amplitude correspond to smaller response time, and vice versa. To compare our results with conventional methods, we calculated the average P300 amplitude at channel 100 for subject #2. This was simply the average singletrial amplitude times the 100th entry of the scalp topography in (3.17). It was found to be 22. mV, compared with the 17.3mV from the ensemble average ERP. Taking into account of the possible latency jitter of P300, the true amplitude could be only larger than 17.3mV. Therefore, we obtained an upper bound of 28% on the positive bias of our average P300 estimate in channel 100. The coefficient of variation, which is defined as the ratio of the standard deviation to the mean of a positive random variable, is used as a measure of dispersion of the estimated amplitude and it was found to be around 0.60. This compares favorably with 0.79 obtained using the simple peakpicking method around its ensemble average peak at 400ms. Although the gain may seem small, we should keep in mind that this variation will incorporate the estimation error as well as that of the underlying change in P300 amplitude itself, because there are systematic changes in P300 amplitude as suggested above. So the estimation variance of our method is reduced by a factor of at least 1.7 from the peakpicking method. For instance, if one half of the total variance of our method came from the underlying P300 amplitude, this roughly means that our method reduced the estimation variance by a factor of 2.5 (assuming additive and uncorrelated estimation error). Of course, the comparison would be much more direct and informative if the P300 amplitude was expected to remain constant. All the above results were obtained using a fixed Gamma template with k = 11, 0 =5. If we change the template, specifically, the spread parameter 0, the estimated amplitude will also change. However, we found that the amplitude estimation is only slightly affected by this change. For instance, the average estimated P300 amplitude in channel 100 for subject #2 was around 20.5mV when 0 = 1 (this is too small for P300, rise time 40ms) and was around 23.8mV when 0 = 8 (this is too large, rise time 320ms). There is less than 8% change from the result (22. mV) obtained with the original template with 0 = 5. This agrees with our earlier findings using simulated ERP data (Li et al., 2008). 5.1.3 Discussions As a straightforward test of the present method, we examined the relationship between target detection performance and features of the P300 component evoked by the targets in an oddball task with rare targets varying in terms of their salience on a trialbytrial basis. In the present case, we replicated and extended a standard result in target detection studies in the visual domain: When target identification is made difficult or saliency is reduced (e.g., by presenting many targets in succession, Gonsalvez and Polich, 2002), P300 amplitude often decreases (Polich et al., 1997). This pattern has been interpreted as reflecting reduced resource allocation to a given target stimulus (Keil et al., 2007). Notably, previous work in this area has typically relied on averages across all trials of an experimental condition, or on block by condition averages across many trials (for a review, see Kok, 2001). The present results suggest that the relationship between response time and P300 amplitude in featurebased attention task is of a continuous nature, rather than a consequence of a bimodal function separating easy and hard trials. The sensitivity of the method was sufficient to demonstrate this linear relationship on a singlesubject level, which is often desirable in clinical studies. In a similar manner, other research questions will benefit from the ability to examine hypotheses as to the time course and distribution of single brain responses, in terms of their magnitude and latency. 5.2 Habituation Study 5.2.1 Materials and Methods EEG data were recorded from subjects during a passive pictureviewing experiment, consisting of 12 alternating phases: the habituation phase and mixed phase. Each phase has 30 trials. During the 30 trials of the habituation phase, the same picture was repeatedly presented 30 times. During the mixed phase, the 30 pictures are all different. Each trial lasts 1600 ms, and there is 600 ms prestimulus, and 1000 ms poststimulus. The scalp electrodes were placed according to the 128channel Geodesic Sensor Nets standards. All 128 channels were referred to channel Cz and were digitally sampled for analysis at 250Hz. A bandpass filter between 0.01Hz and 40Hz was applied to all channels, which were then converted to average reference. To correct for vertical and horizontal ocular artifacts, an eye movement artifact movement correction procedure (Gratton et al., 1983) was applied to EEG recordings. 5.2.2 Estimation Results We assume that the entire ERP may be decomposed into several monophasic components with compact support. We will estimate their parameters (amplitude and latency) one by one, using the Gamma template as in the simulation study. The present study illustrates the application of the method for a single late potential component, and the Gamma is not adapted. Its parameters are selected based on neurophysiology plausibility and are set as k = 5, 0 = 6, corresponding to a rise time of 96ms. In reality, we do not know a priori exactly how many components there are in a single trial, nor do we know when they occur. However, we may be able to estimate these values from singletrial EEG data in the data analysis session. Following the same procedures in Section 5.1.2, we identified 5 distinct peaks after 300ms of the stimulus onset. Assuming that the error in the latency estimation is equally biased and independent from trial to trial and since there are also about 5 local minima for each trial, we conjecture that these peaks correspond to the latencies of 5 distinct components. These components, which may have different origins, are likely to compromise the Late Positive Potentials (LPP). According to Codispoti et al.(2006), the grandaverage of LPP is maximal around 400ms to 500ms after stimulus. We will concentrate on the component with a latency of 500ms to exemplify the methodology. We search between 440ms and 560ms (which correspond to the two neighboring local minima) and set the component latency as the local minimum closest to 500ms. To avoid the influence of EEG outliers from unexpected artifacts, we reject those trials with 3 times or larger amplitude of the minimum one. This will eliminate 14 trials from the total of 360 trials (rejection rate: 4%). The same rejection criterion was applied to the other two subjects, leading to the rejection of 8 (2%) and 33 (9%) trials, respectively. Figure 56 shows the results of estimated scalp topography of the LPP component for 3 subjects. They are similar in the sense that all show large projections in the posterior area. The difference with subject 2 is that the scalp topography shifts its strength a bit to the occipital area. It may be that the pictures shown to the 3 subjects caused some emotional bias. It is also possible that the SNR of the ERP data is too low to allow for a stable estimate of the scalp projection across subjects (note that in habituation phase, the LPP amplitude decreases quickly with the trial index). Figure 57 shows the results of estimated amplitude of the LPP component for 3 subjects. Each point in Fig. 57 stands for the average amplitude over 6 trials with the same index in the same phase (habituation or mixed). It is clear that for the habituation phase, the amplitude diminishes rapidly with the trial index, while for the mixed phase, the amplitude does not show significant decay. To make the figure more intuitive, we also include the best fit (in the least square sense) to the estimated amplitude for both habituation and mixed phase. We fitted a straight line for the mixed phase, while an exponential curve was fitted to the estimated amplitude of the habituation phase. The fitted exponential curve for the habituation phase has a time constant of around 1.5 trials, which suggests that after 3 or 4 trials, the LPP amplitude decreases close to zero. We estimate the SNR for the mixed phase at around 4.1dB. Similar results were obtained with the ERP data from 2 other subjects as shown in Fig. 57 (B) and (C). The fitted exponential curves for the habituation phase for these 2 subjects has a time constant of around 1.5 and 2.0 trials, respectively. The SNR of the mixed phase for these 2 subjects are estimated to be around 7.6dB and 2.4dB respectively. Table 51. Correlation statistics for the 4 subjects: Scalp topography estimation Subject Sample 2 #size Confidence interval Confidence interval Mean Mean (95%) (95%) 1 98 0.520 [0.362, 0.649] 0.393 [0.215, 0.546] 2 85 0.367 [0.167, 0.538] 0.304 [0.097, 0.486] 3 74 0.198 [0.027, 0.404] 0.200 [0.025, 0.406] 4 65 0.446 [0.233, 0.619] 0.359 [0.133, 0.551] Table 52. Regression statistics for response time and estimated amplitude Subjec Sample R Slope Confidence SCorrelation R sope t statistic p value t # size square estimate interval (95%) 98 0.440 0.194 0.712 4.80 <0.0001 85 0.539 0.291 0.310 5.84 <0.0001 74 0.263 0.069 0.105 2.31 0.012 65 0.351 0.123 0.108 2.97 0.002 [1.007, 0.418] [0.416, 0.205] [0.197, 0.014] [0.181, 0.036] 1 2 3 4 Figure 51 Pictures used in the experiment as stimuli x 10 35 A 3 25 S1 5 A 200 250 300 350 400 450 500 550 600 time lag (ms) 200 250 300 350 400 450 500 550 600 B time lag (ms) 3 95r S 200 250 300 350 400 450 500 550 600 time lag (ms) a 3 85 S38 3 75 37 3 65 36 200 250 300 350 400 450 500 550 600 D time lag (ms) Figure 52 Cost function in (3.9) versus time lag for different regularization parameters for subject #2. A) A = 106 B) A = 10 5. C) A = 100. D) A = 102. Regularization parameter that is too small led to ragged cost function and spurious latency estimates; Regularization parameter that is too large led to oversmoothed cost function and missed candidates for latency. VV _,_J x 103 3 2.5 2 S1.5 1 \ 0.5 0 100 200 300 400 500 600 700 time lag(ms) Figure 53 Estimated pdf of time lags corresponding to local minima of the cost function in (39) using the Parzen windowing pdf estimator with a Gaussian kernel size of 4.2. Regularization parameter A = 10 . p I 2 SIi  I Ii i C D Figure 54 Scalp topographies for the four subjects. A) Subject 1. B) Subject 2. C) Subject 3. D) Subject 4. S300 C E 200 S100 0 A 1300 350 400 450 500 550 600 650 Response time (ms) 200 o10 150 0 o S50 o 250 300 350 400 450 500 550 600 650 B Response time (ms) 150 E 100 E50 I W, 400 500 600 700 800 900 1000 Response time (ms) 50 200 300 400 500 600 700 800 900 1000 Response time (ms) Figure 55 Scatter plot of the response time versus the estimated amplitude for each single trial for the four subjects. A) Subject 1. B) Subject 2. C) Subject 3. D) Subject 4. Note that the estimated amplitude is with respect to the EEG data in all the channels as a whole. There appears to be a negative relationship between the response time and the estimated amplitude. 107 C26 25 C 16 os1 S015 (/ {o 2 02 0 25 B 02 0 15 n I02 C Figure 56 Estimated scalp topography for mixed and habituation phase. A) Subject 1. B) Subject 2. C) Subject J021 0 I5 (0 1 702 Figure 56 Estimated scalp topography for mixed and habituation phase. A) Subject 1. B) Subject 2. C) Subject 3. 80 +    60 00 20 0 0 20 0 0 5 10 15 tnal index 10 15 trial index 20 25 30 20 25 30 Figure 57 Estimated amplitude for mixed and habituation phase. A) Subject 1. B) Subject 2. C) Subject 3. Note that the LPP amplitude decreases with trials. 0 Habituation Mixed  expotential it  linear fit 20 25 Habituation Mixed expotential fit linear fit   E 60  40 E Habituation Mixed expotential fit linear fit t 10 15 trial index CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 6.1 Conclusions Traditional ERP analysis has relied on ensemble average over a large number of trials to deal with the typically low SNR environments in EEG data. To analyze ERP on a single event basis, we have introduced a new spatiotemporal filtering method for the problem of singletrial ERP estimation. Our method relies on explicit modeling of ERP components (not the full ERP waveform), and its output is limited to local descriptors (amplitude and latency) of these components. The reason that we model the ERP components instead of the full ERP waveform is to exploit the localization of scalp projection for each single ERP component, which is impossible to do for the entire ERP. Indeed, note that the ensemble ERP in different channels usually have different morphology because there are multiple neural sources originating from different locations of the brain that give rise to different scalp projections. Since one spatial filter can extract effectively only one scalp projection, in order to utilize the spatial information in a meaningful way, only a component based analysis is viable. Concentrating only on latency and amplitude of each component together with optimal spatial filtering presents an alternative to deal with the negative SNR. Moreover, since these are in fact the features of importance in cognitive studies, the methodology has the same descriptive power of traditional approaches. The proposed methodology can be seen as a generalization of Woody's filter (Woody 1967) in the spatial domain for latency estimation. It also obtains an explicit expression for amplitude estimation on a singletrial basis. By design, the method is especially suitable to extract ERP features in the spontaneous EEG activity, in contrast to PCA and ICA which work best for reliable (large) signals. Another distinction is that, unlike most methods based on PCA and ICA, our method utilizes explicitly the timing information, as well as the spatial information. The methodology as presented is based on least squares, but it can be further extended to robust estimation (Li et al. 2007) for better results. Using simulated ERP data, we have shown that although the mismatch between the presumed and synthetic ERP components introduces a bias for both latency and amplitude estimation, the bias for the latency is relatively small and the estimated amplitudes are still comparable across experimental conditions for ERP data with a SNR higher than 20dB. Furthermore, the mismatch of components has minimal influence on the estimation of scalp projection. These all compare favorably with some of the popular methods (Li et al., 2008). Despite its advantages over traditional methods, there are still some issues with our spatiotemporal filtering method. First it is based on the linear generative EEG model in (3.1). While this greatly simplifies the analysis, it may not be adequate to fully describe the complex information processing in the brain. One weak link of the method is that it requires an explicit template that is unknown apriori. Mismatch between the template and the true ERP component waveform brings both bias and larger variance to the estimation of the latency and amplitude that increases with decreasing SNR (Li, et al. 2008). It may be desirable to be able to adapt the template while estimating the model parameters. Another weakness of the method is the assumption of statistical uncorrelatedness among all the ERP components in deriving (3.14). With monophasic waveforms, this is equivalent to the condition that all the ERP components do not overlap in time (but overlap in space is allowed), which is seldom satisfied in practice. Temporal overlap will bring bias to the amplitude estimation and poses a serious problem for the latency estimation, since it works effectively only for monophasic waveforms that are well separated in time. When there is heavy overlap among multiple components (e.g., P300 and possibly other unknown late components), the peak latency estimation based on (3.10) may fail. Therefore, care must be taken not to overinterpret the results of singletrial estimates. A crucial factor for amplitude estimation is a reasonably low SNR (>20dB). This may not be satisfied for some ERP components under certain experimental conditions. Our ability to infer the template accurately, which are selected heuristically from real data, deteriorates with decreasing SNR. As a rule of thumb, we would recommend against the use of the present method for data with SNR less than 15dB. 6.2 Future Research The use of a parametric template (Gamma function) provides the flexibility to change the shape and scale parameters continuously. However, this introduces undesirable bias when there is a mismatch between the template and true ERP component. Using a stochastic formulation, our method may be extended to a noisy template model and potentially the two nuisance Gamma parameters may be extracted from the data also for best fit. It is almost certain that activations of different ERP components overlap in time. If this is the case, the temporal overlap will introduce a bias to the estimation of singletrial scalp projection, because the derivation in (3.15) assumes the uncorrelatedness between the ERP component and all the other sources (including the overlapping ERP components) and unlike the background EEG, these overlapping components are coherent in all the trials. This bias, together with the estimation variance due to finitesample data, constitute the two main sources of error in the estimation of the scalp projection. Note that this in turn will influence the estimation of the amplitude. In Chapter 4, we have proposed an explicit procedure to compensate for the overlapping effects for the amplitude and topography estimation. This assumes that the latency is (relatively) fixed and a fairly accurate knowledge of the shape of all the overlapping ERP components. When this is not the case (particularly when we wish to find the latency change from trial to trial in the presence of component overlap), we need to come up with new procedures to compensate for the overlapping issue. The current method considers the singletrial amplitude of an ERP component as i.i.d. data. It may be advantageous to take into account the dynamics of certain properties of the component with respect to the trial index. For instance, we expect that during the habituation phase, the amplitude of LPP components diminishes rapidly with the number of trials. Using regularization techniques, this apriori information may be incorporated into the proposed singletrial estimation method to provide more stable estimate for the amplitude. In the end, the evolution of the singletrial amplitude with trial index may be inferred with more resolution and more confidence. APPENDIX A PROOF OF VALIDITY OF THE PEAK LATENCY ESTIMATION IN (310) We justify the use of the time lag corresponding to the local minimum of J(r) in (310) as the peak latency. Given the singletrial data matrix X, the peak latency of an ERP component coincides with the local minimum of J(r) if the following conditions are satisfied: (1), the presumed component so and the actual component s have the same morphology; (2), n sso(r) = 0, for i 1,...,N, and r e Ts. (the signal and noise are uncorrelated); (3), X is full rank. Proof: The optimal spatial filter is given by (38). We plug it into (37) and get: J(r) = so(r)' .(CI) s= o(r) .(C C C+I)S (Z) where, C= X XX') X. note that CC = C, so, J(r)= so(r)" C.so(r) +so(r)s (r)= [so(r) s .(a'R la)+so(r) S () where, R = X X' is a positive definite matrix independent of the time lag r. With the constraint that s (r)' s (r) = const, the minimum of the cost function J(r) is achieved when so(r)Ts achieves its maximum, since a'R 'a is positive. This happens when r coincides with the peak latency / of the actual ERP component s. APPENDIX B GAMMA FUNCTION AS AN APPROXIMATION FOR MACROSCOPIC ELECTRIC FIELD The macroscopic electrical field is created from spike trains by a nonlinear generator with a secondorder linear component with real poles (Freeman 1975). Suppose that the transfer function of the secondorder system with stable real poles a, b is: 1 H(s) = (s a)(s b) where, without loss of generality: b < a < 0. Then the impulse response in the time domain is: h(t) = 1 (eat ebt). ab 1 This is also a monophasic waveform with a single mode at t = In(b /a). The rising ab time depends on the relative magnitude of the two real poles. The impulse response can be expanded: h(t) 1 1 (b h(t)= ebt.(e (ab)t ebt. [(ab)t] ab ab n, n! Thus we can see that the impulse response is a sum of infinite weighted Gamma functions. However, it is always possible to find a few dominant terms around the mode, where, t, (a b) = ln(b / a). If we knew the values for a, b, we can choose the shape parameter K of the Gamma function as the largest term, i.e., the integer part of ln(b/a). A special case is when the system has two identical poles. Then, the impulsive response is exactly modeled by a single Gamma function with K = 1, 0 = 1/a. This is also approximately true when the magnitude of one pole is much larger than the other one. APPENDIX C DERIVATION OF THE UPDATE RULE FOR THE CONSTRAINED OPTIMIZATION PROBLEM Using one Lagrange multiplier, we convert the constrained optimization problem in (41) to an unconstrained optimization problem. argmin Xcas + a +a 2\  1) (T,a Note that, Xcas =2 Tr (xcasT).(X _X a.asT) = Tr (XX) 2Tr (Xsa) + C2Tr (assa' ) = Tr (XX' ) 2cTr (a'Xs) + a2'sTr (aar) = Tr (XX) 2coa'Xs + oC2s s a.a Setting the gradient of the Lagrangian function to 0 with respect to a, a, p respectively, we find that the following set of equations holds: 20ss a a 2cXs + 2A (a ao) + 2/a = 0 2a Xs +2ss a a = 0 {allu 1= 0 Solving for a, c, we have: a'Xs sTs LAa + CXs Aao + Xs 2 This is not a closedform solution for the optimal values. However, it can be effectively used as a fixed point update to iteratively find the optimal values. APPENDIX D DERIVATION OF THE UPDATE RULE FOR THE UNCONSTRAINED OPTIMIZATION PROBLEM The unconstrained optimization problem is: argmin Xas +A a/ a a 2 a First we note that: a/a2 "[ 21(/ a0) (l/a ao) = a a/aI + ao ao 2a a,/ a 2 = 22aao/ a 2 Taking the derivative to 0, we get, 2s's*a2Xs+2 (a'aaoa/ a ao/a ) = 0 Or equivalently, SXs + Aa0/ lll a= sTs+ Aa0Ta/ ai. This is not a closedform solution for a. However, it can be effectively used as a fixed point update to iteratively find the optimal values. APPENDIX E MAP SOLUTION FOR THE ADDITIVE MODEL With the assumptions indicated in Section 4.4, the posteriori probability can be rewritten as: K p(ao,o ...K al .aK) 1 =l (ak jk,ao)p(k)P(ao) (al...aK) k=l Given the model, maximization of the posterior probability is equivalent to: K arg max j p(ak ao, k)p(cak)P(ao) a,o K k=1 We assume a uniform (flat) apriori distribution for ok. Since ao is constrained to have a unit norm, its a priori distribution is a Dirac delta function: (1 ao 12) . If we assume that uk is zeromean i.i.d. Gaussian noise with the same covariance matrix d2I across all the trials, maximization of the posterior probability can be further simplified: K K arg max log j p(ak ao,, k)p(a,)= arg max ak ( ak a )+ log0 ( ao 1) ao, U uK k=l ao, 'K k=l This can be converted to a constrained optimization problem: K min J Iak ka k=l S.t. Ia02 =1 Setting the derivative to zero, a necessary condition for minimum is: o = ak o ak. Plug this into the above cost function. We have: K 2 K J= ak oT ak oz k ao o k=l k=l Zak (a ao ao a) ka k=1 K ak T a. ao T )ak kI k=1 K K k T ak a k k T a k=l k=l The first term does not depend on ao, so MAP is equivalent to: arg max af A ao, ao S.t. ao 2=1 K K where A = Ak = k a ak a The matrix A is symmetric, so it can be diagonalized. k=l k=1 The maximum occurs when ao is the normalized eigenvector of the matrix A corresponding to the largest eigenvalue. APPENDIX F NORMALIZED ADDITIVE NOISE MODEL 1: MAP SOLUTION Following the same rationale in model 1, we attempt to derive the MAP solution for the model 2. The difference is in the covariance matrix of conditional probability: p(ak I ao, k) N(0, rk21) So, p(ak ac,)= exp a,a /2) (2z )D 0_k k k akao The MAP becomes: K arg max j p(ak a,,k)p(k)p(ao) ao,01 (K k=1 =argmax ( ak,/k a )+logS(a, 2 i) ao,,0i K k=1 where, we have used the prior distribution for Uk: p(k) = k D This is an improper prior (but still a prior for Bayesian inference). It is mainly motivated by analytical tractability. Similarly as in model 1, we can write the following constrained optimization problem: K argmin Yak,/,a ao, k k=l S.t. Ilaoz =1 We first find a necessary condition for minimum: o = a kT ak/ao ak . T Let Bk = akak Note that Bk is idempotent, i.e., BkBk = Bk ak *ak Plug this into the above cost function. 