Citation
Temporal processing with neural networks

Material Information

Title:
Temporal processing with neural networks the development of the gamma model
Creator:
De Vries, Bert, 1962- ( Dissertant )
Principe, Jose C. ( Thesis advisor )
Childers, Donald G. ( Reviewer )
Taylor, Fred J. ( Reviewer )
Green, David M. ( Reviewer )
van der An, Jan J. ( Reviewer )
Phillips, Winfred M. ( Degree grantor )
Lockhart, Madelyn M. ( Degree grantor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1991
Language:
English
Physical Description:
ix, 148 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Architectural models ( jstor )
Cognitive space ( jstor )
FIR filters ( jstor )
IIR filters ( jstor )
Mathematical variables ( jstor )
Memory ( jstor )
Modeling ( jstor )
Neural networks ( jstor )
Parametric models ( jstor )
Signals ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Neural computers ( lcsh )
Neural networks (Computer science) ( lcsh )
Signal processing ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
This dissertation discusses the problem of processing complex temporal patterns by artificial neural networks. The relatively broad topic of this work is intentional--processing here includes such specialties as system identification, time series prediction, interference canceling and sequence classification. Rather than focusing on a particular application, this research concentrates on the paradigm of time representation in neural network structures. In all temporal processing applications, an essential capacity for a neural net is to store information from the recent past (we refer to this capacity as short term memory). The main contribution of this work is the introduction of a new (neural net) mechanism to store temporal information. This model, the gamma neural model, compares very favorable to competing memory structures, such as the tapped delay line and rirst-order self-recurrent memory units. The gamma memory mechanism is characterized by a cascade of uniform locally self-recurrent delay units. An interesting feature of the gamma memory mechanism is the adaptability of the memory depth and resolution. The gamma model is analyzed and compared with competing neural models. A temporal back propagation training procedure for gamma neural nets is derived. Experiments in time series prediction (electro-encephalogram (EEG) and synthetic chaotic signals), noise removal from a chaotic signal and sytem identification are discussed. In all experiments, the gamma model outperforms competing network architectures. Interestingly, the application of the gamma memory structure is not limited to neural nets. A chapter is devoted to introduce adaline (u) an adaptive filter with gamma memory. Adaline (U) generalizes Widrow's adaptive linear combiner (adaline), the most widely used structure in adaptive signal processing. The signal characteristics and processing applications where adaline (u) improves on the performance of adaline are identified.
Thesis:
Thesis (Ph. D.)--University of Florida, 1991.
Bibliography:
Includes bibliographical references (leaves 143-147)
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Bert De Vries.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Bert De Vries. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
026891607 ( ALEPH )
25541247 ( OCLC )
AJC2112 ( NOTIS )

Downloads

This item has the following downloads:


Full Text











TEMPORAL PROCESSING WITH NEURAL NETWORKS -
THE DEVELOPMENT OF THE GAMMA MODEL

















BY
BERT DE VRIES








A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1991














ACKNOWLEDGMENTS


The chances of mental depression during one's Ph.D. studies are not small.

Poverty and personal, social and intellectual isolation are just a few of the risks of the

"profession". Thus the real-world environment as provided by family, friends and

faculty largely determines the sanity of the student and therefore the quality of the

resulting dissertation. I have been very fortunate in this respect, although my friends,

committee members, let alone my family deserve no blame for the quality of this work.

My Ph.D. supervisor, Dr. Jose C. Principe, has been much more than an advisor.

His limitless supply of ideas, his personal commitment and warm character make him

an ideal supervisor, a fact which many students besides me have recognized. At this

place I wish to thank him for his collaboration and his friendship. In the Spring of 1989

I worked in the Hearing Research Laboratory headed by Dr. David Green. To witness

Dr. Green's approach toward conducting science is one of the best lessons a Ph.D.

student can receive. Dr. Jan van der Aa has been both on my master's and Ph.D.

supervising committee. His commitment to offer outstanding help at any time and his

personal friendship are very much appreciated. Dr. Donald Childers has been very

helpful at several times in guiding the next research steps. Dr. Fred Taylor's willingness

to serve on my doctoral committee is very much appreciated. I have had much support

from Dr. James Keesling from the mathematics department and Dr. Antonio Arroyo

from the electrical engineering department. Dr. Pedro Guedes de Oliviera from the

electrical engineering department of the University of Aveiro in Portugal visited our

laboratory during the 1991 spring semester. He has made significant contributions to

the understanding of the gamma model. His help and friendship is also very much














appreciated.


Several graduate students in the Computational Neuro-Engineering Laboratory

(CNEL) have directly contributed to the work that is presented here. James Kuo

performed the experiments on noise reduction which are discussed in chapter 5. The

experiments on prediction of the Mackey-Glass series was carried out by Alok Rathie.

Curt Lefebvre, Samel Selebi, James Tracey and Mark Goldberg have done significant

work on the gamma model as well.

Furthermore I should thank my friends and the students in our laboratory for

their friendship.

Most of all, I am indebted to my dear parents and my sisters Karin and Marleen.

Their support, encouragement and love cannot be compensated by a few simple lines.

I dedicate this work for what it is worth to their health and happiness.















TABLE OF CONTENTS


CHAPTER 1

THE TEMPORAL PROCESSING PROBLEM AND RESEARCH GOALS

1.1 Introduction........................................................................................... 1
1.2 A Statement of the Problem........................................................ 2
1.3 Research Goals ................................................... ........................... 7
1.4 A Summary of the Next Chapters ........................................................ 8

CHAPTER 2

A REVIEW OF NEURAL NETS FOR TEMPORAL PROCESSING

2.1 Introduction ................................................................... ............ 10
2.2 A Recapitulation of Linear Digital Filters ........................................... 10
2.3 Introduction to Neural Networks ......................................................... 12
2.4 The Adaptive Linear Combiner ...................................................... 16
2.5 Neural Network Paradigms Static Models ..................................... 20
2.5.1 The Continuous M apper ......................................................... 20
2.5.2 The Associative Memory ...................................................... 22
2.6 Neural Network Paradigms Dynamic Nets ..................................... 23
2.6.1 Short Term Memory by Local Positive Feedback ................... 24
2.6.2 Short Term Memory by Delays ............................................ 28
2.6.3 The Sequential Associative Memory ................................... 32
2.7 Other Dynamic Neural Nets ........................................... ........ ... 33
2.8 D discussion ............................................................. ....................... 34

CHAPTER 3

THE GAMMA NEURAL MODEL

3.1 Introduction Convolution Memory versus ARMA Model .................. 35
3.2 The Gamma Memory Model ............................................................ 39
3.3 Characteristics of Gamma Memory .................................... ........... 44
3.3.1 Transformation to s- and z-Domain ........................................ 44















3.3.2 Frequency Domain Analysis................................ ......... 46
3.3.3 Time Domain Analysis ........................................ .......... 48
3.3.4 Discussion ................................................ ....................... 50
3.4 The Gamma Neural Net .............................................................. 51
3.4.1 The M odel ................................................ ........................ 51
3.4.2 The Gamma Model versus the Additive Neural Net .............. 52
3.4.3 The Gamma Model versus the Convolution Model ................ 55
3.4.4 The Gamma Model versus the Concentration-in-Time net ..... 56
3.4.5 The Gamma Model versus the Time Delay Neural Net ......... 58
3.4.6 The Gamma Model versus Adaline ...................................... 58
3.5 D discussion ....................................................... ............................ 59

CHAPTER 4

GRADIENT DESCENT LEARNING IN THE GAMMA NET

4.1 Introduction Learning as an Optimization Problem ...................... 60
4.2 Gradient Computation in Simple Static Networks .......................... 63
4.2.1 Gradient Computation by Direct Numerical Differentiation ... 64
4.2.2 The Backpropagation Procedure ........................................ 65
4.2.3 An Evaluation of the Direct Method versus Backpropagation 70
4.3 Error Gradient Computation in the Gamma Model ......................... 72
4.3.1 The Direct M ethod ............................................ ............ 73
4.3.2 Backpropagation in the Gamma Net ..................................... 76
4.4 The Focused Gamma Net Architecture ........................................... 80
4.4.1 Architecture ............................................... ...................... 82

CHAPTER 5

EXPERIMENTAL RESULTS

5.1 Introduction ...................................................... ........................... 87
5.2 Gamma Net Simulation and Training Issues .................................... 88
5.2.1 Gamma Net Adaptation ...................................... ......... 89
5.3 (Non-)linear Prediction of a Complex Time Series .......................... 92
5.3.1 Prediction/Noise Removal of Sinusoidals contaminated by Gaussian
N oise ................................................... .......................... 92
5.3.2 Prediction of an EEG Sleep Stage Two Segment ................... 95















5.3.3 Prediction of Mackey-Glass chaotic Time series ........ .......... 95
5.4 System Identification .................................................................. 98
5.5 Temporal Pattern Classification Training a Concentration-in-Time Net 99
5.6 Noise Reduction in State Space .......................................................... 102
5.7 D discussion ........................................................................................... 109

CHAPTER 6

THE LINEAR FILTERING PERSPECTIVE

6.1 Introduction .................................................................................
6.2 A Recapitulation of Linear Digital Filter Architectures ..................... 112
6.3 Generalized Feedforward Filters Definitions .................................... 113
6.4 The Adaptive Gamma Filter ............................................... 116
6.4.1 D efinitions ................................................................................ 116
6.4.2 Stability .................................................................................... 117
6.4.3 Memory Depth versus Filter Order ....................................... 118
6.4.4 LMS Adaptation ....................................................................... 118
6.4.5 Wiener-Hopf Equations for the Adaptive Gamma Filter ......... 120
6.5 Experimental Results ....................................................................... 122
6.6 The Gamma Transform A Design and Analysis Tool For Gamma Filters 125
6.7 A Second-order Memory Delay Element ............................................ 129
6.8 D discussion ............................................................................................ 130

CHAPTER 7

CONCLUSIONS AND FUTURE RESEARCH RECOMMENDATIONS

7.1 A Recapitulation of the Research ......................................................... 134
7.2 Ongoing Research Projects .................................................................. 136
7.3 Future Research Directions .................................................................. 138
R EFER EN CES ............................................................................................. 141

BIOGRAPHICAL SKETCH .......................................................................... 145














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


TEMPORAL PROCESSING WITH NEURAL NETWORKS -
THE DEVELOPMENT OF THE GAMMA MODEL



By

Bert de Vries

October 1991

Chairman: Dr. Jose C. Principe

Major Department: Electrical Engineering

This dissertation discusses the problem of processing complex temporal

patterns by artificial neural networks. The relatively broad topic of this work is

intentional processing here includes such specialities as system identification, time

series prediction, interference canceling and sequence classification. Rather than

focusing on a particular application, this research concentrates on the paradigm of time

representation in neural network structures.

In all temporal processing applications, an essential capacity for a neural net is

to store information from the recent past (we refer to this capacity as short term

memory). The main contribution of this work is the introduction of a new (neural net)

mechanism to store temporal information. This model, the gamma neural model,

compares very favorably to competing memory structures, such as the tapped delay line














and first-order self-recurrent memory units. The gamma memory mechanism is

characterized by a cascade of uniform locally self-recurrent delay units. An interesting

feature of the gamma memory mechanism is the adaptability of the memory depth and

resolution.

The gamma model is analyzed and compared with competing neural models. A

temporal backpropagation training procedure for gamma neural nets is derived.

Experiments in time series prediction (electro-encephalogram (EEG) and

synthetic chaotic signals), noise removal from a chaotic signal and system

identification are discussed. In all experiments, the gamma model outperforms

competing network architectures.

Interestingly, the application of the gamma memory structure is not limited to

neural nets. A chapter is devoted to introduce adaline(gl), an adaptive linear filter with

gamma memory. Adaline(tg) generalizes Widrow's adaptive linear combiner (adaline),

the most widely used structure in adaptive signal processing. The signal characteristics

and processing applications where adaline(gi) improves on the performance of adaline

are identified.














CHAPTER 1


THE TEMPORAL PROCESSING PROBLEM AND RESEARCH GOALS


1.1 Introduction

When I started this research somewhere around January 1988, the goal was to

develop a speech recognition system that was characterized by a relatively high degree

of biological plausibility. Thus I spent the first semester of 1988 in the Hearing

Research Center at the University of Florida. This speech system-in-waiting was going

to be developed from the bottom up! First I would develop a neural auditory model that

captures the essential features for speech analysis, followed by new techniques for

neural temporal pattern classification and so on. As time passed by, the topic of this

dissertation has shifted more toward the fundamental problem of time representation in

network structures per se. As a result, a neural network model for processing of

temporal patterns has been developed, and at the time of the writing of this thesis we

are conducting experiments in isolated word recognition and phoneme classification

making use of this new model. As it turns out, the application of the gamma model (this

is how the model is called) is not limited to the temporal classification problem.

Successful experiments were performed as well for problems in noise reduction,

system identification and time series prediction. For these tasks the gamma model

performance is very promising, in fact better than many competing techniques. In this

thesis the development of the gamma model is presented. I try to explain why and how

it works and report on a few experimental results.









1.2 A Statement of the Problem

This dissertation deals with processing of time-varying signals by a neural

network. By a signal we mean a time sequence of patterns and consequently the

adjectives "time-varying" or "temporal" will often be deleted. The temporal context is

always assumed. The 6-dimensional (input) signal to be processed is denoted by s(t)

and for the M-dimensional processed (output) signal we will write x(t). Processing of

s(t) implies the application of a map 0 which transforms s(t) into the signal x(t). In this

work we are interested in processing applications where the past of s(t) affects the

computation of x(t). Thus, 0 is actually a map from s(x) for x < t to x(t). Typical

applications include speech recognition, dynamic system identification and prediction

of a time series. The main problem of this thesis is how to effectively represent the past

of s(t). The following example shows why this is a difficult task.

Consider an isolated-word recognition system 0 which processes the incoming

speech signal in real-time. Assume a 16 word vocabulary and hence the output space

can be described in 4 bits. The speech signal is sampled at 8000 [Hz] and quantized by

8 bits. Then, for a typical word, which last 0.5 seconds, the input signal is represented

by 8000 x 8 x 0.5 = 32, 000 bits. Thus, 4 is a map from 232000 input pattern

combinations to 4 bits. This sequence of 32000 bits obviously contains many redundant

copies of information and is contaminated by irrelevant noise. Since 0 is a real-time

system, it must throw away data as quickly as possible, while preserving enough

information to classify the words. Note the inherent optimization task associated with

this problem: if we throw away too much data there may not be enough information left

for word classification, but yet we do not want to store redundant data and noise for it

will significantly complicate the processing task.

As another example, consider the problem of predicting a discrete time series

s(t) by a model 4. How many samples from the past of s(t) does the model 0 need in









order to reliably predict the next sample s(t+l)? Do we need to store the entire history

of s(t)? Very likely not, and in fact the additional noise associated with a deep memory

will work detrimental on the performance of (.

In general we conclude that before the data stream is submitted to the actual

processing task, first we need to rid the data stream of irrelevant information as much

as possible. Processing of a data stream so as to reduce the amount of data for further

processing is captured by the name pre-processing.


I pre-processing
to L (data reduction)

Figure 1.1 Modular feedforward information processing architecture.




In Figure 1.1 the general architecture for a feedforward information processing

scheme is shown. In this simple scheme two stages are distinguished. The first stage,

the pre-processing stage, serves to reduce the data flow to the second stage. In practice,

pre-processing of a data stream commonly consists of segmentation and feature

extraction. The second stage performs the actual processing goal. For instance in a

pattern recognition environment, the second stage is a classifier. For the purpose of this

discussion it is assumed that the second processing stage is implemented by an adaptive








neural network.

Let the signal s(t) in Figure 1.1 be defined for to 5 t 5 t in the temporal

dimension. At any time, only part of the temporal domain of s(t) is available for further

processing. The selection mask is called a window. The width of the window is denoted

by 8. In some cases before the data sequence is submitted to the actual processing task

), signal features are computed from the windowed data segment. As an example in

word classification, the pitch period provides important information concerning the

excitation source of the speech signal. Note that both (windowed) segmentation and

feature extraction contribute to the data reduction process.

The feature extraction process is usually very problem dependent. For now, let

us concentrate on the implications of the choice of the window width 8. Three

possibilities arise. First, the width 8 of the window equals the duration tf to of the

signal. This case effectively transforms the temporal dimension to an extra spatial

dimension. Since the entire past of the signal is always available, no temporal

processing capabilities are required and the problem is transferred to a static processing

problem. The second possibility concerns 0 < 8 < tf- t, which is called the sliding

window technique. In this case, the selection mask moves in some fashion over the

signal domain so as to cover the input signal space as time evolves. The extreme case

of the sliding window technique occurs when 8 -- 0, that is, only the current signal

values are available for further processing. We will refer to this choice as current-time

processing, and effectively when 8 -4 0 there is no segmentation. Obviously, in this

case the memory has to be moved to the second processing stage.

The choice of the sliding window width influences the system performance. We

identify two problems associated with the selection of 8. First, for large 5. the

dimensionality of the processing system increases. This fact minimally complicates

the neural net training. It has been shown that neural net adaptation time unfortunately









scales worse than proportional with the dimension of the weight vectors (Perugini and

Engeler, 1989). There are more problems associated with large networks, such as the

required increased dimension of the training set. For smaller windows the neural

network may not have enough information to appropriately learn the signal dynamics

because only a fraction of the decision space is available. However, the network

dimension gets progressively smaller which eases the learning requirements in terms

of training set size and number of learning steps.

A second issue that complicates the choice of 8 involves segmentation of non-

stationary signals. Normally the length of the stationary interval is not known a-priori,

and can very well change with time. A large 6 tends to average the time-varying

statistics of non-stationary signals before the signals enter the neural net. A smaller 5

makes the classification very sensitive to the actual signal segment utilized, and tends

to make the classification less robust. The balance is very difficult to achieve and in

general varies with time. The common practice in speech is to bias the selection to

fixed, relatively small segments of approximately 10 milliseconds (Rabiner and

Schafer, 1978).

Apart from the difficulty of choosing the window width, the temporal

resolution of the window is another important pre-processing parameter. We define the

(temporal) resolution R of the window as the number of outputs of the window divided

by the window width (in seconds). As is the case for the width 6, the optimal resolution

is dependent on the processing goal. For instance if the vocabulary size were 1000

instead of 16 in the isolated-word recognition example, the demands on the resolution

of the window would obviously increase.

In speech processing, it is common to determine the window width based on

statistical measures of the input signal. For instance, zero-crossing rate and energy

measures have been used to estimate pseudo-stationary signal segments (Rabiner and









Schafer, 1978). Note that this approach does not use any system performance feedback

to determine the pre-processing parameters. However, as is clear from the foregoing

discussion, optimal values for the pre-processing parameters such as window size and

resolution are a function of the processing goal as expressed by a system performance

criterion. Ideally the representation of the input signal would be adapted by

performance feedback of the total processing system.

This observation forms the basis for the neural net system that is proposed in

this dissertation. The system that I propose stores the signal history in an adaptive

short-term memory structure of a neural net. The capacity of a neural net to store and

compute with information from the recent past is referred to as short term memory. The

architecture of this system is shown in Figure 1.2. The neural short-term memory

mechanism substitutes and obviates a priori signal segmentation. An important

advantage of this approach is that neural network structures can be adapted so as to

optimize a system performance criterion. In the figure, the performance of the system

is measured by the error signal e(t), the difference between a desired output signal d(t)

and the system output x(t). Other measures of system performance are also possible. Of

central importance however is that in this framework the signal representation is

optimized by performance feedback instead of input signal statistics.

It will be shown in this thesis that adaptive pre-processing can be integrated

with information processing in the same neural network framework and

implementation.

Traditionally, linear signal processing is implemented by linear filter structures.

Digital filters can be categorized into two main architectural groups: the finite impulse

response (FIR) filters and infinite impulse response (IIR) filters. FIR filters are

feedforward and the past of the input signal is stored in a tapped delay line. IIR filters

are of recurrent (feedback) nature. As a result, more complicated memory structures
















e(t)


based on feedback are possible. As will be discussed in chapter 2, the same principles

for short term memory hold in neural networks. In fact, from an engineering viewpoint,

neural networks can be considered as a generalized class of non-linear adaptive filters.

The combination of adaptation and non-linearity makes neural nets very versatile

processing architectures that can be applied to a wide range of complex problems.

Indeed the possibility to incorporate the input signal representation in a unified

adaptive framework with the system overall performance is a premier stimulus for the

research reported herein.

1.3 Research Goals

The central theme of this dissertation is the representation of temporal

information in a neural net. The main goal of this research is the development of a

neural network where the input signal representation is optimized adaptively by the

neural net itself. Thus I try to achieve that preprocessing parameters such as window

resolution or depth are adaptively optimized with respect to a performance measure of


adaptive
algorithm

Figure 1.2 Adaptive signal representation using a neural network
processing framework.









the total system.

A literature review, scheduled for chapter 2, will reveal that the main techniques

for temporal processing with neural networks are quite mediocre with respect to the

capacity to adapt to a varying signal environment. Since experimentation with a wide

range of network architecture is still going on, it seems that a consistent framework for

dynamic neural networks is still missing. Based on these premises the research plan is

scheduled as follows:

Design a neural network for temporal processing with adaptive short term

memory.

Develop training algorithms for this neural model.

Evaluate both in theory and by experimentation the applicability of the new

model. In particular we are interested in a comparison with alternative widely used

neural processing architectures.

Determine appropriate application areas for the new model.

1.4 A Summary of the Next Chapters

Chapter 2 starts with a concise review of neural networks. Neural nets are being

studied from a variety of viewpoints. I will take the "electrical engineering view" and

emphasize the relation to linear digital filters. Mechanisms for short term memory in

neural nets are reviewed. The analysis focuses in particular on two widely used

structures, the tapped delay line and the first-order self-recurrent units (context units).

Both mechanisms are shown to have limited applicability. For example, the tapped

delay line has limited fixed memory depth whereas the context units always overwrite

information from the past with more recent information.

In chapter 3 a new framework for storage of past information in neural nets is

introduced. The new memory model, gamma memory, is supported by a mathematical









framework which allows a comparative analysis with respect to competing models. The

gamma memory model is adaptive and unifies several existing models such as the

tapped delay line and the context units. Neural networks that store the past in an

adaptive gamma memory structure are defined as gamma neural models.

Since gamma memory is an adaptive structure, training algorithms for the

gamma model have to developed. This topic is discussed in chapter 4. Attention is

focused on gradient descent adaptive algorithms. The gradient descent algorithm for

the gamma model as described here generalizes backpropagation (for strictly

feedforward neural nets) and the least mean square (LMS) algorithm (for adaptive

linear filters).

In chapter 5 the gamma model is applied to noise reduction, prediction of

complex time series and a system identification problem. The performance of the

gamma model is compared to alternative neural network models. The gamma model

comes out very well in a comparison to the conventional memory models such as the

tapped delay line and the context unit models.

As a linear structure, the gamma memory can be applied in adaptive linear

filters. In chapter 6, a new filter structure, the gamma filter, is introduced. The gamma

filter is a FIR filter where the delay line is substituted by a gamma memory. This new

filter structure provides an interesting medium between FIR and IIR filters. The

adaptive gamma filter is mathematically analyzed and applied in some validation

experiments.

Chapter 7 summarizes the results of this work and makes recommendations for

further research.














CHAPTER 2


A REVIEW OF NEURAL NETS FOR TEMPORAL PROCESSING


2.1 Introduction

In this chapter various neural network architectures for processing of time-

varying patterns are reviewed.

In order to set the framework, we are concerned with extracting information

from a temporal sequence, but let it unspecified whether the processing goal is to

predict a future trend of the time series or to classify the sequence.

First, in the next section linear digital filters are recapitulated. Digital filters are

the basic computational tool for temporal processing. Moreover, the architectural

principles of digital filters underlie most neural network models.

2.2 A Recapitulation of Linear Digital Filters

Linear signal processing is traditionally implemented by linear filters. In a

discrete time environment, linear digital filters are networks of delay elements,

summation element and constant-factor multipliers. Digital filters are distinguished

into two main categories: Feedforward or finite impulse response (FIR) filters and

recurrent or infinite impulse response (IIR) filters.

In a FIR filter, the input signal history is stored in a tapped delay line. The

signals at the taps are referred to as state variables. The output of the FIR filter is a

linear weighted combination of the tap variables (see Figure 2.1). FIR filters are always

stable but note that the depth of the memory is fixed and equals the number of taps of









the delay line.



X,(l) X:(l) X- ^(t)
'O z-1 -- z-1 -- M z-1 -0


W Or W1 V '2 \VK

+ +w
Figure 2.1 The feedforward (FIR) filter.


IIR filters are more complex structures since recurrent connections are also

allowed. In control theory similar linear structures are known as auto-regressive

moving-average (ARMA) systems. A so-called observer canonical form

implementation of the IIR filter is shown in Figure 2.2. The existence of recurrent

connections implies the risk of instability of the system, but increases the

computational power of the system. The state variables xi(t) are a function of both the

lower index state variables (memory by delay as in the FIR filter) as well as higher

indexed variables (memory by feedback). The feedback connections in the IIR model

implies that the depth of the memory is no longer coupled to the number of delay

elements.

Linear filters are widely applied but the processing power is limited. Linear

processing is appropriate for tasks such as removal of signal-independent noise and

rearranging the temporal structure of a signal. For many important tasks linear

processing does not suffice though. Examples are removal of signal-dependent noise,

classification (decision making!) and modeling of a chaotic time series.

Neural networks as an engineering tool are probably best interpreted as a

generalized class of nonlinear adaptive filters. As such, they provide the computational

features that potentially better cope with solving complex non-linear problems. Next,
























Figure 2.2 The recurrent (IIR) filter.


an introduction to neural nets is presented.

2.3 Introduction to Neural Networks

This section contains a brief introduction to neural networks. In the literature

we also find names as connectionist models, parallel distributed processing devices or

artificial neural nets, all denoting the same kind of processing architecture. The

discussion will be of general nature. For a deeper look into some equations and

implementations of neural networks, I like to refer to a paper by Lippmann (1987) and

a book by Simpson (1990). A more thorough look at neural networks is offered in

books by Hertz et al. (1991) and Hecht-Nielsen (1990).

There is not a single best definition for a neural network. Neural net research is

being approached from various viewpoints. Different models range widely in

biological plausibility. In the context of this thesis, an electrical engineering

dissertation, the biological plausibility is not considered a high priority. We are more

interested in the computational properties of a model. In a computer science book I

have seen a definition as short as the following:


- A neural net is a weighted directed graph of simple processors.








As may be clear from a previous discussion, I like to interpret neural nets as a

generalized class of non-linear adaptive filters. The following features of a neural net

processor are typical:

parallel architecture: a weighted network of simple processors.

adaptation: the connection weights are adaptive.

non-linearity: the processor transfer function is in general non-linear.

The mathematical framework for neural nets is non-linear dynamics. In a

continuous-time setting, neural nets are described by a set of differential equations. In

discrete time, the dynamics are described by difference equations. Characteristically,

the constant coefficients of the equations, called weights, adapt when examples of the

problem at hand are presented to the net. Ideally, the adaptation or learning of the

weights is also determined by a differential equation. As mentioned before, neural

networks are non-linear. For one, the computational power of non-linear dynamical

systems far exceeds that of linear systems. Secondly, the non-linearity of neural nets

originates from the fact that it is believed that most interesting primitive cognitive

functions such as associative memory are non-linear. Generally, let

x(t) = [l (t) ... xN(t)] hold the N-dimensional state of a neural net,


w11 ... wIN
w = i an N2-dimensional vector (for a fully connected net) of adaptive
WNI ... WN

weights and I(t) = I' (t) ... IN(f) the external input to the net. Then the system

is completely described by the following set of equations:

dx,
--(t) = fi (x, I, w) El









dwt
dt (t) = gi ,x)(WX

The dynamics for the state x(t) are described by Eq. and Eq.2 describes the adaptation

dynamics. The equilibria of system Eq.1 are computed by


0 = fi (x I, w), WEa2

where x* holds the steady state.

The most widely used neural network model is the so-called additive model described

by


t (t) = -ax,(t) + Y wjxj (t) + l (t) E_ A


The additive model is used in the great majority of practical applications of

neural networks today. Sejnowski provides a biological motivation for the additive
model (Sejnowski, 1981). A flow diagram of the additive model is shown inFigure 2.3.

The state vector xi(t) is affected by a passive decay -axi(t), yielding short term

memory, non-linear neural feedback signals a(wijxj(t)), and an external input Ii(t). The

neuron signal function a() normally is a non-linear function. A typical choice is the

logistic function a (x) = tanh (x). The feedback signals from the net itself are

sometimes shortly denoted by the variable net, that is,

N
neti(t) = wix (t) .
j=1

The system described by Eq.4 is called additive since the weights w are not a

function of the states x. In case w = w (x), the model exhibits mass-action behavior;

such systems are called mass-action, shunting or multiplicative models. In order for

Eq.4 to be computationally interesting, the three dynamic variables I, x and w must






























Figure 2.3 The additive neural network model.


perform over three different time scales. From a neurodynamic viewpoint, we can

interpret x(t) to hold a short term memory (stm) trace and w(t) to process long term

memory traces (Itm). The philosophy behind system Eq.4 as a pattern recognition

device for temporal patterns then basically runs as follows. As time passes by, the Itm

traces w(t) sample and average over time the neuronal activity x(t), thus forming some

kind of template or reference pattern of neural activity. At any time a short term average

of the current external environment I(t) is reflected in the stm traces x(t). The degree of

matching between the stm traces and the Itm traces determines how well the current

environmental input is recognized.

The basic architectural component of neural networks and adaptive signal

processing structures is the adaptive linear combiner. An understanding of the working

of the adaptive linear combiner and the least mean square (LMS) algorithm is essential

for the neural network structures that are surveyed in this thesis. The next section









introduces the adaptive linear combiner.

2.4 The Adaptive Linear Combiner

The adaptive linear combiner or non-recursive adaptive filter is fundamental to

adaptive signal processing and neural network theory and applications. This structure,

normally shortly referred to as adaline (from adaptive linear neuron), was introduced

by Widrow and Hoff in 1960 (Widrow and Hoff, 1960). The adaline structure appears

in some form in nearly all feedforward neural network structures. The processing and

adaptation properties of adaline are well understood and documented in Widrow and

Stearns (1975) and Haykin (1990). In this thesis we will only introduce the properties

that are essential in the context of this work.

The describing equations for adaline are given by

K
y(t) = W wkxk(t),
k=

where xk(t) are the input signals, y(t) the output signal and wk the adaptive parameters

or weights. Adaline is a discrete-time structure, that is, the independent time variable t

runs through the natural numbers to,to+1,... and so on. Adaline is shown in Figure 2.4.


tapped delay
line


Figure 2.4 The adaptive linear combiner structure.








Although the input signals xk(t) may originate from any source, very often the

input signals are generated from a tapped delay line as shown in the figure. For this

case, the adaline structure is similar to a regular transversal FIR filter. In the adaptive
signal processing literature, it is common to define the following vectors

T T
w- [w ...] xi((t)(=[i() ... x (t) E.7

Thus we can write the describing equation for adaline as

y(t) = wTx(t). Eq

Let a desired output signal be given by d(t). d(t) is also referred to as target
signal or teacher signal. The difference between the desired output and actual output is

defined as the (instantaneous) error signal e (t) = d(t) -y (t). Substitution of Eq.8

and squaring leads to the following expression for the instantaneous squared error
signal:

e2(t) = d2 (t) +wT(t)xT(t)w-2d(t)xT (t)w. E.9

An important assumption in the theory of adaptive signal processing is that the signals

e(t), d(t) and x(t) are statistically stationary, that is, their statistical moments are
constant over time. In that case, taking the expected value of Eq.9 yields

E[e2 (t) ] = E[d2 (t)] + wTRw- 2PTw Eq.

where we defined the input correlation matrix R E [x(t)x (t)] and the cross-

correlation vector P E [d (t) x (t) ]. Note that the expression for the mean squared

error 4 is quadratic in the parameters w. The minimal mean squared error is obtained

by setting the error gradient to zero. Differentiating Eq.10 yields for the error
dw


gradient:









= 2Rw- 2P. Eqll
aw

Thus, the optimal weight vector woptis given by


wopt = R-1P. E 2

The expression Eq.12 is known by the name Wiener-Hopf equation or normal equation.

The Wiener-Hopf equation provides an expression for the minimal mean-square-error

weight vector, assuming the stationarity conditions hold. This expression is

fundamental in adaptive signal processing and linear neural network theory. Note that

if the stationarity conditions do not hold, the correlation matrices R and P are time-

varying, and consequently the optimal weight vector is time-varying.

The computation of the correlation matrices R-' and P is usually very expensive,

in particular when the network dimension K is large. Instead it is common to adapt the

weights on a sample-by-sample basis so as to search for the optimal values. As is

apparent from Eq.9, the mean-square-error is quadratic in the weights. Thus, the

performance surface 4 is a (hyper-)paraboloid with a minimum at wopt. A gradient

descent procedure should therefore in theory lead to the optimal weights. The steepest

descent update algorithm adapts the weights as follows:


w(t+l) = w(t)- 1 E13


The step size parameter il controls the rate of adaptation. In the neural net literature, iT

is referred to as the learning rate. Note that adaptation comes naturally to a halt when

the weights are optimal, since at the minimum of the performance surface we have

w = 0. The computation of the error gradients determines the complexity of the

learning algorithm. Widely used and very efficient is the Least Mean Square (LMS)

algorithm. We will now proceed to derive the LMS algorithm for the adaline structure,








as it is the precursor for the widely used backpropagation procedure in neural net

adaptation. At a later stage in this thesis, the backpropagation procedure is derived and
applied to several signal processing problems.