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BIOGRAPHICAL SKETCH Ruijiang Li was born in Shandong, China. He received the B.S. degree in automation with emphasis on systems and control in 2004, from Zhejiang University, Hangzhou, China. Since 2004, he has been working toward his Ph.D. at the Electrical and Computer Engineering Department at the University of Florida, under the supervision of Jose C. Principe. His current research interests include statistical signal processing, machine learning and their applications in biomedical engineering. PAGE 1 1 SPATIOTEMPORAL FILTERING METHODOLOGY FOR SINGLETRIAL EVENTRELATED POTENTIAL COMPONENT ESTIMATION By RUIJIANG LI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Ruijiang Li PAGE 3 3 To all scientists and researchers, who ha ve lived in pursuit of knowledge, and have dedicated themselves to the advancement of science PAGE 4 4 ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor Dr. Jose C. Principe, for his great inspiration and encouragement thr oughout the course of my research. Not just that. He has really become a mentor and guide during pivotal times of my life, which I would have to say regretfully that I did not take full advantage of. One could ask for no more from such an advisor. I wish to thank the members of my committ ee, Dr. John Harris, Dr. Jianbo Gao, and Dr. Mingzhou Ding, for their valuable time and interest in serving on my supervisory committee, as well as their comments, which helped improve the quality of this dissertation. I am grateful for Dr. Andreas Keils expertise on psychology as well as his support, which made our collaboration fruitful. I would like to thank my friends and colle agues at the Computational NeuroEngineering Laboratory. They have made my stay in Florida during the pa st four years an enjoyable experience. Last but not least, I wish to thank my parents, who raised me up. Without them, all is in vain. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............10 CHAPTER 1 INTRODUCTION..................................................................................................................12 1.1 Basic Concepts of the ERP..........................................................................................12 1.1.1 Generation of the ERP.......................................................................................12 1.1.2 The ERP Components........................................................................................13 1.2 Estimation of the ERP..................................................................................................16 2 SINGLETRIAL ERP ESTIMATION...................................................................................19 2.1 SingleTrial ERP Estimation Usi ng SingleChannel Recording.................................19 2.1.1 TimeInvariant Digital Filtering........................................................................20 2.1.2 TimeVarying Wiener Filtering.........................................................................20 2.1.3 Adaptive Filtering..............................................................................................21 2.1.4 Kalman Filtering................................................................................................22 2.1.5 Subspace Projection and Regularization............................................................22 2.1.6 Parametric Modeling..........................................................................................23 2.1.7 Other Methods Using SingleChannel Recording.............................................24 2.2 SingleTrial ERP Estimation Usi ng MultiChannel Recording...................................25 2.2.1 Generative EEG Model......................................................................................25 2.2.2 What Is a Spatial Filter and What Can It Do?...................................................27 2.3 Review of Spatiotempor al Filtering Methods..............................................................28 2.3.1 Principal Component Analysis (PCA)...............................................................28 2.3.2 Independent Component Analysis (ICA)...........................................................32 2.3.3 Spatiotemporal Filtering Methods for the Classification Problem....................36 3 NEW SPATIOTEMPORAL FILTERI NG METHODOLOGY: BASICS.............................40 3.1 Spatial Filter as a Noise Can celler in the Spatial Domain...........................................40 3.2 Deterministic Approach...............................................................................................42 3.2.1 Finding Peak Latency........................................................................................42 3.2.2. Finding Scalp Topography and Peak Amplitude...............................................44 3.3 Stochastic Approach....................................................................................................46 3.4 Simulation Study..........................................................................................................48 3.4.1. Gamma Function as a Template for ERP Component.......................................49 PAGE 6 63.4.2. Generation of Simulated ERP Data...................................................................49 3.4.3 Case Study I: Comparison with Other Methods................................................50 3.4.4 Case Study II: Effects of Mismatch...................................................................53 4 ENHANCEMENTS TO THE BASIC METHOD..................................................................59 4.1 Iteratively Refined Template.......................................................................................59 4.2 Regularization..............................................................................................................61 4.2.1 Constrained Optimization..................................................................................61 4.2.2 Unconstrained Optimization..............................................................................64 4.3 Robust Estimation: the CIM Metric.............................................................................65 4.4 Bayesian Formulations of the Topography Estimation...............................................68 4.4.1 Model 1: Additive Noise Model........................................................................69 4.4.2 Model 2: Normalized Additive Noise Model....................................................70 4.4.3 Model 3: Original Model...................................................................................71 4.4.4 Comparison among the Three Models...............................................................71 4.4.5 Online Estimation..............................................................................................72 4.5 Explicit Compensation for Tem poral Overlap of Components...................................72 5 APPLICATIONS TO COGNITIVE ERP DATA..................................................................90 5.1 Oddball Target Detection.............................................................................................90 5.1.1 Materials and Methods.......................................................................................90 5.1.2 Estimation Results.............................................................................................92 5.1.3 Discussions........................................................................................................98 5.2 Habituation Study........................................................................................................99 5.2.1 Materials and Methods.......................................................................................99 5.2.2 Estimation Results.............................................................................................99 6 CONCLUSIONS AND FUTURE RESEARCH..................................................................110 6.1 Conclusions................................................................................................................110 6.2 Future Research.........................................................................................................112 APPENDIX A PROOF OF VALIDITY OF THE PEAK LATENCY ESTIMATION IN (310)................114 B GAMMA FUNCTION AS AN APPROXIMATION FOR MACROSCOPIC ELECTRIC FIELD...............................................................................................................115 C DERIVATION OF THE UPDATE RULE FOR THE CONSTRAINED OPTIMIZATION PROBLEM..............................................................................................116 D DERIVATION OF THE UPDATE RU LE FOR THE UNCONSTRAINED OPTIMIZATION PROBLEM..............................................................................................117 E MAP SOLUTION FOR THE ADDITIVE MODEL............................................................118 PAGE 7 7 F NORMALIZED ADDITIVE NOISE MODEL 1: MAP SOLUTION.................................120 G NORMALIZED ADDITIVE NOISE MODEL 2: MAP SOLUTION.................................122 LIST OF REFERENCES.............................................................................................................123 BIOGRAPHICAL SKETCH.......................................................................................................135 PAGE 8 8 LIST OF TABLES Table page 31 Latency estimation: mean and standard deviation.............................................................55 32 Amplitude estimation: mean and standard deviation.........................................................55 33 Scalp topography estimation: correlation coefficient........................................................55 34 Effects of mismatch I: SNR = 20dB.................................................................................56 35 Effects of mismatch II: SNR = 10dB...............................................................................56 41 Latency estimation: mean and standard deviation.............................................................75 42 Amplitude estimation: mean and standard deviation.........................................................75 43 Scalp topography estimation: correlation coefficient........................................................75 44 Estimation results for the iterati vely refined template method..........................................76 45 Estimation with MCC for the mismatch case at SNR = 0dB.............................................76 46 Estimation with MCC for the mismatch case at SNR = 20dB.........................................76 47 Amplitude estimation for three Bayesian models..............................................................77 48 Scalp topography estimation for three Bayesian models...................................................77 51 Correlation statistics for the 4 subj ects: Scalp topography estimation............................102 52 Regression statistics for response time and estimated amplitude....................................102 PAGE 9 9 LIST OF FIGURES Figure page 31 Gamma functions with differe nt shapes and scales...........................................................57 32 Waveforms of synthetic and presumed ERP component...................................................58 41 Mean and standard deviation of the es timated amplitude under different SNR conditions..................................................................................................................... ......78 42 The waveforms of the synthetic component, presumed template and refined template under 4 SNR conditions.....................................................................................................79 43 Waveforms of two overlapped comp onents used in regularization...................................80 44 Scalp topography of two overlapped ERP co mponents used in regularization.................81 45 Amplitude and scalp topography estimati on I with regularization (constrained optimization) under 3 SNR conditions..............................................................................82 46 Amplitude and scalp topography estimati on II with regularization (constrained optimization) under 3 SNR conditions..............................................................................84 47 Amplitude and scalp topography estimati on I with regularization (unconstrained optimization) under 3 SNR conditions..............................................................................86 48 Amplitude and scalp topography estimati on II with regularization (unconstrained optimization) under 3 SNR conditions..............................................................................88 51 Pictures used in the experiment as stimuli.......................................................................103 52 Cost function in (3.9) versus time lag fo r different regularization parameters for subject #2..................................................................................................................... ....104 53 Estimated pdf of time lags corresponding to local minima of the cost function in (39) using the Parzen window ing pdf estimator with a Gaussian kernel size of 4.2..........105 54 Scalp topographies for the four subjects..........................................................................106 55 Scatter plot of the response time versus the estimated amplitude for each single trial for the four subjects. .......................................................................................................107 56 Estimated scalp topography for mixed and habituation phase.........................................108 57 Estimated amplitude for mixed and habituation phase....................................................109 PAGE 10 10 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SPATIOTEMPORAL FILTERING METHODOLOGY FOR SINGLETRIAL EVENTRELATED POTENTIAL COMPONENT ESTIMATION By Ruijiang Li December 2008 Chair: Jose Principe Major: Electrical and Computer Engineering Eventrelated potential (ERP) is an importa nt technique for the study of human cognitive function. In analyzing ERP, the fundamental probl em is to extract the waveform specifically related to the brains response to the stimulus from electroen cephalograph (EEG) measurements that also contain the spontaneous EEG, whic h may be contaminated by artifacts. A major difficulty for this problem is the low (typically negative) signaltonoi se ratio (SNR) in EEG data. The most widely used tool analyzing ERP has been to average EEG measurements over an ensemble of trials. Ensemble aver aging is optimal in the least s quare sense provided that the ERP is a deterministic signal. However, over four deca des of research have sh own that the nature of ERP is a stochastic process. In particular the latencies and the amplitudes of the ERP components can have random variation between repetitions of the stimulus. Under these circumstances, estimation of the ERP on a singletr ial basis is desirable. Traditional singletrial estimation methods only consider the time course in a single channel of the EEG. With the advent of dense electrode EEG a number of spatiotemporal filtering methods have been proposed for the singletrial estimation of ERP using multiple channels. PAGE 11 11 In this work, we introduce a new spatiotemporal filtering method for the problem of singletrial ERP component estimation. The met hod relies on modeling of the ERP component local descriptors (latency and amplitude) and thus is tailored to extract faint signals in EEG. The model allows for both amplitude and latency variability in the act ual ERP component. The extracted ERP component is cons trained through a spat ial filter to have minimal distance (with respect to some metric) in the temporal domai n from a template ERP component. The spatial filter may be interpreted as a noi se canceller in the spatial do main. Study with simulated data shows the effectiveness of the proposed method to signal to noise ratios down to 10 dB. The method is also tested in real ERP data from cognitive experiments where the ERP are known to change, and corroborate experiment ally the expected behavior. PAGE 12 12 CHAPTER 1 INTRODUCTION 1.1 Basic Concepts of the ERP Eventrelated potential (ERP) is an impor tant and wellestab lished technique for neuroscientists and psychologists to study human c ognitive function. In this section, we briefly review some of the basic facts and concepts rela ted to ERP generation and analysis (Coles et al. 1995). 1.1.1 Generation of the ERP When a pair of electrodes are attached to th e surface of the human scalp and connected to a differential amplifier, the output of the amplifier reveals a pattern of volta ge variation over time. This voltage variation is known as the electroencephalograph (EEG). The amplitude of the normal EEG varies between approximately V and most of the EEG frequency contents range between 0.5Hz and 40Hz. Here, we do not review the recordi ng techniques of EEG (Ruchkin 1987). If we present a stimulus to a human subj ect while recording the EEG, we can define a period of time (an epoch or a trial) where some of the EEG components are timelocked to the stimulus. Within this epoch, there may be voltage changes that are specifically related to the brains response to the stimulus. These voltage changes constitute the eventrelated potential, or ERP. Although it is not completely understood how the measurements at the scalp relate to the underlying brain activity, the following points app ear to be clear and are generally accepted (Scherg and Picton 1991, Wood 1987). First, ERP recorded from the scalp represents net electrical fields associated with the activity of sizeable neuron populations. These neuron populations act as current sources whose electrical fields pr opagate to the entire scalp through PAGE 13 13 volume conduction. Second, the individual neurons that compromise such a population must be synchronously active and have a certain geom etric configuration to produce measurable potentials at the scalp. In particul ar, these neurons must be configur ed in such a way (usually in a parallel orientation) that their individual fields summa te to yield a dipolar field. Therefore, the ERP recorded at the scalp is se lective of the totality of the brain activity. This is advantageous in that the resultant measur ements would otherwise be so complex as to be difficult or impossible to analyze. On the othe r hand, we should also be aware that there are certainly numerous functionally important neural processes that cannot be detected by the ERP technique. 1.1.2 The ERP Components The issue of ERP components has aroused much controversy among the ERP research community, particularly the ques tion of the definition of an ERP component. Suppose for the moment that we have obtained the ERP usi ng some method. A simple way to define a component is to focus on some feature of the resu lting waveform (for instance, a peak or trough), and this feature becomes the component of interest. Some common features include the amplitude and latency parameters of a particular peak or trough. A major problem with the simple approach mentioned above is component overlap, both spatially and temporally. Since the brain is a conducting medium, activ ity generated in one spatial location may be propagate d through the brain tissue and app ear at other locations. Thus, the waveform we observe by measuring the voltage at the scalp may well be attributed to a variety of different sources in different spatial locations of the brain. One consequence of volume conduction is that there need be no direct correspondence between the timing of the distinctive features of an ERP waveform (peaks and troughs ) and the temporal characteristics of the underlying neural systems. For instance, an ERP p eak with a latency of 300 ms, might reflect the PAGE 14 14 activity of a single neural generator maximally ac tive at that time, or the combined activity of two (or more) neural generators, maximally act ive before and after 300 ms, but with fields summating to a maximum at that time. Due to these ambiguities surrounding the interpretation of peaks and troughs in ERP waveforms, other definitions for ERP components have been proposed. Naatanen and Picton (1987) adopted what might be called the physiol ogical approach to component definition. They proposed that a defining characte ristic of an ERP component is its anatomical source within the brain. According to this defin ition, to measure a particular ER P component, we must have a method of identifying the contributing sources. Dochin (1979, 1981) adopted what might be called the functional approach to ERP component definition, which is concerned more with the information processing operations with which a pa rticular component is correlated. According to this definition, it is entirely possible for a componen t to be identified with a particular feature of the waveform that reflects the activity of multiple gene rators within the br ain, so long as these generators constitute a functionally ho mogeneous system (Coles, et al.1995). Although the above physiological and psychological approaches to component definition seem to be counteractive, for many investigat ors it is more appropr iate to combine both approaches. A classical approach to component definition, was pr oposed by Dochin et al. (1978). They argued that an ERP component should be defined by a combination of its polarity, its characteristic latency, its dist ribution across the scalp and its sensitivity to characteristic experimental manipulations. Noti ce that polarity and scalp distri bution imply a consistency in physiological source, while latency and sensit ivity imply a consistency in psychological function. PAGE 15 15 ERP components can be broadly classified into two types: exogenous and endogenous components. Characteristics of the exogenous components (amplitude, latency and scalp projection) largely depend on the physical properties of sensory stimuli, such as their modality and intensity. On the other hand, endogenous com ponents largely depend on the nature of the subjects interaction with the stimulus. These co mponents vary as a function of such factors as attention, task relevance and the nature of the information proc essing required by the stimulus. The dichotomy of the exogenousendogenous distin ction turned out to be an oversimplified version of the reality. Many early sensory components have been shown to be modifiable by cognitive manipulations (e.g., attention) and many of the later cognitive components have been shown to be influenced by the physical attributes of stimulus (e.g., modality). In what follows, we briefly discuss one particular wellknown ERP component, the P300. For a comprehensive review on other wellknown components, we refer to Coles, et al. 1995. The P300 is probably the most important and th e most studied compone nt of the ERP. It was first described in the 1960s by Sutton et al (1965). The P300 is evoked by a task known as the oddball paradigm. During this task a series of one type of fre quent stimuli is presented to the experimental subject. A different type of nonfrequent (target) stimulus is also presented. The task of the subject is to react to the presence of target s timulus by a given motor response, typically by pressing a button, or just by mental counting to the target stimuli. Virtually any sensory modality (auditory, visual, somatosensor y, olfactory) can be us ed to elicit the P300 response. (Polich 1999). The shape and latency of the P300 differs with each modality. This indicates that the sources generating the P 300 differ and depend on the stimulus modality (Johnson 1989). PAGE 16 16 There are several theories on the neural pro cesses underlying the origin of the P300. The most cited and most criticized theory was proposed by Donchin and Coles (1988a, b). According to their theory, the P300 reflect s a process of context or memo ry updating by which the current model of the environment is modified as a function of incoming information. Several investigators (e.g., Johnson 1986) have pointed out th at the P300 does not appear to be a unitary component, and instead may represent the activity of a widely distributed system which may be more or less coupled depending on the situati on. More information about the underlying neural systems is required before a consensus is atta ined about the functiona l significance of this component. For a more recent review on the research of P300, we refer to Linden, 2005. 