The central idea of the LMS algorithm is to approximate the stochastic gradient

aE [e2 (t) ] e2 (t)
w[e2 W by the instantaneous (time-varying) gradient Note that the
aw aw

instantaneous error gradient is an unbiased estimator of the stochastic gradient, that


is,E = e(t) DE [e.2t) I Substituting e (t) = d(t) x (t) w leads to


ae2 (t) De (t)
S= 2e (t) = -2e (t) x(t). .14
aw jw

Thus, the LMS update equation evaluates to

w(t+ 1) = w(t) +2rle(t)x(t). E

Note how simple the final equation for the LMS algorithm is. The signals e(t) and x(t)
are readily available. The combination of simplicity and accuracy have made the LMS
algorithm the most popular algorithm in adaptive signal processing. Widrow discusses

in his book a number of successful practical applications, such as adaptive
equalization, system identification, adaptive control, inference canceling and adaptive
beamforming (Widrow and Stearns, 1985).

In the next section, the core architectural paradigms for neural nets are

introduced. While adaline enjoys a wide application in neural network architectures,

the inherent linearity limits its computational power substantially. Neural nets in
general are more powerful, since they can be non-linear, recurrent and multi-input-

multi-output systems.









2.5 Neural Network Paradigms Static Models

Over the years two different paradigms have emerged that exploit the dynamics

of system Eq.4 to serve as a non-linear information processor. Nearly all theory and

practice deals with processing of static patterns. I will shortly introduce the two

concepts for the static case, since an understanding of this is essential in order to

comprehend efforts of extension to processing of space-time patterns. The section on

static nets is followed by dynamic nets.

2.5.1 The Continuous Mapper

The first paradigm offers the continuous mapper or many-to-many map. The

computational result of the continuous mapper is a continuous function from an 6-

dimensional input space to an M-dimensional output space. The standard method to

implement such a map is by way of a multi-layer feedforward network. An important

historical (and current) example of the continuous mapper is the perception

architecture (Rosenblatt, 1962). In Figure 2.5, a neural net implementation of the

continuous map is displayed.

The network in Figure 2.5 is feedforward, that is, there are no closed loops in

this structure. If we assume that the neurons are labeled sequentially starting at the

input layer, then the weight matrix w is lower triangular for feedforward nets, since

each neuron receives inputs from nodes with lower index. The states of the neurons are

independent of time. In particular, the additive static neuronal states are described by

the following algebraic relation:

Xi = o(wYjXj) +. Eq16
j
It has been proven that a three-layer network (two hidden layers) in principle is

capable to compute an arbitrary continuous map from the O-dimensional input space to

the real numbers (Hecht-Nielsen, 1987). Although this may be impressive, the problem













output layer





2nd hidden layer





1st hidden layer






Input layer x,


I 17, 1
Figure 2.5 Structure of three-layer feedforward mapper


of finding the correct set of weights may be very hard. The problem of finding good

weights is called the loading problem. Theoretically, a learning mechanism such as

simulated annealing can be used to obtain the map that minimizes the error between the

desired map and the actual (network) implementation. However, simulated annealing

(stochastic optimization) is very slow and in practice, although not perfect, the back-

propagation training procedure has been quite effective for many applications. Back-
propagation involves adaptation of the weights by gradient descent so as to minimize a

performance criterion.









2.5.2 The Associative Memory

The second prototype entails the associative memory or many-to-one map, for

which the Hopfield net is the prime illustration (Hopfield, 1982 and 1984, see Figure

2.6). In terms of topology, the Hopfield net is a recurrent net with symmetric weights

(wj=wji, wi,=O), which enables the association of a Lyapunov (energy) function with

the system dynamics. Using Lyapunov's stability theory, it can be shown that this

system always converges to a point attractor. There is no external input to the

associative memory. The input is the initial state of the neurons x(to). Used as a

processing device, information is stored by locating point attractors at positions in the

state space that correspond to memories. Recognition then consists of settling into the

minimum closest to the initial state vector x(to).












W2N

Figure 2.6 An associative memory neural net the
Hopfield net structure.




Both the continuous mapper and the associative memory work only in a static

pattern environment. Although the Hopfield net processes information by a dynamic

relaxation process, the input pattern (initial state vector) is assumed to be static. Next,

temporal extensions of both the continuous mapper and the associative memory are

discussed. It will become clear that the ideas for computing with time in dynamic









neural nets correspond strongly to linear signal processing theory.

2.6 Neural Network Paradigms Dynamic Nets

The basic neural network model for processing of static patterns is the static

additive model. The activation of the units are computed by

i = o( iwjX) +l, E-.I
j
where xi is the activation of neuron (unit, node) i. The weight factor wi connects node

j to node i. o() is a (non)-linear squashing function and Ii represents the external input.

We assume a system dimensionality of N. Sometimes the shorthand notation

net, = wixi will be used.
J
Static neural nets have no memory. As a result, temporal relations can not be

stored or computed on by static neural nets. In order to process a temporal flow of

information, a neural net needs a short term memory mechanism. Neural network

models with short term memory are called dynamic neural nets. The simplest way to

dx,
add dynamics (memory) to the static model is to add a capacitive term C- to the left-

hand side of Eq.17. After rearrangement of terms, the so-called dynamic additive

model is obtained:

dx.
it = -xi+a( wijxj) +i. E4


This model is mathematically equivalent to the system described by Eq.4, where the

1
time constant is expressed by the decay parameter a = -. Let us look at the biological

picture of neural nets. In nature, the neural time constants are fixed and equal

approximately 5 msecs (Shen, 1989). This number is estimated by assuming an average









action potential rate of 200 per second. Higher rates are quite rare due to the refractory

period of the neurons. However, recognition of a spoken word requires the ability to

remember the contents of a passage for approximately 1 second. To accomplish this,

neural temporal resolution decreases while the "temporal window of susceptibility"

increases toward the cortex. Apparently, the brain is able to modulate temporal

resolution and depth of short-term memory making use of processing units with fixed

small time constants. The dominant biological principles for increasing the time

constants are feedback and delays. These are exactly the same strategies that are used

in digital filters to implement a temporal data buffer. Naturally, neural net researchers

have concentrated on the same concepts of feedback and delays when designing neural

nets for temporal processing applications. Next, the characteristics of both approaches

are analyzed.

2.6.1 Short Term Memory by Local Positive Feedback

The additive model can be extended with a positive state feedback term,

yielding

dx,
C- = X + o (net) + Ii + kx, E19

where k is a positive constant. In the biological literature, such local positive feedback

is often named reverberation, while neural net researchers speak of self-excitation.

Eq.19 can be rewritten as

T dxi
(-k)dt = -xl+a (neti) + I, B2

Y (neti) Ii
where b (neti) k and i -. For = 5 msec and k = 0.995, we get the


new time constant T = 1 = 1 sec. Units that self-excite over a time span that is
1-k








relevant with respect to the processing problem are referred to in the neural net

literature as context units. Several investigators have explored the temporal

computational properties of additive feedforward nets, extended by context units

(Jordan, 1986; Elman, 1990; Mozer, 1989; Stornetta et al., 1988). In Hertz et al. (1991),

neural models of this kind are collectively referred to as sequential nets. In sequential

neural nets, all units are additive and static, apart from the context units. The context

units are of type Eq.20 or a similar model. Sequential neural nets are a kind of

extension of the continuous mapper to the spatiotemporal domain. In Figure 2.7,

various architectural examples are displayed.In 1986, Jordan developed the

architecture as displayed in Figure 2.7a for learning of spatiotemporal maps. The state

of the context units evolve according to

x(t+ 1) = lx (t) +xut(t), E.21

where xout(t) is the state of an output unit. Note that Jordan's architecture makes use of

global recurrent loops (context to hidden to output to context units). As a result, care

must be taken to keep the total system stable. In Jordan (1986), he shows that this

network can successfully mimic co-articulation data. Anderson et al. (1989) have used

this architecture to categorizing a class of English syllables.

Elman (1990) utilizes non-linear self-recurrent hidden units of the type

x (t + 1) = g(o (x (t)) to store the past (Figure 2.7b). This network was able recognize

sequences, and even to produce continuations of sequences. Cleeremans et al. (1989)

showed that this architecture is able to learn and mimic a finite state machine, where

the hidden units represent the internal states of the automaton.

Stornetta et al. (1988) have used recurrent units at the input layer only to

represent a temporally weighted trace of the input signal (Figure 2.7c). There are no

weighted connections from the hidden or output units toward the context units. This

restriction results in several advantages when the network is trained by a gradient


























(a) Jordan's network (b) Elman


outputs


(c) Stornetta et al.


hidden


c xt



I inputs I
Figure 2.7 Various sequential network architectures. (a) Michael Jordan's
architecture feeds the output back to an additional set of recurrent input
units. (b) Elman's structure uses recurrent non-linear hidden units. (c)
Stornetta et al. keep a history trace at the input units. This structure offers
particular advantages when back-propagation learning is used.



descent technique. We will discuss this issue in more detail in chapter 4 on training a

neural net. The author performed successful experiments in recognition of short

sequences. Mozer (1988) and Gori et al. (1989) have also made use of similar









architectural restrictions.

While the positive feedback mechanism is simple and used in biological

information processing, there are two computational problems associated with this

method. First, the new time constant is very sensitive to k. For our example, an increase

of 0.5% in k from k = 0.995 to k = 1 makes the model unstable. The time-varying nature

of biological parameters makes it therefore unlikely that reverberation is the

predominant mechanism for short term memory over long periods. The second

handicap of Eq.20 is that the new model is still governed by first-order dynamics. As a

result, weighting in the temporal domain is limited to a recency gradient (exponential

for linear feedback), that is, the most recent items carry a larger weight than previous

inputs. Note that the analytical solution to Eq.20 can be written as

t -(-s)
x(t) = Je [(net(s)) +I(s)]ds. EQ.22
0

t
Thus, the past input is weighted by a factor e which exponentially decays over time.

For a neural net composed of N neurons, the number of weights in the spatial

domain is O(N2), while the temporal domain is governed only by T. The use of a fixed

passive memory function then implies a limit to how structured the representation of

the past in the net can be. As an example, optimal temporal weighting for the

discrimination of the words "cat" and "mat" will not be a recency but rather a primacy

gradient. Another example, in a time-series analysis framework, the input signals

sometimes change very fast, sometimes slow. For fast changing input, we like the time

constant small, so that the net state can follow the input. For slow moving input, the

time constant may be larger in order to have a deeper memory available. This argument

pleads for the short term memory time constant to be a variable that should be learned

for each neuron and may even be modulated by the input. For physiologic mechanisms









of short-term modulation of r, one may think of adaptation (decreased sensitivity of

receptor neuron to a maintained stimulus) or heterosynaptic facilitation (the ability of

one synapse of a cell to temporarily increase the efficacy of another synapse; see Wong

and Chun (1986) for an application to neural nets).

In conclusion, short term memory by local positive feedback is simple and has

been applied successfully in artificial neural nets. However, reverberation may lead to

instability. Secondly, this mechanism restricts computational flexibility in the temporal

domain. In the next section, short term memory by delays is reviewed.

2.6.2 Short Term Memory by Delays

A general delay mechanism can be represented by temporal convolutions

instead of (instantaneous) multiplicative interactions. Consider the following extension

of the static additive model,

dx t
di -Xi + o w j(t -s) j (s) ds +I. E.23
vJo

We will call this model the (additive) convolution model. In the convolution

model the net input is given by

t
net(t) = :fw j(t-s)xj(s)ds. Eq.24
J0

In a discrete time environment, this translates to

t
neti(t) = wi(t-n)xj(n). Ea.25
J n=O

There is ample biological support for the substitution of weight constants w by time

varying weights w(t). Miller has reviewed experimental evidence that "... cortico-

cortical axonal connections impose a range of conduction delays sufficient to permit









temporal convergence at the single neuron level between signals originating up to 100-

200 msec apart" (Miller, 1987). Several artificial neural net researchers have also

experimented with additive delay models of type Eq.23. However, due to the

complexity of general convolution models, only strong simplifications of the weight

kernels have been proposed.

Lang et al. (1990) used the discrete delay kernels w (t) = wkO (t tk) in the
k

time delay neural network (TDNN).The TDNN architecture is shown in Figure 2.8. The

TDNN, considered the state-of-the-art, is a multilayer feedforward net that is trained

by error backpropagation. The past is represented by tapped delay lines as in FIR

filters. The authors reported excellent results on a phoneme recognition task. A

recognition rate of 98.5% at a phoneme recognition task ("B", "D" and "G") compared

to 93.7% for a hidden Markov model was achieved. Recently, the CMU-group

introduced the TEMPO 2 model, where adaptive gaussian distributed delay kernels

store the past (Bodenhausen and Waibel, 1991). Distributed delay kernels such as used

in the TEMPO 2 model improve on the TDNN with respect to the capture of temporal

context.

Tank and Hopfield (1987) also prewired w(t) as a linear combination of

t
a acll-)
dispersive kernels, in particular w(t) = XWk (t) = kwk() e This
k k

technique was utilized as a preprocessor to a Hopfield net for classification of temporal

patterns. The ideas are illustrated in Figure 2.9. Let an input signal successively

activate I, through 14. The delay between consecutive activations is one time step. If the

delays associated with the weights are as shown in the figure, then the input unit

activations arrive at the output unit at the same time. Thus, the output neuron is very

sensitive to an impulse moving over the input layer in the direction of the time arrow,

while an impulse moving in opposite direction does not activate the output node.


















hidden units


Neural nets of this type, where information of several neurons at different times

integrates at one neuron, were called Concentration-of-Information-in-Time (CIT)

neural nets. The weight factors Wk were non-adaptive and determined a priori. They

successfully built such a system in hardware for an isolated word recognition task. In

particular, the robustness against time warped input signals should be mentioned. In a

later publication successful experiments were reported with adaptive gaussian

distributed delay kernels (Unnikrishnan et al., 1991).

When compared to the first-order context-unit networks, the convolution model

in its general formulation is more flexible in the temporal domain, since the weighting

of the past is not restricted to a recency gradient. However, a high price has to be paid

for the increased flexibility. I identify three complications for the convolution model

when compared to the additive model.

Analysis. The convolution model is described by a set of functional

differential equations (FDE) instead of ordinary differential equations (ODE) for the


Figure 2.8 An example of a Time-Delay Neural Network.





























additive model. Such equations are in general harder to analyze a handicap when we

need to check (or design for) certain system characteristics such as stability and

convergence.

Numerical Simulation. For an N-dimensional convolution model, the required

number of operations to compute the next state for the FDE set scales with O(N2T),

where T is the number of time steps necessary to evaluate the convolution integral

dx
(using Euler method: x(t+h) = x(t)+h d). An N-dimensional additive model

scales by O(N2).

Learning. The weights in the convolution model are the time-varying

parameters w(t). Thus, the dimensionality of the free parameters grows linearly with

time. For a long temporal segment, the large weight vector dimensionality impairs the

ability to train the network.

The two models for incorporating short term memory in neural networks,

positive feedback and delays, have led to a number of architectures that essentially

generalize the continuous mapper to the space-time domain. In the discussion on static


time-varying
S. weight kernels



4_wl output


Figure 2.9 Principle of the Concentration-in-
Time neural net. The output node is tuned to
classify the sequence I-I,2-I,-I4.









neural nets we introduced the associative memory model. Is there also a temporal

extension for this model. Indeed, in particular in the physics community, several

researchers have experimented with temporal associative memories. The principal

ideas of the sequential associative memory are now shortly reviewed.

2.6.3 The Sequential Associative Memory

The sequential associative memory is a (recurrent) dynamical system that stores

memories in attractors (sinks) of zeroth order (point attractors) (Kleinfeld, 1986;

Sompolinsky and Kanter, 1986). Physicists have explored several ways to ignite

attractor transitions under influence of an external stimulus. The most widely used

method consists of forcing a combination of the external signal and the delayed

network state upon the net. The delayed state wants to keep the net in its current state,

while the external input tries to alter the state. As a result, the net state ideally hops

from one stable attractor to another. In a pattern recognition environment, the sequence

of visited states identifies the external input.

Sequential associative memories are theoretically very interesting and moreover

provide a neural explanation of categorical perception, due to the corrective properties

of the basins of attraction. However it has been noticed that these nets may not be very

selective pattern recognizers, in other words, nearly every input (if high enough) will

induce transitions (Amit, 1988). Secondly, the memory capacities of such nets are very

limited. It is the nature of point attractors that falling into a basin of one induces the

forgetting of previous states. Consequently, the 'deepness' of memory is fixed, short

and cannot be modulated. In my opinion, the sequential associative memory may be a

useful module for tasks like central pattern generation (Kleinfeld, 1986), but are not

(yet) flexible enough to encode the varying temporal relations of complex signals such

as speech.









2.7 Other Dynamic Neural Nets

We have discussed dynamic extensions of both the continuous mapper and the

associative memory. However, the architectures that were discussed in this chapter are

not the only viable constructions. A very important class of neural nets are networks

with globally recurrent connections. Note that the memory structures that have been

discussed sofar, the tapped delay line and the context units, are mechanism to store the

activations of local units. Generation of memory by feedback on a global scale has not

been discussed. The difference between local feedback at the unit level and global

feedback at the network level is displayed in Figure 2.10. Globally recurrent nets or

fully recurrent nets are an important area of current dynamic neural net research

(Williams and Zipser, 1989; Gherrity, 1989; Pearlmutter, 1989). The important issues

that confine applications of fully recurrent nets are control of stability and adaptation

problems. In a fully recurrent net, the performance surface is not necessarily convex

which is a severe handicap for gradient-based adaptation methods. Secondly, learning

in recurrent networks has been found to progress much slower than in feedforward nets.


Figure 2.10 An example of a globally recurrent network.


context
unit


output


s(t)

input









2.8 Discussion

In this chapter various neural architectures for temporal processing have been

discussed. The main principles for storage of and computing with a temporal data flow,

delays and feedback, were analyzed in some detail.

The feedforward tapped delay line has been used very successfully in the time-

delay neural net. Yet, the delay period per tap and window width are fixed and must be

chosen a priori. It was already discussed in chapter 1 that such a fixed signal

representation scheme likely leads to sub-optimal system performance.

Most context-unit neural networks do adapt the decay parameter of the context

units. As a result, the depth of memory can be controlled so as to match the goal of

processing. On the other hand, the weighting of the past is always restricted to a

recency gradient. Moreover, context units overwrite the past with new information.

In order to get more processing power, some investigators use globally recurrent

networks. These systems suffer from the same problem as recurrent filters: how do we

control stability during adaptation? Additionally, in particular for moderate to large

networks the currently available training algorithms do not suffice.

Although the importance of the future of global feedback neural nets should not

be underestimated, in this work we will concentrate on developing an improved

architecture for local (unit) short term memory. There are several reasons why this is a

important research direction. First, it will be shown later in this thesis that feedforward

networks of units with local (feedback) memory have some practical advantages over

globally recurrent nets. Stability for one is much easier controlled in such networks.

Secondly, the local short term mechanism that will be developed here does not exclude

global recurrence in the network. In fact, the integration of local feedback and global

feedback may lead to very interesting dynamic network architectures.


- I















CHAPTER 3


THE GAMMA NEURAL MODEL


3.1 Introduction Convolution Memory versus ARMA Model

It was discussed in chapter 2 that a general delay mechanism can be written as

t
net(t) = Jw(t s) x (s)ds Eq2
0

for the continuous time domain and

t
net(t) = w(t-n)x(n) E_.27
n=0

in the discrete time domain. It was mentioned that a problem associated with time-

dependent weight functions is that the number of parameters grows linearly with time.

This presents a severe modeling problem since the number of parameters of freedom

of a system do not always increase linearly with the memory depth of that system.

Although a convolution model is interesting as a biological model for memory by delay

and powerful as a computational model, the previous arguments make this model not

very attractive as a model for engineering applications. It makes sense to investigate

under what conditions an arbitrary time varying kernel w(t) can be adequately

approximated by a fixed dimensional set of constant weights. This problem was studied

by Fargue (1973) and the answer is provided by the following theorem.

Theorem 3. 1 The (scalar) integral equation









t
net(t) = Jw(t-s)x(s)ds E.28
0

can be reduced to a K- dimensional system of ordinary differential

equations with constant coefficients if (and only if) w(t) is a solution of

dK K-w dkw
(t) = ak (t),
dtK k = 0 dtk

where ao, aI,...,aK.1 are constants.

Proof. A constructive proof for sufficiency is provided. The initial conditions

dkw
for Eq.29 are rewritten as ^k (0), where k = 0,...,K-l, and we define the
dt

dkw
variables wk(t) = d (t), k = 0,...,K-1, which allows to rewrite Eq.29 as the
dtk

following set of K first-order differential equations.

dwk
dt (t) = wk+l(t), k = 0,...,K-2,

dwK- K-i
dt (t) = k akwk().
k=0

Next, we introduce the state variables


(t) Wk (t -s) (s) ds, k = 0,...,K-1. E31
0

Note that the system output is given by

t
net(t) = wo (t-s)x(s)ds = xo(t) Eq32
0









The state variables xk(t) can be recursively computed. Differentiating Eq.31 with

respect to t using Leibniz' rule gives

dxk t E
(t) = ffWk(t-S)X(S)ds+wk(O)x(t),
0

which using the recurrence relations from Eq.30 evaluates to

dxk
dt (t) = k+1() + k (t), for k = 0,...,K-2, and


dxK K-1
dt (t) = akk(t) + iK- x(t). a
k=0

Thus, if the weight kernel w(t) is a solution of the recurrence relation Eq.29, then the

integral equation Eq.28 can be reduced to a system of differential equations with

constant coefficients (Eq.34). O (end proof).

The following theorem reveals what is meant by imposing the condition Eq.29

on w(t).

Theorem 3. 2 Solutions of the system

dK K-1 dk
w(t) = ak- (t)
dtK k=0 dtk

k. .tr
can be written as a linear combination of the functions t 'e where

m
1 : i < m,0 < ki < K and I Ki = K. (end theorem).
k=l

The proof of Theorem 3.2 is provided in most textbooks on ordinary differential

equations (e.g. Braun, 1983, page 258). The xi's are the eigenvalues of the system. In

particular, the ji's are the solutions of the characteristic equation of Eq.29,









K-1
SK- I a k = o. EQ.
k=O

m is the number of different eigenvalues and Ki the multiplicity of eigenvalue gi. The

k. 9-t
functions t 'e are the eigenfunctions of system Eq.29, where i enumerates the

various eigenmodes of the system.

In the signal processing and control community, the system described by Eq.32

and Eq.34 is called an auto-regressive moving average (ARMA) model (see Figure

3.1). It was discussed in section 2.2 that the memory of an ARMA system is

represented in the state variables xk(t). It is interesting to observe the relation between

the ARMA model parameters and the convolution model. The auto-regressive

parameters ak are the coefficients of the recurrence relation Eq.29 for w(t). The moving

average parameters ik equal the initial conditions of Eq.29.


Figure 3.1 An ARMA model implementation of a convolution model
with recursive w(t).


In the context of this exposition, I like to think of an ARMA model as a

dynamic model for a memory system. It was just proved that this configuration is


net(t)









equivalent to a convolution memory model if the condition described byEq.29 is

obeyed. Yet, I do not know of neural network models that utilize the full ARMA model

to store the past of x(t). The reason is that the global recurrent loops in the ARMA

model make it difficult to control stability in this configuration. This is particularly true

when the auto-regressive parameters ak are adaptive. There are some substructures of

the ARMA model for which stability is easily controlled. Examples are the feedforward

tapped delay line and the first order autoregressive model (or context unit). These two

structures, which are shaded differently for clarity in Figure 3.1, have been used

extensively in neural networks as a memory mechanism. The virtues and shortcomings

of either approach have already been discussed in chapter 2. In this chapter, a different

approximation to the ARMA memory model is introduced. A look ahead to Figure 3.4

shows that the memory model that will be introduced utilizes a cascade of locally

recursive structures in contrast to the global loops in the full ARMA model. As a result,

this model will provide a more flexible approximation to the ARMA or convolution

model. Yet, the stability conditions will prove to be trivial. In the next section, a

mathematical framework for this new memory model is presented.

3.2 The Gamma Memory Model

Let us consider the following case, a specific subset of the class of functions that

admit Eq.29:

K
w(t) = wkg~(t). E
k=

where

k
g (t) = t le- k = 1,....K, ( > 0). E38
It is easily checked that the kernels gt) (the superscript is dropped) are a solution to
It is easily checked that the kernels gk(t) (the superscript ji is dropped) are a solution to








Eq.29. Since the functions gk(t) are the integrands of the (normalized) F-function

CO
(F(x)- tJ-le-tdt), they will be referred to as gamma kernels. In view of the
0

solutions of Eq.29, the gamma kernels are characterized by the fact that all eigenvalues

are the same, that is, p. = p.. Thus, the gamma kernels are the eigenfunctions of the

following recurrence relation:

d K
(-+pg) g(t) = 0. Ea9


tk
The factor normalizes the area, that is
(k- 1)!

t
gk (s) ds = 1, k=1,2... E.4
0

The shape of the gamma kernels gk(t) is pictured in Figure 3.2 for p. = 0.7.

It is straightforward to derive an equivalent ARMA model for net(t) when w(t) is

constrained by Eq.37. The procedure is similar to the proof of Theorem 3. 1. First, the

kernels gk(t) are written as the following set of first-order differential equations

dg1 dgk
dt= -ig dt k + 1, k = 2,...K. Eq41

Substitution of Eq.37 into Eq.26 yields

K
net (t) = wkk, Eq.42
k=l

where the gamma state variables are defined as

t
xk(t) = gk(t- s) x (s) ds, k= 1,...,K. Eq.43
0











0.7

0.6

0.5 k=l
gk() '
0.4-

0.3- k=2

k=3
0.2k=3 k=4 k=5
k=5


0.1
C ----------..-, ...........
0 2 4 6 8 10 12 14 16
t4p

Figure 3.2 The gamma kernels gk,() for p=0.7

The gamma state variables hold memory traces of the neural states x(t). How are the

variables yk (t) computed? Differentiating Eq.43 leads to


() = J k(t-s)x(s)ds+gk(0)x(t), Ea
0

which, since gk (0) = 0 for k > 2 and gl (0) = I, evaluates to


k
S(t) = xk (t) + xk 1 (t) k = 1...,K, E.4


where we defined x0 (t) x (t) The initial conditions for Eq.45 can be obtained from

evaluating xk (0) = gk (0) x (0), which reduces to

X0(0) = x(0) x1(0) = jx(0)


xk(O) = 0, k = 2,...,K.









Thus, when w(t) admits Eq.37, net(t) can be computed by a K-dimensional

system of ordinary differential equationsEq.45. The following theorem states that the

approximation of arbitrary w(t) by a linear combination of gamma kernels can be made

as close as desired.

Theorem 3. 3 The system gk(t), k=1,2,...,K, is closed in L2 [0, -] .

Theorem 3. 3 is equivalent to the statement that for all w(t) in L2 [0, o] (that is,

00
any w(t) for which f w (t) 2dt exists), for every e > 0, there exists a set of parameters
0

Wk, k = 1,...,K, such that

0 K 2
Jw (t) wkk(t) dt 0 k=1

The proof for this theorem is based upon the completeness of the Laguerre polynomials

and can be found in Szego (1939, page 108, theorem 5.7.1). The foregoing discussion

can be summarized by the following important result.

Theorem 3. 4 The convolution model described by

t
net(t) = Jw(t-s)x(s)ds, Ea.
0

is equivalent to the following system:

K
net(t) = Wkk(t)
k=l

where x0 (t) = x (t), and

dxk
d(t) = -xk(t) + .xk-_ (t),k = 1,...,K, >0. E.49








(end theorem).

The term gamma memory will be reserved to indicate the delay structure

described by Eq.49. The recursive nature of the gamma memory computation is
illustrated in Figure 3.3.



X XI X2 .XK





Figure 3.3 The gamma memory structure.



For the discrete time case, the derivative in Eq.49 is approximated by the first-

order forward difference, that is

dxk
t (t) = Xk (t + 1) Xk(t) E

This approximation is not the most accurate, but it is the simplest. Also, this particular

choice implies that the boundary value p.=l reduces the gamma memory to a tapped
delay line. This feature facilitates the comparison of the discrete gamma model to
tapped delay line structures. Applying Eq.50 leads to the following recurrence relations

for the discrete (time) gamma memory

x0(t) = x (t)

xk(t) = (1- t)Xk(t- 1) + Lxk_(t- ),k= ,...K, and t = to,t,t2... E.51

The time index t now runs through the iteration numbers to,tl,t2,... The discrete gamma

memory structure is displayed in Figure 3.4.





















3.3 Characteristics of Gamma Memory


In this section the gamma memory structure is analyzed both in the time and

frequency domain.

3.3.1 Transformation to s- and z-Domain

Since the recursive relations that generate the gamma state variables xk(t) at

successive taps k are linear, the Laplace transformation can be applied. The (one-sided)

Laplace transform is defined as

00
Xk () = Xk (t)e-stdt = L {Xk () E.52
0

Application of Eq.52 to Eq.49 leads to the following recursive relations for the

generation of the gamma state variables in the s-domain:

X (s) = X(s)


Xk (S) = Xk (s) ,k = ..., K. E


The operator G (s) will be referred to as the gamma delay operator. Note that
s+Rp

Xk(s) can be expressed as a function of the memory input X(s) only. Repeated

application of Eq.53 yields








Xk(s) = ( 9) X(s) = Gk(s)X(s). E.54

It can be verified that Gk (s) = L {gk(t) }.

The system Eq.53 also suggests a hardware implementation of gamma memory,
which is shown in Figure 3.5. It follows that gamma memory can be interpreted as a
tapped low-pass ladder filter. There are two gamma memory parameters: the order K
and bandwidth ji = (RC) -1

x(t) Xl(t) x2(t) XK(t)

R ?R RT
C C C

0-T T .................. T
Figure 3.5 A hardware implementation of gamma memory.

The corresponding frequency domain for discrete-time systems is the z-domain.
It follows from Eq.53 that the z-transform can be found by substitution of s = z-1 in the
Laplace transform. This leads to

Xk+(z) = -(z) k = 1,...,K. E55

The discrete gamma delay operator is G (z) = The transfer function from

memory input X to the kth tap Xk follows from Eq.55:
Gkk
Gk(Z) = Ea(-)"

Inverse z-transformation of Eq.56 leads to the discrete gamma kernels

gk(t) = k(l-)t-k-Jk= 1,.... Kt =k,k+1,... E.
9k W = 11 Ul~) I J' ~ aS








The discrete gamma kernel gk(t) is the impulse response for the kth tap of the discrete

gamma memory model. Eq.57 can be interpreted as follows. In order to get from the
memory input to the kth tap xk(t) in time t, the signal has to take k forward steps and

pass through t-k loops. Each forward step involves a multiplication by p, and a pass
through a loop involves multiplication by 1-p.. The number of different paths from x(t)

to xk(t) in time t equals -~ j1

Next, some of the frequency domain characteristics of the gamma memory will
be analyzed. The analysis will be performed using the discrete model, although similar

properties may be derived for the continuous time version.

3.3.2 Frequency Domain Analysis

The transfer function for the (kth tap of the) discrete gamma memory is given
by Eq.56. The Kth order discrete gamma memory has a Kth order eigenvalue at

z = 1 p. Since a linear discrete-time model is stable when all eigenvalues lie within

the unit circle, it follows that the discrete gamma memory is a stable system when

0 p i 5 2. The group delay and magnitude response for this structure are displayed for

the second tap (k=2) in Figure 3.6 and Figure 3.7 respectively.

The group delay of a filter structure is defined as the negated derivative of the
phase response to the frequency. It provides a measure of the delay in the filter with
respect to the frequency of the input signal. When g=l, gamma memory reduces to a
tapped delay line structure. In this case, all frequencies pass with gain one and the
group delay is 2, the tap index.

When 0 < p < 1, the gamma memory implements a linear Kth order low-pass

filtr. The low frequencies are delayed more than the high frequencies. In fact, the low
frequencies can be delayed by more than the tap index, which is the (maximal) delay
for a tapped delay line. For instance, for g=0.25 at tap k=2, a delay up to 8 can be





















p=0.25







p=0.5



II


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
"0 (D/T1


41


I
[t=1.75/



*/




0 1 0 0 0=i .25 0


I
0 0. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9


Figure 3.6 Group delay for discrete gamma memory at k=2.
Figure 3.6 Group delay for discrete gamma memory at k=2.


d(p
do


x


1





48


achieved for the low frequencies.The cost for the additional delay for low frequencies

is paid for by the high frequencies. The high frequencies are attenuated and the group

delay is less than the tap index. Thus, for 0 < p < 1, the storage of the low frequencies

is favored at a cost for the high frequencies.

The gamma memory behaves as a high pass filter when 1 < pI < 2. As a result,

the high frequencies are delayed by more than the tap index.

3.3.3 Time Domain Analysis

Although the impulse response g9(t) of the kth tap of the gamma memory

extends to infinite time for 0 < pi < 1, it is possible to formulate a mean memory depth

for a given memory structure gk(t). Let us define the mean sampling time tk for the kth

tap as

00
ik tg(t) = Z {tgk (t) = -1 E
t=0 z dz z= 1

1
We also define the mean sampling period Atk (at tap k) as Aik k 1 = The
p.

mean memory depth Dk for a gamma memory of order k then becomes



k i k
i=1

In the following, we drop the subscript when k = K. If we define the resolution Rk as
1
Rk k = p., the following formula arises which is of fundamental importance for the

characterization of gamma memory structures:

K = DR. Eg.60

Formula Eq.60 reflects the possible trade-off of resolution versus memory depth in a













100




IG21




10 1










10-2


In,


IG21


101


- \
... i=l















=0 .25,




0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
P i. (/It


Cl=1.75








1p=l1,


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Do (0)/7t


Figure 3.7 Magnitude response for discrete gamma memory at k=2.









memory structure for fixed dimensionality K. Such a trade-off is not possible in a non-

dispersive tapped delay line, since the fixed choice of pg = 1 sets the depth and

resolution to D = K and R = 1 respectively. However, in the gamma memory, depth

and resolution can be adapted by variation of ip.

In most neural net structures, the number of adaptive parameters is proportional

to the number of taps (K). Thus, when gi = 1, the number of weights is proportional to

the memory depth. Very often this coupling leads to overfitting of the data set (using

parameters to model the noise). The parameter p. provides a means to uncouple the

memory order and depth.

As an example, assume a signal whose dynamics are described by a system with

5 parameters and maximal delay 10, that is, y (t) = f(x (t n,), wi) where i = 1,...,5,

and max,(ni) = 10. If we try to model this signal with an adaline structure, the choice K

= 10 leads to overfitting while K < 10 leaves the network unable to incorporate the

influence ofx(t -10). In an adaline with gamma memory network, the choice K = 5 and

pg = 0.5 leads to 5 free network parameters and mean memory depth of 10, obviously a

better compromise.

3.3.4 Discussion

In this section the gamma memory structure was analyzed in both the time and

frequency domains. When 0 < pi < 1 the storage of the low frequencies is favored over

the high frequencies. In the time domain this translates to a loss of resolution but a win

in memory depth. There are many applications for which this bias towards storing low

frequencies can be exploited. In particular we think of applications where a long

memory depth is required, as in echo cancelation or room equalization.

Next the gamma memory model is incorporated into the additive neural net. The

additive model with adaptive gamma memory provides a unified framework for non-












50

45

40
D 35-
K30 =5
30

25-
K=4
20 \

15 K=3

10 K=2

K = .
0
0.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9


Figure 3.8 Memory depth of discrete gamma memory vs. p.


linear processing with adaptive short term memory capacities.


3.4 The Gamma Neural Net

In this section, the additive neural net is extended with gamma memory. This

new model will be compared to the various competing networks for temporal

processing as described in chapter 2.


3.4.1 The Model

Recall the general model for additive neural nets:


dxi N
S(t) = aixi (t) +oi Wijxj (t) + i (t), i = 1,...,N. EN.


Let us assume that each neuron has the capability to store its past in a gamma

memory structure of (maximal) order K and bandwidth pi. The activation of the kth tap








of neuron i is written as xik(t). The weight wik connects the kth tap of neuron j to neuron

i. The system equations for this model are

dx,
(t)= -aixi(t) + o(Y wijkxjk(t)) +I (t)
j k

for the activations xi(t) and for the taps xik(t)

dxik
dt (t) = xik(t) + i, k (t), k= ,... ,K.

The system described by Eq.62 and Eq.63 will be referred to as the (additive) gamma
neural net or gamma model. In Eq.62 the time constant is processed in the decay

parameter at. Also, for notational convenience we defined xi (t) xio (t) .The structure

of the gamma neural model is displayed in Figure 3.9.

The discrete gamma memory can be applied in discrete-time neural network
models. The discrete gamma neural model is defined as follows:

Xi(t) = (I wijkxjk()) +Ii(t)
j
Xik(t) = (1 -i) ik(t- 1) + ii, k- (t- 1), k= 1,...,K E.65

Note that p=l leads to an additive network where the memory is implemented as a
tapped delay line. A feedforward network of this type is equivalent to the time-delay
neural net. The other extreme at p=0 obviates the gamma memory and reduces the
structure to a "normal" additive model.

Next, the gamma model is compared to previously introduced neural models for
temporal processing.

3.4.2 The Gamma Model versus the Additive Neural Net

Additive neural model are characterized by the fact that the adaptive net









































Figure 3.9 The additive gamma neural model.









parameters are constants, that is, not dependent on the neural activation nor time

dependent. The free parameters in the gamma neural model are pi and wik, hence the

gamma net is an additive model. This is an important result, since the substantial theory

on additive neural nets applies directly to the gamma model. Also, existing learning

procedures such as Hebbian learning and backpropagation apply without restriction to

the gamma net. Since the gamma model is additive, it is possible to express the system

equations Eq.40 or Eq.65 as a Grossberg additive model. This is easiest shown by

rewriting the gamma model equations in matrix form. This conversion will now be

carried out for the continuous time model, since the result hints to an interesting

interpretation of the gamma model. First the node indices i and j are eliminated. We

define the signal vectors xk = [Xlk...,XNk], I = [Ii,..,N]r and parameter matrices a =


Wlk WiNk
diagd(ai), gi = diagN(ii) and wk = .Then, the gamma model equations
WN1k WNNk

can be written in matrix form as

dx0 K
dt (t) =-axo(t) +o w) +I)
=0

dxk
d(t) = xk(t) + xk (t),k=1,...,K. E.66


x0

Next, index k is eliminated. We define the gamma state vector X = X the input


xK











I = the squashing function XZ() = the matrix of decay

0 0


a 0 WW "' ...WK
parameters M = I and the matrix of weights Q = a .Then

0 0 ''

the gamma model evaluates to

dX
=- MX+ 2Y+E, Eq.67


an N(K+1)-dimensional Grossberg additive model.

Notice the form of the weight matrix U. Many entrees are preset to zero; in other

words, the gamma model is a pre-wired additive model. As a result, the pre-wiring of

the gamma model for processing of temporal patterns has rendered this model with less

free weights than a general additive model. This property is important, since both the

learning time and the number of training patterns needed grows with the number of free

weight in a neural net.

3.4.3 The Gamma Model versus the Convolution Model

In chapter 2, I identified theoretical analysis, numerical simulation and learning

as problem areas for the convolution model. It was also shown that the gamma model

is mathematically equivalent to the convolution model for sufficiently large order K.

Next these problems are re-evaluated when the convolution model is expressed as a

gamma model.

Analysis As was discussed in section 3.4.2, the gamma model can be

represented as a pre-wired additive model. Consequently, theoretical results for the









additive model are entirely applicable to the gamma model.

Numerical Simulation Whereas the complexity of numerical integration of the

convolution model scales as O(N2T), the gamma model scales as O(N2K). Thus, as time

progresses the evaluation of the next state for the convolution model involves an

increasing amount of computation. The computational load for the gamma model is

independent of time.

Learning While the weights in the convolution model are time-varying, the

gamma weights are constants. Thus, the dimensionality of the weight vector is

independent of time. This is a very important distinction from an engineering

standpoint, since it allows a direct generalization of conventional learning algorithms

to the gamma model.

3.4.4 The Gamma Model versus the Concentration-in-Time net

In 1987, Tank and Hopfield presented an analog neural net with dispersive delay

kernels for temporal processing (Tank and Hopfield, 1987). The memory taps xk(t) for

this "Concentration-in-Time net" (CITN) are obtained by convolving the input signal

with the kernels


t a (1--)
t
fk(t) = () e ,k=l,...,K,

where a is a positive integer. fk(t) is normalized to have maximal value 1 for t = k. The

degree of dispersion is regulated by parameter a. In Figure 3.10, the kernels fk(t) are

displayed for k=1 to 5 when a = 5.

Although fk(t) visually resembles peak-normalized gamma kernels, it is not

possible to generate the kernelsfk(t) by a recursive set of ordinary differential equations

with constant coefficients, as is the case for the gamma kernels. In fact, differentiating

Eq.68 leads to the following time-varying differential equation forfk(t):














0.9- 1(p f5(

0.8

k(t)0.7 -

0.6

0.5

0.4-

0.3

0.2

0.1
0 .... ....
0 I 2 3 4 5 6 7 8 9 10
0-34 5 6 7 8 9 10 t

Figure 3.10 Tank and Hopfield's delay kernels; a=5, k=1,...,5


dfk 1 1
dt (t) =a(- )k(t). Ea9


While Tank and Hopfield's model shares with the gamma model the capability of

regulating temporal dispersion, only the gamma model provides an additive neural

mechanism for this capacity. As a result, in the gamma model the dispersion control

parameter Ip can be treated as an adaptive weight. In Tank and Hopfield's model, a is

fixed. The relative merits of peak- versus area-normalization of dispersive delay

kernels have not been investigated.

Similar arguments hold when the gamma memory is compared to the adaptive

gaussian distributed delay models such as the TEMPO 2 model (Bodenhausen and

Waibel, 1991) and the gaussian version of the concentration-in-time neural net









(Unnikrishnan et al., 1991). These memory models do offer the advantage of adaptive

dispersion, yet only the gamma memory offers an additive neural mechanism to create

dispersive delays. The other models require evaluation of a convolution integral with

respect to the delay kernels in order to compute the memory traces. Although not a

priority for engineering applications, the gamma memory is biologically plausible,

since there is no (non-neural) external mechanism required to generate delay kernels.

3.4.5 The Gamma Model versus the Time Delay Neural Net

The memory structure in the time delay neural net (TDNN) is a tapped delay

line. In fact, TDNN structures can be created in the gamma memory by fixing p. = 1 in

the discrete gamma model. Thus, the TDNN is a special case of the discrete gamma

model. When 0 < p < 1, the discrete gamma memory implements a tapped dispersive

delay line. The amount of dispersion is regulated by the adaptive memory parameter p..

We discussed that the memory depth of the gamma memory can be estimated by K/lg.

Hence, the memory depth can be adapted independently from the number of taps (and

weights!) in the structure. In the TDNN, the memory depth and the memory order both

equal K. As a result, increasing the memory depth in the TDNN is always coupled with

an increase in the number of weights in the net, which is sometimes not desirable.

3.4.6 The Gamma Model versus Adaline

The simplest discrete-time gamma model is a linear one-layer feedforward

structure with one output unit. The equations for this network are given by

K
y(t) = kxk(t)
k=O

x0(t) = I (t)


xk (t) = (1 ) Xk (t 1) + gxk_- (t- 1), k=l,...,K.









This structure is depicted in Figure 3.11. For gi=1, Widrow's adaline is

obtained. Also, adaline is the simplest (linear, one-layer, one output) implementation

of the time-delay neural net.


y(t)
Figure 3.11 The adaline(p) structure.


The gamma memory generalizes adaline to an adaptive filter with a dispersive

tapped delay line. Adaline with gamma memory will be referred to as adaline(u) or

adaptive gamma filter. Several interesting aspects of this filter are worth a deeper look.

A special chapter (5) will be dedicated to the analysis of adaline(g).

3.5 Discussion

In this chapter the gamma neural model has been developed and analyzed. The

gamma model is characterized by a specific short term memory architecture, the

gamma memory structure. Gamma memory is an adaptive local short term memory

structure. Memory depth and resolution can be altered by variation of a continuous

parameter It. In the next chapter, gradient descent adaptation procedures for the gamma

model are presented.














CHAPTER 4


GRADIENT DESCENT LEARNING IN THE GAMMA NET


4.1 Introduction Learning as an Optimization Problem

Learning in a neural net concerns the modification of the weights of the net so

as to improve performance of the system. The term adaptation will be used as a

synonym for learning. Commonly it is assumed that the performance of the neural net

is expressed by a scalar performance index or total error E. In general we write


ptm
p t m

4P (t) is a cost functional which describes the error measure at output node m E M at

time t [0, T] when pattern p e P is presented to the system. Often the (weighted

1
by ) quadratic deviation from a given target trajectory dP (t) is chosen as the cost.

For this case, Eq.71 evaluates to


E= [dP (t) -x(t)]2 = 2 [e ,(t) ]2
p, t, m p, t, m

where eP (t) df (t) -xPm (t) is the instantaneous error signal which is immediately

measurable at any time.

The learning goal is to minimize E over the system parameters w and I,

constrained by the network state equations x = f(x, I;w, i) This problem has been

studied extensively in optimal control theory (Bryson and Ho, 1975). The most








dE
common approach to search for the minimum of E involves the use of the gradients a-

DE
and -. When E is minimal, these gradients necessarily vanish, that is, at the optimum

we have

DE DE
= =0. E


An algorithmic method is now discussed which searches for the values w and gI

that minimize E. Assume an available set of training pattern pairs (Ip (t), dP (t)) that

adequately represents the problem at hand. This set is referred to as the training set P.

Next, the training set is presented to the network and the activations xp (t) are

recorded. The presentation of the entire pattern set P is called an e-ch or batch. Note

that the availability of x, (t) and dP (t) allows the evaluation of the performance index

E using Eq.72. This measure can be used to determine when to stop training of the

DE DE
system. Next the gradients and are computed and the weights are updated in

the direction of the negative gradients:

DE
Wnew = Wold -IW E-

DE
new = 1old-- T1a

If the learning rate or step size rl is small enough, this update will decrease the

total error E on the next batch run. There are other methods of utilizing the error
gradients to search for the minimum of E. As an example, the successive weight
updates can be made orthogonal to each other, a process which is called conjugate

gradient descent. In this work we are not interested in optimizing the learning process








per se. Our goal will be to generalize gradient descent adaptation to the gamma net and

evaluate the properties of this generalization. The equations Eq.74 and Eq.75

implement an update strategy which is called steepest descent. The process of running

training set epochs followed by a weight update is repeated until the total error E no

longer decreases. Note that if the error surface E = E (w, g) is convex, this procedure

leads to a global optimization of the network performance.

The learning process described above updates the weights only after

presentation of the entire epoch. We call this epochwise learning or learning in batch

mode. A faster training method would be to adapt the weights after each time step t.

BE DE
This involves computation of the time-dependent error gradients (t) and (t

The update equations then become

bE
w (t) = w(t-1) l (t) E.76
aw

BE
A(t) = I(t- 1) -'IDE -- E(.77

This mode is called real-time learning or on-line learning. Real-time learning

converges faster than learning in batch mode, but the updates are no longer in the

BE BE
opposite direction of the total error gradients D- and Thus, even for convex error

surfaces, real time adaptation not necessarily leads to global optimization of the

network performance.

A look at the update equations makes clear that the crucial aspect of the learning

BE aE
process just described is the computation of the error gradients and -. Therefore

the remaining part of this chapter concerns methods of evaluating these gradients.

This chapter is organized as follows. In the next section the literature on









gradient computation in simple neural nets is reviewed. Two different methods will be

evaluated. The direct method computes the gradients by direct numerical

differentiation of the describing system equations. The alternative method, (error)

backpropagation, utilizes the specific network architecture to compute the gradients.

As a result, backpropagation will proof to be a more efficient technique (in terms of

number of operations) than the direct method. However, as we will see when dynamic

networks such as the gamma net are introduced, application of backpropagation is

restricted to short time intervals. The trade-offs of backpropagation versus the direct

method will be evaluated for the gamma net. Finally, a special gamma net architecture,

the focused gamma net, is introduced. This structure is of special interest, since it can

be trained by a fast hybrid learning procedure. Some well-known signal processing

structures such as adaline and the feedforward networks are special cases of the

focused gamma net.

4.2 Gradient Computation in Simple Static Networks

In this section the essentials of error gradient computation in neural nets are

treated. For the time being we deal with the simplest cases, since we only want to

convey the strategy of error gradient computation. The weight update equations for the

gamma model are postponed to section 4.3.

It is assumed that the processing system can be described by a static additive

neural model:


Xi = Oi( 1WijXj) + i. E
j
We will also write neti = wijxj Also, we assume that the states xi are computed by
j increasing index order. Thus, first xl is computed, then x2 and so forth until XN. There

are no temporal dynamics associated with Eq.78. As discussed before, the central task









DE
of all gradient descent adaptation procedures is to compute for all weights. It will
awi
be assumed that the total error measure E can be expressed as


E= X I= e m = -(dm m) 2
meM m m

Thus the training set consists of one static pattern. The learning task is to adapt the

weights such that the mean square error between the target dm and net activation Xm is

BE
minimal when I, is presented to the system. Next two exact algorithms to compute

are presented.

4.2.1 Gradient Computation by Direct Numerical Differentiation

DE
The simplest method to compute the gradients is just by differentiating the

equations Eq.78 and Eq.79. Applying the chainrule to Eq.79 yields

DE axm
I_ = -Eema Ea80
ii m ij

ax
Next the gradient variable P is defined. 3! can be directly computed by
iw i

differentiating the state equation Eq.78. This leads to

Sd(o (netm) Bnetm
1 dnetm aw-U
im ij

= m"'(netm) imxj+ wmnij1 Ea
n
where 8im is the kronecker delta function.

The set of equations Eq.78, Eq.80 and Eq.81 provide a system to compute the








error gradients. Together with an update rule such as

DE
Awij = _- EWi2


they form a neural net learning system.

4.2.2 The Backpropagation Procedure

In contrast to the direct method, the backpropagation method exploits the
specific network structure of neural nets in order to compute the error gradients. As a
result, backpropagation is a computationally more efficient procedure than the direct
method.

Before the backpropagation method is introduced, it is necessary to define more
precisely what is meant by the error gradients. We now proceed by an intermezzo in
order to explain how we define partial (error) derivatives in networks. Consider the
network in Figure 4.1. The state equations for this network are given by

x1 = I1
x2 = W21x + 12
X3 = 32X2
x4 = w422 + W433
x5 = W52X2 + 53X3 + 54x4

It is assumes that the variables xl through x5 are computed in indexed order, that is, first

x], then x2 and so forth until x5. A network whose state variables are computed one at

Ox5
a time in a specified order will be called an ordered network. Let us compute ax2
aX2

3x5
Explicit partial differentiation of the equation for x5 in Eq.83 gives aX2 = w52

However, this only reflects the direct or explicit dependence of xs on x2. x2 also affects

x5 indirectly through the network. Incorporating these indirect or implicit influences








leads to the following expression:

ax5 ax5 ax5
-= W52 +w323 + 42 a4
;x2 (3 4-

Sx5 ax5
= 52+ w32 53+ W43 a + w424


= 52 + 32 (w53 + W43W54) + W42w54 E

This difference between explicit and implicit dependencies in networks has been
treated in a backpropagation context by Werbos (1989). He introduced the term ordered
derivative to denote the total partial derivative (including the network influences). In
this work, whenever we speak of a partial derivative to x the ordered partial derivative
a
is meant, which is denoted by the symbol If we only want to include the explicit

(direct) dependence on x we speak of explicit partial derivative, for which the symbol
ae
Swill be reserved. Werbos (1989) proved the following theorem for ordered
ax
networks:

Theorem 4.1 Consider a network ofN variables xi whose dependencies are
ordered by the list L = [x1, x2, ..., XN] this means that xi only depends
on xi where j < i. Let a performance function E be defined by

E = E(xl, x2,...,N). E.85

aE
Then, the (ordered) partial derivatives a- can be computed by
axi


aE eE aE ex.
I 5 X J
W-i = i xj -xi


(end theorem).









ax5
It can be checked that application of Theorem 4.1 to the computation of in
ax2

the network Eq.83 leads to expression Eq.84. Note that the computation of the

DE
gradients ;- requires knowledge of the error gradients with respect to xj where j > i.

As a result, the error gradients must be computed in descending index order. This

feature has inspired the name backpropagation for algorithmic procedures that make

use of Eq.86 in order to compute the error gradients in neural networks. Here the
intermezzo on partial derivatives ends. We now proceed to derive the backpropagation

method.

Let the network equations and performance index be given by Eq.78 and

Eq.79 respectively. An order or sequence of computation has to be determined for all

variables involved in the state equation Eq.78. Since the set of weights {wij} are

initialized before the state variables x, are evaluated, they can be put at the beginning

of the list. This leads to the following list:

L = [{wij}, X,...,xN] E Z

Next the partial derivatives of the performance index E to all variables in L are

computed, making use of the rule for partial derivatives in ordered networks Eq.86. For

the state variables xi this leads to


E a eE E aexj
ax ax .x axBx
J>1i ji

BE
=-ei+ Iax.xo a'(netj)wji Eq.88
j>i j

aE
For the gradients D we obtain the following expression:
owij










aE E/ E exn
= +7 x
aij iJ n:N n iji

BE
= xGi' (neti)xj EqM


In the backpropagation literature it is customary to define the variables

BE aE
i =- and 5i = = Ei'(net). "E
aii anet, 1.90

Substitution of Eq.90 into Eq.88 and Eq.89 yields for the computation of the error

gradients

Ei ei + w. Ea
j>i

i = Eii'(neti) E 2

aE
W = 8- E93


The set of equations Eq.91, Eq.92 and Eq.93 constitute the backpropagation method to

compute the error gradients.

Now let us see how to apply all these equations to the learning problem. Say we

have a network described by Eq.78 and a training data set consisting of one input

patterns i, and a target pattern dm. To start the learning system, the input pattern is

presented to the net and the state variables xi are evaluated by Eq.78. The variables xi

are computed by increasing index order i. This completes the forward pass. Next, the

error variables ei = di x. are evaluated and stored. The next phase, the backward or

aE
backpropagation pass, computes the variables 8. = by evaluating Eq.91 and
a anetb









Eq.92. The quantities 58 are called backpropagation errors. They measure the

sensitivity of the total error E with respect to an infinitesimal change in neti. It has been

mentioned before that the backpropagation errors are computed in descending index

order, that is, first 5N followed by 5N_1 until 6,. After the backpropagation pass, the

error gradients with respect to the weights are computed by Eq.93. Next, if a steepest

descent update rule is used, the weights are adapted according to

DE
Aw 1 = --11 5-. Ea.94


In order to appreciate the architecture of the backpropagation method, we

rewrite the backpropagation equation Eq.88 as

E = -e + S (.'(net)wji E
I J iii
j>i

Note the structural similarity between the state equation Eq.78 and Eq.88. In the

backpropagation equation the backprop errors EP serve as the network states and -e1 is

the external input. In order to discriminate between the error variables ei and E,, ei is

sometimes referred to as the injection error whereas El (and 5i) are called

backprooagation errors. Since the backprop weights wji connect node j It i (note the

arrow reversal as compared to Eq.78), the backpropagation network structure is the

transposed network of the network during the forward pass. This property is visualized

in Figure 4.2 where the backpropagation structure is drawn for an example feedforward

network. Thus, the backpropagation method makes explicit use of the network

structure in order to compute the error gradients. As a result, the backpropagation

method is computationally more efficient than the direct method. Specifically how the

computational cost of the backpropagation method compares to the direct method will

be evaluated next.

































Figure 4.2 Backpropagation architecture for a static feedforward net.

4.2.3 An Evaluation of the Direct Method versus Backpropagation

There are hardly any results with respect to convergence speed of gradient

descent update rules applied to neural networks. All we can say is that the success of

training a neural net using gradients often depends on the randomly selected initial

weights. It is however interesting to make a comparison of computational complexity

of competing learning strategies. When learning algorithms are compared as for their

time and space resource consumption it will be assumed that the learning process is

carried out on one sequential processor. In order to describe the complexity of

algorithms the notation O((p(n)) is used, which is defined as the set of (positive integer

valued) functions which are less or equal to some constant positive multiple of (p(n).

We will assume that the cost of one operation addition or multiplication carried out

on one processor is 0(1). As an example, the evaluation of the system equations for the









additive net Eq.78 costs O(N2) operations. The reasoning goes as follows. The

evaluation of x, costs O(N) operations since node i is connected to maximal N nodes.

Since we need to evaluate the activations for all i, that is i = 1,...,N, the total cost

becomes O(N2). The cost of storage (space) for the system Eq.78 is O(N2) since the

space requirements are dominated by the (maximal) N2 weights wy.

Now let us evaluate the cost of the error computation by the direct method. The

number of required operations is dominated by the evaluation of the gradients PT. The

computation of a variable PT involves O(N) operations, Since there are maximal N3

variables P? it follows that the total number of operations scale by O(N4). We need

O(N3) space to store P.

As for the computational cost of the backpropagation method, note that the

computation of the backpropagation errors requires evaluation of the (transposed)

network. It was already discussed that evaluation of the network requires O(N2) number

of operations. The space requirements are dominated by the weights wy, hence O(N2)

storage is needed. Thus, both the number of operations (time) and space requirements

for gradient evaluation scale favorably for the backpropagation method in comparison

to the direct method.

The computational complexity is of course only a ballpark measure of the

merits of an algorithm. In particular for neural nets it is important if or to what degree

the algorithm can be carried out on parallel hardware. An interesting property for

network algorithms in this respect is locality. We will say that an algorithm is local if

the states of the network can be computed from information that is locally available at

the site of computation. Locality is not only a property of the biological archetype, it

greatly facilitates implementation in parallel hardware. For instance, the








backpropagation errors are computed by means of the transposed network. As a result,

in a hardware implementation only the direction of the communication paths between

processors need to be reversed. The direct method, on the other hand, is not local. The

gradients (. need to be computed for all nodes n e N and all weight indices

(i,j) e NxN.

As a conclusion, both the computational complexity and the locality criterion

favor the backpropagation method over the direct method for static networks.

Therefore, the direct method should not be used for computation of error gradients in

static nets. However, it will be shown that the situation is more complicated for

dynamic networks. In the next section, the direct method and backpropagation are

extended to the gamma net operating in a temporal environment.

4.3 Error Gradient Computation in the Gamma Model

Since the gamma model can be formulated as a regular additive model, it

follows that both the direct method and the backpropagation procedure can be extended

DE DE
to the gamma net. In this section the error gradients -- and are derived for the
7Wijk i
discrete gamma model as described by


x (t) = a(i (2 ijkx(j))+ Ii(t)
j k

where the gamma state variables are computed by

xik(t) = (1 -9i)xik (t- 1) +gxi, k- 1 (t- 1)

In contrast to the previous section, it is assumed that the activations and target

patterns are time-varying. Thus, the performance index E is defined as

E Z/ (t)
t m









2= [dm (t) xm (t) ] 2
t,m


=2 ,[em(t)]2
t, m

We now proceed to derive the error gradients using the direct method.

4.3.1 The Direct Method

The procedure is similar to the derivation for the static model. Partially
differentiating E to wijk yields


aE
Sijk


Oxm (t)
-- em (t)
t, m ijk


Eq.99


Sxm (mt)
We define the gradient signal Pk (t) --- k (t)
k daw ijk k


can be evaluated by partial


differentiation of Eq.96, which leads to

P1k (t) = m' (netm (t)) [ imxjk (t) + WmnP Eq.100
ne N

where 5i is the Kronecker delta (and remember the notation wmn wmn).

aE
A similar derivation can be applied to obtain the gradients i-. Analogously to

Eq.99 we write


DE axm (t)
-a em (t)
i /t,m i


Eq.101


axm (t)
Applying the chainrule to the partial derivatives -- leads to


ax (t) alnk(t)
nN k xnk (t)


=xm (t)
_^^^)cr


aXm (t)
-ti


axik (t)
X agi


Eq.102









axm (t)
The signal xik follows by differentiation of Eq.96, yielding


aXm (t)
aXik(t) m (netm (t)) Wmik. E103
axik (t) m m mik'

Substitution of Eq.103 into Eq.101 yields for the error gradients

aE
= em (t) m (netm (t) ) YwmikO (t) EQ
t, m k


where we defined ak (t) = i The signals an (t) can be computed on line by
I a-- i I

differentiation of Eq.97, which evaluates to

a (t) = (- i) (t-) + i-(t 1) + [Xi, k 1 (t 1) Xik(t-1)] E 5

aE
The set of equations Eq.99 and Eq.100 provide the gradients whereas
JWijk
DE
Eq.104 and Eq.105 compute the gradients -. A steepest descent adaptive procedure

would use these variables in a update rule of the form

DE
Awik ijk

and an analogous expression for the adaptation of ti. Together with the gamma system

equations Eq.96 and Eq.97 they constitute a gamma model learning system.

The learning system as derived here assumed adaptation in batch mode.
However, this algorithm is easily converted to a real-time learning system. We just
define a time-dependent performance index E, by

Et [em (t)]2 07
meM









Note that since E = _Et, the only change in the formulae is to take out the I from
t t
the error gradients, which reduces the error gradient expressions to

aE
W= aem (t) ijk (t) and E J109
ijk m

SE
-i = em (t) am' (netm (t)) 2Wmikk ()
i m k

The signals P13 (t) and a' (t) are computed by the same equations as in the batch

mode, that is, Eq.100 and Eq.105 respectively. The real time mode for this algorithm is

particularly interesting since the required number of operations is equivalent to the

batch mode algorithm. However, the storage requirements for the real-time mode are

greatly reduced (by factor T, the number of time steps) since we update on-line by

DEt Et
Awijk ( = k and AIi (t) = -T11 il

In addition real-time adaptation usually converges faster than epochwise

updating. Therefore in practice real time updating is used far more that learning in
batch mode. In fact, the real-time mode of the algorithm described here was derived for

recurrent neural nets by Williams and Zipser (1989). They coined the name real time

recurrent learning algorithm (RTRL). We will take over their terminology. Thus, the

direct method for error gradient computation in gamma nets leads to a (special) RTRL

algorithm.

Let us analyze the locality and complexity properties of the RTRL algorithm for

the gamma net. Assume that the number of units in the system equals N. Each unit
stores a history trace of its activation in a gamma memory structure of maximal order

K. The (maximal) number of weights wijk then becomes N2K. The number of memory

parameters equal N. Also, it is assumed that the system is run for T time steps.









The gradient variables a (t) and jk (t) determine the complexity of the

procedure. There are maximal N3K variables jk (t), each of which is evaluated by

Eq.100 at a cost O(N) per time step. Thus the total cost is O(N4K) per time step. It

JE
follows from Eq.104 that the evaluation of D- requires a cost O(NK) per time step.


JE
The cost of evaluating a( (t) is 0(1). Since there are N variables it follows that


the total cost pre time step for memory adaptation id O(N2K). The space costs are

dominated by j (t), requiring O(N3K) memory locations. Note again that the
ijk

gradients ak (t) and pj (t) cannot be computed locally in space, but all computations
I ijk
are local in time since the algorithm is real-time. The results for RTRL and other

algorithm are summarized in Figure 4.6.

4.3.2 Backpropagation in the Gamma Net

In this section the backpropagation procedure as derived in section 4.2.2 is

generalized to the gamma neural net. We start by defining the list L that holds the order

of evaluation of the system variables. It is assumed that the activations Xik (t) are

evaluated in the order as schematically specified by Figure 4.3.
fort = 0 i T do
for t = to T do
fok=OlQmKdo
evaluate xik (t)
end; end;end
Figure 4.3 Evaluation order in gamma model


This leads to the following list:









L = [ {i, {Wijk },X1 (), X11(0), ...,XNK (O),x (1), ...,XNK(T)]. EQ.

The same performance index as defined for the RTRL procedure is used, that is


E=C [e, (t)]2.112
t, m

Recall that in order to compute the error gradients in the backpropagation algorithm,
we make use of Werbos' formula for ordered derivatives, which evaluates for the

activations xik (t) to

e
BE aeE E xjl ()
DE + x ------ E 113
xik (0t) Xik (, j, 1) > (t, i, k) j) xik(t '

The expression (T,j, 1) > (t, i, k) under the summation sign refers to all index

combinations (T,j, 1) that appear after (t, i, k) in the list L. Although cumbersome,

working out Eq.113 is straightforward. In order to simplify Eq.113, the two cases

k = 0 and k 0 have to be considered.

e
JE .Xjl (Q)
First -- is worked out (k = 0). In order to evaluate the factor in
ax, (t) axi (t)

expression Eq.113 we need to find the activations xjl () that explicitly depend on

xi(t). It follows from the gamma system equations Eq.96 and Eq.97 that only

xil (t+ 1) and x (t) (j > i ) directly depend on xi (t) Thus, Eq.113 evaluates to

DE DE DE
axi(t) = ei(t)+piaxi(t+1) + '(netx(t)

Next the formula for ordered derivatives Eq. 113 is evaluated for the tap variables xik(t)

for k = 1,...,K. Applying Eq.113 to the gamma state equations yields








0
JE _aeE aE E vE
DE = + (1 -i)x E +. E + .'(net (t))wjik
Jxik(t) zk(t) +xik(t+l1) ixi, k+l(t+l) j>i kx(t)

Eq.115

DE
The equations Eq.114 and Eq.115 backpropagate the gradients ---. Note
JXik (t)

that the gradients at time t are a function of the gradients at time t+l. Therefore, the
backpropagation system has to be run backwards in time, that is from t=T backwards
to t = 0. In fact, this is also clear when we recall that the list L is run backwards during
the backpropagation pass. For this reason the procedure described here is called
backpropagation-through-time (BPTT). Next the error gradients are computed with
respect to the system parameters by applying Eq.113 to the list L. For the weights wik
we get

aE aE
aw a ,' (neti(t))jk ) Ea 116
ijk txi (t)

SE
and for the gradients ,


DE E ye
x -iXik (t)
-9i t k Oxik (t) ik

_E
= Oik(t) [Xi,k-l(t--) -Xik(t-l E

In Figure 4.4 the set of equations that describe the backpropagation method for
the gamma model is summarized. In Figure 4.4 for convenience the notation

DE DE
- (t) = and 8i (t) = E) is used for the backpropagation errors.
k axik(t) W aneti(t)

The temporal aspect of gamma backpropagation impacts the use of this
procedure substantially. Similarly to regular backpropagation, the backpropagation























Ei(t) =-ei(t)+.piEi (t+1)+ )i. (1)
J>1

Eik() = (-li) Eik(+ ) +P.i,k+ i(+l) + ik. (t)
j>1

8i(t) = Ei(1) i (neti(1))

backpropagation equations run from k = K to 0, i = N to 1, 1 = T to 0,



aE


= Eik(t) [. i, k I(- ) -xik(t- )]


error gradients


Figure 4.4 Backpropagation-through-time equations for the gamma
neural model.


network is of the same complexity as the forward pass net. Note that this algorithm is

not local in time the backpropagation errors can only be computed after a complete

epoch has ended (at t = T). Thus real-time learning is excluded as well. Additionally, it

follows that the states xik(t) and errors ei(t), ei(t) and 8i(t) must be stored for the entire


N K
xi () = oi ijkxjk (1) + li()
j= lk= 0


xik(t) = (- )ik( 1) + pixi, k- (t- I)

state equations run from i = 0 to T, i = I to N, k = 0 to K









epoch. Thus the storage requirements scale by O(NKT+N2K) (first term for xik(t) and

second term for wijk). Obviously this limits the applicability of this algorithm to a

small epoch size T.