1.2 Estimation of the ERP The fundamental problem in the analysis of ER P is to extract the si gnals that are brains specific response to the stimulus from the EEG meas urements that also cont ain noise. By noise, or the background EEG, we mean the electrical activities from hear t, muscles and eye movements as well as the spontaneous brain activiti es that are not related to brains response to the stimulus. A major difficulty with the extraction of ERP is that, in most cases, ERP signals are small (on the order of microvolts) relative to the b ackground EEG (on the order of tens of microvolts) in which they are embedded. For this reason, it is necessary to employ signal processing techniques to estimate the ERP si gnals in the presence of noise. By far the most commonly used techni que has been the averaging of the EEG measurements over an ensemble of timelocked epochs. This is optimal in the mean square sense, given the assumption that the ERP is a de terministic signal timelocked to the stimulus and the additive background EEG is zeromean and uncorrelated with the ERP. PAGE 17 17 However, for over four decades it has been evid ent that the nature of ERP is more or less random. In particular, the amplitudes and latenc ies of the peaks in the ERP can have random variations between repetitions of the stimuli (Brazier 1964). In addition, the variations may be trendlike and the mean of the amplitudes and the latencies can change across the trials. Under these circumstances, the information regarding to these variations in ERP is lost through averaging. Furthermore, the average waveform may not, in fact, resemble the actual ERP waveform that is recorded in an individual trial. The resulting estimates for the ERP, therefore, may not correspond to the underlying neural processes and inferen ce about the cognitive function may be misleading. Estimation of the ERP on a singletrial basis is desired for the situations when the peak amplitude and latency of a particular component change significantly across trials. A major difficulty with singletrial ERP estimation is agai n the very low signaltonoiseratio (SNR) in the singletrial EEG, typically lower than 10dB. Statistically speaking, the aver age ERP, or the sample mean, is an example of the use of the first order statistic s, where only the first order moment of the population parameters is estimated. The next obvious improvement is to use the second order statis tics, i.e., covariance analysis. The most common approach is to form an estimator (filter) with which the unwanted contribution of the background EEG can be filtered out. To find such an estimator, some models or assumptions are imposed on the ERP and b ackground EEG concerning their respective second order statistics. The estimator that satisfies th e minimum mean square criterion can then be derived. The performance of the estimator th en largely depends on how realistic these assumptions are. PAGE 18 18 For historical reasons, these traditional sing letrial estimation met hods only consider the time course of a single recordi ng channel in the EEG. In some cases, simple ERP components, e.g., the brainstem auditory evoked potentials, can be adequately examined using a single channel. However, for most ERPs, simultaneous recording from multiple electrode locations is necessary to disentangle overl apping ERP components on the ba sis of their topographies, to recognize the contribution of ar tifactual potentials to the ER P waveform, and to measure different components in the ERP that may be optima lly recorded at different scalp sites (Picton et al. 2000). Today, highdensity EEG can simultaneously record scalp potentials in up to 256 electrodes. This increased number of sensors and thus increased spatial resolution has created a need for signal processing methods that can simu ltaneously analyze the time series of multiple channels. Recently, various methods have been pr oposed for singletrial analysis that linearly combine the time series in multiple channels to generate a representation of the observed data that is easier to interpret (Chapman and Mc Crary 1995, Makeig et al. 1996, Parra et al. 2002). This linear projection combines the information from all the available sensors into a single channel with reduced interferen ce from other neural sources and may provide a better estimate of the underlying neural activity th an the EEG measurements in a single channel. The linear projection, in this sense, may be called a spatial filter and these methods can be generally called spatiotemporal filtering methods. We will review these and othe r singletrial estimation methods in detail in Chapter 2. PAGE 19 19 CHAPTER 2 SINGLETRIAL ERP ESTIMATION In this chapter, we review the existing methods for singletrial ERP estimation. The methods are broadly categorized into two classes: those based on singlechannel EEG recording and those based on multichannel EEG recording. Methods based on singlechannel recording rely solely on the modeling of temporal char acteristics of the ERP and EEG, while methods based on multichannel recording inve stigate both the spatial and temporal characteristics of the ERP and EEG, and are termed with the genera l notion of spatiotemporal filtering methods. 2.1 SingleTrial ERP Estimation Using SingleChannel Recording For the estimation of ERP using a single channel, all the available information is contained in a single time series () x t: the EEG recording at a certain electrode. We assume that the measurements consist of two parts: the signal of interest (ERP) and additive noise (background EEG), denoted by () st and () nt, respectively. The observation model for the EEG can be written as: ()()() x tstnt (21) In this review, the EEG measurements for a single trial x is a finitelength vector with elements sampled from the origin al continuoustime waveform. When the time series in (21) are interpreted as stochastic processes, x becomes a random vector. Its length equals the number of samples in one trial. In vector form, the observation model is: x=s+n (22) Here we do not attempt to give an exhaus tive review on the topi c of singletrial ERP estimation with a single channe l. For other reviews on the gene ral ERP estimation problem, we refer to Aunon et al. (1981), Ruchkin (1987), McGille m and Aunon (1987), Silva (1993), Karjalainen (1997). PAGE 20 202.1.1 TimeInvariant Digital Filtering Digital filtering is a good place to start for time series analysis. The simplest approach is to design digital filters that have a desired fre quency response. Ruchkin and Glaser (1978) used simple moving average FIR filters to estimate ER P on a single trial basis. More complicated ones may be designed to estimate some partic ular component such as P300 (Farwell, et al. 1993). Wiener filtering may also be used to estimate sing letrial ERP, provided that the power spectra of the ERP and background EEG can be estimated appropriately (Aunon and McGillem 1975, Cerutti, et al. 1987). The major problem with linear timeinvariant filt ering is the fact that the ERP is typically a transient and smooth waveform w ith no periodicity. The spectrum for this kind of signal is not defined properly. Consequently, the spectra of the ERP and background EEG (if they are estimated) were usually found to overlap significantly (Krieger et al. 1995, Spreckelsen and Bromm 1988, Steeger et al. 1983). Thus the application of digita l filters with constant frequency response is not expected to give desirable results in most cases. Th e effects of dig ital filtering are studied in (Ruchkin and Glaser 1978, Maccabee, et al. 1983, Nishida et al. 1993). 2.1.2 TimeVarying Wiener Filtering When the optimal Wiener solution is computed for singletrial EEG data, the filtering becomes timevarying with respect to each trial. In general, these methods necessitate some analytic model for the ERP. Yu and McGillem (1983) introduced what was called the timevarying minimum mean square error filter. A cr ucial task for their method is to obtain a good estimate for the crosscovariance between the ERP and measurements. Under the assumption that the ERP and background EEG are uncorrelate d, the crosscovariance becomes the autocovariance of the ERP signal itself. The ERP is parametrically modeled by a superposition of the PAGE 21 21 components with random location and amplitude. The parameters for the ERP are then calculated from the Wiener solution on a single trial basis. 2.1.3 Adaptive Filtering The use of adaptive filtering for the analysis of singletrial ERP, particularly the use of the least mean square (LMS) algorithm (Widrow 1985), was ex tensively studied during the 1980s (Madhavan et al. 1984, 1986; Vila et al. 1986; Thakor 1987; Doncarli 1988). For these methods, the measurement () x t is selected as the desired signal, and several choices are proposed for the input signal. Thakor (1987) is probably the one of th e most cited works among the adaptive filtering methods for ERP estimation. From the princi ples of adaptive noise cancellation, Thakor proposed a novel way of choosing reference a nd primary inputs. Two sets of singletrial measurements (),()ij x txt (, ij being the trial index) serve as the reference and primary inputs, respectively. The idea is to estimate the primar y input with a set of delayed version of the reference input on which some form of ensemble averaging is performe d. The criticism of Thakors work is summarized in (Madhavan 1988) where the author asse rted that if the ERP signal is assumed to be identical across trials, the above approach does not provide any signaltonoise ratio improvement and dist orts the signal at frequencie s where signal and noise power spectra overlap. Madhavan (1992) proposed a modified adaptive line enhancement method. In this method the prestimulus EEG data are adaptively modeled with an autoregressive (AR) model, which is then used to filter the poststimulus EEG da ta. The notion of modified means that a nonadaptive filter is used to process the poststimulus data. PAGE 22 222.1.4 Kalman Filtering AlNashi (1986) adopted the Kalman filtering approach for the ERP estimation problem. It is assumed that the ERP can be modeled as a deterministic signal with additive random noise. The additive noise is assumed to be an auto regressive moving average (ARMA) process and another ARMA model is used for the background EEG The scalar Kalman filter is then used to predict the singletrial ERP. The basic assumption for AlNashis approach is that the difference between the singletrial ERP and the ensemble av erage is a stationary process. This is not consistent with (Ciganek 1969), which found that the differences are usually larger in late components than in early components. Liberati et al. (1991) model the singletrial ERP as a timevarying AR process using the ensemble average data and mode l the background EEG as a stati onary AR process using the prestimulus data. The AR parameters are then used to create the state and observation equations with the ERP as the unknown states. The singletrial ERP is then estimated using the Kalman filter equations (Kalman 1960). 2.1.5 Subspace Projection and Regularization The subspace projection approach starts with the linear observation model: H x=s+n=+n (23) where, contains the parameters to be estimated a nd the ERP signal is constrained to lie in the subspace spanned by some basis vectors, namely the columns of the matrix H. If the ERP is assumed to consist of positive and negative humps, sampled Gaussian functions may be a good choice for the basis vectors instead of a generic basis (e.g., polynomials). An alternative is to choose the eigenvectors of the EEG data autocorrelation matrix that correspond to the firs t few largest eigenvalues. This is motivated by the fact that the eigenvectors constitute a basis se t with the minimum number of ba sis vectors that are required to PAGE 23 23 model the ERP, assuming that the ERP span s nearly the same subspace with the EEG measurements. The least square solution with th is basis set is equivalent to the principal component regression approach (Lange 1996, Karjalainen 1997). The above two basis sets may be combined in to a single criterion, w ith the Gaussian basis vectors modeling the ERP, and the subspace span ned by the eigenvectors representing the prior information about the problem. This leads to the subspace regularization method, which is closely related to the Bayesian mean square estimation (Karjalainen et al. 1999). The ERP may also be estimated recursively using Kalman filtering (Karjalainen et al. 1996). 2.1.6 Parametric Modeling There is one type of parametric models, which uses damped sinusoids as basis function for the modeling of singletrial ERP. The model for th e ERP with additive noise can be written as: 1()sin()()p t iii i x tAtnt (24) Estimation of the parameters ,,iiiA is a nonlinear problem, which can be solved with an approximation method called Pronys method (Marple 1987). Its use with generalized singular value decomposition was proposed by Ha nsson and Cedholt (1990) and Gansler and Hansson (1991). The Pronys method was utilized by Hansson et al. (1996) for the estimation of singletrial ERP and robust performance was ach ieved for EEG data with SNR>10dB. A more recent improvement, called piecewise Pronys me thod was proposed by Garoosi and Jansen (2000) to deal with nonstationary characteristics of the sinusoids. Another well studied tool that can be used for the analysis of singletrial ERP is the wavelet transform (Daubechies 1992 ). Wavelets provide a tiling of timefrequency space that gives a balance between time and frequency re solution and they can represent both smooth signals and singularities. This makes them suita ble models for the anal ysis of transient and PAGE 24 24 nonstationary signals like th e ERP (Thakor 1993; Schiff et al. 1994; Samar 1995; Coifman 1996; Basar et al. 1999; Effern et al. 2000; Quian and Garcia 2003). The idea is based on a technique called wavel et shrinkage or wavel et denoising, which can automatically select an appropriate subset of basis functions and the corresponding wavelet coefficients. This relies on the property that natura l signals, such as images, neural activity, can be represented by a sparse code compromising onl y a few large wavelet coefficients. Gaussian noise, on the other hand, compromises a full set of wave coefficients whose size depends on the noise variance. By shrinking these noise coeffi cients to zero using a thresholding procedure (Donoho and Johnstone 1994), one can denoise data. However, the application of the wavelet method to singletrial ERP anal ysis requires some form of ensemble averaging in order to derive an optimal wavelet basis set that is tuned to the ERP signal. Sometimes the appropriate selection of the ERP ensemble may be a difficult task due to the effects of internal and exte rnal experimental parameters (Effern et al. 2000). 2.1.7 Other Methods Using SingleChannel Recording Some methods exist that try to explicitly estimate the latencies of the singletrial ERP components. A simple approach is to use cr osscorrelation of the signal with a template waveform and find the maximum point of the correlation (Gratton et al. 1989). Pham et al. (1987) applied a maximum likelihood (ML) method to estimate the latencies of ERP assuming a constant shape and amplitude. The ML method was extended in (Jaskowski and Verleger 1999) incorporating variable amp litude into consideration. Truccolo et al. (2003) devel oped a Bayesian inference framework for estimation of singletrial multicomponent ERP termed differentially variable component analysis(dVCA). Each component is assumed to have a trialinvariant waveform with trialdependent amplitude scaling factors and latency shifts. A Maxi mum a Posteriori solution of this model is implemented via an PAGE 25 25 iterative algorithm from which the components waveform, singletrial amplitude scaling factors and latency shifts are estimated. The method wo rks well for relatively lowfrequency and largeamplitude eventrelated components. 2.2 SingleTrial ERP Estimation Using MultiChannel Recording The use of multichannel recording for the esti mation of singletrial ERP gave rise to a number of spatiotemporal filt ering methods. These methods a ssume, either explicitly or implicitly, a generative EEG model, which we will introduce in the next section. We then explain what is meant by a spatial filter and illustrate what it can do for us in estimating ERP on a singletrial basis. A review on existing spatiote mporal filtering methods is furnished in the following section, where we concentrate on me thods that are based on wellknown statistical principles. 2.2.1 Generative EEG Model We start with the neural ge nerator assumption of EEG data i.e., neuron populations in cortical and subcortical brain tissues act as current sources (Caspers et al. 1980, Sams 1984). Within the EEG frequency range (below 100Hz), br ain tissues can be assu med to be primarily a resistive medium (Reilly 1992). Thus, according to Oh ms law, the electrical potentials collected at each sensor (channel) as a result of volume co nduction, is basically a linear combination of neural current sources (and nonne ural artifacts). The linear generative model for EEG data can be written in matrix form: X=AS (25) or: 1N T ii iX=as (26) where, N is the number of current sources. PAGE 26 26 We denote the singletrial EEG data with a DT matrix X with D channels and Tsamples; S is a NT matrix with each row T is representing the time course of the current density of the i th current source; A is an unknown DN matrix. Strictly speaking, the number of the neural current sources N is necessarily much larger than the number of channels D It is usually assumed that the numbers of sources and sensors are equal for the purpose of convenience. The column vector ia of the matrix A represents the projection of the i th current source to each sensor at the scalp and is called the forward model associated with the source. This scalp projection is generally unknown and depends on the location and orientation of the dipolar current source as well as the conductivity distri bution of the underlying brai n tissues, skull, skin and electrodes (Parra et al. 2005). Thus, if the scalp projection can be estimated, it may provide us some further evidence to the neurophysiological significance of the corresponding estimated source. An equivalent way to write the generative EEG model (25) is to us e the notation of time series: ()() tt xAs (27) where, 1()(),,()T Dtxtxt x is a column vector representing the EEG recordings in D channels; 1()(),,()T Ntstst s is a column vector representi ng the time course of the current density of the sources. A is the same matrix defined as before. There is some degeneracy in the model, i.e ., the scaling factor of the current source T is and its corresponding scalp projection ia. In this case, either the current sources or the scalp projections are constrained to ha ve unit power to avoid ambiguity. We wish to point out that PAGE 27 27 there is no ambiguity whatsoever if we want to extract or eliminate from the EEG data the contribution of the i th current source, i.e., iT iiXas. 2.2.2 What Is a Spatial Filter and What Can It Do? To illustrate our point, we begin with a si mple example. Suppose we measure the EEG from 3 electrodes where only 3 sources are presen t. Using the time series notation (the numbers are selected for illustration purposes): 1123 2123 3123()()2()() ()2()()() ()()()2() x tststst x tststst x tststst (28) We would like to recover each of the three so urces using the EEG measurements from all the available sensors. Sine the measurements ar e linearly related to the current sources, we speculate that this could be done by linearl y combining all the EEG measurements through a weight vector w: 1()()()D T ii i y ttwxtwx (29) In fact, if we select the weight vector in (29) as: [1,3,1]TT w, we will get, 1231()()3()()4() y txtxtxtst (210) which is exactly the first current source with a scaling factor. The other two sources can be recovered by using the weight vectors: [3,1,1],[1,1,3]TT respectively. Of course the above example is simplistic, be cause in reality, there are certainly numerous current sources simultaneously active in the brain and the number of sensors is usually up to a few hundred. We also do not know an y of the elements of the matrix A in general. However, the following point should be clear: by combining the EEG measurements fr om multiple channels with a simple weight vector, we are able to r ecover (or estimate) many sources of interest that PAGE 28 28 could not be recovered using a sing le channel. In principle, the re jection of the interferences can be perfect, as shown in (210) if the coefficients are known. The weight vector w in (29), which operates on the EEG measurements in the sensor space, is called a spatial filter Just like a filter operating in th e time domain, a spatial filter can have either lowpass or highpass characteristics in the spatial fr equency domain. For instance, a spatial filter summing the measurements from a group of neighboring se nsors have a lowpass characteristic; the use of a single channel recording corresponds to a highpass spatial filter with an impulse response attenuating the data from all the other channels to zero. The selection of the spatial filter w is usually based on some constraints or desired characteristics of the output () yt. Different constraints will generally lead to different methods of extracting the outputs. Loosely speaking, ma ximum power of the output s leads to principal component analysis (PCA); statistical indepe ndence among the outputs leads to independent component analysis (ICA); and maximum difference between the outputs leads to linear discriminant analysis (LDA). We will review these and other spatiotemporal filtering methods in the following section. 2.3 Review of Spatiotemporal Filtering Methods In this section, we provide a review on exis ting spatiotemporal filtering methods for singletrial ERP analysis. Particularly, we focus on tw o popular and well established methods, namely, principal component analysis (PCA), independent component analysis (ICA). 