There is another disadvantage associated by deep backpropagation paths. Recall

that the backpropagated error signal traverses the transposed network in reverse

direction. The backprop errors hold an estimation of the sensitivity of the total error

with respect to a change in the local activation. If the system parameters are not close

to the optimal values, the backpropagation pass will soon degrade the accuracy of the

backprop errors. Also, in dense networks, the backprop errors will disperse through the

network and hence degrade other error estimates as well. Thus, for fast adaptation,

backpropagation paths should be kept as short as possible.

In practice, when there is a natural temporal boundary, as in word classification

problems, BPTT is a good choice. For typical real time learning applications, as in

prediction or system identification, most researchers apply RTRL. Yet at this time both

methods are restricted to relatively small problems when processed by sequential

machines. The computational cost of RTRL is excessive for large networks, while the

application of BPTT is hampered by increasing memory requirements over time. The

results of this section have been summarized in Figure 4.6. In the next section methods

to overcome the sharp increase in computational cost when neural nets are used in a

temporal environment are investigated.

4.4 The Focused Gamma Net Architecture

In this chapter general exact gradient descent adaptive procedures for the

gamma neural net have been studied. We have come up with two methods, the

backpropagation-through-time algorithm and the real-time-recurrent-learning

procedure. The renewed interest lately in neural net research has been largely propelled

by the application of the backpropagation method. Indeed, for static networks, the









backpropagation method provides an algorithmic approach to solve a large area of

problems that were previously not approachable due to the amount of computation

involved. This advantage is not so obvious when we generalize BP to dynamic

networks such as the gamma model. This procedure is very restricted in the sense that

the storage requirements grow linearly with time. The alternative method, real-time-

recurrent-learning, imposes a constant (in time) load on the computational resources.

Yet the application of RTRL is restricted to small networks.

So what is then the status of gradient descent learning in dynamic neural nets?

For small applications, the methods described sofar have been applied quite

successfully. Currently, research is concentrated on how to adapt the procedures

described here such that larger problems can be attacked with reasonable

computational cost. Two strategies are prevailing in this search. One area of research

focuses on approximate error gradient computation with reduced complexity as

compared to the exact methods described here. For instance, Williams has developed

the truncated backpropagation-through-time procedure (Williams and Zipser, 1991).

This algorithm is less accurate than BPTT but the memory requirements are constant

over time since the backpropagation pass involves a fixed number of time steps. The

other way to speed up error gradient computation is to prewire the network architecture

in order to reduce the complexity of the learning algorithm. In this section we propose

a restricted gamma net architecture, the focused gamma net. The focused gamma net is

inspired by Mozer's efforts to design an efficient dynamic neural net architecture

(Mozer, 1989).

Next the architecture of the focused gamma net is introduced. Some of the

characteristics of this structure are discussed, followed by a derivation of the error

gradients. The specific net architecture allows a very efficient hybrid approach to

gradient computation.








4.4.1 Architecture

The focused gamma net is schematically drawn in Figure 4.5. Assume the 6-
dimensional input signal I(t). The past of this signal is represented in a gamma memory

structure as described by

xi (t) = I (t) EaQ 18

Xik(t) = (1 )i) xik(t- 1) +PXi, k_- (t- 1)9

where t = 0,...,T, i = 1,...,6 and k = 1,...,K. This layer, the input layer, has 6 memory
parameters ti. The activations in the input layer are mapped onto a set of output nodes

by way of a (non-linear) static strictly feedforward net. The nodes in the feedforward
net are indexed 6+1 through N. Thus, this map can be written as

from feedforward net from input layer

xi(t) = i i Xj(t) + E E ik (1) EQ.120
0+l

For convenience Eq.120 will be written as

i (t) = Oi( ijkXjk(t)) E).121
j
where we have utilized the notation xio (t) xi (t) and wfo -wi.

Similar architectures have been used by Stornetta et al. (1988) and Mozer
(1989). These investigators however only used a first-order memory structure (K = 1).
Mozer analyzed some of the properties of structures of this kind and coined the term
focused backpropagation architecture. It turns out that the focused network
architecture enjoys a number of advantages in comparison to the fully connected

dynamic networks.

Let us first derive the update equations for the weights wik. The












static feedforvard net


backpropagation method will be used. As before, the derivations are based on the

performance index E = [em (t) 2 and the evaluation order is determined by the
t,m
list L = [ { i}, {Wijk}, {Xik(t) } ]. We have already discussed in section 4.2.2 how

to apply backpropagation to feedforward nets. Thus, applying Werbos' formula for
ordered derivatives to the activations xi(t) in the feedforward net leads to the following

backpropagation system:

i (t = -ei (t) + wjisj ( Eq122
j>i

6i(t) = oi'(neti(t))ei(t), Q 123

DE DE
where we defined e (t) (t and 8. (t) -neti (t. Similarly, it follows from
&i (t) i net (t)

DE
section 4.2.2 that the gradients -- can be computed as
W ijk


XN.(I) x,v(I)


gamma memory


Figure 4.5 The focused gamma net architecture








DE
aw ii(t) jk (t). E124
ijk t

Note that since the mapping network is static and feedforward, we do not need to
backpropagate through time in order to find the backprop errors 8i(t). In fact, since the

85 (t) 's are computed in real-time, it is convert Eq.124 into a real-time procedure by

defining

DE
( ijk ( t) (t) xjk

Application of backpropagation to compute the error gradients with respect to

the parameters g, leads to a backpropagation-through-time procedure, since the input

layer is recurrent in nature. In most networks however, the number of memory
parameters is relatively small so it is efficient to use the direct method here. This
procedure has already been derived for the more general gamma nets in section 4.3.1.

DE
Hence, without explanation we derive the error gradients D- as follows -


E DE xm (t) xik (t)
E(t) = Xmt i x I
atti mxm (t) axik (t) X ti


= -eem (t) m' (netm (t)) Iwmik( (t)2
m k


where a (t) xik Ok (t) can be computed by evaluation of Eq.105.


An important property is that the backpropagation path is short since the
feedforward net is static. The errors estimates in the dynamic input layer do not
disperse during training since the gamma memory structures do not have lateral
connections in the input layer. This property is confirmed by considering Eq.105,
which propagates the errors through time. Thus the error estimates do not disperse in









the focused gamma net, which explains the adjective "focused".


gamma net architecture
N units, memory order K RTRL BPTT FOCUSED
T time steps

.space O(N3K) O(NKT) O(ONK)


time O(N4KT) O(N'2KT) O(N'K2T)


space no yes yes

-time yes no yes


Figure4.6 A complexity comparison ofgradient descent
learning procedures for the gamma net.



In this architecture we have taken advantage of the particular characteristics of

both the BPTT and RTRL adaptive procedures. Since the feedforward net is static, the

very efficient backpropagation procedure is used to update the weights wik. The input

layer of the focused net is dynamic however and as a result, application of

backpropagation would introduce the burden of time-dependent storage requirements

and error dispersion during the backward pass. RTRL on the other hand is a real-time

procedure that is tailored to application in small dynamic networks. Thus RTRL is used

in the recurrent input layer to compute the error gradients to the memory parameters pi.

The focused gamma model is not as general as a fully connected architecture.

Thus, certain dynamic input-output maps can not be computed by the focused

architecture. For example, this representation assumes that the output can be encoded

as a static map of (the past of) the input pattern. Yet, some very interesting architectures

can be created in this framework. Mozer (1989) and Stornetta et al. (1988) have

obtained promising results in word recognition experiments using a first-order memory





86


focused gamma net. Note that a linear one-layer focused gamma net generalizes

Widrow's adaline structure.














CHAPTER 5


EXPERIMENTAL RESULTS


5.1 Introduction

In this chapter experimental simulation results for the gamma model are

presented. The goals for the simulation experiments are the following:

1. How does the gamma model perform when it is applied to various temporal

processing protocols. In particular, we are interested in an experimental comparison to

alternative neural network architectures.

2. How well do the adaptation algorithms that were derived in chapter 4

perform? The following questions are interesting in this respect and will be addressed

in this chapter. How well does the gradient descent procedure for the focused gamma

net work? Can we learn the weights w? Can we learn the gamma memory parameters

i? How does the adaptation time for the focused gamma net compare to alternative

neural net models?

With respect to the first goal, simulation experiments for problems in

prediction, system identification, temporal pattern classification and noise reduction

were selected. All experiments were carried out by members of the CNEL group. We

used 386-based DOS personal computers and 68030- and 68040-CPU based NeXT

computers for all simulations. The programming language was C.

For all neural net simulations we used a version of the focused gamma neural

net. The focused gamma net is a very versatile structure as it reduces to a time-delay-

neural-net when It is fixed to 1. Also, a one-layer linear focused gamma net reduces to









adaline(g). More complex architectures are certainly possible but in this work we are

mainly interested in a comparative evaluation of the gamma memory structure per se.

The topic of designing complex globally recurrent neural net architectures with gamma

memory is not addressed here. Also, experimental evaluation of neural networks in

relation to alternative non-neural processing techniques is not presented here (with the

exception of the noise reduction experiments). The latter topic has been studied by a

special DARPA committee (DARPA, 1988).

Before the experimental results are presented, some general practical issues

concerning gamma net simulation and adaptation are discussed.

5.2 Gamma Net Simulation and Training Issues

The system architecture that is used in the experiments is shown in Figure 5.1.

The signal to be processed source signal is denoted by s(t). Both the neural net input

signal and the desired signal d(t) are derived from s(t). The particular form of this

transformation depends on the processing goal. A subset of the neural net states x(t),

the outputs, are measured and compared to the desired signals. The difference signal,

e(t) = d(t) x(t), is called the instantaneous error signal and it is used as the input to the

steepest descent training procedure.


Figure 5.1 Experimental architecture.









5.2.1 Gamma Net Adaptation

The training strategy of the gamma neural net deserves more attention. In all

cases the network parameters w and gp were adapted using the focused backpropagation

method as derived in chapter 4. We used the simple steepest descent update method,

JE
that is, Aw = -r -. In all experiments, we used real-time updating. Thus, the weights
aw

were adapted after each new sample. The stepsize (learning rate) T1 is an important

parameter. For large rl the adaptation algorithm may become unstable, while a small rl

leads to slow adaptation. We were not so much interested in optimizing the speed of

adaptation. A value between 0.01 and 0.1 for q1 provided in all cases a stable adaptation

phase. Another central problem is when to halt adaptation. Let us assume that the

network is trained by presentation of a set of pattern pairs, where each pair consists of

an input pattern and the corresponding target pattern. This set of patterns is called the

training data set. The presentation of all patterns from the training set is called an
epoch. In the experimental setting of this work, we obtain the training set by selecting

an appropriate source signal segment. The performance index (total error) for the

training set as a function of the epoch number provides a good measure as to how well

the neural net is able to model the training set. However, accurate modeling of the

training set is not the goal of adaptation. The idea of adaptation by exemplar patterns

over time is to present a good representation of the problem at hand to the neural net,

which after adaptation is able to extrapolate the information contained in the training

set to new input patterns. Thus, it is a good habit to test how well the neural net is able

to generalize to an additional set of patterns that are not used for adaptation. This set

of patterns is called the validation set. In general, whereas the total error for the training

set decreases as adaptation progresses, this is not necessarily the case for the

performance index of the validation set (Hecht-Nielsen, 1990). In practice it has been

found that gradient descent procedures first adapt to the gross features of the training









set. As training progresses, the system starts to adapt to the finer features of the training

set. The fine details of the training set very often do not represent features of the

problem, as is the case when the training data is corrupted by noise. When the system

starts to adapt to model the training data specific noise, the total error for the validation

data usually increases. This is the time when adaptation should be stopped. In this way,

the stop criterion detects when the adaptation process has reached a maximal

performance with respect to extrapolation to other patterns (not from the training set)

that are representative for the problem task. In all of the experiments that are presented

here we have used this strategy to determine when to stop training.

As an example, consider the learning curves as displayed in Figure 5.2. The

normalized square error for both the training and validation set is plotted as a function

of the epoch number. This example was taken from an elliptic filter modeling

experiment that will be discussed in section 5.4. In our experiments we stop training -

or detect convergence if the normalized error for the validation data set increases over

four consecutive epochs.



03


convergence
O"8 detected
0 ?6
Etrain
Eval ]
2 3 6 & 10 12 II Itb 1 :2
---- epoch no.

Figure 5.2 Normalized total error for training set and validation
set in sinusoidal prediction experiment. Convergence is detected
after 4 successive increases of error in validation set.



Another important issue when considering neural net training is the adaptation









time. The adaptation time is the number of patterns (or epochs for batch learning) that

have to be presented to the neural net before the weights converge. There is not much

theory about the adaptation time of backpropagation algorithms. However the question

whether and how the value of gI affects the adaptation time can be experimentally

tackled. This problem was studied for a third-order elliptic filter modeling problem,

which is covered in more detail in section 5.4. The architecture was an adaline(.)

structure with K=3. The adaptation time expressed in the number of samples was

measured as a function of gp and the results are plotted in Figure 5.3. The plot shows

that the adaptation time is nearly unaffected if gI is greater than 0.2. This is rather good

news, although we have not been able to establish explicit formulae for the adaptation

time dependence on p.





#of
samples

80-

60-

40-

20
I I I I I I I I
0.1 0.2 0.5


Figure 5.3 Adaptation time as a function of p for the
elliptic filter modelling experiment in section 5.4



In the next sections the experimental results with respect to application of the

gamma model to temporal processing problems are discussed.









5.3 (Non-)linear Prediction of a Complex Time Series

5.3.1 Prediction/Noise Removal of Sinusoidals contaminated by Gaussian Noise

We constructed an input signal consisting of a sum of sinusoids, contaminated

by additive white gaussian noise (AWGN). Specifically, I(t) was described by

I(t) = sin (7t (0.06t + 0.1)) + 3sin (t (0.12t + 0.45)) + 1.5sin (t (0.2t+ 0.34))
+ sin (7t (0.4t+ 0.67)) +AWGN .

The signal-to-noise ratio is 10 dB. This signal is shown in Figure 5.4.


--t I ---

Figure 5.4 (a) The sinusoidal signal plus AWGN (SNR = 10 dB). (b) Power
spectrum of the contaminated signal.


The processing goal was to predict the next sample of the sum of sinusoidals.

Hence, the processing problem involves a combination of prediction and noise

cancelation. The processing system was adaline(ji). The goals of this experiment are

the following:

1. Determine the optimal system performance as a function of g for 0 < p. < 1

and K. Note that this implies a comparison of the gamma memory structure versus the

tapped delay line (for (t=1) and the context-unit memories (for K=1).

2. Can the system parameters wk and tt be adapted to converge to the optimal

values?




Full Text
xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008217300001datestamp 2009-02-16setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Temporal processing with neural networks dc:creator De Vries, Bertdc:publisher Bert de Vriesdc:date 1991dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082173&v=0000125541247 (oclc)001709827 (alephbibnum)dc:source University of Floridadc:language English



PAGE 1

7(0325$/ 352&(66,1* :,7+ 1(85$/ 1(7:25.6 7+( '(9(/230(17 2) 7+( *$00$ 02'(/ %< %(57 '( 95,(6 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

$&.12:/('*0(176 7KH FKDQFHV RI PHQWDO GHSUHVVLRQ GXULQJ RQHf¬V 3K' VWXGLHV DUH QRW VPDOO 3RYHUW\ DQG SHUVRQDO VRFLDO DQG LQWHOOHFWXDO LVRODWLRQ DUH MXVW D IHZ RI WKH ULVNV RI WKH f¯SURIHVVLRQf°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f¬V DSSURDFK WRZDUG FRQGXFWLQJ VFLHQFH LV RQH RI WKH EHVW OHVVRQV D 3K' VWXGHQW FDQ UHFHLYH 'U -DQ YDQ GHU $D KDV EHHQ ERWK RQ P\ PDVWHUf¬V DQG 3K' VXSHUYLVLQJ FRPPLWWHH +LV FRPPLWPHQW WR RIIHU RXWVWDQGLQJ KHOS DW DQ\ WLPH DQG KLV SHUVRQDO IULHQGVKLS DUH YHU\ PXFK DSSUHFLDWHG 'U 'RQDOG &KLOGHUV KDV EHHQ YHU\ KHOSIXO DW VHYHUDO WLPHV LQ JXLGLQJ WKH QH[W UHVHDUFK VWHSV 'U )UHG 7D\ORUf¬V ZLOOLQJQHVV WR VHUYH RQ P\ GRFWRUDO FRPPLWWHH LV YHU\ PXFK DSSUHFLDWHG , KDYH KDG PXFK VXSSRUW IURP 'U -DPHV .HHVOLQJ IURP WKH PDWKHPDWLFV GHSDUWPHQW DQG 'U $QWRQLR $UUR\R IURP WKH HOHFWULFDO HQJLQHHULQJ GHSDUWPHQW 'U 3HGUR *XHGHV GH 2OLYLHUD IURP WKH HOHFWULFDO HQJLQHHULQJ GHSDUWPHQW RI WKH 8QLYHUVLW\ RI $YHLUR LQ 3RUWXJDO YLVLWHG RXU ODERUDWRU\ GXULQJ WKH VSULQJ VHPHVWHU +H KDV PDGH VLJQLILFDQW FRQWULEXWLRQV WR WKH XQGHUVWDQGLQJ RI WKH JDPPD PRGHO +LV KHOS DQG IULHQGVKLS LV DOVR YHU\ PXFK X

PAGE 3

DSSUHFLDWHG 6HYHUDO JUDGXDWH VWXGHQWV LQ WKH &RPSXWDWLRQDO 1HXUR(QJLQHHULQJ /DERUDWRU\ &1(/f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

PAGE 4

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

PAGE 5

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fOLQHDU 3UHGLFWLRQ RI D &RPSOH[ 7LPH 6HULHV 3UHGLFWLRQ1RLVH 5HPRYDO RI 6LQXVRLGDOV FRQWDPLQDWHG E\ *DXVVLDQ 1RLVH 3UHGLFWLRQ RI DQ ((* 6OHHS 6WDJH 7ZR 6HJPHQW Y

PAGE 6

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

PAGE 7

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f 7KH PDLQ FRQWULEXWLRQ RI WKLV ZRUN LV WKH LQWURGXFWLRQ RI D QHZ QHXUDO QHWf PHFKDQLVP WR VWRUH WHPSRUDO LQIRUPDWLRQ 7KLV PRGHO WKH JDPPD QHXUDO PRGHO FRPSDUHV YHU\ IDYRUDEO\ WR FRPSHWLQJ PHPRU\ VWUXFWXUHV VXFK DV WKH WDSSHG GHOD\ OLQH 9OO

PAGE 8

DQG ILUVWRUGHU VHOIUHFXUUHQW PHPRU\ XQLWV 7KH JDPPD PHPRU\ PHFKDQLVP LV FKDUDFWHUL]HG E\ D FDVFDGH RI XQLIRUP ORFDOO\ VHOIUHFXUUHQW GHOD\ XQLWV $Q LQWHUHVWLQJ IHDWXUH RI WKH JDPPD PHPRU\ PHFKDQLVP LV WKH DGDSWDELOLW\ RI WKH PHPRU\ GHSWK DQG UHVROXWLRQ 7KH JDPPD PRGHO LV DQDO\]HG DQG FRPSDUHG ZLWK FRPSHWLQJ QHXUDO PRGHOV $ WHPSRUDO EDFNSURSDJDWLRQ WUDLQLQJ SURFHGXUH IRU JDPPD QHXUDO QHWV LV GHULYHG ([SHULPHQWV LQ WLPH VHULHV SUHGLFWLRQ HOHFWURHQFHSKDORJUDP ((*f DQG V\QWKHWLF FKDRWLF VLJQDOVf QRLVH UHPRYDO IURP D FKDRWLF VLJQDO DQG V\VWHP LGHQWLILFDWLRQ DUH GLVFXVVHG ,Q DOO H[SHULPHQWV WKH JDPPD PRGHO RXWSHUIRUPV FRPSHWLQJ QHWZRUN DUFKLWHFWXUHV ,QWHUHVWLQJO\ WKH DSSOLFDWLRQ RI WKH JDPPD PHPRU\ VWUXFWXUH LV QRW OLPLWHG WR QHXUDO QHWV $ FKDSWHU LV GHYRWHG WR LQWURGXFH DGDOLQHMLf DQ DGDSWLYH OLQHDU ILOWHU ZLWK JDPPD PHPRU\ $GDOLQH_Lf JHQHUDOL]HV :LGURZf¬V DGDSWLYH OLQHDU FRPELQHU DGDOLQHf WKH PRVW ZLGHO\ XVHG VWUXFWXUH LQ DGDSWLYH VLJQDO SURFHVVLQJ 7KH VLJQDO FKDUDFWHULVWLFV DQG SURFHVVLQJ DSSOLFDWLRQV ZKHUH DGDOLQH_Lf LPSURYHV RQ WKH SHUIRUPDQFH RI DGDOLQH DUH LGHQWLILHG 9OOO

PAGE 9

&+$37(5 7+( 7(0325$/ 352&(66,1* 352%/(0 $1' 5(6($5&+ *2$/6 ,QWURGXFWLRQ :KHQ , VWDUWHG WKLV UHVHDUFK VRPHZKHUH DURXQG -DQXDU\ WKH JRDO ZDV WR GHYHORS D VSHHFK UHFRJQLWLRQ V\VWHP WKDW ZDV FKDUDFWHUL]HG E\ D UHODWLYHO\ KLJK GHJUHH RI ELRORJLFDO SODXVLELOLW\ 7KXV , VSHQW WKH ILUVW VHPHVWHU RI LQ WKH +HDULQJ 5HVHDUFK &HQWHU DW WKH 8QLYHUVLW\ RI )ORULGD 7KLV VSHHFK V\VWHPLQZDLWLQJ ZDV JRLQJ WR EH GHYHORSHG IURP WKH ERWWRP XS )LUVW , ZRXOG GHYHORS D QHXUDO DXGLWRU\ PRGHO WKDW FDSWXUHV WKH HVVHQWLDO IHDWXUHV IRU VSHHFK DQDO\VLV IROORZHG E\ QHZ WHFKQLTXHV IRU QHXUDO WHPSRUDO SDWWHUQ FODVVLILFDWLRQ DQG VR RQ $V WLPH SDVVHG E\ WKH WRSLF RI WKLV GLVVHUWDWLRQ KDV VKLIWHG PRUH WRZDUG WKH IXQGDPHQWDO SUREOHP RI WLPH UHSUHVHQWDWLRQ LQ QHWZRUN VWUXFWXUHV SHU VH $V D UHVXOW D QHXUDO QHWZRUN PRGHO IRU SURFHVVLQJ RI WHPSRUDO SDWWHUQV KDV EHHQ GHYHORSHG DQG DW WKH WLPH RI WKH ZULWLQJ RI WKLV WKHVLV ZH DUH FRQGXFWLQJ H[SHULPHQWV LQ LVRODWHG ZRUG UHFRJQLWLRQ DQG SKRQHPH FODVVLILFDWLRQ PDNLQJ XVH RI WKLV QHZ PRGHO $V LW WXUQV RXW WKH DSSOLFDWLRQ RI WKH JDPPD PRGHO WKLV LV KRZ WKH PRGHO LV FDOOHGf LV QRW OLPLWHG WR WKH WHPSRUDO FODVVLILFDWLRQ SUREOHP 6XFFHVVIXO H[SHULPHQWV ZHUH SHUIRUPHG DV ZHOO IRU SUREOHPV LQ QRLVH UHGXFWLRQ V\VWHP LGHQWLILFDWLRQ DQG WLPH VHULHV SUHGLFWLRQ )RU WKHVH WDVNV WKH JDPPD PRGHO SHUIRUPDQFH LV YHU\ SURPLVLQJ LQ IDFW EHWWHU WKDQ PDQ\ FRPSHWLQJ WHFKQLTXHV ,Q WKLV WKHVLV WKH GHYHORSPHQW RI WKH JDPPD PRGHO LV SUHVHQWHG , WU\ WR H[SODLQ ZK\ DQG KRZ LW ZRUNV DQG UHSRUW RQ D IHZ H[SHULPHQWDO UHVXOWV

PAGE 10

$ 6WDWHPHQW RI WKH 3UREOHP 7KLV GLVVHUWDWLRQ GHDOV ZLWK SURFHVVLQJ RI WLPHYDU\LQJ VLJQDOV E\ D QHXUDO QHWZRUN %\ D VLJQDO ZH PHDQ D WLPH VHTXHQFH RI SDWWHUQV DQG FRQVHTXHQWO\ WKH DGMHFWLYHV f¯WLPHYDU\LQJf° RU f¯WHPSRUDOf° ZLOO RIWHQ EH GHOHWHG 7KH WHPSRUDO FRQWH[W LV DOZD\V DVVXPHG 7KH AGLPHQVLRQDO LQSXWf VLJQDO WR EH SURFHVVHG LV GHQRWHG E\ VWf DQG IRU WKH 0GLPHQVLRQDO SURFHVVHG RXWSXWf VLJQDO ZH ZLOO ZULWH [Wf 3URFHVVLQJ RI VWf LPSOLHV WKH DSSOLFDWLRQ RI D PDS i ZKLFK WUDQVIRUPV VWf LQWR WKH VLJQDO [Wf ,Q WKLV ZRUN ZH DUH LQWHUHVWHG LQ SURFHVVLQJ DSSOLFDWLRQV ZKHUH WKH SDVW RI VWf DIIHFWV WKH FRPSXWDWLRQ RI [Wf 7KXV Mf LV DFWXDOO\ D PDS IURP VWf IRU bW WR [Wf 7\SLFDO DSSOLFDWLRQV LQFOXGH VSHHFK UHFRJQLWLRQ G\QDPLF V\VWHP LGHQWLILFDWLRQ DQG SUHGLFWLRQ RI D WLPH VHULHV 7KH PDLQ SUREOHP RI WKLV WKHVLV LV KRZ WR HIIHFWLYHO\ UHSUHVHQW WKH SDVW RI VWf 7KH IROORZLQJ H[DPSOH VKRZV ZK\ WKLV LV D GLIILFXOW WDVN &RQVLGHU DQ LVRODWHGZRUG UHFRJQLWLRQ V\VWHP i ZKLFK SURFHVVHV WKH LQFRPLQJ VSHHFK VLJQDO LQ UHDOWLPH $VVXPH D ZRUG YRFDEXODU\ DQG KHQFH WKH RXWSXW VSDFH FDQ EH GHVFULEHG LQ ELWV 7KH VSHHFK VLJQDO LV VDPSOHG DW >+]@ DQG TXDQWL]HG E\ ELWV 7KHQ IRU D W\SLFDO ZRUG ZKLFK ODVW VHFRQGV WKH LQSXW VLJQDO LV UHSUHVHQWHG E\ [ [ ELWV 7KXV LV D PDS IURP LQSXW SDWWHUQ FRPELQDWLRQV WR ELWV 7KLV VHTXHQFH RI ELWV REYLRXVO\ FRQWDLQV PDQ\ UHGXQGDQW FRSLHV RI LQIRUPDWLRQ DQG LV FRQWDPLQDWHG E\ LUUHOHYDQW QRLVH 6LQFH LV D UHDOWLPH V\VWHP LW PXVW WKURZ DZD\ GDWD DV TXLFNO\ DV SRVVLEOH ZKLOH SUHVHUYLQJ HQRXJK LQIRUPDWLRQ WR FODVVLI\ WKH ZRUGV 1RWH WKH LQKHUHQW RSWLPL]DWLRQ WDVN DVVRFLDWHG ZLWK WKLV SUREOHP LI ZH WKURZ DZD\ WRR PXFK GDWD WKHUH PD\ QRW EH HQRXJK LQIRUPDWLRQ OHIW IRU ZRUG FODVVLILFDWLRQ EXW \HW ZH GR QRW ZDQW WR VWRUH UHGXQGDQW GDWD DQG QRLVH IRU LW ZLOO VLJQLILFDQWO\ FRPSOLFDWH WKH SURFHVVLQJ WDVN $V DQRWKHU H[DPSOH FRQVLGHU WKH SUREOHP RI SUHGLFWLQJ D GLVFUHWH WLPH VHULHV VWf E\ D PRGHO Mf +RZ PDQ\ VDPSOHV IURP WKH SDVW RI VWf GRHV WKH PRGHO -f QHHG LQ

PAGE 11

RUGHU WR UHOLDEO\ SUHGLFW WKH QH[W VDPSOH VLOf" 'R ZH QHHG WR VWRUH WKH HQWLUH KLVWRU\ RI VWf 9HU\ OLNHO\ QRW DQG LQ IDFW WKH DGGLWLRQDO QRLVH DVVRFLDWHG ZLWK D GHHS PHPRU\ ZLOO ZRUN GHWULPHQWDO RQ WKH SHUIRUPDQFH RI _f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

PAGE 12

QHXUDO QHWZRUN /HW WKH VLJQDO VWf LQ )LJXUH EH GHILQHG IRU W4WW\ LQ WKH WHPSRUDO GLPHQVLRQ $W DQ\ WLPH RQO\ SDUW RI WKH WHPSRUDO GRPDLQ RI VWf LV DYDLODEOH IRU IXUWKHU SURFHVVLQJ 7KH VHOHFWLRQ PDVN LV FDOOHG D ZLQGRZ 7KH ZLGWK RI WKH ZLQGRZ LV GHQRWHG E\ ,Q VRPH FDVHV EHIRUH WKH GDWD VHTXHQFH LV VXEPLWWHG WR WKH DFWXDO SURFHVVLQJ WDVN ! VLJQDO IHDWXUHV DUH FRPSXWHG IURP WKH ZLQGRZHG GDWD VHJPHQW $V DQ H[DPSOH LQ ZRUG FODVVLILFDWLRQ WKH SLWFK SHULRG SURYLGHV LPSRUWDQW LQIRUPDWLRQ FRQFHUQLQJ WKH H[FLWDWLRQ VRXUFH RI WKH VSHHFK VLJQDO 1RWH WKDW ERWK ZLQGRZHGf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f§! WKHUH LV QR VHJPHQWDWLRQ 2EYLRXVO\ LQ WKLV FDVH WKH PHPRU\ KDV WR EH PRYHG WR WKH VHFRQG SURFHVVLQJ VWDJH 7KH FKRLFH RI WKH VOLGLQJ ZLQGRZ ZLGWK LQIOXHQFHV WKH V\VWHP SHUIRUPDQFH :H LGHQWLI\ WZR SUREOHPV DVVRFLDWHG ZLWK WKH VHOHFWLRQ RI )LUVW IRU ODUJH WKH GLPHQVLRQDOLW\ RI WKH SURFHVVLQJ V\VWHP LQFUHDVHV 7KLV IDFW PLQLPDOO\ FRPSOLFDWHV WKH QHXUDO QHW WUDLQLQJ ,W KDV EHHQ VKRZQ WKDW QHXUDO QHW DGDSWDWLRQ WLPH XQIRUWXQDWHO\

PAGE 13

VFDOHV ZRUVH WKDQ SURSRUWLRQDO ZLWK WKH GLPHQVLRQ RI WKH ZHLJKW YHFWRUV 3HUXJLQL DQG (QJHOHU f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f $SDUW IURP WKH GLIILFXOW\ RI FKRRVLQJ WKH ZLQGRZ ZLGWK WKH WHPSRUDO UHVROXWLRQ RI WKH ZLQGRZ LV DQRWKHU LPSRUWDQW SUHSURFHVVLQJ SDUDPHWHU :H GHILQH WKH WHPSRUDOf UHVROXWLRQ 5 RI WKH ZLQGRZ DV WKH QXPEHU RI RXWSXWV RI WKH ZLQGRZ GLYLGHG E\ WKH ZLQGRZ ZLGWK LQ VHFRQGVf $V LV WKH FDVH IRU WKH ZLGWK WKH RSWLPDO UHVROXWLRQ LV GHSHQGHQW RQ WKH SURFHVVLQJ JRDO )RU LQVWDQFH LI WKH YRFDEXODU\ VL]H ZHUH LQVWHDG RI LQ WKH LVRODWHGZRUG UHFRJQLWLRQ H[DPSOH WKH GHPDQGV RQ WKH UHVROXWLRQ RI WKH ZLQGRZ ZRXOG REYLRXVO\ LQFUHDVH ,Q VSHHFK SURFHVVLQJ LW LV FRPPRQ WR GHWHUPLQH WKH ZLQGRZ ZLGWK EDVHG RQ VWDWLVWLFDO PHDVXUHV RI WKH LQSXW VLJQDO )RU LQVWDQFH ]HURFURVVLQJ UDWH DQG HQHUJ\ PHDVXUHV KDYH EHHQ XVHG WR HVWLPDWH SVHXGRVWDWLRQDU\ VLJQDO VHJPHQWV 5DELQHU DQG

PAGE 14

6FKDIHU f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f WKH GLIIHUHQFH EHWZHHQ D GHVLUHG RXWSXW VLJQDO GWf DQG WKH V\VWHP RXWSXW [Wf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f ILOWHUV DQG LQILQLWH LPSXOVH UHVSRQVH +5f ILOWHUV ),5 ILOWHUV DUH IHHGIRUZDUG DQG WKH SDVW RI WKH LQSXW VLJQDO LV VWRUHG LQ D WDSSHG GHOD\ OLQH +5 ILOWHUV DUH RI UHFXUUHQW IHHGEDFNf QDWXUH $V D UHVXOW PRUH FRPSOLFDWHG PHPRU\ VWUXFWXUHV

PAGE 15

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

PAGE 16

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f¯HOHFWULFDO HQJLQHHULQJ YLHZf° DQG HPSKDVL]H WKH UHODWLRQ WR OLQHDU GLJLWDO ILOWHUV 0HFKDQLVPV IRU VKRUW WHUP PHPRU\ LQ QHXUDO QHWV DUH UHYLHZHG 7KH DQDO\VLV IRFXVHV LQ SDUWLFXODU RQ WZR ZLGHO\ XVHG VWUXFWXUHV WKH WDSSHG GHOD\ OLQH DQG WKH ILUVWRUGHU VHOIUHFXUUHQW XQLWV FRQWH[W XQLWVf %RWK PHFKDQLVPV DUH VKRZQ WR KDYH OLPLWHG DSSOLFDELOLW\ )RU H[DPSOH WKH WDSSHG GHOD\ OLQH KDV OLPLWHG IL[HG PHPRU\ GHSWK ZKHUHDV WKH FRQWH[W XQLWV DOZD\V RYHUZULWH LQIRUPDWLRQ IURP WKH SDVW ZLWK PRUH UHFHQW LQIRUPDWLRQ ,Q FKDSWHU D QHZ IUDPHZRUN IRU VWRUDJH RI SDVW LQIRUPDWLRQ LQ QHXUDO QHWV LV LQWURGXFHG 7KH QHZ PHPRU\ PRGHO JDPPD PHPRU\ LV VXSSRUWHG E\ D PDWKHPDWLFDO

PAGE 17

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f DQG WKH OHDVW PHDQ VTXDUH /06f DOJRULWKP IRU DGDSWLYH OLQHDU ILOWHUVf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

PAGE 18

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f ILOWHUV DQG UHFXUUHQW RU LQILQLWH LPSXOVH UHVSRQVH I+5O ILOWHUV ,Q D ),5 ILOWHU WKH LQSXW VLJQDO KLVWRU\ LV VWRUHG LQ D WDSSHG GHOD\ OLQH 7KH VLJQDOV DW WKH WDSV DUH UHIHUUHG WR DV VWDWH YDULDEOHV 7KH RXWSXW RI WKH ),5 ILOWHU LV D OLQHDU ZHLJKWHG FRPELQDWLRQ RI WKH WDS YDULDEOHV VHH )LJXUH f ),5 ILOWHUV DUH DOZD\V VWDEOH EXW QRWH WKDW WKH GHSWK RI WKH PHPRU\ LV IL[HG DQG HTXDOV WKH QXPEHU RI WDSV RI

PAGE 19

WKH GHOD\ OLQH +5 ILOWHUV DUH PRUH FRPSOH[ VWUXFWXUHV VLQFH UHFXUUHQW FRQQHFWLRQV DUH DOVR DOORZHG ,Q FRQWURO WKHRU\ VLPLODU OLQHDU VWUXFWXUHV DUH NQRZQ DV DXWRUHJUHVVLYH PRYLQJDYHUDJH $50$ V\VWHPV $ VRFDOOHG REVHUYHU FDQRQLFDO IRUP LPSOHPHQWDWLRQ RI WKH +5 ILOWHU LV VKRZQ LQ )LJXUH 7KH H[LVWHQFH RI UHFXUUHQW FRQQHFWLRQV LPSOLHV WKH ULVN RI LQVWDELOLW\ RI WKH V\VWHP EXW LQFUHDVHV WKH FRPSXWDWLRQDO SRZHU RI WKH V\VWHP 7KH VWDWH YDULDEOHV [^Wf DUH D IXQFWLRQ RI ERWK WKH ORZHU LQGH[ VWDWH YDULDEOHV PHPRU\ EY GHOD\ DV LQ WKH ),5 ILOWHUf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f DQG PRGHOLQJ RI D FKDRWLF WLPH VHULHV 1HXUDO QHWZRUNV DV DQ HQJLQHHULQJ WRRO DUH SUREDEO\ EHVW LQWHUSUHWHG DV D JHQHUDOL]HG FODVV RI QRQOLQHDU DGDSWLYH ILOWHUV $V VXFK WKH\ SURYLGH WKH FRPSXWDWLRQDO IHDWXUHV WKDW SRWHQWLDOO\ EHWWHU FRSH ZLWK VROYLQJ FRPSOH[ QRQOLQHDU SUREOHPV 1H[W

PAGE 20

DQ LQWURGXFWLRQ WR QHXUDO QHWV LV SUHVHQWHG ,QWURGXFWLRQ WR 1HXUDO 1HWZRUNV 7KLV VHFWLRQ FRQWDLQV D EULHI LQWURGXFWLRQ WR QHXUDO QHWZRUNV ,Q WKH OLWHUDWXUH ZH DOVR ILQG QDPHV DV FRQQHFWLRQLVW PRGHOV SDUDOOHO GLVWULEXWHG SURFHVVLQJ GHYLFHV RU DUWLILFLDO QHXUDO QHWV DOO GHQRWLQJ WKH VDPH NLQG RI SURFHVVLQJ DUFKLWHFWXUH 7KH GLVFXVVLRQ ZLOO EH RI JHQHUDO QDWXUH )RU D GHHSHU ORRN LQWR VRPH HTXDWLRQV DQG LPSOHPHQWDWLRQV RI QHXUDO QHWZRUNV , OLNH WR UHIHU WR D SDSHU E\ /LSSPDQQ f DQG D ERRN E\ 6LPSVRQ f $ PRUH WKRURXJK ORRN DW QHXUDO QHWZRUNV LV RIIHUHG LQ ERRNV E\ +HUW] HW DO f DQG +HFKW1LHOVHQ f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

PAGE 21

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f IM&M If M&M\LfO KROG WKH $GLPHQVLRQDO VWDWH RI D QHXUDO QHW Z Z :,1 , , :1 fµ :11 DQ $AGLPHQVLRQDO YHFWRU IRU D IXOO\ FRQQHFWHG QHWf RI DGDSWLYH ZHLJKWV DQG Wf >M AWf@ WKH H[WHUQDO LQSXW WR WKH QHW 7KHQ WKH V\VWHP LV FRPSOHWHO\ GHVFULEHG E\ WKH IROORZLQJ VHW RI HTXDWLRQV G;M GW Ic[,Zf (T

PAGE 22

MU JcMZ[f 0O 7KH G\QDPLFV IRU WKH VWDWH [Wf DUH GHVFULEHG E\ (TO DQG (T GHVFULEHV WKH DGDSWDWLRQ G\QDPLFV 7KH HTXLOLEULD RI V\VWHP (TO DUH FRPSXWHG E\ Ic[r Zf (J ZKHUH [rr KROGV WKH VWHDG\ VWDWH 7KH PRVW ZLGHO\ XVHG QHXUDO QHWZRUN PRGHO LV WKH VRFDOOHG DGGLWLYH PRGHO GHVFULEHG E\ G[ I 1 ‘! n mc¯fD :LM;MLWf fµ Wf n GW (J 7KH DGGLWLYH PRGHO LV XVHG LQ WKH JUHDW PDMRULW\ RI SUDFWLFDO DSSOLFDWLRQV RI QHXUDO QHWZRUNV WRGD\ 6HMQRZVNL SURYLGHV D ELRORJLFDO PRWLYDWLRQ IRU WKH DGGLWLYH PRGHO 6HMQRZVNL f $ IORZ GLDJUDP RI WKH DGGLWLYH PRGHO LV VKRZQ LQ)LJXUH 7KH VWDWH YHFWRU [Wf LV DIIHFWHG E\ D SDVVLYH GHFD\ D[AWf \LHOGLQJ VKRUW WHUP PHPRU\ QRQOLQHDU QHXUDO IHHGEDFN VLJQDOV FZcM;MWff DQG DQ H[WHUQDO LQSXW 7KH QHXURQ VLJQDO IXQFWLRQ Jf QRUPDOO\ LV D QRQOLQHDU IXQFWLRQ $ W\SLFDO FKRLFH LV WKH ORJLVWLF IXQFWLRQ D [f WDQN [f 7KH IHHGEDFN VLJQDOV IURP WKH QHW LWVHOI DUH VRPHWLPHV VKRUWO\ GHQRWHG E\ WKH YDULDEOH QHW WKDW LV 1 QHWLWf @7 ZcM;MWf (IOJ 7KH V\VWHP GHVFULEHG E\ (T LV FDOOHG DGGLWLYH VLQFH WKH ZHLJKWV Z DUH QRW D IXQFWLRQ RI WKH VWDWHV [ ,Q FDVH Z Z [f WKH PRGHO H[KLELWV PDVVDFWLRQ EHKDYLRU VXFK V\VWHPV DUH FDOOHG PDVVDFWLRQ VKXQWLQJ RU PXOWLSOLFDWLYH PRGHOV ,Q RUGHU IRU (T WR EH FRPSXWDWLRQDOO\ LQWHUHVWLQJ WKH WKUHH G\QDPLF YDULDEOHV [ DQG Z PXVW

PAGE 23

SHUIRUP RYHU WKUHH GLIIHUHQW WLPH VFDOHV )URP D QHXURG\QDPLF YLHZSRLQW ZH FDQ LQWHUSUHW [^Wf WR KROG D VKRUW WHUP PHPRU\ VWPf WUDFH DQG ZWf WR SURFHVV ORQJ WHUP PHPRU\ WUDFHV OWPf 7KH SKLORVRSK\ EHKLQG V\VWHP (T DV D SDWWHUQ UHFRJQLWLRQ GHYLFH IRU WHPSRUDO SDWWHUQV WKHQ EDVLFDOO\ UXQV DV IROORZV $V WLPH SDVVHV E\ WKH OWP WUDFHV ZWf VDPSOH DQG DYHUDJH RYHU WLPH WKH QHXURQDO DFWLYLW\ [Wf WKXV IRUPLQJ VRPH NLQG RI WHPSODWH RU UHIHUHQFH SDWWHUQ RI QHXUDO DFWLYLW\ $W DQ\ WLPH D VKRUW WHUP DYHUDJH RI WKH FXUUHQW H[WHUQDO HQYLURQPHQW OLWf LV UHIOHFWHG LQ WKH VWP WUDFHV [^Wf 7KH GHJUHH RI PDWFKLQJ EHWZHHQ WKH VWP WUDFHV DQG WKH OWP WUDFHV GHWHUPLQHV KRZ ZHOO WKH FXUUHQW HQYLURQPHQWDO LQSXW LV UHFRJQL]HG 7KH EDVLF DUFKLWHFWXUDO FRPSRQHQW RI QHXUDO QHWZRUNV DQG DGDSWLYH VLJQDO SURFHVVLQJ VWUXFWXUHV LV WKH DGDSWLYH OLQHDU FRPELQHU $Q XQGHUVWDQGLQJ RI WKH ZRUNLQJ RI WKH DGDSWLYH OLQHDU FRPELQHU DQG WKH OHDVW PHDQ VTXDUH /06f DOJRULWKP LV HVVHQWLDO IRU WKH QHXUDO QHWZRUN VWUXFWXUHV WKDW DUH VXUYH\HG LQ WKLV WKHVLV 7KH QH[W VHFWLRQ

PAGE 24

LQWURGXFHV WKH DGDSWLYH OLQHDU FRPELQHU 7KH $GDSWLYH /LQHDU &RPELQHU 7KH DGDSWLYH OLQHDU FRPELQHU RU QRQUHFXUVLYH DGDSWLYH ILOWHU LV IXQGDPHQWDO WR DGDSWLYH VLJQDO SURFHVVLQJ DQG QHXUDO QHWZRUN WKHRU\ DQG DSSOLFDWLRQV 7KLV VWUXFWXUH QRUPDOO\ VKRUWO\ UHIHUUHG WR DV DGDOLQH IURP DGDSWLYH OLQHDU QHXURQf ZDV LQWURGXFHG E\ :LGURZ DQG +RII LQ :LGURZ DQG +RII f 7KH DGDOLQH VWUXFWXUH DSSHDUV LQ VRPH IRUP LQ QHDUO\ DOO IHHGIRUZDUG QHXUDO QHWZRUN VWUXFWXUHV 7KH SURFHVVLQJ DQG DGDSWDWLRQ SURSHUWLHV RI DGDOLQH DUH ZHOO XQGHUVWRRG DQG GRFXPHQWHG LQ :LGURZ DQG 6WHDUQV f DQG +D\NLQ f ,Q WKLV WKHVLV ZH ZLOO RQO\ LQWURGXFH WKH SURSHUWLHV WKDW DUH HVVHQWLDO LQ WKH FRQWH[W RI WKLV ZRUN 7KH GHVFULELQJ HTXDWLRQV IRU DGDOLQH DUH JLYHQ E\ . \Wf ZN[NA! (T N ZKHUH [NWf DUH WKH LQSXW VLJQDOV \Wf WKH RXWSXW VLJQDO DQG ZN WKH DGDSWLYH SDUDPHWHUV RU ZHLJKWV $GDOLQH LV D GLVFUHWHWLPH VWUXFWXUH WKDW LV WKH LQGHSHQGHQW WLPH YDULDEOH W UXQV WKURXJK WKH QDWXUDO QXPEHUV DQG VR RQ $GDOLQH LV VKRZQ LQ )LJXUH

PAGE 25

$OWKRXJK WKH LQSXW VLJQDOV [NWf PD\ RULJLQDWH IURP DQ\ VRXUFH YHU\ RIWHQ WKH LQSXW VLJQDOV DUH JHQHUDWHG IURP D WDSSHG GHOD\ OLQH DV VKRZQ LQ WKH ILJXUH )RU WKLV FDVH WKH DGDOLQH VWUXFWXUH LV VLPLODU WR D UHJXODU WUDQVYHUVDO ),5 ILOWHU ,Q WKH DGDSWLYH VLJQDO SURFHVVLQJ OLWHUDWXUH LW LV FRPPRQ WR GHILQH WKH IROORZLQJ YHFWRUV (R 7KXV ZH FDQ ZULWH WKH GHVFULELQJ HTXDWLRQ IRU DGDOLQH DV \Wf Z7[Wf (T /HW D GHVLUHG RXWSXW VLJQDO EH JLYHQ E\ GWf GWf LV DOVR UHIHUUHG WR DV WDUJHW VLJQDO RU WHDFKHU VLJQDO 7KH GLIIHUHQFH EHWZHHQ WKH GHVLUHG RXWSXW DQG DFWXDO RXWSXW LV GHILQHG DV WKH LQVWDQWDQHRXVf HUURU VLJQDO HWf GWf f§ \Wf 6XEVWLWXWLRQ RI (T DQG VTXDULQJ OHDGV WR WKH IROORZLQJ H[SUHVVLRQ IRU WKH LQVWDQWDQHRXV VTXDUHG HUURU VLJQDO HWf G^Wf Z7[Wf[7 WfZGWf[7 Wf Z (T $Q LPSRUWDQW DVVXPSWLRQ LQ WKH WKHRU\ RI DGDSWLYH VLJQDO SURFHVVLQJ LV WKDW WKH VLJQDOV HWf GWf DQG [Wf DUH VWDWLVWLFDOO\ VWDWLRQDU\ WKDW LV WKHLU VWDWLVWLFDO PRPHQWV DUH FRQVWDQW RYHU WLPH ,Q WKDW FDVH WDNLQJ WKH H[SHFWHG YDOXH RI (T \LHOGV (>HWf@ (>GWf@ Z75Z37Z (R 7 ZKHUH ZH GHILQHG WKH LQSXW FRUUHODWLRQ PDWUL[ 5 (>[Wf[ Wf @ DQG WKH FURVVn FRUUHODWLRQ YHFWRU 3 (>GWf[Wf@ 1RWH WKDW WKH H[SUHVVLRQ IRU WKH PHDQ VTXDUHG HUURU _ LV TXDGUDWLF LQ WKH SDUDPHWHUV Z 7KH PLQLPDO PHDQ VTXDUHG HUURU LV REWDLQHG JUDGLHQW

PAGE 26

A ,5ZO3 µZ (D-, 7KXV WKH RSWLPDO ZHLJKW YHFWRU Z4SWLV JLYHQ E\ ZRSW 5aOSfµ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n DQG 3 LV XVXDOO\ YHU\ H[SHQVLYH LQ SDUWLFXODU ZKHQ WKH QHWZRUN GLPHQVLRQ . LV ODUJH ,QVWHDG LW LV FRPPRQ WR DGDSW WKH ZHLJKWV RQ D VDPSOHE\VDPSOH EDVLV VR DV WR VHDUFK IRU WKH RSWLPDO YDOXHV $V LV DSSDUHQW IURP (T WKH PHDQVTXDUHHUURU LV TXDGUDWLF LQ WKH ZHLJKWV 7KXV WKH SHUIRUPDQFH VXUIDFH b LV D K\SHUfSDUDERORLG ZLWK D PLQLPXP DW ZRSW $ JUDGLHQW GHVFHQW SURFHGXUH VKRXOG WKHUHIRUH LQ WKHRU\ OHDG WR WKH RSWLPDO ZHLJKWV 7KH VWHHSHVW GHVFHQW XSGDWH DOJRULWKP DGDSWV WKH ZHLJKWV DV IROORZV GW ZW f YY f U_A (TMD 7KH VWHS VL]H SDUDPHWHU U_ FRQWUROV WKH UDWH RI DGDSWDWLRQ ,Q WKH QHXUDO QHW OLWHUDWXUH 7_ LV UHIHUUHG WR DV WKH OHDUQLQJ UDWH 1RWH WKDW DGDSWDWLRQ FRPHV QDWXUDOO\ WR D KDOW ZKHQ WKH ZHLJKWV DUH RSWLPDO VLQFH DW WKH PLQLPXP RI WKH SHUIRUPDQFH VXUIDFH ZH KDYH A 7KH FRPSXWDWLRQ RI WKH HUURU JUDGLHQWV A GHWHUPLQHV WKH FRPSOH[LW\ RI WKH OHDUQLQJ DOJRULWKP :LGHO\ XVHG DQG YHU\ HIILFLHQW LV WKH /HDVW 0HDQ 6TXDUH /06f DOJRULWKP :H ZLOO QRZ SURFHHG WR GHULYH WKH /06 DOJRULWKP IRU WKH DGDOLQH VWUXFWXUH

PAGE 27

DV LW LV WKH SUHFXUVRU IRU WKH ZLGHO\ XVHG EDFNSURSDJDWLRQ SURFHGXUH LQ QHXUDO QHW DGDSWDWLRQ $W D ODWHU VWDJH LQ WKLV WKHVLV WKH EDFNSURSDJDWLRQ SURFHGXUH LV GHULYHG DQG DSSOLHG WR VHYHUDO VLJQDO SURFHVVLQJ SUREOHPV 7KH FHQWUDO LGHD RI WKH /06 DOJRULWKP LV WR DSSUR[LPDWH WKH VWRFKDVWLF JUDGLHQW G(>HWf@ GHWf A E\ WKH LQVWDQWDQHRXV WLPHYDU\LQJf JUDGLHQW f§Af§ 1RWH WKDW WKH LQVWDQWDQHRXV HUURU JUDGLHQW LV DQ XQELDVHG HVWLPDWRU RI WKH VWRFKDVWLF JUDGLHQW WKDW LV ( UD GZ G(>HWf@ W A 6XEVWLWXWLQJ H Wf GWf [ WfZ OHDGV WR D GZ HWf GHWf HWf[Wf (D 7KXV WKH /06 XSGDWH HTXDWLRQ HYDOXDWHV WR ZOf Z Wf U?H Wf[Wf (T 1RWH KRZ VLPSOH WKH ILQDO HTXDWLRQ IRU WKH /06 DOJRULWKP LV 7KH VLJQDOV HWf DQG [Wf DUH UHDGLO\ DYDLODEOH 7KH FRPELQDWLRQ RI VLPSOLFLW\ DQG DFFXUDF\ KDYH PDGH WKH /06 DOJRULWKP WKH PRVW SRSXODU DOJRULWKP LQ DGDSWLYH VLJQDO SURFHVVLQJ :LGURZ GLVFXVVHV LQ KLV ERRN D QXPEHU RI VXFFHVVIXO SUDFWLFDO DSSOLFDWLRQV VXFK DV DGDSWLYH HTXDOL]DWLRQ V\VWHP LGHQWLILFDWLRQ DGDSWLYH FRQWURO LQIHUHQFH FDQFHOLQJ DQG DGDSWLYH EHDPIRUPLQJ :LGURZ DQG 6WHDUQV f ,Q WKH QH[W VHFWLRQ WKH FRUH DUFKLWHFWXUDO SDUDGLJPV IRU QHXUDO QHWV DUH LQWURGXFHG :KLOH DGDOLQH HQMR\V D ZLGH DSSOLFDWLRQ LQ QHXUDO QHWZRUN DUFKLWHFWXUHV WKH LQKHUHQW OLQHDULW\ OLPLWV LWV FRPSXWDWLRQDO SRZHU VXEVWDQWLDOO\ 1HXUDO QHWV LQ JHQHUDO DUH PRUH SRZHUIXO VLQFH WKH\ FDQ EH QRQOLQHDU UHFXUUHQW DQG PXOWLLQSXW PXOWLRXWSXW V\VWHPV

PAGE 28

1HXUDO 1HWZRUN 3DUDGLJPV 6WDWLF 0RGHOV 2YHU WKH \HDUV WZR GLIIHUHQW SDUDGLJPV KDYH HPHUJHG WKDW H[SORLW WKH G\QDPLFV RI V\VWHP (T WR VHUYH DV D QRQOLQHDU LQIRUPDWLRQ SURFHVVRU 1HDUO\ DOO WKHRU\ DQG SUDFWLFH GHDOV ZLWK SURFHVVLQJ RI VWDWLF SDWWHUQV , ZLOO VKRUWO\ LQWURGXFH WKH WZR FRQFHSWV IRU WKH VWDWLF FDVH VLQFH DQ XQGHUVWDQGLQJ RI WKLV LV HVVHQWLDO LQ RUGHU WR FRPSUHKHQG HIIRUWV RI H[WHQVLRQ WR SURFHVVLQJ RI VSDFHWLPH SDWWHUQV 7KH VHFWLRQ RQ VWDWLF QHWV LV IROORZHG E\ G\QDPLF QHWV 7KH &RQWLQXRXV 0DSSHU 7KH ILUVW SDUDGLJP RIIHUV WKH FRQWLQXRXV PDSSHU RU PDQYWRPDQY PDS 7KH FRPSXWDWLRQDO UHVXOW RI WKH FRQWLQXRXV PDSSHU LV D FRQWLQXRXV IXQFWLRQ IURP DQ GLPHQVLRQDO LQSXW VSDFH WR DQ 0GLPHQVLRQDO RXWSXW VSDFH 7KH VWDQGDUG PHWKRG WR LPSOHPHQW VXFK D PDS LV E\ ZD\ RI D PXOWLOD\HU IHHGIRUZDUG QHWZRUN $Q LPSRUWDQW KLVWRULFDO DQG FXUUHQWf H[DPSOH RI WKH FRQWLQXRXV PDSSHU LV WKH SHUFHSWURQ DUFKLWHFWXUH 5RVHQEODWW f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rfµ R ; ZLM[S fµ (DML ,W KDV EHHQ SURYHQ WKDW D WKUHHOD\HU QHWZRUN WZR KLGGHQ OD\HUVf LQ SULQFLSOH LV FDSDEOH WR FRPSXWH DQ DUELWUDU\ FRQWLQXRXV PDS IURP WKH LGLPHQVLRQDO LQSXW VSDFH WR WKH UHDO QXPEHUV +HFKW1LHOVHQ f $OWKRXJK WKLV PD\ EH LPSUHVVLYH WKH SUREOHP

PAGE 29

RI ILQGLQJ WKH FRUUHFW VHW RI ZHLJKWV PD\ EH YHU\ KDUG 7KH SUREOHP RI ILQGLQJ JRRG ZHLJKWV LV FDOOHG WKH ORDGLQJ SUREOHP 7KHRUHWLFDOO\ D OHDUQLQJ PHFKDQLVP VXFK DV VLPXODWHG DQQHDOLQJ FDQ EH XVHG WR REWDLQ WKH PDS WKDW PLQLPL]HV WKH HUURU EHWZHHQ WKH GHVLUHG PDS DQG WKH DFWXDO QHWZRUNf LPSOHPHQWDWLRQ +RZHYHU VLPXODWHG DQQHDOLQJ VWRFKDVWLF RSWLPL]DWLRQf LV YHU\ VORZ DQG LQ SUDFWLFH DOWKRXJK QRW SHUIHFW WKH EDFN SURSDJDWLRQ WUDLQLQJ SURFHGXUH KDV EHHQ TXLWH HIIHFWLYH IRU PDQ\ DSSOLFDWLRQV %DFN SURSDJDWLRQ LQYROYHV DGDSWDWLRQ RI WKH ZHLJKWV E\ JUDGLHQW GHVFHQW VR DV WR PLQLPL]H D SHUIRUPDQFH FULWHULRQ

PAGE 30

7KH $VVRFLDWLYH 0HPRU\ 7KH VHFRQG SURWRW\SH HQWDLOV WKH DVVRFLDWLYH PHPRU\ RU PDQYWRRQH PDS IRU ZKLFK WKH +RSILHOG QHW LV WKH SULPH LOOXVWUDWLRQ +RSILHOG DQG VHH )LJXUH f ,Q WHUPV RI WRSRORJ\ WKH +RSILHOG QHW LV D UHFXUUHQW QHW ZLWK V\PPHWULF ZHLJKWV Z ZML Zf± f ZKLFK HQDEOHV WKH DVVRFLDWLRQ RI D /\DSXQRY HQHUJ\f IXQFWLRQ ZLWK WKH V\VWHP G\QDPLFV 8VLQJ /\DSXQRYf¬V VWDELOLW\ WKHRU\ LW FDQ EH VKRZQ WKDW WKLV V\VWHP DOZD\V FRQYHUJHV WR D SRLQW DWWUDFWRU 7KHUH LV QR H[WHUQDO LQSXW WR WKH DVVRFLDWLYH PHPRU\ 7KH LQSXW LV WKH LQLWLDO VWDWH RI WKH QHXURQV [Wf 8VHG DV D SURFHVVLQJ GHYLFH LQIRUPDWLRQ LV VWRUHG E\ ORFDWLQJ SRLQW DWWUDFWRUV DW SRVLWLRQV LQ WKH VWDWH VSDFH WKDW FRUUHVSRQG WR PHPRULHV 5HFRJQLWLRQ WKHQ FRQVLVWV RI VHWWOLQJ LQWR WKH PLQLPXP FORVHVW WR WKH LQLWLDO VWDWH YHFWRU [Wf %RWK WKH FRQWLQXRXV PDSSHU DQG WKH DVVRFLDWLYH PHPRU\ ZRUN RQO\ LQ D VWDWLF SDWWHUQ HQYLURQPHQW $OWKRXJK WKH +RSILHOG QHW SURFHVVHV LQIRUPDWLRQ E\ D G\QDPLF UHOD[DWLRQ SURFHVV WKH LQSXW SDWWHUQ LQLWLDO VWDWH YHFWRUf LV DVVXPHG WR EH VWDWLF 1H[W WHPSRUDO H[WHQVLRQV RI ERWK WKH FRQWLQXRXV PDSSHU DQG WKH DVVRFLDWLYH PHPRU\ DUH GLVFXVVHG ,W ZLOO EHFRPH FOHDU WKDW WKH LGHDV IRU FRPSXWLQJ ZLWK WLPH LQ G\QDPLF

PAGE 31

QHXUDO QHWV FRUUHVSRQG VWURQJO\ WR OLQHDU VLJQDO SURFHVVLQJ WKHRU\ 1HXUDO 1HWZRUN 3DUDGLJPV '\QDPLF 1HWV 7KH EDVLF QHXUDO QHWZRUN PRGHO IRU SURFHVVLQJ RI VWDWLF SDWWHUQV LV WKH VWDWLF DGGLWLYH PRGHO 7KH DFWLYDWLRQ RI WKH XQLWV DUH FRPSXWHG E\ D ; ZLM[3 f¬ (R-O ML ZKHUH [W LV WKH DFWLYDWLRQ RI QHXURQ XQLW QRGHf 7KH ZHLJKW IDFWRU ZWM FRQQHFWV QRGH M WR QRGH L Ff LV D QRQfOLQHDU VTXDVKLQJ IXQFWLRQ DQG UHSUHVHQWV WKH H[WHUQDO LQSXW :H DVVXPH D V\VWHP GLPHQVLRQDOLW\ RI 1 6RPHWLPHV WKH VKRUWKDQG QRWDWLRQ QHW 9 :; ZLOO EH XVHG -L 6WDWLF QHXUDO QHWV KDYH QR PHPRU\ $V D UHVXOW WHPSRUDO UHODWLRQV FDQ QRW EH VWRUHG RU FRPSXWHG RQ E\ VWDWLF QHXUDO QHWV ,Q RUGHU WR SURFHVV D WHPSRUDO IORZ RI LQIRUPDWLRQ D QHXUDO QHW QHHGV D VKRUW WHUP PHPRU\ PHFKDQLVP 1HXUDO QHWZRUN PRGHOV ZLWK VKRUW WHUP PHPRU\ DUH FDOOHG G\QDPLF QHXUDO QHWV 7KH VLPSOHVW ZD\ WR G;M DGG G\QDPLFV PHPRU\f WR WKH VWDWLF PRGHO LV WR DGG D FDSDFLWLYH WHUP [f§ WR WKH OHIW KDQG VLGH RI (T $IWHU UHDUUDQJHPHQW RI WHUPV WKH VRFDOOHG G\QDPLF DGGLWLYH PRGHO LV REWDLQHG G[ AMWf« [L D&AZLM[Mf,L (T-O M 7KLV PRGHO LV PDWKHPDWLFDOO\ HTXLYDOHQW WR WKH V\VWHP GHVFULEHG E\ (T ZKHUH WKH WLPH FRQVWDQW LV H[SUHVVHG E\ WKH GHFD\ SDUDPHWHU D /HW XV ORRN DW WKH ELRORJLFDO ; SLFWXUH RI QHXUDO QHWV ,Q QDWXUH WKH QHXUDO WLPH FRQVWDQWV [ DUH IL[HG DQG HTXDO DSSUR[LPDWHO\ PVHFV 6KHQ f 7KLV QXPEHU LV HVWLPDWHG E\ DVVXPLQJ DQ DYHUDJH

PAGE 32

DFWLRQ SRWHQWLDO UDWH RI SHU VHFRQG +LJKHU UDWHV DUH TXLWH UDUH GXH WR WKH UHIUDFWRU\ SHULRG RI WKH QHXURQV +RZHYHU UHFRJQLWLRQ RI D VSRNHQ ZRUG UHTXLUHV WKH DELOLW\ WR UHPHPEHU WKH FRQWHQWV RI D SDVVDJH IRU DSSUR[LPDWHO\ VHFRQG 7R DFFRPSOLVK WKLV QHXUDO WHPSRUDO UHVROXWLRQ GHFUHDVHV ZKLOH WKH f¯WHPSRUDO ZLQGRZ RI VXVFHSWLELOLW\f°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f§ [ * QHWf fµ N[c (T ZKHUH N LV D SRVLWLYH FRQVWDQW ,Q WKH ELRORJLFDO OLWHUDWXUH VXFK ORFDO SRVLWLYH IHHGEDFN LV RIWHQ QDPHG UHYHUEHUDWLRQ ZKLOH QHXUDO QHW UHVHDUFKHUV VSHDN RI VHOIH[FLWDWLRQ (T FDQ EH UHZULWWHQ DV [ G[ Mf§ [ADLQHWLf (T4 DQHWcf ,c ZKHUH FU QHWcf f§\f§S DQG \f§\ )RU [ PVHF DQG N ZH JHW WKH [ QHZ WLPH FRQVWDQW [ f§ VHF 8QLWV WKDW VHOIH[FLWH RYHU D WLPH VSDQ WKDW LV f¯ .!