2.3.1 Principal Component Analysis (PCA) From the early days of cognitive ERP resear ch, principal component analysis (PCA) was already proposed as a linear, multivariate data reduction approach (Donchin, 1966). Since then, PCA has been one of most widely used tools among psychologists for ERP analysis (Glaser and PAGE 29 29 Ruchkin, 1976; Donchin and Heffley, 1978; Mo cks and Verleger 1991; Chapman and McCrary 1995; van Boxtel 1998). By identifying unique vari ance patterns in a given set of ERP data, PCA decomposes the variance structure of the observed data into a set of latent variables that ideally correspond to the individual ERP components. In the ERP research community, these latent variables are usually called fact ors instead of components to avoid confusion with the ERP components. Among the vast literatures on PCA applied to ERP analysis, one classical method using the PCAVarimax strategy, is particularly popular an d is the primary analytic tool for many ERP researchers (Gaillard and Ritter 1983). The method treats the recorded potential at a given time of the EEG epoch as variables. The domain of the observations is taken to be the Cartesian product of the recording channels, experimental conditions, participants. Suppose we have T samples in a given EEG epoch, D recording channels, C experimental conditions and P participants. The data matrix for this method has a dimension of TDCP This particular arrangement of the data matrix leads to the socalled temporal PCA approach, which gives orthogonal factors (eigenvectors of the covariance matrix). The PCA solution is then followed by the Varimax rotation (Kaiser, 1955). The Varimax rotation is an orthogonal rotation that aims to maximize the values that are large for a factor and minimize the values that are small (by maximizing the fourth power the factor). Th is corresponds to the maximum compactness criterion, which will make the new factors have a small number of large values and a large number of zero (or small) values. This is reas onable for ERP estimation because for the most part, ERP components appear to be monophasic and compact in time. PAGE 30 30 It is easy to see that the above PCAVarimax ap proach is a spatiotemporal filtering method. We denote ,cpX as the singletrial EEG data defined in (25) from c th experimental condition and p th participant. Then the covariance matrix is: 21TD CXX (211) where, 111PC cp pcPCXX (212) Formally, PCA is equivalent to the singular value decomposition (SVD) of the data matrix defined in (212), which is the average EEG data matrix for all experimental conditions and all participants. Suppose we have the SVD of X as follows: TUV X (213) where, UV are orthogonal matrices of dimension DD and TT and contain the leftsingular and rightsingular vectors, respectively. contains the singular values of the data matrix. Equation (213) can be equivalently written as: TTUV X (214) The Varimax rotation procedure simply adds another DD orthogonal matrix R multiplied on the both sides of (214): TT R URV X (215) The right side of (215) is a DT matrix, whose rows can be seen as the factors extracted by the PCAVarimax approach. We define a DD matrix: PAGE 31 311 T T T D R U w W w (216) We further denote: 1 T T T D R V y Y y (217) Thus, we have: YWX (218) or: ,TT ii y wX for 1,, iD (219) This is the familiar form for the spatial filter defined in (29), which is now written in matrix form. Clearly, the matrix W is an orthogonal matrix. So PCA finds a number of (D) outputs that are uncorrelate d with the constraint that the sp atial filters are orthogonal. On the other hand, the PCAVarimax method searches for outputs that are maximally compact in time while still constraining the spat ial filters to be orthogonal. In the context of the generative EEG model, PCA basically assumes that there are equal number of sources and channels. If we multiply 1 W on both sides of (218), we get: 1 XWY (220) Thus, the rows of the output matrix Y contain the time course of the current sources, while the columns of the matrix 1 W constitute the scalp projections of the corresponding sources. This means that the scalp projections for th e underlying current source s are orthogonal to each other, which is a high ly dubious assumption. PAGE 32 32 Due to the above problem, an oblique rota tion like Promax (Hendrickson and White 1964) has been proposed as a postprocessing stage after Varimax, to relax the orthogonality constraint on the scalp projections. St udies with both simulated and real dataset have shown that temporal PCA with Promax extracted markedly more accurate ERP components (Dien 1998). An alternative approach to the popular temporal PCA is the spatial PCA (Duffy et al. 1990, Donchin 1997, Spencer, et al. 1999), which treats the recorded potential at a given channel as variables. The EEG data matrix is formed with channel as one dimension, and time by experimental condition by participant as the ot her dimension. The same rotation procedures follow as in temporal PCA. However, spatia l PCA still assumes orthogonality of the scalp projections. Two other welldocumented problems for th e PCA approach are the misallocation of variance (Wood and McCarthy 1984) a nd the issue of latency jitter. Dien (1998) using extensive simulations, has argued that spa tial PCA as a complement to temporal PCA, together with parallel analysis (Horn, 1965) to identify noise factors, and oblique rotation to allow for correlated factors, can address thes e and other shortcomings of PCA. More recent developments include a combined spatial and temporal PC A approach that is successfully applied to real ERP data extracting known ERP components (Spencer et al. 1999, 2001). Dien et al. (2005) have presented a st andard protocol to optimi ze the performance of PCA when it is applied to ERP datasets, recommendi ng the use of covariance matrix over correlation matrix, and Promax rotation over Varimax rotation, etc. 2.3.2 Independent Component Analysis (ICA) ICA was originally proposed to solve th e blind source separation (BSS) problem, to recover a number of source signa ls after they are linearly mi xed and observed in a number of sensors, while assuming as little as possible a bout the mixing process and the individual sources PAGE 33 33 (Comon 1994). The most basic ICA model assu mes linear and instantaneous mixing, which means that the source signals arrive at the se nsors without time delay and are mixed in the sensors linearly with other source signals. This basic ICA model naturally fits into the generative EEG model in (25), which we repeat here, a ssuming that the sources and measurements are random vectors: x=As (221) In the ICA literature, the observation x is called mixtures, and the unknown matrix A is called mixing matrix. Usually it is assumed th at the number of sources is equal to the number of sensors. In this case, the mixing matrix becomes a square matrix. The key assumption used in ICA to solve the B SS problem is that the time courses of the sources are as statistically independent as possibl e. Statistical independence means that the joint probability density function (pdf) of the outputs factorizes. For the linear instantaneous BSS problem, the solution is in the form of a square demixing matrix W, specifying spatial filters that linearly invert the mixing pr ocess. If the mixing matrix is invertible, the outputs should be identical to the original source signals, excep t for scaling and permutation indeterminacies (Comon 1994): y Wx (222) There are a multitude of algorithms that ha ve been proposed to solve the basic ICA problem, among which, Infomax (Bell and Sejnowski 1995, Lee et al. 1999), FastICA (Hyvarinen and Oja, 1997), JADE (C ardoso, 1999), SOBI (Belouchrani et al. 1997) are probably the most widely used. Some algorithms are base d on the canonical informationtheoretic contrast function for ICA, i.e., mutual information, or it s approximations (Infomax, FastICA, etc.); others utilize higherorder statistics of the data (e.g., fo rthorder cumulant) to perform source separation PAGE 34 34 (JADE); still others make use of the difference in the temporal spectra of the source signals (SOBI). For a more detailed review on ICA and it s applications to BSS problems, we refer to the following: Cardoso (1998); Hyvarinen et al. (2001); Roberts and Ever son (2001); Cichocki and Amari (2002); James and Hesse (2004); Choi et al. (2005). Review papers comparing different ICA algorithms and their relationships are also available: Hy varinen (1999); Lee et al. (2000). We notice that while PCA requires the spatial fi lters to be orthogonal, here in the case of ICA, there is no more constraint on the spatial filters (or the demixing matrix W). On the other hand, while PCA only uses secondorder statistics (the covari ance matrix), to decorrelate outputs, ICA imposes a much stronger condition, i.e., independence on the outputs. The fact that ICA tries to factorize the joint pdf of the outputs implies that all the higherorder statistics (HOS) are taken into consideration by ICA. This mean s that for nonGaussian data, the structures contained in HOS (e.g., kurtosis), while totally ignored by PCA, may be captured by ICA. Since many natural signals are nonGau ssian distributed (e.g., spe ech signals usually follow a Laplacian distribution), ICA may be more suitable for this and other applications than PCA. Since Makeig et al. (1996) published their seminal pape r on the application of ICA to EEG data, there have been numerous studies during th e last decade dedicated to this research topic (Makeig et al. 1997, 1999, 2002, 2004; Vigario et al. 1998, 2000; Jung et al. 1999, 2000, 2001; Delorme et al. 2002, 2003, 2007; Debener et al. 2005). Until now, ICA and its variants still remains a powerful tool for the analysis of EEG and ERP data. The application of ICA to th e study of EEG data requires th at the following conditions be satisfied: (1), statistical independence of all the underlying neural source si gnals; (2), their linear instantaneous mixing at the sensors; (3), the stationarity of the mixing process. PAGE 35 35 Since most of the energy in EEG data lies below 100Hz, the quasistatic approximation of Maxwell equations holds. So there is (virtually) no propagation delay of the electrical potentials from the neuronal sources to the sensors thr ough volume conduction. Thus the assumption of instantaneous mixing is valid. The linearity of the mixing follows from the Maxwell equations as well. The stationarity of the mixing process co rresponds to a constant mixing matrix. For the dipole source model, this means that the dipola r neuronal sources should have fixed locations and orientations. Although there is no reason to believe that thes e neuronal sources are spatially fixed over time, for those that are involved in a specific in formation processing task and therefore are of interest to ERP researchers, they should at least ha ve a relatively stable configuration or a stable scalp projection, wh ich is congruent with the definition of ERP components as proposed by Fabiani et al. (1987). We have seen that conditions (2) and (3) ar e approximately valid for EEG data. The most debatable and perhaps perplexing condition is the first one: statistical independence of all the underlying neural source signals. Th e independence criterion applies solely to the amplitudes of the source signals, and does not correspond to any consideration of the morphology or physiology of the neural structures However, the different nature of the sources originated from completely different mechanisms often yields signals that appear to be statistically independent. Particularly, analysis of the dist ributions of artifacts such as th e cardiac cycle, ocular activity has shown the statistical inde pendence assumption approximately holds (Vigario 2000). Although ICA continues to be a useful tool fo r EEG and ERP analysis, there are also some limitations to it. First, ICA can decompose up to (or at most) D sources from data recorded at D scalp electrodes (D may be ranged from several dozen to a few hundred). On one hand, the researcher has to analyze the extracted D components one by one (including the time course and PAGE 36 36 scalp projection), which is laborious when is D large and the results are subject to interpretation. If he/she chooses to analyze onl y a part of the all the component s, the subsequent analysis is correlated with the retention criterion. (Note that PCA also has this problem). On the other hand, the effective number of statistically independent signals contributing to scalp EEG is almost certainly much larger than the number of electrodes D. Using simulated EEG data, Makeig et al. (2000) has found that given a large number of sources with a limited number of available channels, ICA algorithm can accurately identify a few relatively large sources but fails to reliably extract smaller and briefly active sources. This suggests that ICA decomposition in high dimensional space is an illposed problem. Second, the assumption of sta tistical independence used by ICA is violated when the training dataset is too small or separate topographi cally distinguishable phenomena nearly always cooccur in the data (Li and Principe, 2006) In the latter case, simulations show that ICA may derive a single component accounting for the cooccurring phenomena, along with additional components accounting for their separate activities (Makeig et al. 2000). These limitations imply that the results obtaine d with ICA must be va lidated by researchers using behavioral and/or physiological eviden ce before their functional significance can be correctly interpreted. Current research on app lications of ICA is focused on incorporating domainspecific knowledge into the ICA fram ework. Recently there has been work on combining ICA with the Bayesian approach (Tsai, et al. 2006) or with th e regularization technique (Hesse and James, 2006). 2.3.3 Spatiotemporal Filtering Method s for the Classification Problem It is worthwhile to point out a related but different approa ch, which is the (supervised) singletrial EEG classification pr oblem. It is generally less difficult than the (unsupervised) singletrial estimation problem in the sense that the availability of label information for PAGE 37 37 classification facilitates learning. Many spatiote mporal filtering methods have been proposed for the singletrial EEG classification problem, which include, but not limited to, common spatial patterns (CSP) (Ramoser et al., 2000), common sp atiospectral patterns (CSSP) (Lemm et al., 2005), linear discriminant analysis (Parra et al., 2002), bilinear di scriminant component analysis (Dyrholm et al., 2007). The common spatial patterns met hod was initially proposed by Koles et al. (1990) to classify normal versus abnormal EEG (Koles et al. 1994). The method has been used for singletrial EEG classification in braincomputer in terface (BCI) systems (MllerGerking et al. 1999; Ramoser et al. 2000). Given the singletrial EEG data for two diffe rent experimental conditions, the CSP method decomposes the EEG data into spatial patterns which maximize the difference between the two conditions. The spatial filters are designed such that the variances of the outputs are optimal (in the leastsquare sense) for the discriminati on of the two conditions. This is realized by simultaneously diagonalizing the two covariance matrices of th e EEG associated with the two experimental conditions. The two resulting diagonal matrices (containing the eigenvalues for the two covariance matrices) add up to the identity ma trix. Thus, the spatial filters that give the (n, an integer) largest variance in their outputs (ass ociated with the largest eigenvalues) for one condition, will accordingly give the (n) smallest variance in their ou tputs for the other condition; and vice versa. It is along these directions that the la rgest differences between the two conditions lie. In (MllerGerking et al. 1999), the CSPs are called th e source distribution matrix (equivalent to the mixing matrix in ICA), and the spatially filtered outputs are claimed to be the source signals, although the EEG data were tempor ally bandpass filtered between 830Hz prior to analysis. Para et al. (2005) showed that the simultaneous diagonalization of the covariance PAGE 38 38 matrices is equivalent to the generalized eige nvalue decomposition, and according to Parra and Sajda (2003), the CSP method is in fact es timating the independent components of the temporally filtered EEG data. The original CSP method does not take into account the temporal information of the filtered EEG data. In light of this, Lemm et al. (2005) proposed an algor ithm called common spatiospectral pattern (CSSP), which utilized the me thod of delay embedding and extended the CSP algorithm to the state space (with only one tapdelay). Dornhege et al. (2006) futher improved the CSSP algorithm by optimizing an arbitrary fin iteimpulse response (F IR) filter within the CSP framework. The overfitting of the spectral f ilter is controlled by a regularizing sparsity constraint. The CSP method and its variants all use the relevant oscill atory brain activity for EEG classification. Sometimes it is more appropriate to use coherent evoked potentials (of lowpass nature) instead. Para et al. (2002) proposed a spatiotemporal filtering method using conventional linear discrimination to compute the optimal spatial filters for singletrial detection in EEG. Specifically, the search for the optimal spatial filter given the singletrial EEG data as in (27), is based on constraining the output () yt to be maximally discrimi nating between two different experimental conditions. The optimality criterion is restricted to a prespecified time interval, i.e., the time corresponding to a number of samples prior to an explicit button push. After finding the optimal spatial filter using c onventional logistic regr ession, the output is av eraged within that period of time to obtain a more robust feature. Th e detection performance is then evaluated using receiver operating characteristic (ROC ) analysis on a singletrial basis. Unlike other conventional methods such as ICA, where the scalp projections are given directly by inverting the demixing ma trix containing all the spatial filters, here since there is only PAGE 39 39 one spatial filter and one output, other techniques ha ve to be sought in order to estimate the scalp projection associated with that output. Parra et al. did this by projecting the EEG data to the discriminating output () y t assuming that the output is uncorre lated with all other brain sources, and found that the discrimination model captured information direc tly related to the underlying cortical activity. The method was improved in (Luo and Sajda, 2006), where the prespecified time interval is allowed to be different and opt imized for each EEG channel. This effectively defines a discrimination traject ory in the EEG sensor space. PAGE 40 40 CHAPTER 3 NEW SPATIOTEMPORAL FILTERI NG METHODOLOGY: BASICS In this chapter, we propose a new spatio temporal filtering method for singletrial ERP estimation. The method relies on modeling of the ER P component descriptors and thus is tailored to extract small signals in EEG. The model allows for both amplitude and latency variability in the actual ERP component. We constrain through a spatial filter w the extracted ERP component to have minimal distance (with respect to some metric) in the temporal domain from a presumed ERP component. Note that we do not constrain the entire ERPs, but instead a single ERP component. We maintain the point in the next section that the spatial filter may be interpreted as a noise canceller in the spatial domain. We then introduce two approaches for the proposed method: the deterministic appr oach and the stochastic approach. 3.1 Spatial Filter as a Noise Ca nceller in the Spatial Domain Since the method deals with one ERP component at a time, we wish to distinguish between signal and noise instead of using the general term sources. To accommodate this distinction, we rewrite the generative EEG model in (26) as follows: 1N TT ii iX=as+bn (31) where, s is the time course of the ER P component to be extracted, in denotes noise in general. The distinction between signal a nd noise is somewhat arbitrary, e.g., when P300 is the signal of interest, N100 will become noise in the model. Note that for notational convenience, we have assumed the effective number of sources to be 1N The EEG model in (31) can in turn be rewritten as: 1N TT s ooioioi i X=as+bn (32) PAGE 41 41 where, oa, os, oib, oin are the normalized versions (with respective to some norm, e.g., 2l) of their counterparts in (31) and 1ii The scalars s i may be seen respectively as the overall contributions of the signal and noise to the singletrial EEG data. In the case of independent noise, we may define the SNR for the si ngletrial EEG data (note that it is different from SNR in a single channel.) as: 2 120logN si iSNR (33) The vectors oa, 'ib, represent the scalp topography of the corresponding signal and noise. For a meaningful ERP component, it must have a stable scalp topography oa. Thus, we may assume that oa is fixed for all trials. We also assume that the waveform of a particular ERP component os (dimensionless) remains the same for all trials, although its amplitude s may change across trials. Next, we claim the following lemma, which is ba sically a direct consequence of the linear generative EEG model in (31). Lemma: There exists a spatial filter w, that will completely reject the interference from the first 1D largest noise in the output when it is applied to the singletrial EEG data, if, 11det0D abb (34) Further, the extracted ERP component will approach th e actual ERP component if, s D (35) Such a spatial filter wcan be found by taking the first row of the inverse of th e matrix in (34). Note that (34) implies that, (,)0,1,...,1iangleiDab and, (,)0,11ijangleijD bb (36) PAGE 42 42 which means that at leas t the scalp topography of th e source and the first 1 D noise should not be the same or very similar to each other from a computational perspective. Here, we wish to stress the point that the spa tial filter specified in the above lemma may be interpreted as a noise canceller in the spatial dom ain. It may or may not be the optimal spatial filter for enhancing the SNR in the extracted co mponent. In addition, the SNR enhancement due to the spatial filter increases m onotonically with the number of ch annels (electrodes) if the EEG data were measured in those channels. This mean s that the more channels we use to record the EEG, the higher SNR we will get (theoretically) in the extracted component. Note that the lemma is an existence statement, it does not tell us how to find such a optimal spatial filter. This will be the subject of the next two sections. 3.2 Deterministic Approach Most ERP components are monophasic waveform s with compact support in time. The morphology of the waveform can be consider ed relatively fixed due to the common cytoarchitecture of the neocorte x and similar neuron populations, but may vary in both its peak latency and amplitude from trial to trial. Based on this, we assume that a particular ERP component can be modeled by a fixed dimensionless template (e.g. no physical unit), in the temporal domain, denoted by ()ols(where l is the unknown peak late ncy), multiplied by an unknown and possibly variable amplitude s across trials. We attempt to estimate the variable peak latency and amplitude on a singletrial basis. 