PAGE 33

UHOHYDQW ZLWK UHVSHFW WR WKH SURFHVVLQJ SUREOHP DUH UHIHUUHG WR LQ WKH QHXUDO QHW OLWHUDWXUH DV FRQWH[W XQLWV 6HYHUDO LQYHVWLJDWRUV KDYH H[SORUHG WKH WHPSRUDO FRPSXWDWLRQDO SURSHUWLHV RI DGGLWLYH IHHGIRUZDUG QHWV H[WHQGHG E\ FRQWH[W XQLWV -RUGDQ (OPDQ 0R]HU 6WRUQHWWDHW DO f ,Q +HUW] HW DO f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f ?L[Wf[RXWWf (T ZKHUH [RXWf LV WKH VWDWH RI DQ RXWSXW XQLW 1RWH WKDW -RUGDQf¬V DUFKLWHFWXUH PDNHV XVH RI JOREDO UHFXUUHQW ORRSV FRQWH[W WR KLGGHQ WR RXWSXW WR FRQWH[W XQLWVf $V D UHVXOW FDUH PXVW EH WDNHQ WR NHHS WKH WRWDO V\VWHP VWDEOH ,Q -RUGDQ f KH VKRZV WKDW WKLV QHWZRUN FDQ VXFFHVVIXOO\ PLPLF FRDUWLFXODWLRQ GDWD $QGHUVRQ HW DO f KDYH XVHG WKLV DUFKLWHFWXUH WR FDWHJRUL]LQJ D FODVV RI (QJOLVK V\OODEOHV (OPDQ f XWLOL]HV QRQOLQHDU VHOIUHFXUUHQW KLGGHQ XQLWV RI WKH W\SH [ W f SD r f WR VWRUH WKH SDVW )LJXUH Ef 7KLV QHWZRUN ZDV DEOH UHFRJQL]H VHTXHQFHV DQG HYHQ WR SURGXFH FRQWLQXDWLRQV RI VHTXHQFHV &OHHUHPDQV HW DO f VKRZHG WKDW WKLV DUFKLWHFWXUH LV DEOH WR OHDUQ DQG PLPLF D ILQLWH VWDWH PDFKLQH ZKHUH WKH KLGGHQ XQLWV UHSUHVHQW WKH LQWHUQDO VWDWHV RI WKH DXWRPDWRQ 6WRUQHWWD HW DO f KDYH XVHG UHFXUUHQW XQLWV DW WKH LQSXW OD\HU RQO\ WR UHSUHVHQW D WHPSRUDOO\ ZHLJKWHG WUDFH RI WKH LQSXW VLJQDO )LJXUH Ff 7KHUH DUH QR ZHLJKWHG FRQQHFWLRQV IURP WKH KLGGHQ RU RXWSXW XQLWV WRZDUG WKH FRQWH[W XQLWV 7KLV UHVWULFWLRQ UHVXOWV LQ VHYHUDO DGYDQWDJHV ZKHQ WKH QHWZRUN LV WUDLQHG E\ D JUDGLHQW

PAGE 34

RXWSXWV KLGGHQ QRGHV Df -RUGDQf¬V QHWZRUN Ef (OPDQ RXWSXWV Ff 6WRUQHWWD HW DO )LJXUH 9DULRXV VHTXHQWLDO QHWZRUN DUFKLWHFWXUHV Df 0LFKDHO -RUGDQf¬V DUFKLWHFWXUH IHHGV WKH RXWSXW EDFN WR DQ DGGLWLRQDO VHW RI UHFXUUHQW LQSXW XQLWV Ef (OPDQf¬V VWUXFWXUH XVHV UHFXUUHQW QRQOLQHDU KLGGHQ XQLWV Ff 6WRUQHWWD HW DO NHHS D KLVWRU\ WUDFH DW WKH LQSXW XQLWV 7KLV VWUXFWXUH RIIHUV SDUWLFXODU DGYDQWDJHV ZKHQ EDFNSURSDJDWLRQ OHDUQLQJ LV XVHG GHVFHQW WHFKQLTXH :H ZLOO GLVFXVV WKLV LVVXH LQ PRUH GHWDLO LQ FKDSWHU RQ WUDLQLQJ D QHXUDO QHW 7KH DXWKRU SHUIRUPHG VXFFHVVIXO H[SHULPHQWV LQ UHFRJQLWLRQ RI VKRUW VHTXHQFHV 0R]HU f DQG *RUL HW DO f KDYH DOVR PDGH XVH RI VLPLODU

PAGE 35

DUFKLWHFWXUDO UHVWULFWLRQV :KLOH WKH SRVLWLYH IHHGEDFN PHFKDQLVP LV VLPSOH DQG XVHG LQ ELRORJLFDO LQIRUPDWLRQ SURFHVVLQJ WKHUH DUH WZR FRPSXWDWLRQDO SUREOHPV DVVRFLDWHG ZLWK WKLV PHWKRG )LUVW WKH QHZ WLPH FRQVWDQW LV YHU\ VHQVLWLYH WR N )RU RXU H[DPSOH DQ LQFUHDVH RI b LQ N IURP N WR N PDNHV WKH PRGHO XQVWDEOH 7KH WLPHYDU\LQJ QDWXUH RI ELRORJLFDO SDUDPHWHUV PDNHV LW WKHUHIRUH XQOLNHO\ WKDW UHYHUEHUDWLRQ LV WKH SUHGRPLQDQW PHFKDQLVP IRU VKRUW WHUP PHPRU\ RYHU ORQJ SHULRGV 7KH VHFRQG KDQGLFDS RI (T LV WKDW WKH QHZ PRGHO LV VWLOO JRYHUQHG E\ ILUVWRUGHU G\QDPLFV $V D UHVXOW ZHLJKWLQJ LQ WKH WHPSRUDO GRPDLQ LV OLPLWHG WR D UHFHQF\ JUDGLHQW H[SRQHQWLDO IRU OLQHDU IHHGEDFNf WKDW LV WKH PRVW UHFHQW LWHPV FDUU\ D ODUJHU ZHLJKW WKDQ SUHYLRXV LQSXWV 1RWH WKDW WKH DQDO\WLFDO VROXWLRQ WR (T FDQ EH ZULWWHQ DV W &Vf [Wf ?H r >EQHWVff,Vf@GV (T W 7KXV WKH SDVW LQSXW LV ZHLJKWHG E\ D IDFWRU H ZKLFK H[SRQHQWLDOO\ GHFD\V RYHU WLPH )RU D QHXUDO QHW FRPSRVHG RI 1 QHXURQV WKH QXPEHU RI ZHLJKWV LQ WKH VSDWLDO GRPDLQ LV 2A1f ZKLOH WKH WHPSRUDO GRPDLQ LV JRYHUQHG RQO\ E\ [ 7KH XVH RI D IL[HG SDVVLYH PHPRU\ IXQFWLRQ WKHQ LPSOLHV D OLPLW WR KRZ VWUXFWXUHG WKH UHSUHVHQWDWLRQ RI WKH SDVW LQ WKH QHW FDQ EH $V DQ H[DPSOH RSWLPDO WHPSRUDO ZHLJKWLQJ IRU WKH GLVFULPLQDWLRQ RI WKH ZRUGV f¯FDWf° DQG f¯PDWf°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

PAGE 36

RI VKRUWWHUP PRGXODWLRQ RI [ RQH PD\ WKLQN RI DGDSWDWLRQ GHFUHDVHG VHQVLWLYLW\ RI UHFHSWRU QHXURQ WR D PDLQWDLQHG VWLPXOXVf RU KHWHURV\QDSWLF IDFLOLWDWLRQ WKH DELOLW\ RI RQH V\QDSVH RI D FHOO WR WHPSRUDULO\ LQFUHDVH WKH HIILFDF\ RI DQRWKHU V\QDSVH VHH :RQJ DQG &KXQ f IRU DQ DSSOLFDWLRQ WR QHXUDO QHWVf ,Q FRQFOXVLRQ VKRUW WHUP PHPRU\ E\ ORFDO SRVLWLYH IHHGEDFN LV VLPSOH DQG KDV EHHQ DSSOLHG VXFFHVVIXOO\ LQ DUWLILFLDO QHXUDO QHWV +RZHYHU UHYHUEHUDWLRQ PD\ OHDG WR LQVWDELOLW\ 6HFRQGO\ WKLV PHFKDQLVP UHVWULFWV FRPSXWDWLRQDO IOH[LELOLW\ LQ WKH WHPSRUDO GRPDLQ ,Q WKH QH[W VHFWLRQ VKRUW WHUP PHPRU\ E\ GHOD\V LV UHYLHZHG 6KRUW 7HUP 0HPRU\ EY 'HOD\V $ JHQHUDO GHOD\ PHFKDQLVP FDQ EH UHSUHVHQWHG E\ WHPSRUDO FRQYROXWLRQV LQVWHDG RI LQVWDQWDQHRXVf PXOWLSOLFDWLYH LQWHUDFWLRQV &RQVLGHU WKH IROORZLQJ H[WHQVLRQ RI WKH VWDWLF DGGLWLYH PRGHO (T :H ZLOO FDOO WKLV PRGHO WKH DGGLWLYHf FRQYROXWLRQ PRGHO ,Q WKH FRQYROXWLRQ PRGHO WKH QHW LQSXW LV JLYHQ E\ QHWMWf
PAGE 37

WHPSRUDO FRQYHUJHQFH DW WKH VLQJOH QHXURQ OHYHO EHWZHHQ VLJQDOV RULJLQDWLQJ XS WR PVHF DSDUWf° 0LOOHU f 6HYHUDO DUWLILFLDO QHXUDO QHW UHVHDUFKHUV KDYH DOVR H[SHULPHQWHG ZLWK DGGLWLYH GHOD\ PRGHOV RI W\SH (T +RZHYHU GXH WR WKH FRPSOH[LW\ RI JHQHUDO FRQYROXWLRQ PRGHOV RQO\ VWURQJ VLPSOLILFDWLRQV RI WKH ZHLJKW NHUQHOV KDYH EHHQ SURSRVHG /DQJ HW DO f XVHG WKH GLVFUHWH GHOD\ NHUQHOV Z "f AZ!IF W f§ WNf LQ WKH N WLPH GHOD\ QHXUDO QHWZRUN 7'11f7KH 7'11 DUFKLWHFWXUH LV VKRZQ LQ )LJXUH 7KH 7'11 FRQVLGHUHG WKH VWDWHRIWKHDUW LV D PXOWLOD\HU IHHGIRUZDUG QHW WKDW LV WUDLQHG E\ HUURU EDFNSURSDJDWLRQ 7KH SDVW LV UHSUHVHQWHG E\ WDSSHG GHOD\ OLQHV DV LQ ),5 ILOWHUV 7KH DXWKRUV UHSRUWHG H[FHOOHQW UHVXOWV RQ D SKRQHPH UHFRJQLWLRQ WDVN $ UHFRJQLWLRQ UDWH RI b DW D SKRQHPH UHFRJQLWLRQ WDVN f¯%f° f¯'f° DQG f¯*f°f FRPSDUHG WR b IRU D KLGGHQ 0DUNRY PRGHO ZDV DFKLHYHG 5HFHQWO\ WKH &08JURXS LQWURGXFHG WKH 7(032 PRGHO ZKHUH DGDSWLYH JDXVVLDQ GLVWULEXWHG GHOD\ NHUQHOV VWRUH WKH SDVW %RGHQKDXVHQ DQG :DLEHO f 'LVWULEXWHG GHOD\ NHUQHOV VXFK DV XVHG LQ WKH 7(032 PRGHO LPSURYH RQ WKH 7'11 ZLWK UHVSHFW WR WKH FDSWXUH RI WHPSRUDO FRQWH[W 7DQN DQG +RSILHOG f DOVR SUHZLUHG ZWf DV D OLQHDU FRPELQDWLRQ RI GLVSHUVLYH NHUQHOV LQ SDUWLFXODU Z Wf Wf MYYf H . fµ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

PAGE 38

1HXUDO QHWV RI WKLV W\SH ZKHUH LQIRUPDWLRQ RI VHYHUDO QHXURQV DW GLIIHUHQW WLPHV LQWHJUDWHV DW RQH QHXURQ ZHUH FDOOHG &RQFHQWUDWLRQRI,QIRUPDWLRQLQ7LPH &,7f QHXUDO QHWV 7KH ZHLJKW IDFWRUV ZN ZHUH QRQDGDSWLYH DQG GHWHUPLQHG D SULRUL 7KH\ VXFFHVVIXOO\ EXLOW VXFK D V\VWHP LQ KDUGZDUH IRU DQ LVRODWHG ZRUG UHFRJQLWLRQ WDVN ,Q SDUWLFXODU WKH UREXVWQHVV DJDLQVW WLPH ZDUSHG LQSXW VLJQDOV VKRXOG EH PHQWLRQHG ,Q D ODWHU SXEOLFDWLRQ VXFFHVVIXO H[SHULPHQWV ZHUH UHSRUWHG ZLWK DGDSWLYH JDXVVLDQ GLVWULEXWHG GHOD\ NHUQHOV 8QQLNULVKQDQ HW DO f :KHQ FRPSDUHG WR WKH ILUVWRUGHU FRQWH[WXQLW QHWZRUNV WKH FRQYROXWLRQ PRGHO LQ LWV JHQHUDO IRUPXODWLRQ LV PRUH IOH[LEOH LQ WKH WHPSRUDO GRPDLQ VLQFH WKH ZHLJKWLQJ RI WKH SDVW LV QRW UHVWULFWHG WR D UHFHQF\ JUDGLHQW +RZHYHU D KLJK SULFH KDV WR EH SDLG IRU WKH LQFUHDVHG IOH[LELOLW\ , LGHQWLI\ WKUHH FRPSOLFDWLRQV IRU WKH FRQYROXWLRQ PRGHO ZKHQ FRPSDUHG WR WKH DGGLWLYH PRGHO $QDO\VLV 7KH FRQYROXWLRQ PRGHO LV GHVFULEHG E\ D VHW RI IXQFWLRQDO GLIIHUHQWLDO HTXDWLRQV )'(f LQVWHDG RI RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV 2'(f IRU WKH

PAGE 39

DGGLWLYH PRGHO 6XFK HTXDWLRQV DUH LQ JHQHUDO KDUGHU WR DQDO\]H D KDQGLFDS ZKHQ ZH QHHG WR FKHFN RU GHVLJQ IRUf FHUWDLQ V\VWHP FKDUDFWHULVWLFV VXFK DV VWDELOLW\ DQG FRQYHUJHQFH 1XPHULFDO 6LPXODWLRQ )RU DQ $GLPHQVLRQDO FRQYROXWLRQ PRGHO WKH UHTXLUHG QXPEHU RI RSHUDWLRQV WR FRPSXWH WKH QH[W VWDWH IRU WKH )'( VHW VFDOHV ZLWK ^17f ZKHUH 7 LV WKH QXPEHU RI WLPH VWHSV QHFHVVDU\ WR HYDOXDWH WKH FRQYROXWLRQ LQWHJUDO G[ XVLQJ (XOHU PHWKRG [W Kf [Wf KAf $Q AGLPHQVLRQDO DGGLWLYH PRGHO VFDOHV E\ 2L1f /HDUQLQJ 7KH ZHLJKWV LQ WKH FRQYROXWLRQ PRGHO DUH WKH WLPHYDU\LQJ SDUDPHWHUV ZWf 7KXV WKH GLPHQVLRQDOLW\ RI WKH IUHH SDUDPHWHUV JURZV OLQHDUO\ ZLWK WLPH )RU D ORQJ WHPSRUDO VHJPHQW WKH ODUJH ZHLJKW YHFWRU GLPHQVLRQDOLW\ LPSDLUV WKH DELOLW\ WR WUDLQ WKH QHWZRUN 7KH WZR PRGHOV IRU LQFRUSRUDWLQJ VKRUW WHUP PHPRU\ LQ QHXUDO QHWZRUNV SRVLWLYH IHHGEDFN DQG GHOD\V KDYH OHG WR D QXPEHU RI DUFKLWHFWXUHV WKDW HVVHQWLDOO\ JHQHUDOL]H WKH FRQWLQXRXV PDSSHU WR WKH VSDFHWLPH GRPDLQ ,Q WKH GLVFXVVLRQ RQ VWDWLF

PAGE 40

QHXUDO QHWV ZH LQWURGXFHG WKH DVVRFLDWLYH PHPRU\ PRGHO ,V WKHUH DOVR D WHPSRUDO H[WHQVLRQ IRU WKLV PRGHO ,QGHHG LQ SDUWLFXODU LQ WKH SK\VLFV FRPPXQLW\ VHYHUDO UHVHDUFKHUV KDYH H[SHULPHQWHG ZLWK WHPSRUDO DVVRFLDWLYH PHPRULHV 7KH SULQFLSDO LGHDV RI WKH VHTXHQWLDO DVVRFLDWLYH PHPRU\ DUH QRZ VKRUWO\ UHYLHZHG 7KH 6HTXHQWLDO $VVRFLDWLYH 0HPRU\ 7KH VHTXHQWLDO DVVRFLDWLYH PHPRU\ LV D UHFXUUHQWf G\QDPLFDO V\VWHP WKDW VWRUHV PHPRULHV LQ DWWUDFWRUV VLQNVf RI ]HURWK RUGHU SRLQW DWWUDFWRUVf .OHLQIHOG 6RPSROLQVN\ DQG .DQWHU f 3K\VLFLVWV KDYH H[SORUHG VHYHUDO ZD\V WR LJQLWH DWWUDFWRU WUDQVLWLRQV XQGHU LQIOXHQFH RI DQ H[WHUQDO VWLPXOXV 7KH PRVW ZLGHO\ XVHG PHWKRG FRQVLVWV RI IRUFLQJ D FRPELQDWLRQ RI WKH H[WHUQDO VLJQDO DQG WKH GHOD\HG QHWZRUN VWDWH XSRQ WKH QHW 7KH GHOD\HG VWDWH ZDQWV WR NHHS WKH QHW LQ LWV FXUUHQW VWDWH ZKLOH WKH H[WHUQDO LQSXW WULHV WR DOWHU WKH VWDWH $V D UHVXOW WKH QHW VWDWH LGHDOO\ KRSV IURP RQH VWDEOH DWWUDFWRU WR DQRWKHU ,Q D SDWWHUQ UHFRJQLWLRQ HQYLURQPHQW WKH VHTXHQFH RI YLVLWHG VWDWHV LGHQWLILHV WKH H[WHUQDO LQSXW 6HTXHQWLDO DVVRFLDWLYH PHPRULHV DUH WKHRUHWLFDOO\ YHU\ LQWHUHVWLQJ DQG PRUHRYHU SURYLGH D QHXUDO H[SODQDWLRQ RI FDWHJRULFDO SHUFHSWLRQ GXH WR WKH FRUUHFWLYH SURSHUWLHV RI WKH EDVLQV RI DWWUDFWLRQ +RZHYHU LW KDV EHHQ QRWLFHG WKDW WKHVH QHWV PD\ QRW EH YHU\ VHOHFWLYH SDWWHUQ UHFRJQL]HUV LQ RWKHU ZRUGV QHDUO\ HYHU\ LQSXW LI KLJK HQRXJKf ZLOO LQGXFH WUDQVLWLRQV $PLW f 6HFRQGO\ WKH PHPRU\ FDSDFLWLHV RI VXFK QHWV DUH YHU\ OLPLWHG ,W LV WKH QDWXUH RI SRLQW DWWUDFWRUV WKDW IDOOLQJ LQWR D EDVLQ RI RQH LQGXFHV WKH IRUJHWWLQJ RI SUHYLRXV VWDWHV &RQVHTXHQWO\ WKH f«GHHSQHVVf¬ RI PHPRU\ LV IL[HG VKRUW DQG FDQQRW EH PRGXODWHG ,Q P\ RSLQLRQ WKH VHTXHQWLDO DVVRFLDWLYH PHPRU\ PD\ EH D XVHIXO PRGXOH IRU WDVNV OLNH FHQWUDO SDWWHUQ JHQHUDWLRQ .OHLQIHOG f EXW DUH QRW \HWf IOH[LEOH HQRXJK WR HQFRGH WKH YDU\LQJ WHPSRUDO UHODWLRQV RI FRPSOH[ VLJQDOV VXFK DV VSHHFK

PAGE 41

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f 7KH LPSRUWDQW LVVXHV WKDW FRQILQH DSSOLFDWLRQV RI IXOO\ UHFXUUHQW QHWV DUH FRQWURO RI VWDELOLW\ DQG DGDSWDWLRQ SUREOHPV ,Q D IXOO\ UHFXUUHQW QHW WKH SHUIRUPDQFH VXUIDFH LV QRW QHFHVVDULO\ FRQYH[ ZKLFK LV D VHYHUH KDQGLFDS IRU JUDGLHQWEDVHG DGDSWDWLRQ PHWKRGV 6HFRQGO\ OHDUQLQJ LQ UHFXUUHQW QHWZRUNV KDV EHHQ IRXQG WR SURJUHVV PXFK VORZHU WKDQ LQ IHHGIRUZDUG QHWV

PAGE 42

'LVFXVVLRQ ,Q WKLV FKDSWHU YDULRXV QHXUDO DUFKLWHFWXUHV IRU WHPSRUDO SURFHVVLQJ KDYH EHHQ GLVFXVVHG 7KH PDLQ SULQFLSOHV IRU VWRUDJH RI DQG FRPSXWLQJ ZLWK D WHPSRUDO GDWD IORZ GHOD\V DQG IHHGEDFN ZHUH DQDO\]HG LQ VRPH GHWDLO 7KH IHHGIRUZDUG WDSSHG GHOD\ OLQH KDV EHHQ XVHG YHU\ VXFFHVVIXOO\ LQ WKH WLPH GHOD\ QHXUDO QHW
PAGE 43

&+$37(5 7+( *$00$ 1(85$/ 02'(/ ,QWURGXFWLRQ &RQYROXWLRQ 0HPRU\ YHUVXV $50$ 0RGHO ,W ZDV GLVFXVVHG LQ FKDSWHU WKDW D JHQHUDO GHOD\ PHFKDQLVP FDQ EH ZULWWHQ DV (R IRU WKH FRQWLQXRXV WLPH GRPDLQ DQG QHWWf A Z W Qf [ Qf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f FDQ EH DGHTXDWHO\ DSSUR[LPDWHG E\ D IL[HG GLPHQVLRQDO VHW RI FRQVWDQW ZHLJKWV 7KLV SUREOHP ZDV VWXGLHG E\ )DUJXH f DQG WKH DQVZHU LV SURYLGHG E\ WKH IROORZLQJ WKHRUHP 7KHRUHP 7KH VFDODUf LQWHJUDO HTXDWLRQ

PAGE 44

QHWWf -Z r f§ f‘r 2 (T FDQ EH UHGXFHG WR D . GLPHQVLRQDO V\VWHP RI RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV ZLWK FRQVWDQW FRHIILFLHQWV LI DQG RQO\ LIf ZWf LV D VROXWLRQ RI G.Z . N GZ GU . m! N N N GW (R ZKHUH D DcD.c DUH FRQVWDQWV 3URRI $ FRQVWUXFWLYH SURRI IRU VXIILFLHQF\ LV SURYLGHG 7KH LQLWLDO FRQGLWLRQV GNZ IRU (T DUH UHZULWWHQ DV ZN f§M f ZKHUH N DQG ZH GHILQH WKH GWN GN?Y YDULDEOHV ZIf f§U Wf N ZKLFK DOORZV WR UHZULWH (T DV WKH GWN IROORZLQJ VHW RI . ILUVWRUGHU GLIIHUHQWLDO HTXDWLRQV GA GW Wf ZN OWf N GZ.? GW . ; DNZN: N 1H[W ZH LQWURGXFH WKH VWDWH YDULDEOHV W [NWf _ZAWf[f N 1RWH WKDW WKH V\VWHP RXWSXW LV JLYHQ E\ W QHWWf MZ4AWVf[VfGV ;T¯f (T4 (T (T

PAGE 45

7KH VWDWH YDULDEOHV [NWf FDQ EH UHFXUVLYHO\ FRPSXWHG 'LIIHUHQWLDWLQJ (T ZLWK UHVSHFW WR W XVLQJ /HLEQL]f¬ UXOH JLYHV ILWf MM:NWVf[VfGV ZNf[Wf (IOL R ZKLFK XVLQJ WKH UHFXUUHQFH UHODWLRQV IURP (T HYDOXDWHV WR G[ A [NL ZN[WfIRUN .DQG G[. GW Wf . ; mrrrrf r1.O[: N (T 7KXV LI WKH ZHLJKW NHUQHO ZWf LV D VROXWLRQ RI WKH UHFXUUHQFH UHODWLRQ (T WKHQ WKH LQWHJUDO HTXDWLRQ (T FDQ EH UHGXFHG WR D V\VWHP RI GLIIHUHQWLDO HTXDWLRQV ZLWK FRQVWDQW FRHIILFLHQWV (Tf ’ HQG SURRIf 7KH IROORZLQJ WKHRUHP UHYHDOV ZKDW LV PHDQW E\ LPSRVLQJ WKH FRQGLWLRQ (T RQ ZWf 7KHRUHP 6ROXWLRQV RI WKH V\VWHP G.Z . GW : ? < . f f¯ GNZ r NaG" (J N SL FDQ EH ZULWWHQ DV D OLQHDU FRPELQDWLRQ RI WKH IXQFWLRQV W f«H ZKHUH P L P NL . DQG A .c . HQG WKHRUHPf N 7KH SURRI RI 7KHRUHP LV SURYLGHG LQ PRVW WH[WERRNV RQ RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV HJ %UDXQ SDJH f 7KH SnV DUH WKH HLJHQYDOXHV RI WKH V\VWHP ,Q SDUWLFXODU WKH SV DUH WKH VROXWLRQV RI WKH FKDUDFWHULVWLF HTXDWLRQ RI (T

PAGE 46

.? ?L. ; DN9N (F8 N P LV WKH QXPEHU RI GLIIHUHQW HLJHQYDOXHV DQG .c WKH PXOWLSOLFLW\ RI HLJHQYDOXH _LI 7KH N SU IXQFWLRQV W OH n DUH WKH HLJHQIXQFWLRQV RI V\VWHP (T ZKHUH L HQXPHUDWHV WKH YDULRXV HLJHQPRGHV RI WKH V\VWHP ,Q WKH VLJQDO SURFHVVLQJ DQG FRQWURO FRPPXQLW\ WKH V\VWHP GHVFULEHG E\ (T DQG (T LV FDOOHG DQ DXWRUHJUHVVLYH PRYLQJ DYHUDJH $50$f PRGHO VHH )LJXUH f ,W ZDV GLVFXVVHG LQ VHFWLRQ WKDW WKH PHPRU\ RI DQ $50$ V\VWHP LV UHSUHVHQWHG LQ WKH VWDWH YDULDEOHV [NWf ,W LV LQWHUHVWLQJ WR REVHUYH WKH UHODWLRQ EHWZHHQ WKH $50$ PRGHO SDUDPHWHUV DQG WKH FRQYROXWLRQ PRGHO 7KH DXWRUHJUHVVLYH SDUDPHWHUV DN DUH WKH FRHIILFLHQWV RI WKH UHFXUUHQFH UHODWLRQ (T IRU ZWf 7KH PRYLQJ DYHUDJH SDUDPHWHUV ZN HTXDO WKH LQLWLDO FRQGLWLRQV RI (T ,Q WKH FRQWH[W RI WKLV H[SRVLWLRQ , OLNH WR WKLQN RI DQ $50$ PRGHO DV D G\QDPLF PRGHO IRU D PHPRU\ V\VWHP ,W ZDV MXVW SURYHG WKDW WKLV FRQILJXUDWLRQ LV

PAGE 47

HTXLYDOHQW WR D FRQYROXWLRQ PHPRU\ PRGHO LI WKH FRQGLWLRQ GHVFULEHG E\(T LV REH\HG
PAGE 48

(T 6LQFH WKH IXQFWLRQV JN^Wf DUH WKH LQWHJUDQGV RI WKH QRUPDOL]HGf 7IXQFWLRQ RR U[f MW[aOHaWGWf WKH\ ZLOO EH UHIHUUHG WR DV JDPPD NHUQHOV ,Q YLHZ RI WKH VROXWLRQV RI (T WKH JDPPD NHUQHOV DUH FKDUDFWHUL]HG E\ WKH IDFW WKDW DOO HLJHQYDOXHV DUH WKH VDPH WKDW LV S S 7KXV WKH JDPPD NHUQHOV DUH WKH HLJHQIXQFWLRQV RI WKH IROORZLQJ UHFXUUHQFH UHODWLRQ MW ?Lf.JWf %D0 X$ 7KH IDFWRU f§f§f§ QRUPDOL]HV WKH DUHD WKDW LV r f W  (J 7KH VKDSH RI WKH JDPPD NHUQHOV JNWf LV SLFWXUHG LQ )LJXUH IRU S ,W LV VWUDLJKWIRUZDUG WR GHULYH DQ HTXLYDOHQW $50$ PRGHO IRU QHWWf ZKHQ ZUf LV FRQVWUDLQHG E\ (T 7KH SURFHGXUH LV VLPLODU WR WKH SURRI RI 7KHRUHP )LUVW WKH NHUQHOV JNWf DUH ZULWWHQ DV WKH IROORZLQJ VHW RI ILUVWRUGHU GLIIHUHQWLDO HTXDWLRQV GJL GJ ?LJN ?LJNBYN . (]$O 6XEVWLWXWLRQ RI (T LQWR (T \LHOGV . QHWWf e ZN[N (T N ZKHUH WKH JDPPD VWDWH YDULDEOHV DUH GHILQHG DV W [NWf ?JN^WVf[VfGVN (T

PAGE 49

7KH JDPPD VWDWH YDULDEOHV KROG PHPRU\ WUDFHV RI WKH QHXUDO VWDWHV [Wf +RZ DUH WKH YDULDEOHV \ If FRPSXWHG" 'LIIHUHQWLDWLQJ (T OHDGV WR Wf cAWNWVf[VfGV JNf[Lf (J ZKLFK VLQFH JN f IRU N ! DQG JM f S HYDOXDWHV WR GKF f§ Wf ?L[NWf 9;NB M N . (J ZKHUH ZH GHILQHG MF Wf [ Wf 7KH LQLWLDO FRQGLWLRQV IRU (T FDQ EH REWDLQHG IURP HYDOXDWLQJ [N f JN f [ f ZKLFK UHGXFHV WR rTf [f [A f S[f Af N . (D

PAGE 50

7KXV ZKHQ ZWf DGPLWV (T QHWWf FDQ EH FRPSXWHG E\ D LIGLPHQVLRQDO V\VWHP RI RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV(T 7KH IROORZLQJ WKHRUHP VWDWHV WKDW WKH DSSUR[LPDWLRQ RI DUELWUDU\ ZWf E\ D OLQHDU FRPELQDWLRQ RI JDPPD NHUQHOV FDQ EH PDGH DV FORVH DV GHVLUHG 7KHRUHP 7KH V\VWHP JNWf N O. LV FORVHG LQ / > rr@ 7KHRUHP LV HTXLYDOHQW WR WKH VWDWHPHQW WKDW IRU DOO ZWf LQ / > rr@ WKDW LV DQ\ ZWf IRU ZKLFK _Z Wf?GW H[LVWVf IRU HYHU\ H ! WKHUH H[LVWV D VHW RI SDUDPHWHUV ZN N VXFK WKDW RR . :Wf e ZNJNWf N O GWe (R 7KH SURRI IRU WKLV WKHRUHP LV EDVHG XSRQ WKH FRPSOHWHQHVV RI WKH /DJXHUUH SRO\QRPLDOV DQG FDQ EH IRXQG LQ 6]HJR SDJH WKHRUHP f 7KH IRUHJRLQJ GLVFXVVLRQ FDQ EH VXPPDUL]HG E\ WKH IROORZLQJ LPSRUWDQW UHVXOW 7KHRUHP 7KH FRQYROXWLRQ PRGHO GHVFULEHG E\ W QHWWf MZWVf[VfGV (T LV HTXLYDOHQW WR WKH IROORZLQJ V\VWHP . QHW Wf e!bf N ZKHUH [4Wf [ Wf DQG G[ A
PAGE 51

HQG WKHRUHPf 7KH WHUP JDPPD PHPRU\ ZLOO EH UHVHUYHG WR LQGLFDWH WKH GHOD\ VWUXFWXUH GHVFULEHG E\ (T 7KH UHFXUVLYH QDWXUH RI WKH JDPPD PHPRU\ FRPSXWDWLRQ LV LOOXVWUDWHG LQ )LJXUH )RU WKH GLVFUHWH WLPH FDVH WKH GHULYDWLYH LQ (T LV DSSUR[LPDWHG E\ WKH ILUVW RUGHU IRUZDUG GLIIHUHQFH WKDW LV G[N a[NW f [NWf (IO-4 7KLV DSSUR[LPDWLRQ LV QRW WKH PRVW DFFXUDWH EXW LW LV WKH VLPSOHVW $OVR WKLV SDUWLFXODU FKRLFH LPSOLHV WKDW WKH ERXQGDU\ YDOXH S O UHGXFHV WKH JDPPD PHPRU\ WR D WDSSHG GHOD\ OLQH 7KLV IHDWXUH IDFLOLWDWHV WKH FRPSDULVRQ RI WKH GLVFUHWH JDPPD PRGHO WR WDSSHG GHOD\ OLQH VWUXFWXUHV $SSO\LQJ (T OHDGV WR WKH IROORZLQJ UHFXUUHQFH UHODWLRQV IRU WKH GLVFUHWH WLPHf JDPPD PHPRU\ [4Wf [Wf [NWf O?Lf[NWOf?/[NBOWOfN O.DQGW WWLW!f§ (T 7KH WLPH LQGH[ W QRZ UXQV WKURXJK WKH LWHUDWLRQ QXPEHUV LUU} 7KH GLVFUHWH JDPPD PHPRU\ VWUXFWXUH LV GLVSOD\HG LQ )LJXUH

PAGE 52

&KDUDFWHULVWLFV RI *DPPD 0HPRU\ ,Q WKLV VHFWLRQ WKH JDPPD PHPRU\ VWUXFWXUH LV DQDO\]HG ERWK LQ WKH WLPH DQG IUHTXHQF\ GRPDLQ 7UDQVIRUPDWLRQ WR V DQG ]'RPDLQ 6LQFH WKH UHFXUVLYH UHODWLRQV WKDW JHQHUDWH WKH JDPPD VWDWH YDULDEOHV [NWf DW VXFFHVVLYH WDSV N DUH OLQHDU WKH /DSODFH WUDQVIRUPDWLRQ FDQ EH DSSOLHG 7KH RQHVLGHGf /DSODFH WUDQVIRUP LV GHILQHG DV RR ;NVf M[NWfHaVLGW /^[NWf` (T $SSOLFDWLRQ RI (T WR (T OHDGV WR WKH IROORZLQJ UHFXUVLYH UHODWLRQV IRU WKH JHQHUDWLRQ RI WKH JDPPD VWDWH YDULDEOHV LQ WKH VGRPDLQ ;Vf ;Vf [NVf 7[NLVnff¬N f¬f¬. (VO . 6 -. 8 7KH RSHUDWRU * Vf A A ZLOO EH UHIHUUHG WR DV WKH JDPPD GHOD\ RSHUDWRU 1RWH WKDW ;NVf FDQ EH H[SUHVVHG DV D IXQFWLRQ RI WKH PHPRU\ LQSXW ;Vf RQO\ 5HSHDWHG DSSOLFDWLRQ RI (T \LHOGV