3.2.1 Finding Peak Latency Since we do not know the peak latency in a single trial, we denote the template as ()o s with a variable time lag parameter and slide it one lag a time to search for the peak latency. The search for the optimal filter w could be realized by mini mizing some distance measure PAGE 43 43 between the spatially filtered output T wX and the assumed waveform ()o s for the particular ERP component. We propose the following cost f unction based on secondord er statistics (SOS): 2 2min()TT owwXs (37) Note that the above optimi zation is with respect to w only, with fixed. The optimal solution for w is given by: 1()()()T o wXXXs (38) Obviously, the optimal spatial filter w depends on which ERP component is to be extracted, and also is a function of the variable time lag From (37) and (38), we obtain the cost solely as a function of the time lag : 2 1 2()()TTT oJ sXXXXI (39) The peak latency of the ERP component can be set as the time lag where the local minimum of () J occurs within the meaningful range of peak latencies (S ) for that particular component (provided that its wavefo rm is monophasic) i.e., argmin()SlJ (310) The estimated ERP component is then (this need not be normalized): ()()T sll y Xw (311) It can be shown that under cert ain conditions, the solution in (310) is identical to the true peak latency of the ERP component (Appendix A) Exact match between the modeled and actual ERP component is not a necessary condition for the solution in (310) to be correct. For instance, it is easy to show that when bot h components are symmetric wavefo rms, then (310) also gives the correct latency. PAGE 44 443.2.2. Finding Scalp Topogr aphy and Peak Amplitude In the following, we make the index for trial number k explicit. Denote the estimated ERP component for kth trial by (the peak latenc y depends on the trial number): ()()kkkskkll yy (312) We can absorb the scalar k into a variable scalp topography: kko aa (313) In order to estimate the unknown scalp topograp hy and amplitude of the ERP component, we assume that the ERP component is uncorrela ted with all the noise sources. Replacing the dimensionless ERP component in (32) by it s estimate in (311) and multiplying () s kkly on both sides of (32), we will get an estimate for th e singletrial scalp t opography (the cross terms ()T oiskkl ny vanish because of the uncorrelatedness assumption): () ()()kskk k T s kkskkl ll Xy a yy (314) Taking the normalized version (note that ka is in Volts ) we have, k ok T kk a a aa (315) Ideally, the normalized oka should be the same as the dimensionless scalp topography oa However, in lowSNR EEG data, the above esti mation in (314) is ve ry poor, due to the interference from background activit y in the finitesample data. To estimate the scalp topography for a stable ERP component, we pr opose the following cost function: 2 2 1 minoK ook k aaa (316) PAGE 45 45 This corresponds to a maximum likelihood (ML) estimator for oa under the assumption that each entry of the normalized singletrial scal p topography is an independent identically distributed (i.i.d.) Gaussian random variable. The optimal so lution for (316) is a simple average of the estimated singletrial scalp topographies for all K trials. Taking the normalized version, the following estimate for oa is obtained: 11 211 KK ookok kkKKaaa (317) Notice that (317) is in fact a weighted averag e of the estimates in (314). We also point out that (317) is different from summing up directly (314) for all trials since the peak latency parameter is involved and it ch anges from trial to trial. In the ideal case, the two vectors oa and ka are identical except for a scaling factor, which is exactly the unknown amplitude k associated with the ERP component in the kth trial. Replacing their respective estimates in (314), we can find k using again a SOS criterion: 2 2 minkkkoaa (318) Simple calculation leads to the following estimate for the amplitude: T kok aa (319) This estimate involves information from all the available channels. In order to eliminate the indeterminacies of the linear generative EEG mo del, we set the peak amplitude as the maximum of the estimated ERP component in (312), i.e., ()kskkl y (All the amplitudes in the rest of the paper refer to this quantity). Accordingly, th e contribution of the ER P component to the EEG data may be computed by: () s soskklXay (320) PAGE 46 46 These estimates for the scalp topography, peak latencies and amplitudes of ERP component can be used to analyze its psychol ogical significance on a singletrial basis. Note that we do not directly compute the amplitude from the estimated component, nor do we measure it in any single channel. Instead, th e amplitude is computed in (319) indirectly through an inner product of two scalp topograp hies, which involves information from all available channels. These estimates for the scal p topography, peak latencies and amplitudes of ERP component can then be used to analyze their psychological significance on a singletrial basis. We wish to point out that in contrast to all the spatiotem poral methods mentioned before, where only one, representative spatia l filter (or matrix consisted of spatial filters), is computed given all the EEG data, here the optimal spatial f ilter is computed on a singletrial basis, i.e., given a singletrial EEG data matrix, we can get a spatial filter, as in (38). The reason we did it in this way is that, we believe th at in theory the optimal spatial f ilter that is designed to extract small signals should change from trial to trial. Note in (32) the noise sources are sorted in decreasing order of their power. It is likely that the noisy sources that have relatively large power change drastically across trials. In effect, th is will change the conf iguration in (32) and accordingly, the optimal spatial filter will also change. 3.3 Stochastic Approach In the deterministic approach, the ERP componen t is considered to be a deterministic signal except for a random latency and amplitude across tria ls. It does not take into account the intrinsic error in the modeling of the ERP component itself. Here we propose a stochastic approach for the spatiotemporal filtering method. The idea is to constrain the extracted ERP component to be close to the presumed component with respect to some statistic. PAGE 47 47 We still use the generative EEG model as in (31 ), but here both the signal and the noise are interpreted as stochastic processes. The approach utilizes the following observation model to search for the optimal spatial filter: T Xws+v (321) where, s is the actual ERP component and v is the observation noi se appearing in the spatially filtered EEG data. They are both random vectors with each entry as a sample within a certain time interval from the corresponding stochastic processes. We assume that the ERP component is ge nerated from the following additive model: s= g +u (322) where, g is a fixed signal with a certain morphol ogy serving as the template for the ERP component, and u represents the model uncertainties of the ERP component and is assumed to be independent of th e observation noise v. Given the above model in (321) and (322) the optimal spatial filter may be found by maximizing the loglikelihood of the filtered EEG data T y Xw : max()log()Lp w yg (323) or, max()log()Lp wu+v (324) The loglikelihood function has a simple form under the assumption that u and v are zeromean Gaussian stochastic processes with covariance matrices M DCC respectively, and they are independent of each other, since u+v is nothing but another Gaussi an stochastic process with covariance matrix M D CC In this case, maximizing the log likelihood of the (transformed) observed data given the observati on model and the template of the ERP component yields the following solution for the optimal spatial filter: PAGE 48 48 1argminT TT oMD wwXw g CCXw g (325) This is a quadratic form of w, so it has a closedfrom solution: 1 11 T o wX XX g (326) where, M D CC (327) The estimate for the ERP component is then: T oo y Xw (328) We point out that (325) suggests that in the Gaussi an assumption, observational uncertainties and model uncertainties simply co mbine by addition of thei r respective covariance matrices. We also note that if M C and D C are both chosen as identity matrix (i.e., incorporating the least amount of a priori information into the model), then the solution in (326) essentially reduces to (38) in the deterministic approach. Th e optimality of the spatial filter will depend on how we choose the two covariance matrices. The estimation of the scalp topography and amplitude follow the same procedures as described in (315) (320). 3.4 Simulation Study We present in this section a simulation st udy with synthetic data and real EEG data recorded from subjects during a passive pictureviewing experiment. The goal of the simulation study is to evaluate the latency and amplitude precision of synt hetically generated transients immersed in real EEG backgr ound with different SNRs and wa veform mismatching conditions. PAGE 49 49 3.4.1. Gamma Function as a Template for ERP Component Lange et al. (1996) have used a Gaussian function as the template for an ERP component. Here, we prefer the Gamma function for the shap e of the synthetic ERP component because this is a very flexible function for waveform modeling and has been used extensively in neurophysiological modeling (Koch et al., 1983 ; Patterson et al., 1992). Freeman(1975) argued that the macroscopic EEG electri cal field is created from spik e trains by a nonlinear generator with a secondorder linear com ponent with real poles. Accordi ng to this model, the impulse response of the system is a monophasic waveform with a single mode, where the rising time depends on the relative magnitude of the tw o real poles (Appendix B). This may be approximated by a Gamma function, which is expressed by: 1()exp(/), 0kgtcttt (329) where, 0 k is a shape parameter, 0 is a scale parameter and c is a normalizing constant. The Gamma function is a monophasic waveform with the mode at (1),(1)tkk It has a short rise time and a longer tail for small k, and approximates a symmetric waveform for large k. This makes it a good candidate for modeling both early and late ERP components that tend to be symmetric in the ear ly components and have longer ta ils in the late components. Figure 31 shows four Gamma functions with different shape and scale parameters. 3.4.2. Generation of Simulated ERP Data EEG data were recorded from subjects dur ing a passive pictureviewing experiment, consisting of 12 alternating phases: the habitu ation phase and mixed phase. Each phase has 30 trials. During the 30 trials of the habituation pha se, the same picture was repeatedly presented 30 times. During the mixed phase, the 30 pictures are all different. Each trial lasts 1600 ms, and there is 600 ms prestimulus and 1000 ms poststimulus. PAGE 50 50 The scalp electrodes were placed according to the 128channel Geodesic Sensor Nets standards. All 128 channels were referred to channel Cz and were digitally sampled for analysis at 250Hz. A bandpass filter between 0.01Hz and 40H z was applied to all channels, which were then converted to average reference. Ocular ar tifacts (eye movement) were corrected with EOG recordings. The scalp topography of the synthetic ERP comp onent is chosen as the normalized P300 scalp topography from another st udy (Li, et al., 2006). The si mulation data were created by taking the superposition of the 600ms (150 samples) prestimulus data from 120 trials as the background EEG data and the scalp projecte d Gamma waveform as a proxy for the ERP component. The SNR levels given the background EEG data can be easily adjusted by modifying the normalizing constant c in (17). We define the SNR gi ven the singletrial EEG data as: 20log ()/s TSNR TrT XX (330) Note that since the actual ERP component is a nonstati onary signal, the magnitude s is taken be its peak amplitude. This is different from the conventional definition of SNR. The SNR levels given the background EEG data can be easily adjusted by modifying the normalizing constant c in (329). 3.4.3 Case Study I: Comparison with Other Methods We will test the performance of the proposed spatiotemporal filtering method at varying SNR levels (from 20dB to 12dB), where we have access to the actual (synthetic) ERP component. Two scenarios will be investigated: on e where there is an exact match between the synthetic and the ERP component template and th e other where there is a mismatch between the PAGE 51 51 two components. For the case of exact match, we use the parameters 3, 13 k for both ERP components. For the mismatch case, the synthetic ERP component remains the same, but the template has a different waveform with parameters 5, 6k Figure 32(a) shows the waveforms of the two components for the mismat ch case. We fixed the peak latency of the synthetic component at 200ms for all the 120 trials. For Woodys filter, we selected channel Pz for analysis, and use the initial ensemble average as the template, avoiding the iter ative update on the templa te (since the true late ncy is fixed, this is the bestcase scenario for Woodys filter). We search around the true la tency within 100ms for maximum correlation. We estimate the peak amplitude by taking the average of the peak value and its two adjacent values (corresponding to a noncausal low pass filter with cutoff frequency of 12Hz). For spatial PCA, we select the eigenv ector which has the maximal correlation with the P300 scalp topography (Note this is an ideal cas e for PCA, since in reality we do not know exactly the true scalp topography, nor the exact time course). The simulation results are summarized in Table 31, 32 and 33, which show the estimation mean and standard deviation for the estimated latency and the ratio between estimated and true peak amplitude, as well as the correlation coefficient between estimated and true scalp topography. Since PCA does not give an explicit estimate for latency and amplitude, we will omit the latency estimation and only com pute the amplitude ratio between the estimated ERP component and the synthetic ERP component. First, we note that the singletrial estimation of the peak latency is very stable in the case of exact match. Notably, for EEG data with SN R higher than 4dB, the method estimates the latency correctly for all the tria ls. Second, the amplitude estimate for the exact case is also stable but is more variable than the latency es timation. The mean amplitude approaches to one PAGE 52 52 and the standard deviation decreases to zero as the SNR increases. We may say that in the case of exact match between the model and the component, the estimator for the amplitude is asymptotically unbiased and asymptotically consistent with increasing SNR. The mismatch between the model and the generated component effectively introduces a bias in the estimation of the latency for realistic SNRs. From Table 31, we can see that the mean latency approaches to 188ms, yielding a bias of around 3 samples and the standard deviation is around 9ms (around 2 sa mples). However, the mean does converge to its true value (200ms) and the variance does approach to zero with increasing SNR, although very slowly. For instance, for SNR as high as 40dB and 60dB, the es timated latency has a meanstd statistic of 193 4ms and 200 0.7ms, respectively. Therefore, empirically we can see that the estimation for latency under the mismatch case is also asymptotically unbiased and asymptotically consistent with increasing SNR. The mismatch of components introduced a bias in the estimation of the amplitude for realistic SNRs, which is partly due to the difference in the waveforms of the synthetic component and templa te. The estimated amplitude has a statistic of 1.0458 0.0075 and 1.0015 0.0002 at a SNR of 40dB and 60dB, respectively. Thus in the case of mismatch, the estimator for the amplitude is also asymptotically unbiased and asymptotically consistent with respect to SNR, although the convergence is much slower than the exact match case. Of course, the estimation variance does increase notably as SNR decreases. But as evident from Table 42, our method at 12dB still gives a estimation variance smaller than Woodys method at 0dB. Table 31 and 32 clearly show the advantage of using spatial information, in contrast to the Woody filter based on singlechannel analysis. Specifically, the estimation variance of Woody filter for both latency and amplitude are much larger for realistic (negative) SNR conditions. PAGE 53 53 PCA overestimates the amplitude of the ERP co mponent for low SNR data. In contrast to our method, PCA gives a statistica lly significant bias even at 0dB from the baseline at 12dB (pvalue less than 0.0001). This means that varyi ng SNR (below 0dB) imposes a serious problem on the application of PCA in low SNR conditions. Finally, the simulated mismatch of compone nts affects the estimation of the scalp topography negligibly for the proposed method wh en SNR is higher than 10dB. In fact, the estimation for scalp topography with mismatch w ith our method at 20 dB is comparable to PCA at 4 dB. 3.4.4 Case Study II: Effects of Mismatch The second simulation concerns the effects of the mismatch on the estimation, specifically mismatch in the spread parameter which is the most important. We use the same synthetic component as before and vary the spread parameter with K fixed. Fig. 32(b) shows the waveforms of the synthetic and 3 of the template s, including a Gaussian with a spread of 20. The results are summarized in Table 34 and 35, for SNR = 20dB and 10dB, respectively. We have included a new quantification for the amplit ude estimation: coefficient of variation (CV), which is defined as the ratio of the standard de viation to the mean of a random variable. It is used as a measure of dispersion of the estimated amplitude (since its true mean is fixed at 1). From Table 34 and 35, we can see that at the same SNR leve l, the mean and variance of the estimated amplitude systematically change with respect to the spread parameter, i.e., larger spread parameter gives smaller amplitude. The de gree of variability in the amplitude estimation (measured by coefficients of variation) for mismatch cases exceeds the exact match case by less than 10% except at 10dB for a spread parameter of 7. In some cases, CV is even smaller than the exact match case, which gives a better estimate, but only in terms of the amplitude. This is important because although the estimated amplitudes differ in the mismatch case, as long as we PAGE 54 54 use the same template, these amplitudes on average will always be magnified or attenuated by a constant factor at a certain SNR level. Intuitiv ely, this means that given a fixed template and varying SNR (>20dB), the dominant source of variability in amplitude estimation mainly comes from the estimation variance (not bias) and this variability (measured by CV) is wellbounded from above for a range of spread parameters. Therefore, these amplitude estimates may still be effectively compared across expe rimental conditions as long as the same (meaningful) template is used and the SNR of the ERP data does not fall below 20dB. However, the mismatch clearly introduces a bias in the latency estimation, which may be as large as 50ms in absolute terms. This may or may not be signif icant depending on the applications. We can also see from the two tabl es that choosing a highe r spread parameter will lead to a slightly bette r estimation for the scalp topography. But of course, this comes at the cost of much worse latency estimation. PAGE 55 55 Table 31. Latency estimation: mean and standard deviation SNR (dB) Woody Exact match Mismatch 20 191 60 202 10 190 16 16 190 60 201 7 189 14 12 190 61 201 5 188 11 8 191 63 200 3 188 10 4 195 60 200 2 188 10 0 195 54 200 1 187 10 4 199 45 200 0 187 10 8 199 32 200 0 188 10 12 200 15 200 0 188 9 True latency: 200ms. Table 32. Amplitude estimation: mean and standard deviation SNR (dB) Woody PCA Exact match Mismatch 20 0.22 9.32 10.7 2.12 1.68 2.20 2.22 2.93 16 0.07 5.95 6.80 1.35 1.34 1.29 1.71 1.75 12 0.40 3.84 4.33 0.86 1.18 0.77 1.49 1.05 8 0.36 2.60 2.78 0.55 1.10 0.47 1.34 0.64 4 0.53 1.75 1.86 0.36 1.06 0.29 1.26 0.40 0 0.64 1.14 1.35 0.23 1.04 0.18 1.23 0.25 4 0.80 0.74 1.13 0.13 1.02 0.12 1.20 0.16 8 0.90 0.45 1.05 0.08 1.01 0.07 1.19 0.10 12 0.98 0.24 1.02 0.05 1.01 0.05 1.18 0.07 True amplitude: 1 Table 33. Scalp topography esti mation: correlation coefficient SNR (dB) PCA Exact match Mismatch 20 0.568 0.829 0.759 16 0.579 0.910 0.854 12 0.601 0.959 0.926 8 0.649 0.982 0.966 4 0.759 0.993 0.986 0 0.906 0.997 0.994 4 0.979 0.999 0.998 8 0.996 1.000 0.999 12 1.000 1.000 1.000 PAGE 56 56 Table 34. Effects of mi smatch I: SNR = 20dB Spread parameter Latency(ms) Amplitude Coefficient of Variation CC. of scalp topography 7 150 20 2.32 3.02 1.30 0.68 9 175 16 2.18 2.74 1.25 0.73 11 191 12 1.93 2.51 1.30 0.79 13 202 10 1.68 2.20 1.31 0.83 15 215 11 1.43 1.87 1.31 0.87 17 228 15 1.24 1.60 1.29 0.88 19 246 18 1.07 1.41 1.31 0.87 20 (Gaussian) 210 22 2.14 2.89 1.35 0.76 Table 35. Effects of mi smatch II: SNR = 10dB Spread parameter Latency(ms) Amplitude Coefficient of Variation CC. of scalp topography 7 151 21 1.30 0.92 0.71 0.93 9 175 16 1.34 0.78 0.58 0.95 11 190 8 1.24 0.69 0.55 0.96 13 200 4 1.14 0.60 0.53 0.97 15 211 5 1.03 0.53 0.51 0.98 17 220 7 0.94 0.46 0.49 0.98 19 229 9 0.86 0.41 0.47 0.99 20 (Gaussian) 208 11 1.30 0.78 0.60 0.95 PAGE 57 57 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 timeamplitude K=3,theta=2 K=5,theta=1 K=2,theta=2 K=9,theta=1 Figure 31 Gamma functions with different shapes and scales. PAGE 58 58 A 0 100 200 300 400 500 600 0.2 0 0.2 0.4 0.6 0.8 1 time (ms)amplitude presumed ERP component synthetic ERP component B 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K=3, theta=13 K=3, theta=7 K=3, theta=19 Gaussian Figure 32 Waveforms of synthe tic and presumed ERP compone nt. A) Synthetic component Gamma: K= 3, 13; presumed component: Gamma: K= 5, 6. B) Synthetic component Gamma: K= 3, 13; presumed components: Gamma: K= 3, 7, and 19, Gaussian with a spread of 20. PAGE 59 59 CHAPTER 4 ENHANCEMENTS TO THE BASIC METHOD In chapter 3, we developed the basic spatio temporal filtering method for singletrial ERP component estimation. In this chapter, we will co nsider some modifications to the basic method. Some serves as a heuristic postprocessing techni que (Section 4.1: iterativ ely refined template); some aims to deal with large salient EEG artifacts (S ection 4.2: robust estimation); some utilizes the a priori knowledge on the scalp topography of the ERP component (Section 4.3: regularization); some provides alte rnative formulations for our prev ious results and also derives new ones (Section 4.4: Bayesian formulations of the topography es timation); still others try to deal with the interference from other overl apping ERP components (Section 4.5: explicit compensation for temporal overlap). The details are presented below. 4.1 Iteratively Refined Template The deterministic approach of our estimation in Chapter 3 uses a fixed waveform for the template, regardless of the SNR. The stochastic approach incorporates so me degree of variability in the template, but it is still im plicit. We would like to explicitly utilize the posterior information from the data to update or refine our a priori assumed template. Intuitively, this should improve our estimation at least for high SNR conditions. We use the estimated scalp topography oa as a spatial filter. The output is optimal in the sense that it has the largest correlation coe fficient with the actual component with the uncorrelated noise assumption. (Of course, we use the estimate as a proxy for the true topography. Note that it is different from w ). The refined template is the ensemble average of spatially filtered data, with oa as the filter. The results are show n in Table 41, 42 and 43 (with one iteration of refining). PAGE 60 60 We can see that for the latency estimation, th e refined template method has a larger bias below 12dB than the original template, but impr oves quickly and approaches to the exact match case for positive SNR conditions. For the amplitude estimation, the refined template consistently beats the original template for all SNR conditions in terms of both bias and variance. It also approaches to the exact match case for positiv e SNR conditions. Also see Figure 41. For the scalp projection estimation, the refined template is worse below 8dB but is very close to the exact match case above 8dB. Figure 42 shows the waveforms of the synthetic component, original template and refined template at 4 SNR conditions: 20dB, 12dB, 0dB and 12dB. The spatially filtered ensemble average is still quite erratic below 10dB. That possibly accounts for the worse performance of refined template for very low SNR conditions. So, there exists a critical SNR below which, using the spatially filtered ensemble average will probably worsen performance (in terms of scalp topography, but not amplitude) compared with the orig inal template. For this particular data set, it would be safe to use a refined template as long as the SNR is above 10dB. With 1 iteration, there comes 2, 3 and infinity. The natural question then is: Will it converge? If so, what does it converge to? Theoretic ally, these are difficult questions. Aside from the variable latency parameter, the scalp topography is still nonlinearly related to the data. However, we can experimentally determine the li miting results. These are shown in Table 44 for 3 SNR conditions. The algorithm co nverges within 10 (sometimes 2) iterations. Compared with the first iteration, the estimation for latency and scalp projection barely ch anges, but there is a reduced bias and variance in terms of amplitude for negative SNR conditions. Of course, all these results are a function of the mismatch in the waveforms and the number of trials we use to compute the ensemble aver age (if we had 1000 trials instead of 120, it would PAGE 61 61 be a different story). At this point, it seems that, what matters the most to deal with negative SNR is to accumulate more data, either in space or in time. 4.2 Regularization Sometimes, we have some a priori knowledge of the scalp topography of the ERP component. For instance, the P300 component us ually has a large posit ive projection around Pz area. In these cases, we should utilize that information and in corporate it into our model. 4.2.1 Constrained Optimization Assuming that the ERP component latency has been estimated, we attempt to minimize the following cost function with respect to the amplitude and scalp topography a (which is constrained to have a unit norm in l2): 2 2 0 2 ,argminT FaXasaa (41) S.t., 21 a where, is a regularization parameter and 0a is a normalized vector representing the a priori knowledge of the scalp topog raphy of the ERP component. F denotes the Frobenius norm of a matrix. The reason that we chose this norm will be evident later: the minimization of this norm gave the same solution as (3.14) which was derived with the uncorrelatedness assumption. In Appendix C we derive a fixpoint update equation: 0 0 2T T aXs ss aXs a aXs (42) Particularly, when 0 (no regularization), the optimal solution becomes: PAGE 62 62 2 2T Xs ss Xs a Xs (43) When the optimal solution is: 00T T aXs ss aa (44) When takes intermediate values, the scalp t opography is a weighted average of the two extreme case solutions. This is a real singletrial estimation scheme in that the amplitude is computed from one singletrial data matrix. For the case of no regul arization, it is differe nt from our original formulation, where the amplitude is the inner product between Xs and the normalized average scalp topographies from all the trials. The original formulati on makes the reasonable assumption that the ERP component has a fixed scalp topography and utilizes that information. We demonstrate the effectiveness of regulariza tion to deal with the interference from other overlapping (possibly unknown, if th ey are all known, we can explic itly compensate for thatSee Section 4.5) ERP components. Specifically, we w ill investigate the effect of regularization on the estimation of amplitude and scalp topogr aphy under well controlled conditions. We assume that there are 2 overlapping ERP components and their la tencies are fixed and known. Their waveforms are shown in Figure 43. They are both Gamma functions with the same parameters3,10 K with a peak interval of 80ms. The two ERP components have a correlation coefficient of 0.36. We use template s that are exactly matched with the synthetic components. These components are projected to spontaneous EEG data to generate simulated PAGE 63 63 ERP data. Their scalp topogra phies are shown in the Figur e 44. We use that for 0a (exact a priori knowledge). We then find the optimal solution of for a given The fixed point update always converges within 2 steps. We summarize the results for 3 SNR condi tions (12dB, 12dB, 20dB), shown in Figure 45. For hi gh SNR (12dB) data, regularizat ion brings little difference. Since the overlapping ERP compon ents have the exact opposite s calp topography, it is expected that the estimated amplitude is smaller than 1 for high SNR data. It converges to around 0.42 and 0.41 for component 1 and 2 for large respectively. Note the huge bias in the estimated amplitude for low SNR (especially 20dB) without regularization. Bu t it converges to as small as 0.69 and 0.27 for component 1 and 2 for large respectively. This demonstrates the necessity of regularization for the constrained optim ization problem. A reas onable regularization parameter for all the SNR conditions is between 104 and 105, where an unbiased estimation for amplitude could be achieved. Also notice that th ere is a hump for the sta ndard deviation of the estimated amplitude. Interestingly, this is near the reflection point of the mean amplitude. In practice, this can give us some hint for finding a reasonable regul arization parameter. Note that for these choices of the correlation coefficient of scalp t opography is already very close to 1. This is an example when the overlapping compon ents have negative correlation on the scalp. What about positive correlated components? Figure 46 shows the results for the same components, except that now they have the same scalp topography. The estimated amplitude is expected to be larger than 1 for high SNR data. It converges to 1.6 for both components for large Note the huge bias for low SNR (especially 20dB). It converges to 1.8 and 1.7 for component 1 and 2 for large respectively. This also de monstrates the benefits of regularization for the constrained optimization pr oblem: using a sufficiently large regularization PAGE 64 64 parameter in this case can reduce the bias of the amplitude estimation, while the variance are not affected very much. As expected, for large the correlation coefficient of scalp topography converges to 1. 4.2.2 Unconstrained Optimization Parallel with the above constrained optimizati on, we can also frame the problem into an unconstrained optimization one. 22 0 2 2argminT FaXasaaa (45) Note that the optimization variable a contains the amplitude parameter as well as the topography information. As show n in Appendix D, a fixedpointed update can be obtained: 0 2 3 0 2 TT Xsaa a ssaaa (46) Particularly, when 0 (no regularization), the optimal solution is the same as our original solution in (314). When the optimal solution is not unique (any scaled version of 0a can be a solution). The pr oblem becomes illposed. We have obtained the re gularized solution for singletrial scalp topography. The same procedures for estimating the amplitude follow: ta ke the average of normaliz ed singletrial scalp topography as our estimate for the overall scalp t opography, then the amplitude for a particular trial is just the inner product between this vector and the corr esponding scalp topography. We test the performance of regularization unde r the same conditions as in the constrained optimization. We find the optimal solution for a given Fixed point update usually converges within 10 steps. We summarize the results for 3 SNR conditi ons (12dB, 12dB, 20dB) shown in Figure 47 and 48. As in constrained optimization, for high SNR (12dB) data, regulariz ation has little effect PAGE 65 65 on the estimation. The amplitude converges to around the same valu es as before (0.4 for opposite topography and 1.6 for the same topography). The va riance is not affected much for all the SNR conditions, either. As expected, the scal p topography gets monot onically better as increases. The difference is that the estimated amplitude mean increases monot onically (except for a small interval) with increasing regularization parameter This translates to a larger bias (particularly for low SNR data) when the overlapping components have positively correlated topographies. The estimated amplitude becomes unstable for large There is no evidence at this point that regularization can benef it the estimation for general scalp topography configurations (both positive and negative t opography correlations). So the unconstrained optimization formulation need not be regulariz ed, at least for overlapping components with positively correlated topographies. This lends support to our original solution in (3.14), which is exactly the unregularized solution to th e unconstrained optimization problem here. We also point out that, unlik e the constrained optimization problem, here we utilize the reasonable assumption that an ERP component has a fixed scalp topography. So it does not suffer from the huge bias problem in the constrai ned optimization. For inst ance, at 20dB without regularization, the estimated mean amplitude is around 2.3 and 1.9 for the two components with positively correlated topographies, a modest increase from 1.6 at 12dB. 4.3 Robust Estimation: the CIM Metric We have seen in Section 4.2 that the estima tion for singletrial scalp topography in (3.14) can be found equivalently from the minimization the following criterion: 2argminkT kkk FaXay (47) where, F denotes the Frobenius norm. PAGE 66 66 It is evident from (314) th at the estimate for singletria l scalp topography bears a linear relationship with the EEG data. Because of the noisy nature of EEG (particularly large salient artifacts), this gives a noisy estimate for the singletrial scalp topogra phy (with large variance) and in turn translates into a noisy estimate for the singletrial amplitude in (318). We would like to derive a robust estimator in order to reduce the effects of large EEG artifacts. We can replace the Frobenius norm in (47) with other norms (e.g., l1 norm) or metrics. Here we will consider a special metric: co rrentropy induced metric (CIM) proposed by Liu et al (2007). First we introduce what is called corrent ropy. Given two scalar random variables X and Y, correntropy is defined as: (,)()hhVXYEkXY (48) where, ()hkXY is the Gaussian kernel (h is the kernel size), 2 21() ()exp 2 2hXY kXY h h (49) The correntropy function is a localized similar ity measure in the join t probability space, which is controlled by the bandwidth parameter h (also called kernel size in kernel methods). It induces a metric (CIM) in the sample space which behaves like the 2l norm when the sample point is close to the origin (relative to the kern el size); when the sample point gets further apart from the origin, the metric is similar to the 1l norm and eventually saturates and approaches to the 0l norm (Liu, 2007). As such, CIM practically incorporates the 2l norm as a special case (if h is chosen to be sufficiently large). Minimization of CIM is equivalent to the maximum correntropy criterion (MCC). It has been shown that MCC has a close relation to Mestimation (Huber, 1981) and since correntropy PAGE 67 67 is inherently insensitive to outliers, MCC is esp ecially suitable for reject ing impulsive noise (Liu, 2007). Now, treating each entry of the matrix in (47) as a realization of a random variable, we can write our new cost function as: 2argminkT kkk CIMaXay (410) The nuisance parameter h (kernel size) should be tuned to the data (most notably to the standard deviation). Here, we use the Silvermans rule as a baseline to quantify different values of kernel sizes that we use in the simulation. It is given by 0.21.06hdN where N is the number of samples, and d the standard deviation of the data (Silverman, 1986). While minimization of the Forbenius norm has a closedform solution, minimization of CIM does not. So we have to search for a local minimum, using the st andard gradient descent method. The convergence to a certain local minimum is guaranteed by adopting a stopping criterion that the change in the correlations of the estimate and the MSE so lution between the previous and current iteration is less than 610. Table 45 and 46 summarize the estimation re sults for 2 SNR conditions (0dB and 20dB) for the mismatch case. We also include the MSE solution for comp arison. The results are mixed: MCC gives a slightly higher variance than MSE fo r the estimation of amplitude, but the bias is marginally reduced; it also gives a more accurate estimate for the scalp topography. We also notice that at 0dB, the results of MCC barely change from MSE. Intuitively, when the SNR becomes sufficiently large, the optimal MCC so lution should converge to the MSE solution (both agree with the true values of the parameters). These differences in the results are by no m eans statistically significant. We venture 2 reasons why the MCC results do not change very much from MSE. PAGE 68 68 First, the EEG data are already preprocessed and relatively clean. Large artifacts have already been removed. The resul ting distribution is not far from Gaussian. So MSE should give a solution already close to optimal. MCC has its edge when there is large noise, especially impulse noise. Strictly speaking, it is only optimal (in the sense of maximum likelihood) for one particular type of distribution, ju st as MSE is strictly optimal for Gaussian distribution. It is not clear that how these two criteria compare when the data distribution changes in a neighborhood of their optimal ones. In reality (when EEG da ta are usually preprocessed and artifacts are removed), there is no reason to believe that MCC will outperform MSE uniformly. Another less compelling reason concerns the optimization process associated with MCC. The initial condition of MCC is set to be the MSE solution (starting a random initial condition will seldom beat MSE). When the kernel size is small, the performance surface is highly irregular, so the optimization will never go fa r from MSE solution (it is stuck around the local minimum near the MSE solution). When kernel size is sufficiently large, it is easy to see that MCC approaches to MSE. Only intermediate values of kernel size will produce somewhat different results from MSE. This is seen in both SNR levels, though less evident for 0dB. There are two other cost functions in the estima tion of amplitude that use the MSE criterion, i.e., (3.14) and (3.16), which can also be replaced by MCC. Li et al (2007) have tested its performance and the improvement was shown to be marginal. In practice, one has to weigh the small improve ment in the performan ce of MCC against its high computational cost (and no guarant ee of convergence to global optima). 4.4 Bayesian Formulations of the Topography Estimation The amplitude estimation consists of 3 steps. First we estimate the singletrial scalp topography ka (either by the uncorrelated assumption or equivalently through the minimization PAGE 69 69 of the Frobenius norm of kkk Xa y ). Then we compute the normalized scalp topography oa as the normalized version of a weighted aver age of the singletrial scalp topography ka (the weights being their respective l2 norm). The third step is the minimization of the l2 norm 2kko aa, which gives the optimal singletrial amplitude as the inner product between ka and oa We have seen that the first step gives a n ear optimal solution (as opposed to CIM) if the EEG data have been cleaned. We have also shown that in the third step, using other metrics (e.g., CIM) gives a marginally better results than using the l2 norm. Here we will investigate other alternatives to the estimation of the normalized scalp topography oa in the second step in (3.17). We maintain that after estima ting the singletrial s calp topography in the first step, we treat ka as known and given. Then we ask the question: what is the best es timate for the normalized scalp topography oa given ka for K trials. This is a divideand conquer approach and simplifies matters. Naturally the problem is best formulated in a Bayesian framework. Of course, the formulation will depend on the model we assume for the data. Next, we will present three different models and compare their performance. 4.4.1 Model 1: Additive Noise Model kkok aau (411) where, k is the singletrial amplitude, and ku is the error (model uncertainty). This model is consistent with our linear gene rative EEG model. We shall assume conditional independence between ,iio aa and ,jjo aa for any ij Of course, we also assume i and PAGE 70 70 j are independent for any ij and they are all independen t of the normalized scalp topography oa. We wish to maximize a posteriori probability (MAP): 111 ,...argmax(,......)oKoKKp aaaa (412) It can be shown (Appendix E) that the MAP solution occurs when oa is the normalized eigenvector of the matrix A corresponding to the largest eigenvalue, where, 11KK T kkk kk AAaa (413) Now we have solved the MAP problem under mode l 1. But, there is a weakness: the model error has a constant covariance across all trials. This is a dubious assumption. Intuitively, when the data are noisy, the variance in ku will increase accordingly. We should somehow normalize the data in our m odel. This leads to model 2. 4.4.2 Model 2: Normalized Additive Noise Model k ok k a au (414) The left hand side is in fact a proxy for oka in our linear generative EEG model, except that it may not be normalized. ku is again zeromean i.i.d. Gaussian noise, but now with unit norm. The model can also be written as: kkokkk aau Again, it can be shown (Appendix F) that the MAP solution occurs when oa is the normalized eigenvector of the matrix B corresponding to the largest eigenvalue, where, 11T KK kk k T kk kk aa BB aa (415) PAGE 71 71 Note that the components kB is a normalized version of kA in that the trace of kB is always 1. We also note that, we can treat k only as a normalizing factor not necessarily as the singletrial amplitude. In this case, we separate the estimation of k and oa As before, we can compute the amplitude by: T kok aa 4.4.3 Model 3: Original Model Our last model is actually simpler. We do not include the unknown amplitude in this stage and only consider the estimation of oa The posteriori probability is simply 1argmax(...)ooKpaaaa. Given the model: 2k ok k a au a Appendix G shows that the MAP solution actually coincides with our original solution in (3.17). Th e singletrial amplitude estimation is the same as before. 4.4.4 Comparison among the Three Models We compare the amplitude and scalp topography estimation for the three models. The results are summarized in Table 47 and 48. We can s ee that model 3 (original solution) consistently gave the best results among the three under di fferent SNR conditions. The topography estimation with model 1 is poor at negative SNR conditions. Model 2 is an obvious improvement. Note that its amplitude estimation has a huge bias and variance for low SNR data. This may be due to the improper prior we assign to k (it assign large probability to large values) in the derivation of the MAP estimator. We also incl uded the amplitude estimation with the traditiona l inner product. There is a significant improvement in both the bias and variance, especially in low SNR conditions. PAGE 72 72 4.4.5 Online Estimation Sometimes we wish to know the si ngletrial parameters after recording each trial. It is then necessary to obtain an online estimation method. We assume that the stimulus onset time is known to us. Again, the problem is best formulated in a Bayesian framewor k in order to utilize all the information in previous trials. In fact, it is trivial given the above analysis. Here, we adopt model 3. After the first trial, set: (1) 11 2o aaa, 11 2 a At each trial, we store the running average: 2 1K kkk kcaa When finishing recording trial K+1, we update the topography estimate: 11 (1) 2 11 2 2KKK K o KKK caa a caa, and the amplitude for the newly recorded trial: (1) 11TK KKo aa 4.5 Explicit Compensation for Temporal Overlap of Components In developing our basic method in Chapte r 3, we assumed that ERP components are uncorrelated with each other. In reality, this is seldom satisfied. Because ERP components have relatively stable waveforms and latency, when th ey overlap in time, there will generally be a nonzero correlation among them. Here we are ma inly concerned with the situation where ERP components overlap in time, but latency ji tter across trials is relatively small. Consider two ERP components overlapping in time (it is easy to generalize to multiple components). We assume that th e time courses of the components 1s and 2s are known from physiological knowledge. We also as sume that the latency is give n or can be estimated, e.g., from ensemble average, and it is relatively fi xed. For the purpose of amplitude estimation, PAGE 73 73 latency jitter is considered here as a minor i ssue compared with the possibly heavy overlap of components. Note that, in the case of two overl apping (correlated) comp onents, we lose the ability to estimate the latency simply from the cost function in (39). If we assume that all other components are un correlated with these tw o components (or have negligible temporal overlap with them), then we can compensate for the correlation (due to temporal overlap) to get an unbi ased estimate for both components amplitudes. We write the linear generative EEG model in this case: 1122 1N TTT ii iX=as+asbn (416) We wish to estimate the scalp topographies for the two overlapping components. With the uncorrelated assumption, we can get two set of equations: 1112211 1122222 TT TT ass+assXs ass+assXs (417) Solve for 1a and 2a : 212 1 2 12 121 2 2 12DC DDC DC DDC XsXs a XsXs a (418) where, ,(1,2)T iiiDi ss and 1221 TTC ssss If 0C, we have the same solution as before: ,1,2i i T iii Xs a ss (419) The procedures for estimating the amplitude ar e the same as before. The above technique assumes that accurate estimates of the waveform s of the components are available, since the crosscorrelation in (41 8) depends on the tails of the overl apping components. This restricts its PAGE 74 74 applicability in practice. But if the research ers believe that the ER P components are heavily overlapped and are fairly certain of their wave forms, this technique should serve as a first attempt to reduce the bias in the estimation. PAGE 75 75 Table 41. Latency estimation: mean and standard deviation SNR (dB) Exact match Mismatch Refined template 20 202 10 190 16 182 12 16 201 7 189 14 183 13 12 201 5 188 11 187 12 8 200 3 188 10 191 12 4 200 2 188 10 197 6 0 200 1 187 10 199 4 4 200 0 187 10 200 1 8 200 0 188 10 200 1 12 200 0 188 9 200 1 True latency: 200ms. Table 42. Amplitude estimation: mean and standard deviation SNR (dB) Exact match Mismatch Refined template 20 1.68 2.20 2.22 2.93 2.17 2.73 16 1.34 1.29 1.71 1.75 1.61 1.76 12 1.18 0.77 1.49 1.05 1.33 1.05 8 1.10 0.47 1.34 0.64 1.18 0.59 4 1.06 0.29 1.26 0.40 1.11 0.34 0 1.04 0.18 1.23 0.25 1.06 0.20 4 1.02 0.12 1.20 0.16 1.04 0.12 8 1.01 0.07 1.19 0.10 1.02 0.07 12 1.01 0.05 1.18 0.07 1.01 0.05 True amplitude: 1 Table 43. Scalp topography esti mation: correlation coefficient SNR (dB) Exact match Mismatch Refined template 20 0.829 0.759 0.644 16 0.910 0.854 0.772 12 0.959 0.926 0.891 8 0.982 0.966 0.963 4 0.993 0.986 0.987 0 0.997 0.994 0.995 4 0.999 0.998 0.998 8 1.000 0.999 0.999 12 1.000 1.000 1.000 PAGE 76 76 Table 44. Estimation results for the iteratively refined template method Latency estimation Amplitude estimation Scalp topography SNR (dB) Refined template Iterative Refined Refined template Iterative Refined Refined template Iterative Refined 20 182 12 183 12 2.17 2.73 1.83 2.36 0.644 0.644 12 187 12 187 12 1.33 1.05 1.23 1.01 0.891 0.879 0 199 4 199 2 1.06 0.20 1.06 0.20 0.995 0.995 Table 45. Estimation with MCC for the mismatch case at SNR = 0dB Kernel size (multiples of h) Amplitude Correlation coefficient of Scalp projection 0.5 1.22 0.25 0.994 1 1.22 0.26 0.994 2 1.22 0.27 0.995 5 1.22 0.26 0.995 10 1.22 0.25 0.994 20 1.23 0.25 0.994 MSE 1.23 0.25 0.994 Table 46. Estimation with MCC for the mismatch case at SNR = 20dB Kernel size (multiples of h) Amplitude Correlation coefficient of Scalp projection 0.5 2.22 2.95 0.758 1 2.21 3.05 0.754 2 2.17 3.18 0.786 5 2.18 3.14 0.789 10 2.21 3.00 0.770 20 2.22 2.93 0.759 MSE 2.22 2.94 0.759 PAGE 77 77 Table 47. Amplitude estimation for three Bayesian models SNR (dB) Model 1 Model 2 Model 2 (2) Model 3 (original) 20 0.38 1.30 12.2 378 1.95 3.79 2.22 2.93 12 0.86 0.85 9.96 60 1.42 1.34 1.49 1.05 0 1.20 0.26 1.66 0.26 1.23 0.25 1.23 0.25 12 1.18 0.07 1.21 0.07 1.18 0.07 1.18 0.07 Table 48. Scalp topography estima tion for three Bayesian models SNR (dB) Model 1 Model 2 Model 3 (original) 20 0.188 0.642 0.759 12 0.558 0.841 0.926 0 0.971 0.994 0.994 12 0.998 1.000 1.000 PAGE 78 78 20 15 10 5 0 5 10 15 1 1.5 2 2.5 SNR (dB)mean exact match mismatch refined template 20 15 10 5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 SNR (dB)standard deviation exact match mismatch refined template Figure 41 Mean and standard deviation of the estimated amplitude under different SNR conditions. The refined template method appr oaches to the exact match case for high SNR conditions. PAGE 79 79 Figure 42 The waveforms of the synthetic compone nt, presumed template and refined template under 4 SNR conditions. A) 20dB. B) 12dB. C) 0dB. D) 12dB. The refined template appears erratic for low SNR and approach es to the synthetic component for high SNR. A B C D 0 50 100 150 10 5 0 5 10 sample synthetic presumed refined 0 50 100 150 6 4 2 0 2 4 6 8 10 sample synthetic presumed refined 0 50 100 150 2 0 2 4 6 8 10 sample synthetic presumed refined 0 50 100 150 2 0 2 4 6 8 10 sample synthetic presumed refined PAGE 80 80 0 50 100 150 0 1 2 3 4 5 6 7 8 9 10 Figure 43 Waveforms of two overlapped compone nts used in regulari zation. The correlation coefficient between the two waveforms is around 0.36. PAGE 81 81 0 20 40 60 80 100 120 140 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 channel numbertopography Figure 44 Scalp topography of tw o overlapped ERP components used in regularization. PAGE 82 82 A Figure 45 Amplitude and scalp topography es timation I with regularization (constrained optimization) under 3 SNR conditions. A) 12dB. B) 12dB. C) 20dB. The overlapping ERP components have the ex act opposite scalp topography. So the estimated amplitude is smaller than 1 fo r high SNR data. It converges to 0.42 and 0.41 for component 1 and 2 for large respectively. Notice the huge bias for low SNR (especially 20dB) without regularization. But it conve rges to as small as 0.69 and 0.27 for component 1 and 2 for large respectively. 