PAGE 53

^L N ;NVf 7Uf rrf *NVf;Vf (D . V ; . ,W FDQ EH YHULILHG WKDW *NVf /^JNWf` 7KH V\VWHP (T DOVR VXJJHVWV D KDUGZDUH LPSOHPHQWDWLRQ RI JDPPD PHPRU\ ZKLFK LV VKRZQ LQ )LJXUH ,W IROORZV WKDW JDPPD PHPRU\ FDQ EH LQWHUSUHWHG DV D WDSSHG ORZSDVV ODGGHU ILOWHU 7KHUH DUH WZR JDPPD PHPRU\ SDUDPHWHUV WKH RUGHU . DQG EDQGZLGWK S "&f [Wf ;MWf ;Wf ra9?$f§ZYf§ 5 5 [N $:A 5 R )LJXUH $ KDUGZDUH LPSOHPHQWDWLRQ RI JDPPD PHPRU\ 7KH FRUUHVSRQGLQJ IUHTXHQF\ GRPDLQ IRU GLVFUHWHWLPH V\VWHPV LV WKH ]GRPDLQ ,W IROORZV IURP (T WKDW WKH ]WUDQVIRUP FDQ EH IRXQG E\ VXEVWLWXWLRQ RI V ]O LQ WKH /DSODFH WUDQVIRUP 7KLV OHDGV WR 6IFO ]f f§U;N]f N GO2 (T 7KH GLVFUHWH JDPPD GHOD\ RSHUDWRU LV *]f PHPRU\ LQSXW ; WR WKH N[K WDS ;N IROORZV IURP (T 7KH WUDQVIHU IXQFWLRQ IURP *$]f ] SLf f (T ,QYHUVH ]WUDQVIRUPDWLRQ RI (T OHDGV WR WKH GLVFUHWH JDPPD NHUQHOV JN WN W .N?f N W NN (De/

PAGE 54

7KH GLVFUHWH JDPPD NHUQHO JNWf LV WKH LPSXOVH UHVSRQVH IRU WKH NWK WDS RI WKH GLVFUHWH JDPPD PHPRU\ PRGHO (T FDQ EH LQWHUSUHWHG DV IROORZV ,Q RUGHU WR JHW IURP WKH PHPRU\ LQSXW WR WKH NWK WDS [NWf LQ WLPH W WKH VLJQDO KDV WR WDNH N IRUZDUG VWHSV DQG SDVV WKURXJK WN ORRSV (DFK IRUZDUG VWHS LQYROYHV D PXOWLSOLFDWLRQ E\ ML DQG D SDVV WKURXJK D ORRS LQYROYHV PXOWLSOLFDWLRQ E\ S 7KH QXPEHU RI GLIIHUHQW SDWKV IURP [LWf WR [NWf LQ WLPH W HTXDOV _Me B r 1H[W VRPH RI WKH IUHTXHQF\ GRPDLQ FKDUDFWHULVWLFV RI WKH JDPPD PHPRU\ ZLOO EH DQDO\]HG 7KH DQDO\VLV ZLOO EH SHUIRUPHG XVLQJ WKH GLVFUHWH PRGHO DOWKRXJK VLPLODU SURSHUWLHV PD\ EH GHULYHG IRU WKH FRQWLQXRXV WLPH YHUVLRQ )UHTXHQF\ 'RPDLQ $QDO\VLV 7KH WUDQVIHU IXQFWLRQ IRU WKH WK WDS RI WKHf GLVFUHWH JDPPD PHPRU\ LV JLYHQ E\ (T 7KH .WK RUGHU GLVFUHWH JDPPD PHPRU\ KDV D .WK RUGHU HLJHQYDOXH DW ] _$ 6LQFH D OLQHDU GLVFUHWHWLPH PRGHO LV VWDEOH ZKHQ DOO HLJHQYDOXHV OLH ZLWKLQ WKH XQLW FLUFOH LW IROORZV WKDW WKH GLVFUHWH JDPPD PHPRU\ LV D VWDEOH V\VWHP ZKHQ M[ 7KH JURXS GHOD\ DQG PDJQLWXGH UHVSRQVH IRU WKLV VWUXFWXUH DUH GLVSOD\HG IRU WKH VHFRQG WDS N f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f GHOD\ IRU D WDSSHG GHOD\ OLQH )RU LQVWDQFH IRU S DW WDS N D GHOD\ XS WR FDQ EH

PAGE 55

« FRMW )LJXUH *URXS GHOD\ IRU GLVFUHWH JDPPD PHPRU\ DW N

PAGE 56

DFKLHYHG IRU WKH ORZ IUHTXHQFLHV7KH FRVW IRU WKH DGGLWLRQDO GHOD\ IRU ORZ IUHTXHQFLHV LV SDLG IRU E\ WKH KLJK IUHTXHQFLHV 7KH KLJK IUHTXHQFLHV DUH DWWHQXDWHG DQG WKH JURXS GHOD\ LV OHVV WKDQ WKH WDS LQGH[ 7KXV IRU S WKH VWRUDJH RI WKH ORZ IUHTXHQFLHV LV IDYRUHG DW D FRVW IRU WKH KLJK IUHTXHQFLHV 7KH JDPPD PHPRU\ EHKDYHV DV D KLJK SDVV ILOWHU ZKHQ S $V D UHVXOW WKH KLJK IUHTXHQFLHV DUH GHOD\HG E\ PRUH WKDQ WKH WDS LQGH[ 7LPH 'RPDLQ $QDO\VLV $OWKRXJK WKH LPSXOVH UHVSRQVH JNWf RI WKH NWK WDS RI WKH JDPPD PHPRU\ H[WHQGV WR LQILQLWH WLPH IRU S LW LV SRVVLEOH WR IRUPXODWH D PHDQ PHPRU\ GHSWK IRU D JLYHQ PHPRU\ VWUXFWXUH JNWf /HW XV GHILQH WKH PHDQ VDPSOLQJ WLPH WN IRU WKH $WK WDS DV ;n6MIF: =:nf!_ ]G*N]f W G] ] N :H DOVR GHILQH WKH PHDQ VDPSOLQJ SHULRG $WN DW WDS Nf DV $WN WNWNBO 7KH PHDQ PHPRU\ GHSWK 'N IRU D JDPPD PHPRU\ RI RUGHU N WKHQ EHFRPHV N (D-L L A ,Q WKH IROORZLQJ ZH GURS WKH VXEVFULSW ZKHQ N . ,I ZH GHILQH WKH UHVROXWLRQ 5N DV 5N L S WKH IROORZLQJ IRUPXOD DULVHV ZKLFK LV RI IXQGDPHQWDO LPSRUWDQFH IRU WKH $WN FKDUDFWHUL]DWLRQ RI JDPPD PHPRU\ VWUXFWXUHV . '5 (R )RUPXOD (T UHIOHFWV WKH SRVVLEOH WUDGHRII RI UHVROXWLRQ YHUVXV PHPRU\ GHSWK LQ D

PAGE 57

)LJXUH 0DJQLWXGH UHVSRQVH IRU GLVFUHWH JDPPD PHPRU\ DW N

PAGE 58

PHPRU\ VWUXFWXUH IRU IL[HG GLPHQVLRQDOLW\ . 6XFK D WUDGHRII LV QRW SRVVLEOH LQ D QRQ GLVSHUVLYH WDSSHG GHOD\ OLQH VLQFH WKH IL[HG FKRLFH RI S VHWV WKH GHSWK DQG UHVROXWLRQ WR ' . DQG 5 UHVSHFWLYHO\ +RZHYHU LQ WKH JDPPD PHPRU\ GHSWK DQG UHVROXWLRQ FDQ EH DGDSWHG E\ YDULDWLRQ RI S ,Q PRVW QHXUDO QHW VWUXFWXUHV WKH QXPEHU RI DGDSWLYH SDUDPHWHUV LV SURSRUWLRQDO WR WKH QXPEHU RI WDSV .f 7KXV ZKHQ S WKH QXPEHU RI ZHLJKWV LV SURSRUWLRQDO WR WKH PHPRU\ GHSWK 9HU\ RIWHQ WKLV FRXSOLQJ OHDGV WR RYHUILWWLQJ RI WKH GDWD VHW XVLQJ SDUDPHWHUV WR PRGHO WKH QRLVHf 7KH SDUDPHWHU S SURYLGHV D PHDQV WR XQFRXSOH WKH PHPRU\ RUGHU DQG GHSWK $V DQ H[DPSOH DVVXPH D VLJQDO ZKRVH G\QDPLFV DUH GHVFULEHG E\ D V\VWHP ZLWK SDUDPHWHUV DQG PD[LPDO GHOD\ WKDW LV \Wf I[ W Qcf Zcf ZKHUH L DQG PDMFQf ,I ZH WU\ WR PRGHO WKLV VLJQDO ZLWK DQ DGDOLQH VWUXFWXUH WKH FKRLFH . OHDGV WR RYHUILWWLQJ ZKLOH . OHDYHV WKH QHWZRUN XQDEOH WR LQFRUSRUDWH WKH LQIOXHQFH RI [W f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

PAGE 59

OLQHDU SURFHVVLQJ ZLWK DGDSWLYH VKRUW WHUP PHPRU\ FDSDFLWLHV 7KH *DPPD 1HXUDO 1HW ,Q WKLV VHFWLRQ WKH DGGLWLYH QHXUDO QHW LV H[WHQGHG ZLWK JDPPD PHPRU\ 7KLV QHZ PRGHO ZLOO EH FRPSDUHG WR WKH YDULRXV FRPSHWLQJ QHWZRUNV IRU WHPSRUDO SURFHVVLQJ DV GHVFULEHG LQ FKDSWHU 7KH 0RGHO 5HFDOO WKH JHQHUDO PRGHO IRU DGGLWLYH QHXUDO QHWV G;f« 1 ? Wf D[L D¯ < ZLM[Mnf , fµ L (T0 Y L A /HW XV DVVXPH WKDW HDFK QHXURQ KDV WKH FDSDELOLW\ WR VWRUH LWV SDVW LQ D JDPPD PHPRU\ VWUXFWXUH RI PD[LPDOf RUGHU . DQG EDQGZLGWK S 7KH DFWLYDWLRQ RI WKH tWK WDS

PAGE 60

RI QHXURQ L LV ZULWWHQ DV [LNWf 7KH ZHLJKW ZL-N FRQQHFWV WKH NWK WDS RI QHXURQ M WR QHXURQ 7KH V\VWHP HTXDWLRQV IRU WKLV PRGHO DUH G[‘ D[8f Rn=-rc-c;MNWff,cf (De M N IRU WKH DFWLYDWLRQV [cWf DQG IRU WKH WDSV [LN^Wf A ?LL[LNWfL/L[L-BOWfN O. (Fs 7KH V\VWHP GHVFULEHG E\ (T DQG (T ZLOO EH UHIHUUHG WR DV WKH DGGLWLYHf JDPPD QHXUDO QHW RU JDPPD PRGHO ,Q (T WKH WLPH FRQVWDQW LV SURFHVVHG LQ WKH GHFD\ SDUDPHWHU Dc $OVR IRU QRWDWLRQDO FRQYHQLHQFH ZH GHILQHG [Wf Wf 7KH VWUXFWXUH RI WKH JDPPD QHXUDO PRGHO LV GLVSOD\HG LQ )LJXUH 7KH GLVFUHWH JDPPD PHPRU\ FDQ EH DSSOLHG LQ GLVFUHWHWLPH QHXUDO QHWZRUN PRGHOV 7KH GLVFUHWH JDPPD QHXUDO PRGHO LV GHILQHG DV IROORZV [Wf F@7 <-ZLMN[MN rff (T ML N [LN O?Lcf[cN¯Of?/[L-FBOWOfN O. (T 1RWH WKDW S O OHDGV WR DQ DGGLWLYH QHWZRUN ZKHUH WKH PHPRU\ LV LPSOHPHQWHG DV D WDSSHG GHOD\ OLQH $ IHHGIRUZDUG QHWZRUN RI WKLV W\SH LV HTXLYDOHQW WR WKH WLPHGHOD\ QHXUDO QHW 7KH RWKHU H[WUHPH DW _L REYLDWHV WKH JDPPD PHPRU\ DQG UHGXFHV WKH VWUXFWXUH WR D f¯QRUPDOf° DGGLWLYH PRGHO 1H[W WKH JDPPD PRGHO LV FRPSDUHG WR SUHYLRXVO\ LQWURGXFHG QHXUDO PRGHOV IRU WHPSRUDO SURFHVVLQJ 7KH *DPPD 0RGHO YHUVXV WKH $GGLWLYH 1HXUDO 1HW $GGLWLYH QHXUDO PRGHO DUH FKDUDFWHUL]HG E\ WKH IDFW WKDW WKH DGDSWLYH QHW

PAGE 62

SDUDPHWHUV DUH FRQVWDQWV WKDW LV QRW GHSHQGHQW RQ WKH QHXUDO DFWLYDWLRQ QRU WLPH GHSHQGHQW 7KH IUHH SDUDPHWHUV LQ WKH JDPPD QHXUDO PRGHO DUH M[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cf _L GLDJLID/Wf DQG ZN 7KHQ WKH JDPPD PRGHO HTXDWLRQV FDQ EH ZULWWHQ LQ PDWUL[ IRUP DV GW ;A ?L[NBO fr G[N (R [? 1H[W LQGH[ N LV HOLPLQDWHG :H GHILQH WKH JDPPD VWDWH YHFWRU ; WKH LQSXW

PAGE 63

, , WKH VTXDVKLQJ IXQFWLRQ eef WKH PDWUL[ RI GHFD\ D crc e k e SDUDPHWHUV 0 9 DQG WKH PDWUL[ RI ZHLJKWV 4 D 9 ? ? f± ; WKH JDPPD PRGHO HYDOXDWHV WR A 0;4<( (T DQ 1. f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

PAGE 64

DGGLWLYH PRGHO DUH HQWLUHO\ DSSOLFDEOH WR WKH JDPPD PRGHO 1XPHULFDO 6LPXODWLRQ :KHUHDV WKH FRPSOH[LW\ RI QXPHULFDO LQWHJUDWLRQ RI WKH FRQYROXWLRQ PRGHO VFDOHV DV 2LN37f WKH JDPPD PRGHO VFDOHV DV 2LNI.f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f 7KH PHPRU\ WDSV [NWf IRU WKLV f¯&RQFHQWUDWLRQLQ7LPH QHWf° &,71f DUH REWDLQHG E\ FRQYROYLQJ WKH LQSXW VLJQDO ZLWK WKH NHUQHOV W D DO-f aNf H N (VO0 ZKHUH D LV D SRVLWLYH LQWHJHU LV QRUPDOL]HG WR KDYH PD[LPDO YDOXH IRU W N 7KH GHJUHH RI GLVSHUVLRQ LV UHJXODWHG E\ SDUDPHWHU D ,Q )LJXUH WKH NHUQHOV DUH GLVSOD\HG IRU N WR ZKHQ D $OWKRXJK YLVXDOO\ UHVHPEOHV SHDNQRUPDOL]HG JDPPD NHUQHOV LW LV QRW SRVVLEOH WR JHQHUDWH WKH NHUQHOV E\ D UHFXUVLYH VHW RI RUGLQDU\ GLIIHUHQWLDO HTXDWLRQV ZLWK FRQVWDQW FRHIILFLHQWV DV LV WKH FDVH IRU WKH JDPPD NHUQHOV ,Q IDFW GLIIHUHQWLDWLQJ (T OHDGV WR WKH IROORZLQJ WLPHYDU\LQJ GLIIHUHQWLDO HTXDWLRQ IRU

PAGE 65

)LJXUH 7DQN DQG +RSILHOGf¬V GHOD\ NHUQHOV FF N GIN L L A DMfr: (O :KLOH 7DQN DQG +RSILHOGf«V PRGHO VKDUHV ZLWK WKH JDPPD PRGHO WKH FDSDELOLW\ RI UHJXODWLQJ WHPSRUDO GLVSHUVLRQ RQO\ WKH JDPPD PRGHO SURYLGHV DQ DGGLWLYH QHXUDO PHFKDQLVP IRU WKLV FDSDFLW\ $V D UHVXOW LQ WKH JDPPD PRGHO WKH GLVSHUVLRQ FRQWURO SDUDPHWHU _L FDQ EH WUHDWHG DV DQ DGDSWLYH ZHLJKW ,Q 7DQN DQG +RSILHOGf¬V PRGHO D LV IL[HG 7KH UHODWLYH PHULWV RI SHDN YHUVXV DUHDQRUPDOL]DWLRQ RI GLVSHUVLYH GHOD\ NHUQHOV KDYH QRW EHHQ LQYHVWLJDWHG 6LPLODU DUJXPHQWV KROG ZKHQ WKH JDPPD PHPRU\ LV FRPSDUHG WR WKH DGDSWLYH JDXVVLDQ GLVWULEXWHG GHOD\ PRGHOV VXFK DV WKH 7(032 PRGHO %RGHQKDXVHQ DQG :DLEHO f DQG WKH JDXVVLDQ YHUVLRQ RI WKH FRQFHQWUDWLRQLQWLPH QHXUDO QHW

PAGE 66

8QQLNULVKQDQ HW DO f 7KHVH PHPRU\ PRGHOV GR RIIHU WKH DGYDQWDJH RI DGDSWLYH GLVSHUVLRQ \HW RQO\ WKH JDPPD PHPRU\ RIIHUV DQ DGGLWLYH QHXUDO PHFKDQLVP WR FUHDWH GLVSHUVLYH GHOD\V 7KH RWKHU PRGHOV UHTXLUH HYDOXDWLRQ RI D FRQYROXWLRQ LQWHJUDO ZLWK UHVSHFW WR WKH GHOD\ NHUQHOV LQ RUGHU WR FRPSXWH WKH PHPRU\ WUDFHV $OWKRXJK QRW D SULRULW\ IRU HQJLQHHULQJ DSSOLFDWLRQV WKH JDPPD PHPRU\ LV ELRORJLFDOO\ SODXVLEOH VLQFH WKHUH LV QR QRQQHXUDOf H[WHUQDO PHFKDQLVP UHTXLUHG WR JHQHUDWH GHOD\ NHUQHOV 7KH *DPPD 0RGHO YHUVXV WKH 7LPH 'HOD\ 1HXUDO 1HW 7KH PHPRU\ VWUXFWXUH LQ WKH WLPH GHOD\ QHXUDO QHW 7'11f LV D WDSSHG GHOD\ OLQH ,Q IDFW 7'11 VWUXFWXUHV FDQ EH FUHDWHG LQ WKH JDPPD PHPRU\ E\ IL[LQJ S LQ WKH GLVFUHWH JDPPD PRGHO 7KXV WKH 7'11 LV D VSHFLDO FDVH RI WKH GLVFUHWH JDPPD PRGHO :KHQ S WKH GLVFUHWH JDPPD PHPRU\ LPSOHPHQWV D WDSSHG GLVSHUVLYH GHOD\ OLQH 7KH DPRXQW RI GLVSHUVLRQ LV UHJXODWHG E\ WKH DGDSWLYH PHPRU\ SDUDPHWHU S :H GLVFXVVHG WKDW WKH PHPRU\ GHSWK RI WKH JDPPD PHPRU\ FDQ EH HVWLPDWHG E\ .O?L +HQFH WKH PHPRU\ GHSWK FDQ EH DGDSWHG LQGHSHQGHQWO\ IURP WKH QXPEHU RI WDSV DQG ZHLJKWVf LQ WKH VWUXFWXUH ,Q WKH 7'11 WKH PHPRU\ GHSWK DQG WKH PHPRU\ RUGHU ERWK HTXDO . $V D UHVXOW LQFUHDVLQJ WKH PHPRU\ GHSWK LQ WKH 7'11 LV DOZD\V FRXSOHG ZLWK DQ LQFUHDVH LQ WKH QXPEHU RI ZHLJKWV LQ WKH QHW ZKLFK LV VRPHWLPHV QRW GHVLUDEOH 7KH *DPPD 0RGHO YHUVXV $GDOLQH 7KH VLPSOHVW GLVFUHWHWLPH JDPPD PRGHO LV D OLQHDU RQHOD\HU IHHGIRUZDUG VWUXFWXUH ZLWK RQH RXWSXW XQLW 7KH HTXDWLRQV IRU WKLV QHWZRUN DUH JLYHQ E\ . \ ;bf N [Wf ,Wf ;NWf O?Lf[NWOf ?L[NBOWOf (IOLO4

PAGE 67

7KLV VWUXFWXUH LV GHSLFWHG LQ )LJXUH )RU S O :LGURZf¬V DGDOLQH LV REWDLQHG $OVR DGDOLQH LV WKH VLPSOHVW OLQHDU RQHOD\HU RQH RXWSXWf LPSOHPHQWDWLRQ 7KH JDPPD PHPRU\ JHQHUDOL]HV DGDOLQH WR DQ DGDSWLYH ILOWHU ZLWK D GLVSHUVLYH WDSSHG GHOD\ OLQH $GDOLQH ZLWK JDPPD PHPRU\ ZLOO EH UHIHUUHG WR DV DGDOLQHISO RU DGDSWLYH JDPPD ILOWHU 6HYHUDO LQWHUHVWLQJ DVSHFWV RI WKLV ILOWHU DUH ZRUWK D GHHSHU ORRN $ VSHFLDO FKDSWHU f ZLOO EH GHGLFDWHG WR WKH DQDO\VLV RI DGDOLQHSf 'LVFXVVLRQ ,Q WKLV FKDSWHU WKH JDPPD QHXUDO PRGHO KDV EHHQ GHYHORSHG DQG DQDO\]HG 7KH JDPPD PRGHO LV FKDUDFWHUL]HG E\ D VSHFLILF VKRUW WHUP PHPRU\ DUFKLWHFWXUH WKH JDPPD PHPRU\ VWUXFWXUH *DPPD PHPRU\ LV DQ DGDSWLYH ORFDO VKRUW WHUP PHPRU\ VWUXFWXUH 0HPRU\ GHSWK DQG UHVROXWLRQ FDQ EH DOWHUHG E\ YDULDWLRQ RI D FRQWLQXRXV SDUDPHWHU S ,Q WKH QH[W FKDSWHU JUDGLHQW GHVFHQW DGDSWDWLRQ SURFHGXUHV IRU WKH JDPPD PRGHO DUH SUHVHQWHG

PAGE 68

&+$37(5 *5$',(17 '(6&(17 /($51,1* ,1 7+( *$00$ 1(7 ,QWURGXFWLRQ /HDUQLQJ DV DQ 2SWLPL]DWLRQ 3UREOHP /HDUQLQJ LQ D QHXUDO QHW FRQFHUQV WKH PRGLILFDWLRQ RI WKH ZHLJKWV RI WKH QHW VR DV WR LPSURYH SHUIRUPDQFH RI WKH V\VWHP 7KH WHUP DGDSWDWLRQ ZLOO EH XVHG DV D V\QRQ\P IRU OHDUQLQJ &RPPRQO\ LW LV DVVXPHG WKDW WKH SHUIRUPDQFH RI WKH QHXUDO QHW LV H[SUHVVHG E\ D VFDODU SHUIRUPDQFH LQGH[ RU WRWDO HUURU ( ,Q JHQHUDO ZH ZULWH (RQ S W P If LV D FRVW IXQFWLRQDO ZKLFK GHVFULEHV WKH HUURU PHDVXUH DW RXWSXW QRGH PH 0 DW WLPH WH > 7@ ZKHQ SDWWHUQ SH 3 LV SUHVHQWHG WR WKH V\VWHP 2IWHQ WKH ZHLJKWHG E\ TXDGUDWLF GHYLDWLRQ IURP D JLYHQ WDUJHW WUDMHFWRU\ GSP Wf LV FKRVHQ DV WKH FRVW )RU WKLV FDVH (T HYDOXDWHV WR e , .nf¯nf@ L [ .@ Oe/ S W P S W P ZKHUH HSP Wf GSP Wf [SP Wf LV WKH LQVWDQWDQHRXV HUURU VLJQDO ZKLFK LV LPPHGLDWHO\ PHDVXUDEOH DW DQ\ WLPH 7KH OHDUQLQJ JRDO LV WR PLQLPL]H ( RYHU WKH V\VWHP SDUDPHWHUV Z DQG S FRQVWUDLQHG E\ WKH QHWZRUN VWDWH HTXDWLRQV [ I[ ,Z Sf 7KLV SUREOHP KDV EHHQ VWXGLHG H[WHQVLYHO\ LQ RSWLPDO FRQWURO WKHRU\ %U\VRQ DQG +R f 7KH PRVW

PAGE 69

G( FRPPRQ DSSURDFK WR VHDUFK IRU WKH PLQLPXP RI ( LQYROYHV WKH XVH RI WKH JUDGLHQWV A G( DQG :KHQ ( LV PLQLPDO WKHVH JUDGLHQWV QHFHVVDULO\ YDQLVK WKDW LV DW WKH RSWLPXP ZH KDYH G( B G( GZ -, (J $Q DOJRULWKPLF PHWKRG LV QRZ GLVFXVVHG ZKLFK VHDUFKHV IRU WKH YDOXHV Z DQG _L WKDW PLQLPL]H ( $VVXPH DQ DYDLODEOH VHW RI WUDLQLQJ SDWWHUQ SDLUV " Wf GSP Wff WKDW DGHTXDWHO\ UHSUHVHQWV WKH SUREOHP DW KDQG 7KLV VHW LV UHIHUUHG WR DV WKH WUDLQLQJ VHW 3 1H[W WKH WUDLQLQJ VHW LV SUHVHQWHG WR WKH QHWZRUN DQG WKH DFWLYDWLRQV [" Wf DUH UHFRUGHG 7KH SUHVHQWDWLRQ RI WKH HQWLUH SDWWHUQ VHW 3 LV FDOOHG DQ HSRFK RU EDWFK 1RWH WKDW WKH DYDLODELOLW\ RI [" Wf DQG GSP Wf DOORZV WKH HYDOXDWLRQ RI WKH SHUIRUPDQFH LQGH[ ( XVLQJ (T 7KLV PHDVXUH FDQ EH XVHG WR GHWHUPLQH ZKHQ WR VWRS WUDLQLQJ RI WKH G( G( V\VWHP 1H[W WKH JUDGLHQWV A DQG WKH GLUHFWLRQ RI WKH QHJDWLYH JUDGLHQWV DUH FRPSXWHG DQG WKH ZHLJKWV DUH XSGDWHG LQ G( ZQHZ ZROGaA (T G( (T AQHZ a AROG AMf¬ ,I WKH OHDUQLQJ UDWH RU VWHS VL]H UL LV VPDOO HQRXJK WKLV XSGDWH ZLOO GHFUHDVH WKH WRWDO HUURU ( RQ WKH QH[W EDWFK UXQ 7KHUH DUH RWKHU PHWKRGV RI XWLOL]LQJ WKH HUURU JUDGLHQWV WR VHDUFK IRU WKH PLQLPXP RI ( $V DQ H[DPSOH WKH VXFFHVVLYH ZHLJKW XSGDWHV FDQ EH PDGH RUWKRJRQDO WR HDFK RWKHU D SURFHVV ZKLFK LV FDOOHG FRQMXJDWH JUDGLHQW GHVFHQW ,Q WKLV ZRUN ZH DUH QRW LQWHUHVWHG LQ RSWLPL]LQJ WKH OHDUQLQJ SURFHVV

PAGE 70

SHU VH 2XU JRDO ZLOO EH WR JHQHUDOL]H JUDGLHQW GHVFHQW DGDSWDWLRQ WR WKH JDPPD QHW DQG HYDOXDWH WKH SURSHUWLHV RI WKLV JHQHUDOL]DWLRQ 7KH HTXDWLRQV (T DQG (T LPSOHPHQW DQ XSGDWH VWUDWHJ\ ZKLFK LV FDOOHG VWHHSHVW GHVFHQW 7KH SURFHVV RI UXQQLQJ WUDLQLQJ VHW HSRFKV IROORZHG E\ D ZHLJKW XSGDWH LV UHSHDWHG XQWLO WKH WRWDO HUURU ( QR ORQJHU GHFUHDVHV 1RWH WKDW LI WKH HUURU VXUIDFH ( (Z [f LV FRQYH[ WKLV SURFHGXUH OHDGV WR D JOREDO RSWLPL]DWLRQ RI WKH QHWZRUN SHUIRUPDQFH 7KH OHDUQLQJ SURFHVV GHVFULEHG DERYH XSGDWHV WKH ZHLJKWV RQO\ DIWHU SUHVHQWDWLRQ RI WKH HQWLUH HSRFK :H FDOO WKLV HSRFKZLVH OHDUQLQJ RU OHDUQLQJ LQ EDWFK PRGH $ IDVWHU WUDLQLQJ PHWKRG ZRXOG EH WR DGDSW WKH ZHLJKWV DIWHU HDFK WLPH VWHS W G( G( 7KLV LQYROYHV FRPSXWDWLRQ RI WKH WLPHGHSHQGHQW HUURU JUDGLHQWV A Wf DQG fµ 7KH XSGDWH HTXDWLRQV WKHQ EHFRPH G( ZWf ZLOf QA (R G( Q 0 Uf nQA (T 7KLV PRGH LV FDOOHG UHDOWLPH OHDUQLQJ RU RQOLQH OHDUQLQJ 5HDOWLPH OHDUQLQJ FRQYHUJHV IDVWHU WKDQ OHDUQLQJ LQ EDWFK PRGH EXW WKH XSGDWHV DUH QR ORQJHU LQ WKH G( G( RSSRVLWH GLUHFWLRQ RI WKH WRWDO HUURU JUDGLHQWV A DQG A 7KXV HYHQ IRU FRQYH[ HUURU VXUIDFHV UHDO WLPH DGDSWDWLRQ QRW QHFHVVDULO\ OHDGV WR JOREDO RSWLPL]DWLRQ RI WKH QHWZRUN SHUIRUPDQFH $ ORRN DW WKH XSGDWH HTXDWLRQV PDNHV FOHDU WKDW WKH FUXFLDO DVSHFW RI WKH OHDUQLQJ G( G( SURFHVV MXVW GHVFULEHG LV WKH FRPSXWDWLRQ RI WKH HUURU JUDGLHQWV A DQG A 7KHUHIRUH WKH UHPDLQLQJ SDUW RI WKLV FKDSWHU FRQFHUQV PHWKRGV RI HYDOXDWLQJ WKHVH JUDGLHQWV 7KLV FKDSWHU LV RUJDQL]HG DV IROORZV ,Q WKH QH[W VHFWLRQ WKH OLWHUDWXUH RQ

PAGE 71

JUDGLHQW FRPSXWDWLRQ LQ VLPSOH QHXUDO QHWV LV UHYLHZHG 7ZR GLIIHUHQW PHWKRGV ZLOO EH HYDOXDWHG 7KH GLUHFW PHWKRG FRPSXWHV WKH JUDGLHQWV E\ GLUHFW QXPHULFDO GLIIHUHQWLDWLRQ RI WKH GHVFULELQJ V\VWHP HTXDWLRQV 7KH DOWHUQDWLYH PHWKRG HUURUnf EDFNSURSDJDWLRQ XWLOL]HV WKH VSHFLILF QHWZRUN DUFKLWHFWXUH WR FRPSXWH WKH JUDGLHQWV $V D UHVXOW EDFNSURSDJDWLRQ ZLOO SURRI WR EH D PRUH HIILFLHQW WHFKQLTXH LQ WHUPV RI QXPEHU RI RSHUDWLRQVf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f K 3T ML :H ZLOO DOVR ZULWH QHW A ZLM[M fµ $OVR ZH DVVXPH WKDW WKH VWDWHV [ DUH FRPSXWHG E\ ML LQFUHDVLQJ LQGH[ RUGHU 7KXV ILUVW [c LV FRPSXWHG WKHQ [ DQG VR IRUWK XQWLO [1 7KHUH DUH QR WHPSRUDO G\QDPLFV DVVRFLDWHG ZLWK (T $V GLVFXVVHG EHIRUH WKH FHQWUDO WDVN

PAGE 72

%( RI DOO JUDGLHQW GHVFHQW DGDSWDWLRQ SURFHGXUHV LV WR FRPSXWH Af§ IRU DOO ZHLJKWV ,W ZLOO 2:M EH DVVXPHG WKDW WKH WRWDO HUURU PHDVXUH ( FDQ EH H[SUHVVHG DV (T PH 0 P P 7KXV WKH WUDLQLQJ VHW FRQVLVWV RI RQH VWDWLF SDWWHUQ 7KH OHDUQLQJ WDVN LV WR DGDSW WKH ZHLJKWV VXFK WKDW WKH PHDQ VTXDUH HUURU EHWZHHQ WKH WDUJHW GP DQG QHW DFWLYDWLRQ [Q LV %( PLQLPDO ZKHQ ,c LV SUHVHQWHG WR WKH V\VWHP 1H[W WZR H[DFW DOJRULWKPV WR FRPSXWH af§ GZLM DUH SUHVHQWHG *UDGLHQW &RPSXWDWLRQ EY 'LUHFW 1XPHULFDO 'LIIHUHQWLDWLRQ %( 7KH VLPSOHVW PHWKRG WR FRPSXWH WKH JUDGLHQWV af§ LV MXVW E\ GLIIHUHQWLDWLQJ WKH 2:M HTXDWLRQV (T DQG (T $SSO\LQJ WKH FKDLQUXOH WR (T \LHOGV %( %Z X ‘= %[ P P%Zn P LM (T %[ 1H[W WKH JUDGLHQW YDULDEOH _" V LV GHILQHG " FDQ EH GLUHFWO\ FRPSXWHG E\ GLIIHUHQWLDWLQJ WKH VWDWH HTXDWLRQ (T 7KLV OHDGV WR GF QHW f %QHWP P f§ [ 9 GQHWP %Z P LM rPQHWQf ; < LP ;W PQALM QP (T ZKHUH P LV WKH NURQHFNHU GHOWD IXQFWLRQ 7KH VHW RI HTXDWLRQV (T (T DQG (T SURYLGH D V\VWHP WR FRPSXWH WKH