10 2 10 3 10 4 10 5 106 107 0 0.1 0.2 0.3 0.4 0.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 10 6 107 0.996 5 0.997 0.997 5 0.998 0.998 5 0.999 0.999 5 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 0.1 0.2 0.3 0.4 0.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 10 6 107 0.998 6 0.998 8 0.999 0.999 2 0.999 4 0.999 6 0.999 8 1 REG PARAMETER: LAMDA correlation coefficien t PAGE 83 83 B C Figure 45 Continued 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 2.5 3 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 2.5 3 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 2 4 6 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.4 0.5 0.6 0.7 0.8 0.9 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 1 2 3 4 5 6 7 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.4 0.5 0.6 0.7 0.8 0.9 1 REG PARAMETER: LAMDA correlation coefficien t PAGE 84 84 A Figure 46 Amplitude and scalp topography es timation II with regularization (constrained optimization) under 3 SNR conditions. A) 12dB. B) 12dB. C) 20dB. The overlapping ERP components have the sa me scalp topography. So the estimated amplitude is larger than 1 for high SNR da ta. It converges to 1.6 for both components for large Notice the huge bias for low SNR (especially 20dB) without regularization. It converges to 1.8 an d 1.7 for component 1 and 2 for large respectively. 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 10 6 107 0.999 7 0.999 8 0.999 8 0.999 9 0.999 9 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 10 6 107 0.999 9 0.999 9 0.999 9 0.999 9 1 1 1 1 1 REG PARAMETER: LAMDA correlation coefficien t PAGE 85 85 B C Figure 46 Continued 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.95 0.96 0.97 0.98 0.99 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.985 0.99 0.995 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 1 2 3 4 5 6 7 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 10 5 106 107 0 1 2 3 4 5 6 7 REG PARAMETER: LAMDAamplitude mean std 102 103 104 105 106 10 7 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 REG PARAMETER: LAMDA correlation coefficien t PAGE 86 86 A Figure 47 Amplitude and scalp topography estimation I with regularization (unconstrained optimization) under 3 SNR conditions. A) 12dB. B) 12dB. C) 20dB. The overlapping ERP components have the ex act opposite scalp topography. So the estimated amplitude is smaller than 1 fo r high SNR data. The estimated amplitude mean generally increases with increasing while the variance is not affected much. It becomes unstable for large 10 2 10 3 10 4 105 106 0 0.1 0.2 0.3 0.4 0.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 106 0.996 5 0.997 0.997 5 0.998 0.998 5 0.999 0.999 5 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 0.1 0.2 0.3 0.4 0.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 106 0.998 6 0.998 8 0.999 0.999 2 0.999 4 0.999 6 0.999 8 1 REG PARAMETER: LAMDA correlation coefficien t PAGE 87 87 B C Figure 47 Continued 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficient 10 2 10 3 10 4 105 106 0 2 4 6 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.4 0.5 0.6 0.7 0.8 0.9 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 2 4 6 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.4 0.5 0.6 0.7 0.8 0.9 1 REG PARAMETER: LAMDA correlation coefficien t PAGE 88 88 A Figure 48 Amplitude and scalp topography estimation II with regularization (unconstrained optimization) under 3 SNR conditions. A) 12dB. B) 12dB. C) 20dB. The overlapping ERP components have the sa me scalp topography. So the estimated amplitude is larger than 1 for high SN R data. The estimated amplitude mean generally increases with increasing while the variance is not affected much. It becomes unstable for large 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 106 0.999 7 0.999 8 0.999 8 0.999 9 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 106 0.999 9 0.999 9 0.999 9 0.999 9 1 1 REG PARAMETER: LAMDA correlation coefficien t PAGE 89 89 B C Figure 48 Continued 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.95 0.96 0.97 0.98 0.99 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 0.5 1 1.5 2 2.5 3 3.5 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.985 0.99 0.995 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 2 4 6 8 10 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.8 0.85 0.9 0.95 1 REG PARAMETER: LAMDA correlation coefficien t 10 2 10 3 10 4 105 106 0 2 4 6 8 REG PARAMETER: LAMDAamplitude mean std 102 103 104 10 5 10 6 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 REG PARAMETER: LAMDA correlation coefficien t PAGE 90 90 CHAPTER 5 APPLICATIONS TO COGNITIVE ERP DATA In this chapter, we apply the spatiotemporal filtering method proposed in Chapter 3 to the singletrial ERP estimation problem in two different experiments. The first application is an oddball target detection task with different pictures as stimuli, where the difficulty of the task or saliency of the stimuli leads to decreased P300 amplitude. The second one is the habituation study where the subjects were repeatedly presen ted identical pictures and the amplitude of certain ERP components is expected to d ecrease rapidly with respect to trials. 5.1 Oddball Target Detection 5.1.1 Materials and Methods Because we were interested in singletrial, si nglesubject analyses of amplitude and latency, we selected 4 participants that met a minimum signaltonoise ratios based on their averaged ERPs, from a pilot study (n=8) on implicit cont ent processing during feat ure selection. They were righthanded according to the Edinburgh Handedness Questionnaire and all had normal or corrected vision. Stimuli consisted of pictures from the Inte rnational Affective Pict ure System, depicting adventure scenes, emotionally neutral social intera ctions, erotica, attack scenes, and mutilations. Their color content was manipulated such that they contained only shades of green or shades of red, and for each, color brightness was systemati cally manipulated to yield one bright and one dim version (Figure 51). All pi ctures were presented for 200 ms on the center of a 21inch monitor, situated 1.5 m in front of the subjects. From this vi ewing distance the checkerboards subtended 4.0 deg. x 4.0 deg. of visual angle. A fixation cross was always present, even when no picture was presented on the screen. Target stimu li (p = 0.25) were defined for each experimental block (see below) by a combination of color a nd brightness.. All pictur es were presented in PAGE 91 91 randomized order, with an inte rstimulusinterval varying rand omly between 1000 to 1500 ms in 4 blocks of 120 trials each. One block lasted 7 min. on average. At the beginning of each block subjects were instructed to atte nd either to the bright/dark green or red pictures and to press the space bar of the computer keyboard when they dete cted a target. The target color and brightness were designated in counterbal anced order. Furthermore, the responding hand was changed half way through the experiment, and the sequen ce of hand usage was counterbalanced across subjects. Subjects were also instructed to avoi d blinks and eyemovements and to maintain gaze onto the central fixation cross. Practice trials were provided for each subject for each condition to make sure that every subject had fully understood the task. EEG was recorded continuously from 257 el ectrodes using an Electrical Geodesics (EGI) EEG system and digitized at a rate of 250 Hz, using Cz as a recording reference. Impedances were kept below 50 k as recommended for the Elec trical Geodesics high inputimpedance amplifiers. A subset of EGI net electrodes lo cated at the outer canthi as well as above and below the right eye was used to determine ho rizontal and vertical Electrooculogram (EOG). All channels were preprocesse d online by means of 0.1 Hz highpass and 100 Hz lowpass filtering. Epochs of 1000 ms (280 ms pre, 720 ms pos tstimulus) were obtained for each picture from the continuously recorded EEG, relative to picture onset. The mean voltage of a 120msec segment preceding startle probe ons et was subtracted as the baselin e. In a first step, data were lowpass filtered at a frequency of 40 Hz (24 dB / octave) and then submitted to the procedure proposed by (Junghfer et al., 2000), which uses statis tical parameters to exclude channels and trials that are contaminated with artifacts. This procedure resulted in rejection of trials that were contaminated with artifacts (including ocular artifacts). Artifacts were also evaluated by visual PAGE 92 92 inspection and respective trials were rejected. R ecording artifacts were fi rst detected using the recording reference (i.e. Cz). Subsequently, gl obal artifacts were detected using the average reference and distinct sensors fr om particular trials were rem oved interactively, based on the distribution of their mean amplit ude, standard deviation and maximum slope. Data at eliminated electrodes were replaced with a st atistically weighted spherical sp line interpolation from the full channel set. The mean number of approximated ch annels across conditions and subjects was 20. With respect to the spatial arrangement of th e approximated sensors, it was ensured that the rejected sensors were not located within one region of the scalp, as this would make interpolation for this area invalid. Spherical spline interp olation was used throughout both for approximation of sensors and illustration of voltage maps (Junghfer et al., 1997). Single epochs with excessive eyemovements and blinks or more than 30 channels containing artifacts in the time interval of interest were discarded. The vali dity of this procedure was further tested by visually inspecting the ver tical and horizontal EOG as computed from a subset of the electrodes that were part of the electrode ne t. Subsequently, data were arithmetically transformed to the average refe rence, which was used for all analyses. After artifact correction an average of 69 % of the trials were retained in the analyses. The present analysis highlighted the most reliable signal avai lable in this featurebased target identification task, which is the P300 component in response to a target stimulus (defin ed by a combination of color and brightness, irrespective of picture cont ent). Thus, all subseque nt analyses focused on amplitude and latency estimates for single tr ials belonging to the target condition. 5.1.2 Estimation Results The present study illustrates th e application of the method for a single late potential component. In reality, we do not know a priori how many ERP com ponents there are in a single PAGE 93 93 trial recording, nor do we know exactly when they occur. However, we may be able to estimate these values from singletrial EEG data in the data analysis session. This is a good time to mention one technical requirement of our latency estimation. The singletrial latency is estimated from the cost function in (3.9), which involves the inversion of the matrix TXX In reality, this matrix is usually illconditioned for densearray EEG data (it will certainly be rankdeficient if there are any bad channels which were linearly interpolated from other channels.). This poses a computational problem in practice. Thus the solution in (3.9) somehow has to be regularized. Here, we adopt a simple approach and ad d a regularization term I (0 ) to the matrix TXX before taking the matrix invers ion operation. The regularization parameter acted as a smoother to the cost function in (3.9). Generally, the solution is rather irregular without regula rization, leading to too many local minima and spurious candidates for singletrial latencies due to large noise. With increasing the cost function becomes smoother. This is clearly seen in Fig. 52, which show s the cost function in (3.9) for four different for a particular trial from subject 2. With a smoot h cost function, we can avoid the dilemma of choosing the right latency from too many candidates. Now we have to select an a ppropriate value (or a meaningful range) for the regularization parameter A good value for is one that achieves a balan ce between two extremes: too few and too many local minima. The idea is this: for a particular we group all the candidates for singletrial latencies (time lags corresponding to local minima) together and perform 1D density estimation on these candidates. We count the number of modes (peaks) from the estimated probability density function (pdf). If this number is close to the average number of candidates for each trial, then the regularization parameter is at least internally consistent. Otherwise, it will contradict with itself and should not be used. PAGE 94 94 We illustrate our point using the results from one subject. Fig. 53 shows the estimated pdf of the candidates for singletrial latency from 200ms up to 600ms after stimulus onset when the regularization parameter equals 510 We used the Parzen windo wing pdf estimator (Parzen, 1962) with a Gaussian kernel size of 4.2. The ke rnel size was selected according to Silvermans rule (Silverman, 1986), which is given by 0.21.06 hN, where N is the number of samples, and is the standard deviation of the data. The nu mber of peaks depends on the kernel size, but we found that a kernel size between 0.5h and 2h will give the same number of peaks in the estimated pdf for this data. We can see that the pdf consists of 4 modes (peaks) after 200ms of the stimulus onset. There are 418 local minima and 102 trials in total, so the average number of local minima for each trial is about 4.1 (very clos e to the number of peaks in estimated pdf). This indicates that =510 gives an internally consistent estimate for latency. We can repeat the above procedures for a wi de range of regularization parameters and compute the ratio of the number of peaks in esti mated pdf to the average number of local minima for each trial. For instance, the ratio was computed as around 4.75, 1.03, 0.96, 0.74 for the 4 regularization parameters in Fi g. 52 respectively. Clearly, th e first and last regularization parameter should not be used since they generate selfcontradictory results. It is interesting to note that for a wide range of regularization parameters (from 510 to 010), the results are quite similar. This can also be seen from Fig.52, where both cost functions display 4 local minima and all time lags are near to their counterparts. For practical purposes, we can select any value from this range as a regularization parameter. We were primarily interested in the P300 comp onent, preferably the largest one. From the ensemble average, we know that the maximu m ERP occurred around 380ms after stimulus onset. In Fig. 53, the estimated pdf displays a m ode around 420ms. Thus, we searched around this PAGE 95 95 latency and set the singletrial peak latency as th e one that was closest to it. The mode of latency is 360ms, 420ms and 400ms for the ot her three subjects, respectively. We should point out that since there is about 1 local minimum per mode, the search need not be around the true mode for latency (we do not know this anyway). The results would be almost the same as long as the estimated mode is not skewed to its two neighboring true modes. Figure 54 shows the scalp topographies for the four subjects plotted using EEGLAB (Delorme et al., 2004). As expected for a P300 topography, it has a large positive topography around the Pz area. To evaluate the singletrial estimation of the scalp topography, we compute the correlation between the singletrial scalp t opography in (3.15) and the overall normalized scalp topography (3.17). For comparison, we also compute the correlation between the singletrial scalp topography in (3.15) and the scalp topography obtained from ensemble average for each subject. We name these two correlations 1r and 2r respectively. Statistical inference based directly on the correla tion itself is difficult since its di stribution is complicated. A popular approach is to first apply the Fisher Z transformation to correlati on and then do inference on the transformed variable. The Fisher Z transform is given by (Fisher, 1915): 1 0.5ln 1r Z r (51) Z has a simpler distribution and it converges mo re quickly to a normal distribution. We can calculate the mean and confidence interval of Z based on the correlation, if we assume that the estimation error in (315) is a no rmal distribution. The statistics of the correlation can be easily obtained from the inverse transform of (51). The results are summarized in Table 51. We can see that there is a moderate amount of correlation between the singletria l and overall scalp topography (the average mean correlation for 4 subjects is around 0.40) although the mean co rrelation is lower for su bject 3 at around 0.20. PAGE 96 96 There is a small degradation in mean correlation when the overall scalp topography is computed from the ensemble average. This is expected sinc e the estimate in (317) is close to the ensemble averaged estimate. The correlation between these two estimates for the four subjects are: 0.80, 0.85, 0.89, 0.79 respectively. To evaluate the effectiveness of the singletrial amplitude estimation, we related our estimates to a behavioral measure of target iden tification: response time in target trials. Response time was selected because task difficulty was re latively low, and theref ore error rate did not show pronounced variability, with only limited numbers of misses (mean of 3.9 % across 4 participants) and false alarms (mean of 1.2 % across 4 participants). Thus, response time was used as a measure of target identification, with short response times i ndicating facilitated discrimination and long response times indicating di fficulties with identifica tion in a given trial. Using these measures, we were interested in the relationship between P300 amplitude and response time, expecting that trials in which participants found disc rimination relatively easy (short RT trials) should be associated with greater P300 amplitude, which also indicates successful encoding of the target features and prep aration for responding to a target that has been identified. There seems to be little relationship between the response time and estimated singletrial peak latency. The correlation coe fficients between thes e two for the four s ubjects are: 0.022, 0.248, 0.168 and 0.093 respectively. However, ther e were reliable negative correlations between the response time and estimated singletrial amp litude. Figure 5 shows the scatter plot of the response time versus the estimated amplitude for each single trial for the four subjects. To evaluate the statistical significan ce of the results, we performe d linear regression on the response time and estimated singletrial amplitude for the four subjects. The results are summarized in PAGE 97 97 Table 52. the negative slope parameter estimated from linear regression is statistically significant under a signif icance level of 0.05 fo r all the four subject s, which supports our hypothesis that larger amplitude correspond to smaller response time, and vice versa. To compare our results with conventional methods, we calculated the average P300 amplitude at channel 100 for subject #2. This was simply the average singletrial amplitude times the 100th entry of the scalp topog raphy in (3.17). It was found to be 22.1mV, compared with the 17.3mV from the ensemble average ERP. Taking in to account of the possible latency jitter of P300, the true amplitude could be only larger than 17.3mV. Therefore, we obtained an upper bound of 28% on the positive bias of our average P300 estimate in channel 100. The coefficient of variation, which is defined as the ratio of the standard deviation to the mean of a positive random variable, is used as a measure of disp ersion of the estimated amplitude and it was found to be around 0.60. This compares favorably wi th 0.79 obtained using the simple peakpicking method around its ensemble average peak at 400ms. Although the gain may seem small, we should keep in mind that this variation will incorp orate the estimation error as well as that of the underlying change in P300 amplitude itself, because there are systematic changes in P300 amplitude as suggested above. So the estimation va riance of our method is re duced by a factor of at least 1.7 from the peakpicking method. For inst ance, if one half of th e total variance of our method came from the underlying P300 amplitude, this roughly means that our method reduced the estimation variance by a factor of 2.5 (assu ming additive and uncorrelated estimation error). Of course, the comparison would be much more direct and informative if the P300 amplitude was expected to remain constant. All the above results were obtained using a fixed Gamma template with 11, 5 k If we change the template, specifi cally, the spread parameter the estimated amplitude will also PAGE 98 98 change. However, we found that the amplitude estimation is only slightly affected by this change. For instance, the average estimated P 300 amplitude in channel 100 for subject #2 was around 20.5mV when 1 (this is too small for P300, ri se time 40ms) and was around 23.8mV when 8 (this is too large, rise time 320ms). Th ere is less than 8% change from the result (22.1mV) obtained with the original template with 5 This agrees with our earlier findings using simulated ERP data (Li et al., 2008). 5.1.3 Discussions As a straightforward test of the present method, we examined the relationship between target detection performance and feat ures of the P300 component e voked by the targets in an oddball task with rare targets varying in terms of thei r salience on a trialbytria l basis. In the present case, we replicated and extended a standard resu lt in target detection studies in the visual domain: When target identification is made diffi cult or saliency is reduced (e.g., by presenting many targets in succession, Gonsalvez and Po lich, 2002), P300 amplitude often decreases (Polich et al., 1997). This pattern has been interpreted as reflecti ng reduced resource allocation to a given target stimulus (Keil et al., 2007). Notably, previous work in this area has typically relied on averages across all trials of an experimental condition, or on block by condition averages across many trials (for a review, see Kok, 2001). The present result s suggest that the relationship between response time and P300 amplitude in feat urebased attention task is of a continuous nature, rather than a conseque nce of a bimodal function separating easy and hard trials. The sensitivity of the method was sufficient to demonstrate this linear relationship on a singlesubject level, which is often desirable in clinical studi es. In a similar manner, other research questions will benefit from the ab ility to examine hypotheses as to th e time course and distribution of single brain responses, in terms of their magnitude and latency. PAGE 99 99 5.2 Habituation Study 5.2.1 Materials and Methods EEG data were recorded from subjects dur ing a passive pictureviewing experiment, consisting of 12 alternating phases: the habitu ation phase and mixed phase. Each phase has 30 trials. During the 30 trials of the habituation pha se, the same picture was repeatedly presented 30 times. During the mixed phase, the 30 pictures are all different. Each trial lasts 1600 ms, and there is 600 ms prestimulus and 1000 ms poststimulus. The scalp electrodes were placed according to the 128channel Geodesic Sensor Nets standards. All 128 channels were referred to channel Cz and were digitally sampled for analysis at 250Hz. A bandpass filter between 0.01Hz and 40H z was applied to all channels, which were then converted to average referenc e. To correct for vertical and hor izontal ocular artifacts, an eye movement artifact movement correction proced ure (Gratton et al., 1983) was applied to EEG recordings. 5.2.2 Estimation Results We assume that the entire ERP may be d ecomposed into several monophasic components with compact support. We will estimate their pa rameters (amplitude and latency) one by one, using the Gamma template as in the simula tion study. The present study illustrates the application of the method for a single late poten tial component, and the Gamma is not adapted. Its parameters are selected based on ne urophysiology plausibility and are set as 5, 6 k corresponding to a rise time of 96ms. In reality, we do not know a priori exactly how many components there are in a singletrial, nor do we know when they occur. However, we may be able to estimate these values from singletrial EEG data in the data analysis se ssion. Following the same procedures in Section 5.1.2, we identified 5 distinct peaks after 300ms of the stimulus onset. Assuming that the error in PAGE 100 100 the latency estimation is equally biased and inde pendent from trial to trial and since there are also about 5 local minima for each trial, we conjecture that these peaks correspond to the latencies of 5 distinct components. These components, whic h may have different origins, are likely to compromise the Late Po sitive Potentials (LPP). Accordi ng to Codispoti et al.(2006), the grandaverage of LPP is maximal around 400ms to 500ms after stimulus. We will concentrate on the component with a latency of 500ms to ex emplify the methodology. We search between 440ms and 560ms (which correspond to the two neighboring local minima) and set the component latency as the local minimum closest to 500ms. To avoid the influence of EEG outliers from unexpected artifacts, we reject those trials with 3 times or larger amplitude of the minimum one. This will eliminate 14 trials from the total of 360 trials (rejection rate: 4%). The same rejection criterion wa s applied to the other two subjects, leading to the rejection of 8 (2%) and 33 (9%) trials, respectively. Figure 56 shows the results of estimated scalp topography of the LPP component for 3 subjects. They are similar in the sense that all show large projections in the posterior area. The difference with subject 2 is that the scalp topography shifts its strength a b it to the occipital area. It may be that the pictures shown to the 3 subject s caused some emotional bi as. It is also possible that the SNR of the ERP data is too low to allo w for a stable estimate of the scalp projection across subjects (note that in habituation phase, the LPP amplitude decreases quickly with the trial index). Figure 57 shows the results of estimated amplitude of the LPP component for 3 subjects. Each point in Fig. 57 stands for the average am plitude over 6 trials with the same index in the same phase (habituation or mixed). It is clear that for the habituation phase, the amplitude diminishes rapidly with the trial index, while for the mixed phase, the amplitude does not show PAGE 101 101 significant decay. To make the figure more intuitiv e, we also include the best fit (in the least square sense) to the estimated amplitude for both habituation and mixed phase. We fitted a straight line for the mixed phase, while an exponential curve was fitted to the estimated amplitude of the habituation phase. The fitted exponential curve for the habituation phase has a time constant of around 1.5 trials, which suggests that after 3 or 4 trials, the LPP amplitude decreases close to zero. We es timate the SNR for the mixed phase at around 4.1dB. Similar results were obtained with the ERP data from 2 ot her subjects as shown in Fig. 57 (B) and (C). The fitted exponential curves for the habituation phase for these 2 subjects has a time constant of around 1.5 and 2.0 trials, respectively. The SNR of the mixed phase for these 2 subjects are estimated to be around 7.6dB and 2.4dB respectively. PAGE 102 102 Table 51. Correlation statistics for th e 4 subjects: Scalp topography estimation 1r 2r Subject # Sample size Mean Confidence interval (95%) Mean Confidence interval (95%) 1 98 0.520 [0.362, 0.649] 0.393 [0.215, 0.546] 2 85 0.367 [0.167, 0.538] 0.304 [0.097, 0.486] 3 74 0.198 [0.027, 0.404] 0.200 [0.025, 0.406] 4 65 0.446 [0.233, 0.619] 0.359 [0.133, 0.551] Table 52. Regression statistics for response time and estimated amplitude Subjec t # Sample size Correlation R square Slope estimate t statisticp value Confidence interval (95%) 1 98 0.440 0.194 0.712 4.80 <0.0001 [1.007, 0.418] 2 85 0.539 0.291 0.310 5.84 <0.0001 [0.416, 0.205] 3 74 0.263 0.069 0.105 2.31 0.012 [0.197, 0.014] 4 65 0.351 0.123 0.108 2.97 0.002 [0.181, 0.036] PAGE 103 103 Figure 51 Pictures used in the experiment as stimuli PAGE 104 104 Figure 52 Cost function in (3.9) versus time lag for different regularization parameters for subject #2. A) 610. B) 510 C) 010 D) 210. Regularization parameter that is too small led to ragged cost function and spurious latency estimates; Regularization parameter that is too larg e led to oversmoothed cost function and missed candidates for latency. A B C D 200 250 300 350 400 450 500 550 600 0.5 1 1.5 2 2.5 3 3.5 x 10 5 time lag (ms) cost function 200 250 300 350 400 450 500 550 600 0 0.5 1 1.5 2 2.5 3 3.5 x 104 time lag (ms)cost function 200 250 300 350 400 450 500 550 600 0.4 0.5 0.6 0.7 0.8 0.9 time lag (ms) cost function 200 250 300 350 400 450 500 550 600 3.6 3.65 3.7 3.75 3.8 3.85 3.9 3.95 time lag (ms)cost function PAGE 105 105 100 200 300 400 500 600 700 0 0.5 1 1.5 2 2.5 3 x 103 time lag(ms)pdf Figure 53 Estimated pdf of time lags corresponding to local minima of the cost function in (39) using the Parzen windowing pdf estimator with a Gaussian kernel size of 4.2. Regularization parameter 510 PAGE 106 106 A B C D Figure 54 Scalp topographies for the four subjects. A) Subject 1. B) Subject 2. C) Subject 3. D) Subject 4. PAGE 107 107 Figure 55 Scatter plot of the response time vers us the estimated amplit ude for each single trial for the four subjects. A) Subject 1. B) Subjec t 2. C) Subject 3. D) Subject 4. Note that the estimated amplitude is with respect to the EEG data in all the channels as a whole. There appears to be a negative relati onship between the response time and the estimated amplitude. A B C D 300 350 400 450 500 550 600 650 100 0 100 200 300 400 Response time (ms) estimated amplitude 250 300 350 400 450 500 550 600 650 100 50 0 50 100 150 200 Response time (ms)estimated amplitude 300 400 500 600 700 800 900 1000 100 50 0 50 100 150 200 250 Response time (ms) estimated amplitude 200 300 400 500 600 700 800 900 1000 50 0 50 100 150 200 Response time (ms)estimated amplitude PAGE 108 108 A B C Figure 56 Estimated scalp topography for mixe d and habituation phase. A) Subject 1. B) Subject 2. C) Subject 3. PAGE 109 109 0 5 10 15 20 25 30 40 20 0 20 40 60 80 100 trial indexamplitude(mV) Habituation Mixed expotential fit linear fit A 0 5 10 15 20 25 30 20 10 0 10 20 30 40 50 60 70 80 trial indexamplitude(mV) Habituation Mixed expotential fit linear fit B 0 5 10 15 20 25 30 20 0 20 40 60 80 100 120 trial indexamplitude(mV) Habituation Mixed expotential fit linear fit C Figure 57 Estimated amplitude for mixed and habitu ation phase. A) Subject 1. B) Subject 2. C) Subject 3. Note that the LPP amplitude decreases with trials. PAGE 110 110 CHAPTER 6 CONCLUSIONS AND FUTURE RESEARCH 6.1 Conclusions Traditional ERP analysis has relied on ensemble average over a large number of trials to deal with the typically low SNR environments in EEG data. To analyze ERP on a single event basis, we have introduced a new spatiotemporal filtering method for the problem of singletrial ERP estimation. Our method relies on explicit modeling of ERP components (not the full ERP waveform), and its output is limited to local descriptors (amplitude and latency) of these components. The reason that we model the ERP co mponents instead of the full ERP waveform is to exploit the localization of scalp projection for each sing le ERP component, which is impossible to do for the entire ERP. Indeed, note that the ensemble ERP in different channels usually have different morphology because there are multiple neural sources originating from different locations of the brain that give rise to different scalp pr ojections. Since one spatial filter can extract effectively only one scalp projection, in or der to utilize the spat ial information in a meaningful way, only a component based analysis is viable. C oncentrating only on latency and amplitude of each component togeth er with optimal spa tial filtering presents an alternative to deal with the negative SNR. Moreover, since th ese are in fact the feat ures of importance in cognitive studies, the methodology has the same descriptive power of traditional approaches. The proposed methodology can be seen as a generalization of Woodys filter (Woody 1967) in the spatial domain for latency estimation. It also obtains an explicit expression for amplitude estimation on a singletrial basis. By design, th e method is especially su itable to extract ERP features in the spontaneous EEG activity, in contrast to PCA and ICA which work best for reliable (large) signals. Another distinction is that, unlike most methods based on PCA and ICA, our method utilizes explicitly the timing informa tion, as well as the spatial information. The PAGE 111 111 methodology as presented is based on least square s, but it can be furt her extended to robust estimation (Li et al. 2007) for better results. Using simulated ERP data, we have show n that although the mi smatch between the presumed and synthetic ERP components introd uces a bias for both latency and amplitude estimation, the bias for the latency is relatively small and the estimated amplitudes are still comparable across experimental conditions fo r ERP data with a SNR higher than 20dB. Furthermore, the mismatch of components has minimal influence on the estimation of scalp projection. These all compare favorably with some of the popular methods (Li et al., 2008). Despite its advantages over traditional met hods, there are still some issues with our spatiotemporal filtering method. First it is base d on the linear generativ e EEG model in (3.1). While this greatly simplifies the analysis, it may not be adequate to fully describe the complex information processing in the brain. One weak link of the method is that it requires an explicit template that is unknown a priori. Mismatch between the template and the true ERP component waveform brings both bias and larger variance to the estimation of the latency and amplitude that increases with decreasing SNR (Li, et al. 2008). It may be desirable to be able to adapt the template while estimating the model paramete rs. Another weakness of the method is the assumption of statistical unco rrelatedness among all the ERP components in deriving (3.14). With monophasic waveforms, this is equivalent to the condition that all the ERP components do not overlap in time (but overlap in space is allo wed), which is seldom satisfied in practice. Temporal overlap will bring bias to the amplitude estimation and poses a serious problem for the latency estimation, since it works effectively only for monophasic waveforms that are well separated in time. When there is heavy ove rlap among multiple components (e.g., P300 and possibly other unknown late componen ts), the peak latency estimation based on (3.10) may fail. PAGE 112 112 Therefore, care must be taken not to overint erpret the results of singletrial estimates. A crucial factor for amplitude estimation is a reasonably low SNR (>20dB). This may not be satisfied for some ERP components under certain e xperimental conditions. Our ability to infer the template accurately, which are selected heur istically from real data, deteriorates with decreasing SNR. As a rule of thumb, we woul d recommend against the us e of the present method for data with SNR less than 15dB. 6.2 Future Research The use of a parametric template (Gamma func tion) provides the flexibility to change the shape and scale parameters continuously. However, this introduces undesirable bias when there is a mismatch between the template and true ERP component. Using a stochastic formulation, our method may be extended to a noisy template model and potentially the two nuisance Gamma parameters may be extracted from the data also for best fit. It is almost certain that activations of different ERP components overlap in time. If this is the case, the temporal overlap will introduce a bias to the estimation of single trial scalp projection, because the derivation in (3.15) assumes the uncorrelatedness between the ERP component and all the other sources (includi ng the overlapping ERP compon ents) and unlike the background EEG, these overlapping components are coherent in all the trials. This bi as, together with the estimation variance due to finitesample data, constitute the two main sources of error in the estimation of the scalp projection. Note that th is in turn will influence the estimation of the amplitude. In Chapter 4, we have proposed an explicit procedure to compensate for the overlapping effects for the amplitude and topography estimation. This assumes that the latency is (relatively) fixed and a fairly accurate knowledge of the shape of all the overlapping ERP components. When this is not the case (particular ly when we wish to find the latency change PAGE 113 113 from trial to trial in the presence of com ponent overlap), we need to come up with new procedures to compensate for the overlapping issue. The current method considers the singletri al amplitude of an ERP component as i.i.d. data. It may be advantageous to take into account th e dynamics of certain properties of the component with respect to the trial index. For instance, we expect that during the habituation phase, the amplitude of LPP components diminishes rapidly with the number of trials. Using regularization techniques, this a priori information may be incorporated into the proposed singletrial estimation method to provide more stable estimate for the amplitude. In the end, the evolution of the singletrial amplitude with trial index may be inferred with more resolution and more confidence. PAGE 114 114 APPENDIX A PROOF OF VALIDITY OF THE PEAK LATENCY ESTIMATION IN (310) We justify the use of the time lag corresponding to the local minimum of ()J in (310) as the peak latency. Given the singletrial data matrix X the peak latency of an ERP component coincides with the local minimum of ()J if the following condi tions are satisfied: (1), the presumed component os and the actual component s have the same morphology; (2), ()0T io ns for 1,...,, iN and S (the signal and noi se are uncorrelated); (3), X is full rank. Proof: The optimal spatial filter is give n by (38). We plug it into (37) and get: 2 2()()()()()TTTT oooJ sCIsCCCCIs where, 1 TT CXXXX note that T CCC, so, 2 1()()()()()()()()()TTTTT oooooooJ sCsssssaRass where, T RXX is a positive definite matrix independent of the time lag With the constraint that ()()T ooconst ss the minimum of the cost function () J is achieved when ()T o ss achieves its maximum, since 1 TaRa is positive. This happens when coincides with the peak latency l of the actual ERP component s. PAGE 115 115 APPENDIX B GAMMA FUNCTION AS AN APPROXIMATI ON FOR MACROSCOPIC ELECTRIC FIELD The macroscopic electrical field is created from spike trains by a nonlinear generator with a secondorder linear component w ith real poles (Freeman 1975) Suppose that the transfer function of the secondorder syst em with stable real poles a, b is: 1 () ()() Hs sasb where, without loss of generality: 0 ba Then the impulse response in the time domain is: 1 ()atbthtee ab This is also a monophasic wavefo rm with a sing le mode at 01 ln(/) tba ab The rising time depends on the relative magn itude of the two real poles. The impulse response can be expanded: () 1() 11 ()1 !n btabtbt nabt hteee ababn Thus we can see that the impulse response is a sum of infinite weighted Gamma functions. However, it is always possible to find a few dominant terms ar ound the mode, where, 0()ln(/) tabba If we knew the values for a, b we can choose the shape parameter K of the Gamma function as the largest term, i.e., the integer part of ln(/) ba A special case is when the system has two id entical poles. Then, the impulsive response is exactly modeled by a single Gamma function with K = 1, 1/ a This is also approximately true when the magnitude of one pole is much larger than the other one. PAGE 116 116 APPENDIX C DERIVATION OF THE UPDATE RULE FO R THE CONSTRAINED OPTIMIZATION PROBLEM Using one Lagrange multiplier, we convert the constrained optimization problem in (41) to an unconstrained optimization problem. 2 22 0 22 ,argmin1T F aXasaaa Note that, 2 2 2 22 2 2T TTT F TTTT TTTT TTTTTr TrTrTr TrTrTr Tr XasXasXas XXXsaassa XXaXsssaa XXaXsssaa Setting the gradient of the Lagrangian function to 0 with respect to ,, a respectively, we find that the following set of equations holds: 2 0 2 222220 220 10T TTT ssaXsaaa aXsssaa a Solving for a, we have: 0 0 2 T T aXs ss aXs a aXs This is not a closedform solution for the optimal values. However, it can be effectively used as a fixed point update to itera tively find the optimal values. PAGE 117 117 APPENDIX D DERIVATION OF THE UPDATE RULE FOR THE UNCONSTRAINED OPTIMIZATION PROBLEM The unconstrained optimization problem is: 22 0 2 2argminT FaXasaaa First we note that: 2 000 222 2 2 000 22 0 22 22T TTT T aaaaaaaaa aaaaaaaa aaa Taking the derivative to 0, we get, 3 00 222220TT ssaXsaaaaaa Or equivalently, 0 2 3 0 2 TT Xsaa a ssaaa This is not a closedform solution for a However, it can be effectively used as a fixed point update to iteratively find the optimal values. PAGE 118 118 APPENDIX E MAP SOLUTION FOR TH E ADDITIVE MODEL With the assumptions indicated in Section 4.4, the posteriori probability can be rewritten as: 111 1(,......)(,)()()(...)K oKKkkokoK kpppppaaaaaaaa Given the model, maximization of the pos terior probability is equivalent to: 1,... 1argmax(,)()()oKK kokko kpppaaaa We assume a uniform (flat) a priori distribution for k Since oa is constrained to have a unit norm, its a priori distribution is a Di rac delta function: 21o a If we assume that ku is zeromean i.i.d. Gaussian noise with the same covariance matrix 2dI across all the trials, maximization of the posterior probab ility can be further simplified: 1 12 22 ,...,... 1 1argmaxlog(,)()argmaxlog1oKoKK K kokokkoo k kpp aaaaaaaa This can be converted to a c onstrained optimization problem: 2 2 1minK kko kJaa S.t. 21o a Setting the derivative to zero, a ne cessary condition for minimum is: T kok aa Plug this into the above cost function. We have: PAGE 119 119 22 22 11 1 1 11 KK TT kokokook kk K T TTT kooook k K TT kook k KK TTT kkokko kkJ aaaaaaaa aIaaIaaa aIaaa aaaaaa The first term does not depend on oa, so MAP is equivalent to: argmaxoT ooaaAa S.t. 21o a where 11 KK T kkk kkAAaa The matrix A is symmetric, so it can be diagonalized. The maximum occurs when oa is the normalized eigenvector of the matrix A corresponding to the largest eigenvalue. PAGE 120 120 APPENDIX F NORMALIZED ADDITIVE NOISE MODEL 1: MAP SOLUTION Following the same rationale in model 1, we attempt to derive the MAP solution for the model 2. The difference is in the covariance matrix of conditional probability: 2(,)~0,kokkpNaaI So, 2 /2 21 (,)exp2 2kokkko D D kp aaaa The MAP becomes: 1 1,... 1 2 22 ,... 1argmax(,)()() argmaxlog1oK oKK kokko k K kkoo kppp a aaaa aaa where, we have used the prior distribution for k : () D kkp This is an improper prior (but still a prior for Bayesian inference). It is mainly motivated by analytical tractability. Similarly as in mode l 1, we can write the following constrained optimization problem: 2 2 1argminokK kko kaaa S.t. 21o a We first find a necessary condition for minimum: TT kkkok aaaa Let T kk k T kk aa B aa Note that kB is idempotent, i.e., kkk BBB Plug this into the above cost function. We have: PAGE 121 121 2 2 1 1 1argmin argmin argmin argmaxo o o oK ko k K T oko k K T oko k T ooK a a a aIBa aIBa aBa aBa where, 11 T KK kk k T kk kk aa BB aa Again, the maximum occurs when oa is the normalized eigenvector of the matrix B corresponding to the largest eigenvalue. 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Since 2004, he has been working toward his Ph.D. at the Electrical and Computer Engineering Department at the University of Florida, under th e supervision of Jose C. Principe. His current research interests include statistical signal proces sing, machine learning and their applications in biomedical engineering. 