PAGE 73

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f GHULYDWLYHV LQ QHWZRUNV &RQVLGHU WKH QHWZRUN LQ )LJXUH 7KH VWDWH HTXDWLRQV IRU WKLV QHWZRUN DUH JLYHQ E\ r f¯ K [ ZO[O, [ :r r Z[ Z[ r Zr Zr Zr (T ,W LV DVVXPHV WKDW WKH YDULDEOHV [c WKURXJK [ DUH FRPSXWHG LQ LQGH[HG RUGHU WKDW LV ILUVW ;@ WKHQ [ DQG VR IRUWK XQWLO [V $ QHWZRUN ZKRVH VWDWH YDULDEOHV DUH FRPSXWHG RQH DW r D WLPH LQ D VSHFLILHG RUGHU ZLOO EH FDOOHG DQ RUGHUHG QHWZRUN /HW XV FRPSXWH af§ G[ [ ([SOLFLW SDUWLDO GLIIHUHQWLDWLRQ RI WKH HTXDWLRQ IRU [ LQ (T JLYHV Z +RZHYHU WKLV RQO\ UHIOHFWV WKH GLUHFW RU H[SOLFLW GHSHQGHQFH RI [ RQ [ [ DOVR DIIHFWV [ LQGLUHFWO\ WKURXJK WKH QHWZRUN ,QFRUSRUDWLQJ WKHVH LQGLUHFW RU LPSOLFLW LQIOXHQFHV

PAGE 74

OHDGV WR WKH IROORZLQJ H[SUHVVLRQ G[ G[ G[ Af¯A AAA Z Z G[? G[ f° f° G7M f° G7 Z f° f° f°Z! ZZ (R 7KLV GLIIHUHQFH EHWZHHQ H[SOLFLW DQG LPSOLFLW GHSHQGHQFLHV LQ QHWZRUNV KDV EHHQ WUHDWHG LQ D EDFNSURSDJDWLRQ FRQWH[W E\ :HUERV f +H LQWURGXFHG WKH WHUP RUGHUHG GHULYDWLYH WR GHQRWH WKH WRWDO SDUWLDO GHULYDWLYH LQFOXGLQJ WKH QHWZRUN LQIOXHQFHVf ,Q WKLV ZRUN ZKHQHYHU ZH VSHDN RI D SDUWLDO GHULYDWLYH WR [ WKH RUGHUHG SDUWLDO GHULYDWLYH LV PHDQW ZKLFK LV GHQRWHG E\ WKH V\PERO A ,I ZH RQO\ ZDQW WR LQFOXGH WKH H[SOLFLW GLUHFWf GHSHQGHQFH RQ [ ZH VSHDN RI H[SOLFLW SDUWLDO GHULYDWLYH IRU ZKLFK WKH V\PERO UfH A ZLOO EH UHVHUYHG :HUERV f SURYHG WKH IROORZLQJ WKHRUHP IRU RUGHUHG QHWZRUNV 7KHRUHP &RQVLGHU D QHWZRUN RI -9 YDULDEOHV [c ZKRVH GHSHQGHQFLHV DUH RUGHUHG E\ WKH OLVW / >;M [}[1@ WKLV PHDQV WKDW [ RQO\ GHSHQGV RQ ;M ZKHUH M L /HW D SHUIRUPDQFH IXQFWLRQ ( EH GHILQHG E\ ( (;S; f§[1f (T G( 7KHQ WKH RUGHUHGf SDUWLDO GHULYDWLYHV FDQ EH FRPSXWHG E\ G( Gr( Y G( Gr[M G[c G[c ;IG[M ;G[c n (R HQG WKHRUHPf

PAGE 75

G[ WKH QHWZRUN (T OHDGV WR H[SUHVVLRQ (T 1RWH WKDW WKH FRPSXWDWLRQ RI WKH G( JUDGLHQWV UHTXLUHV NQRZOHGJH RI WKH HUURU JUDGLHQWV ZLWK UHVSHFW WR [c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cM` DUH LQLWLDOL]HG EHIRUH WKH VWDWH YDULDEOHV [W DUH HYDOXDWHG WKH\ FDQ EH SXW DW WKH EHJLQQLQJ RI WKH OLVW 7KLV OHDGV WR WKH IROORZLQJ OLVW / >^ZcM`[ (D0 1H[W WKH SDUWLDO GHULYDWLYHV RI WKH SHUIRUPDQFH LQGH[ ( WR DOO YDULDEOHV LQ / DUH FRPSXWHG PDNLQJ XVH RI WKH UXOH IRU SDUWLDO GHULYDWLYHV LQ RUGHUHG QHWZRUNV (T )RU WKH VWDWH YDULDEOHV [c WKLV OHDGV WR %( Gr(\ G[ G[ MInG;M G; G; (R )RU WKH JUDGLHQWV f§ ZH REWDLQ WKH IROORZLQJ H[SUHVVLRQ

PAGE 76

G( GH[Q ;AM (T ,Q WKH EDFNSURSDJDWLRQ OLWHUDWXUH LW LV FXVWRPDU\ WR GHILQH WKH YDULDEOHV G( G( H AUf§ DQG RVA HD QHWf r G[c r GQHWL r } Y (R 6XEVWLWXWLRQ RI (T LQWR (T DQG (T \LHOGV IRU WKH FRPSXWDWLRQ RI WKH HUURU JUDGLHQWV H f§H \ Z L n $! M M!L HRInQHUIf G( V Af§ RD -L (J (J (R 7KH VHW RI HTXDWLRQV (T (T DQG (T FRQVWLWXWH WKH EDFNSURSDJDWLRQ PHWKRG WR FRPSXWH WKH HUURU JUDGLHQWV 1RZ OHW XV VHH KRZ WR DSSO\ DOO WKHVH HTXDWLRQV WR WKH OHDUQLQJ SUREOHP 6D\ ZH KDYH D QHWZRUN GHVFULEHG E\ (T DQG D WUDLQLQJ GDWD VHW FRQVLVWLQJ RI RQH LQSXW SDWWHUQV I DQG D WDUJHW SDWWHUQ GP 7R VWDUW WKH OHDUQLQJ V\VWHP WKH LQSXW SDWWHUQ LV SUHVHQWHG WR WKH QHW DQG WKH VWDWH YDULDEOHV rfµ DUH HYDOXDWHG E\ (T 7KH YDULDEOHV [c DUH FRPSXWHG E\ LQFUHDVLQJ LQGH[ RUGHU L 7KLV FRPSOHWHV WKH IRUZDUG SDVV 1H[W WKH HUURU YDULDEOHV Hc Gc [c DUH HYDOXDWHG DQG VWRUHG 7KH QH[W SKDVH WKH EDFNZDUG RU r G( EDFNSURSDJDWLRQ SDVV FRPSXWHV WKH YDULDEOHV R A E\ HYDOXDWLQJ (T DQG

PAGE 77

(T 7KH TXDQWLWLHV DUH FDOOHG EDFNSURSDJDWLRQ HUURUV 7KH\ PHDVXUH WKH VHQVLWLYLW\ RI WKH WRWDO HUURU ( ZLWK UHVSHFW WR DQ LQILQLWHVLPDO FKDQJH LQ QHWc ,W KDV EHHQ PHQWLRQHG EHIRUH WKDW WKH EDFNSURSDJDWLRQ HUURUV DUH FRPSXWHG LQ GHVFHQGLQJ LQGH[ RUGHU WKDW LV ILUVW A IROORZHG E\ i1B M XQWLO M $IWHU WKH EDFNSURSDJDWLRQ SDVV WKH HUURU JUDGLHQWV ZLWK UHVSHFW WR WKH ZHLJKWV DUH FRPSXWHG E\ (T 1H[W LI D VWHHSHVW GHVFHQW XSGDWH UXOH LV XVHG WKH ZHLJKWV DUH DGDSWHG DFFRUGLQJ WR G( $ZX Qt³ (DAW LM ,Q RUGHU WR DSSUHFLDWH WKH DUFKLWHFWXUH RI WKH EDFNSURSDJDWLRQ PHWKRG ZH UHZULWH WKH EDFNSURSDJDWLRQ HTXDWLRQ (T DV H H < Dn QHWfZ( (T r O$!M -O M -!W 1RWH WKH VWUXFWXUDO VLPLODULW\ EHWZHHQ WKH VWDWH HTXDWLRQ (T DQG (T ,Q WKH EDFNSURSDJDWLRQ HTXDWLRQ WKH EDFNSURS HUURUV efµ VHUYH DV WKH QHWZRUN VWDWHV DQG Hc LV WKH H[WHUQDO LQSXW ,Q RUGHU WR GLVFULPLQDWH EHWZHHQ WKH HUURU YDULDEOHV Hc DQG HI H LV VRPHWLPHV UHIHUUHG WR DV WKH LQMHFWLRQ HUURU ZKHUHDV DQG f DUH FDOOHG EDFNSURSDJDWLRQ HUURUV 6LQFH WKH EDFNSURS ZHLJKWV :Mc FRQQHFW QRGH M L QRWH WKH DUURZ UHYHUVDO DV FRPSDUHG WR (Tf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

PAGE 78

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ff LV XVHG ZKLFK LV GHILQHG DV WKH VHW RI SRVLWLYH LQWHJHU YDOXHGf IXQFWLRQV ZKLFK DUH OHVV RU HTXDO WR VRPH FRQVWDQW SRVLWLYH PXOWLSOH RI FSQf :H ZLOO DVVXPH WKDW WKH FRVW RI RQH RSHUDWLRQ DGGLWLRQ RU PXOWLSOLFDWLRQ FDUULHG RXW RQ RQH SURFHVVRU LV f $V DQ H[DPSOH WKH HYDOXDWLRQ RI WKH V\VWHP HTXDWLRQV IRU WKH

PAGE 79

DGGLWLYH QHW (T FRVWV 2L1f RSHUDWLRQV 7KH UHDVRQLQJ JRHV DV IROORZV 7KH HYDOXDWLRQ RI [c FRVWV 21f RSHUDWLRQV VLQFH QRGH L LV FRQQHFWHG WR PD[LPDO 1 QRGHV 6LQFH ZH QHHG WR HYDOXDWH WKH DFWLYDWLRQV IRU DOO L WKDW LV L WKH WRWDO FRVW EHFRPHV 2L1f 7KH FRVW RI VWRUDJH VSDFHf IRU WKH V\VWHP (T LV 2L1f VLQFH WKH VSDFH UHTXLUHPHQWV DUH GRPLQDWHG E\ WKH PD[LPDOf 1 ZHLJKWV :\ 1RZ OHW XV HYDOXDWH WKH FRVW RI WKH HUURU FRPSXWDWLRQ E\ WKH GLUHFW PHWKRG 7KH QXPEHU RI UHTXLUHG RSHUDWLRQV LV GRPLQDWHG E\ WKH HYDOXDWLRQ RI WKH JUDGLHQWV "" 7KH FRPSXWDWLRQ RI D YDULDEOH LQYROYHV 2L1f RSHUDWLRQV 6LQFH WKHUH DUH PD[LPDO 1 YDULDEOHV A LW IROORZV WKDW WKH WRWDO QXPEHU RI RSHUDWLRQV VFDOH E\ 2L1f :H QHHG 2L1f VSDFH WR VWRUH 3r1 \ $V IRU WKH FRPSXWDWLRQDO FRVW RI WKH EDFNSURSDJDWLRQ PHWKRG QRWH WKDW WKH FRPSXWDWLRQ RI WKH EDFNSURSDJDWLRQ HUURUV UHTXLUHV HYDOXDWLRQ RI WKH WUDQVSRVHGf QHWZRUN ,W ZDV DOUHDG\ GLVFXVVHG WKDW HYDOXDWLRQ RI WKH QHWZRUN UHTXLUHV 2L1f QXPEHU RI RSHUDWLRQV 7KH VSDFH UHTXLUHPHQWV DUH GRPLQDWHG E\ WKH ZHLJKWV :\ KHQFH 2L1f VWRUDJH LV QHHGHG 7KXV ERWK WKH QXPEHU RI RSHUDWLRQV WLPHf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

PAGE 80

EDFNSURSDJDWLRQ HUURUV DUH FRPSXWHG E\ PHDQV RI WKH WUDQVSRVHG QHWZRUN $V D UHVXOW LQ D KDUGZDUH LPSOHPHQWDWLRQ RQO\ WKH GLUHFWLRQ RI WKH FRPPXQLFDWLRQ SDWKV EHWZHHQ SURFHVVRUV QHHG WR EH UHYHUVHG 7KH GLUHFW PHWKRG RQ WKH RWKHU KDQG LV QRW ORFDO 7KH JUDGLHQWV " QHHG WR EH FRPSXWHG IRU DOO QRGHV QH 1 DQG DOO ZHLJKW LQGLFHV LMf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f ?/c.cWNBWOf (J ,Q FRQWUDVW WR WKH SUHYLRXV VHFWLRQ LW LV DVVXPHG WKDW WKH DFWLYDWLRQV DQG WDUJHW SDWWHUQV DUH WLPHYDU\LQJ 7KXV WKH SHUIRUPDQFH LQGH[ ( LV GHILQHG DV W P

PAGE 81

 YPf«frPUQ A WP NM-XPXf@ (RAD W P :H QRZ SURFHHG WR GHULYH WKH HUURU JUDGLHQWV XVLQJ WKH GLUHFW PHWKRG 7KH 'LUHFW 0HWKRG 7KH SURFHGXUH LV VLPLODU WR WKH GHULYDWLRQ IRU WKH VWDWLF PRGHO 3DUWLDOO\ GLIIHUHQWLDWLQJ ( WR ZLMN \LHOGV G( BY G[!^Wf GZ GZQ (T LMN W P nLMN G[P :H GHILQH WKH JUDGLHQW VLJQDO f§ Af§ FDQ EH HYDOXDWHG E\ SDUWLDO LMN GLIIHUHQWLDWLRQ RI (T ZKLFK OHDGV WR }m}nff 6LP[MN: Q H 1 (J ZKHUH cP LV WKH .URQHFNHU GHOWD DQG UHPHPEHU WKH QRWDWLRQ ZPQ ZPQf G( $ VLPLODU GHULYDWLRQ FDQ EH DSSOLHG WR REWDLQ WKH JUDGLHQWV $QDORJRXVO\ WR (T ZH ZULWH DV B Y -[PA W P (TO4O DrPR $SSO\LQJ WKH FKDLQUXOH WR WKH SDUWLDO GHULYDWLYHV f§f§ OHDGV WR G[P: B Y YAPB G[QN: B a tONWf QH1NG[QN: r r G+I (R

PAGE 82

7KH VLJQDO G[cNWf IROORZV E\ GLIIHUHQWLDWLRQ RI (T \LHOGLQJ G[LNLf RM QHWPWffZPcN PLNn (T 6XEVWLWXWLRQ RI (T LQWR (T \LHOGV IRU WKH HUURU JUDGLHQWV G( 67 nf ! n! fµ W P N (Tf±O4 G[LNWf ZKHUH ZH GHILQHG DrLf f§f§ 7KH VLJQDOV Df°r Wf FDQ EH FRPSXWHG RQ OLQH E\ GLIIHUHQWLDWLRQ RI (T ZKLFK HYDOXDWHV WR RWIf O-OfDILOf-OLDrLOf (t G( 7KH VHW RI HTXDWLRQV (T DQG (T SURYLGH WKH JUDGLHQWV A ZKHUHDV GZLMN G( (T DQG (T FRPSXWH WKH JUDGLHQWV $ VWHHSHVW GHVFHQW DGDSWLYH SURFHGXUH ZRXOG XVH WKHVH YDULDEOHV LQ D XSGDWH UXOH RI WKH IRUP D G( $Z 7L nr GZ LMN (T DQG DQ DQDORJRXV H[SUHVVLRQ IRU WKH DGDSWDWLRQ RI SI 7RJHWKHU ZLWK WKH JDPPD V\VWHP HTXDWLRQV (T DQG (T WKH\ FRQVWLWXWH D JDPPD PRGHO OHDUQLQJ V\VWHP 7KH OHDUQLQJ V\VWHP DV GHULYHG KHUH DVVXPHG DGDSWDWLRQ LQ EDWFK PRGH +RZHYHU WKLV DOJRULWKP LV HDVLO\ FRQYHUWHG WR D UHDOWLPH OHDUQLQJ V\VWHP :H MXVW GHILQH D WLPHGHSHQGHQW SHUIRUPDQFH LQGH[ ( E\ 6 >m0U -6TO4 nPH 0

PAGE 83

1RWH WKDW VLQFH ( WKH RQO\ FKDQJH LQ WKH IRUPXODH LV WR WDNH RXW WKH A IURP W W WKH HUURU JUDGLHQWV ZKLFK UHGXFHV WKH HUURU JUDGLHQW H[SUHVVLRQV WR G(W A 236r :DQG (V20 LMN P ( 67 em} nf 9 }m} nff !:rf« (V2IIO 7KH VLJQDOV DQG RILWf DUH FRPSXWHG E\ WKH VDPH HTXDWLRQV DV LQ WKH EDWFK PRGH WKDW LV (T DQG (T UHVSHFWLYHO\ 7KH UHDO WLPH PRGH IRU WKLV DOJRULWKP LV SDUWLFXODUO\ LQWHUHVWLQJ VLQFH WKH UHTXLUHG QXPEHU RI RSHUDWLRQV LV HTXLYDOHQW WR WKH EDWFK PRGH DOJRULWKP +RZHYHU WKH VWRUDJH UHTXLUHPHQWV IRU WKH UHDOWLPH PRGH DUH JUHDWO\ UHGXFHG E\ IDFWRU 7 WKH QXPEHU RI WLPH VWHSVf VLQFH ZH XSGDWH RQOLQH E\ G(W G(W $ZMW: AA DQG $+nf 7OJcc (V20 LMN ,Q DGGLWLRQ UHDOWLPH DGDSWDWLRQ XVXDOO\ FRQYHUJHV IDVWHU WKDQ HSRFKZLVH XSGDWLQJ 7KHUHIRUH LQ SUDFWLFH UHDO WLPH XSGDWLQJ LV XVHG IDU PRUH WKDW OHDUQLQJ LQ EDWFK PRGH ,Q IDFW WKH UHDOWLPH PRGH RI WKH DOJRULWKP GHVFULEHG KHUH ZDV GHULYHG IRU UHFXUUHQW QHXUDO QHWV E\ :LOOLDPV DQG =LSVHU f 7KH\ FRLQHG WKH QDPH UHDO WLPH UHFXUUHQW OHDUQLQJ DOJRULWKP 575/f :H ZLOO WDNH RYHU WKHLU WHUPLQRORJ\ 7KXV WKH GLUHFW PHWKRG IRU HUURU JUDGLHQW FRPSXWDWLRQ LQ JDPPD QHWV OHDGV WR D VSHFLDOf 575/ DOJRULWKP /HW XV DQDO\]H WKH ORFDOLW\ DQG FRPSOH[LW\ SURSHUWLHV RI WKH 575/ DOJRULWKP IRU WKH JDPPD QHW $VVXPH WKDW WKH QXPEHU RI XQLWV LQ WKH V\VWHP HTXDOV 1 (DFK XQLW VWRUHV D KLVWRU\ WUDFH RI LWV DFWLYDWLRQ LQ D JDPPD PHPRU\ VWUXFWXUH RI PD[LPDO RUGHU . 7KH PD[LPDOf QXPEHU RI ZHLJKWV ZLMN WKHQ EHFRPHV 79$n 7KH QXPEHU RI PHPRU\ SDUDPHWHUV HTXDO 1 $OVR LW LV DVVXPHG WKDW WKH V\VWHP LV UXQ IRU 7 WLPH VWHSV

PAGE 84

7KH JUDGLHQW YDULDEOHV DeWf DQG 3Af GHWHUPLQH WKH FRPSOH[LW\ RI WKH SURFHGXUH 7KHUH DUH PD[LPDO 1. YDULDEOHV ! HDFK RI ZKLFK LV HYDOXDWHG E\ (T DW D FRVW 1f SHU WLPH VWHS 7KXV WKH WRWDO FRVW LV 1.f SHU WLPH VWHS ,W G( IROORZV IURP (T WKDW WKH HYDOXDWLRQ RI Af§ UHTXLUHV D FRVW 21.f SHU WLPH VWHS GIW G( 7KH FRVW RI HYDOXDWLQJ RU Wf LV f 6LQFH WKHUH DUH 1 YDULDEOHV LW IROORZV WKDW WKH WRWDO FRVW SUH WLPH VWHS IRU PHPRU\ DGDSWDWLRQ LG 2L(3.f 7KH VSDFH FRVWV DUH GRPLQDWHG E\ 3AUf UHTXLULQJ 1.f PHPRU\ ORFDWLRQV 1RWH DJDLQ WKDW WKH JUDGLHQWV Dr Lf DQG Wf FDQQRW EH FRPSXWHG ORFDOO\ LQ VSDFH EXW DOO FRPSXWDWLRQV DUH ORFDO LQ WLPH VLQFH WKH DOJRULWKP LV UHDOWLPH 7KH UHVXOWV IRU 575/ DQG RWKHU DOJRULWKP DUH VXPPDUL]HG LQ )LJXUH %DFNSURSDHDWLRQ LQ WKH *DPPD 1HW ,Q WKLV VHFWLRQ WKH EDFNSURSDJDWLRQ SURFHGXUH DV GHULYHG LQ VHFWLRQ LV JHQHUDOL]HG WR WKH JDPPD QHXUDO QHW :H VWDUW E\ GHILQLQJ WKH OLVW / WKDW KROGV WKH RUGHU RI HYDOXDWLRQ RI WKH V\VWHP YDULDEOHV ,W LV DVVXPHG WKDW WKH DFWLYDWLRQV [LNWf DUH HYDOXDWHG LQ WKH RUGHU DV VFKHPDWLFDOO\ VSHFLILHG E\ )LJXUH IRU W WR 7 GR IRU L WR 1 GR IRU N WR . GR HYDOXDWH [LN: HQGHQGHQG )LJXUH (YDOXDWLRQ RUGHU LQ JDPPD PRGHO 7KLV OHDGV WR WKH IROORZLQJ OLVW

PAGE 85

/f§ > ^-$` ! ^Z\MIF` -&M ff;WI.fL -IM f[QN7f @ (TO 7KH VDPH SHUIRUPDQFH LQGH[ DV GHILQHG IRU WKH 575/ SURFHGXUH LV XVHG WKDW LV e !Pnf@ (T W P 5HFDOO WKDW LQ RUGHU WR FRPSXWH WKH HUURU JUDGLHQWV LQ WKH EDFNSURSDJDWLRQ DOJRULWKP ZH PDNH XVH RI :HUERVf¬ IRUPXOD IRU RUGHUHG GHULYDWLYHV ZKLFK HYDOXDWHV IRU WKH DFWLYDWLRQV [LNWf WR H G( GH( A G( G[MW[f A 7 A 77 < f§[ (DO G[LN Wf G[LN Wf [-nNf G;Mc [f G[LN 7KH H[SUHVVLRQ [ M Of ! W Nf XQGHU WKH VXPPDWLRQ VLJQ UHIHUV WR DOO LQGH[ FRPELQDWLRQV [M Of WKDW DSSHDU DIWHU L L Nf LQ WKH OLVW / $OWKRXJK FXPEHUVRPH ZRUNLQJ RXW (T LV VWUDLJKWIRUZDUG ,Q RUGHU WR VLPSOLI\ (T WKH WZR FDVHV N DQG Nr KDYH WR EH FRQVLGHUHG G( G[Qbf )LUVW Nf§UU LV ZRUNHG RXW N f ,Q RUGHU WR HYDOXDWH WKH IDFWRU P G[c f f¬ G[c Wf H[SUHVVLRQ (T ZH QHHG WR ILQG WKH DFWLYDWLRQV ;MO [f WKDW H[SOLFLWO\ GHSHQG RQ [cWf ,W IROORZV IURP WKH JDPPD V\VWHP HTXDWLRQV (T DQG (T WKDW RQO\ [cc W f DQG ;M Wf M!Lf GLUHFWO\ GHSHQG RQ [c Wf 7KXV (T HYDOXDWHV WR G( G( G( Af§U HWf OL f§ \ DQHWWffZ[ A(TO G[cWf nZ +nD[QUf IM. S G[MWf 1H[W WKH IRUPXOD IRU RUGHUHG GHULYDWLYHV (TO LV HYDOXDWHG IRU WKH WDS YDULDEOHV [LNWf IRU N $SSO\LQJ (TO WR WKH JDPPD VWDWH HTXDWLRQV \LHOGV

PAGE 86

G( G[cNWf G( G( %( A G[A-77f er nm!f MLW)M: (J G( 7KH HTXDWLRQV (T DQG (T EDFNSURSDJDWH WKH JUDGLHQWV Af§f§ 1RWH R[LMF^Wf WKDW WKH JUDGLHQWV DW WLPH W DUH D IXQFWLRQ RI WKH JUDGLHQWV DW WLPH U 7KHUHIRUH WKH EDFNSURSDJDWLRQ V\VWHP KDV WR EH UXQ EDFNZDUGV LQ WLPH WKDW LV IURP W 7 EDFNZDUGV WR W ,Q IDFW WKLV LV DOVR FOHDU ZKHQ ZH UHFDOO WKDW WKH OLVW / LV UXQ EDFNZDUGV GXULQJ WKH EDFNSURSDJDWLRQ SDVV )RU WKLV UHDVRQ WKH SURFHGXUH GHVFULEHG KHUH LV FDOOHG EDFNSURSDJDWLRQWKURXJKWLPH %377f 1H[W WKH HUURU JUDGLHQWV DUH FRPSXWHG ZLWK UHVSHFW WR WKH V\VWHP SDUDPHWHUV E\ DSSO\LQJ (T WR WKH OLVW / )RU WKH ZHLJKWV ZLMN ZH JHW G( B G( A < Af§WW2 QHW$Wff[LUWf GZMN AG[W^Wf Y -N. n (R DQG IRU WKH JUDGLHQWV G( G( G( W¯G[cNm (D ,Q )LJXUH WKH VHW RI HTXDWLRQV WKDW GHVFULEH WKH EDFNSURSDJDWLRQ PHWKRG IRU WKH JDPPD PRGHO LV VXPPDUL]HG ,Q )LJXUH IRU FRQYHQLHQFH WKH QRWDWLRQ G( G( ecNWf A DQG c Wf A LV XVHG IRU WKH EDFNSURSDJDWLRQ HUURUV 7KH WHPSRUDO DVSHFW RI JDPPD EDFNSURSDJDWLRQ LPSDFWV WKH XVH RI WKLV SURFHGXUH VXEVWDQWLDOO\ 6LPLODUO\ WR UHJXODU EDFNSURSDJDWLRQ WKH EDFNSURSDJDWLRQ

PAGE 87

)LJXUH %DFNSURSDJDWLRQWKURXJKWLPH HTXDWLRQV IRU WKH JDPPD cQHXUDO PRGHO QHWZRUN LV RI WKH VDPH FRPSOH[LW\ DV WKH IRUZDUG SDVV QHW 1RWH WKDW WKLV DOJRULWKP LV QRW ORFDO LQ WLPH WKH EDFNSURSDJDWLRQ HUURUV FDQ RQO\ EH FRPSXWHG DIWHU D FRPSOHWH HSRFK KDV HQGHG DW W 7f 7KXV UHDOWLPH OHDUQLQJ LV H[FOXGHG DV ZHOO $GGLWLRQDOO\ LW IROORZV WKDW WKH VWDWHV [LNWf DQG HUURUV HWf HcWf DQG f PXVW EH VWRUHG IRU WKH HQWLUH

PAGE 88

HSRFK 7KXV WKH VWRUDJH UHTXLUHPHQWV VFDOH E\ 2L1.7WI.f ILUVW WHUP IRU [LNWf DQG VHFRQG WHUP IRU ZL-Nf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
PAGE 89

EDFNSURSDJDWLRQ PHWKRG SURYLGHV DQ DOJRULWKPLF DSSURDFK WR VROYH D ODUJH DUHD RI SUREOHPV WKDW ZHUH SUHYLRXVO\ QRW DSSURDFKDEOH GXH WR WKH DPRXQW RI FRPSXWDWLRQ LQYROYHG 7KLV DGYDQWDJH LV QRW VR REYLRXV ZKHQ ZH JHQHUDOL]H %3 WR G\QDPLF QHWZRUNV VXFK DV WKH JDPPD PRGHO 7KLV SURFHGXUH LV YHU\ UHVWULFWHG LQ WKH VHQVH WKDW WKH VWRUDJH UHTXLUHPHQWV JURZ OLQHDUO\ ZLWK WLPH 7KH DOWHUQDWLYH PHWKRG UHDOWLPH UHFXUUHQWOHDPLQJ LPSRVHV D FRQVWDQW LQ WLPHf ORDG RQ WKH FRPSXWDWLRQDO UHVRXUFHV
PAGE 90

$UFKLWHFWXUH 7KH IRFXVHG JDPPD QHW LV VFKHPDWLFDOO\ GUDZQ LQ )LJXUH $VVXPH WKH GLPHQVLRQDO LQSXW VLJQDO If 7KH SDVW RI WKLV VLJQDO LV UHSUHVHQWHG LQ D JDPPD PHPRU\ VWUXFWXUH DV GHVFULEHG E\ [LWf ,LWf (D-,6 ;LNWf ¯A[ALWOf ?¯c[LWNLWOf %DGLO ZKHUH W U L OL DQG N 7KLV OD\HU WKH LQSXW OD\HU KDV L PHPRU\ SDUDPHWHUV ; 7KH DFWLYDWLRQV LQ WKH LQSXW OD\HU DUH PDSSHG RQWR D VHW RI RXWSXW QRGHV E\ ZD\ RI D QRQOLQHDUf VWDWLF VWULFWO\ IHHGIRUZDUG QHW 7KH QRGHV LQ WKH IHHGIRUZDUG QHW DUH LQGH[HG LO WKURXJK 1 7KXV WKLV PDS FDQ EH ZULWWHQ DV IURP IHHGIRUZDUG QHW IURP LQSXW OD\HU L M N X )RU FRQYHQLHQFH (T ZLOO EH ZULWWHQ DV r r ( ;ZLMN[MN "fff¬ %DGLO ML N ZKHUH ZH KDYH XWLOL]HG WKH QRWDWLRQ [L4 Wf [ Wf DQG ZAT ZcM 6LPLODU DUFKLWHFWXUHV KDYH EHHQ XVHG E\ 6WRUQHWWD HW DO f DQG 0R]HU f 7KHVH LQYHVWLJDWRUV KRZHYHU RQO\ XVHG D ILUVWRUGHU PHPRU\ VWUXFWXUH . f 0R]HU DQDO\]HG VRPH RI WKH SURSHUWLHV RI VWUXFWXUHV RI WKLV NLQG DQG FRLQHG WKH WHUP IRFXVHG EDFNSURSDJDWLRQ DUFKLWHFWXUH ,W WXUQV RXW WKDW WKH IRFXVHG QHWZRUN DUFKLWHFWXUH HQMR\V D QXPEHU RI DGYDQWDJHV LQ FRPSDULVRQ WR WKH IXOO\ FRQQHFWHG rmnf rL ; 9If ZLMN[MNA (R G\QDPLF QHWZRUNV /HW XV ILUVW GHULYH WKH XSGDWH HTXDWLRQV IRU WKH ZHLJKWV ZLMN 7KH

PAGE 91

EDFNSURSDJDWLRQ PHWKRG ZLOO EH XVHG $V EHIRUH WKH GHULYDWLRQV DUH EDVHG RQ WKH SHUIRUPDQFH LQGH[ ( L A >HP Wf @ DQG WKH HYDOXDWLRQ RUGHU LV GHWHUPLQHG E\ WKH W P OLVW / > ^IL` ^ZcMN` Wf ` @ :H KDYH DOUHDG\ GLVFXVVHG LQ VHFWLRQ KRZ WR DSSO\ EDFNSURSDJDWLRQ WR IHHGIRUZDUG QHWV 7KXV DSSO\LQJ :HUERVf¬ IRUPXOD IRU RUGHUHG GHULYDWLYHV WR WKH DFWLYDWLRQV [cWf LQ WKH IHHGIRUZDUG QHW OHDGV WR WKH IROORZLQJ EDFNSURSDJDWLRQ V\VWHP H Wf HL ZMILM (TM!c Dn QHWW ffH (Tf± DV ZKHUH ZH GHILQHG e Nf§UU DQG aU 6LPLODUO\ LW IROORZV IURP G[cWf nY GQHWWf G( VHFWLRQ WKDW WKH JUDGLHQWV A FDQ EH FRPSXWHG DV

PAGE 92

( f§ AUf (DLMN W 1RWH WKDW VLQFH WKH PDSSLQJ QHWZRUN LV VWDWLF DQG IHHGIRUZDUG ZH GR QRW QHHG WR EDFNSURSDJDWH WKURXJK WLPH LQ RUGHU WR ILQG WKH EDFNSURS HUURUV ,Q IDFW VLQFH WKH Lf f«V DUH FRPSXWHG LQ UHDOWLPH LW LV FRQYHUW (T LQWR D UHDOWLPH SURFHGXUH E\ GHILQLQJ %H 8fa%:[-N8f (DLMN $SSOLFDWLRQ RI EDFNSURSDJDWLRQ WR FRPSXWH WKH HUURU JUDGLHQWV ZLWK UHVSHFW WR WKH SDUDPHWHUV L OHDGV WR D EDFNSURSDJDWLRQWKURXJKWLPH SURFHGXUH VLQFH WKH LQSXW OD\HU LV UHFXUUHQW LQ QDWXUH ,Q PRVW QHWZRUNV KRZHYHU WKH QXPEHU RI PHPRU\ SDUDPHWHUV LV UHODWLYHO\ VPDOO VR LW LV HIILFLHQW WR XVH WKH GLUHFW PHWKRG KHUH 7KLV SURFHGXUH KDV DOUHDG\ EHHQ GHULYHG IRU WKH PRUH JHQHUDO JDPPD QHWV LQ VHFWLRQ G( +HQFH ZLWKRXW H[SODQDWLRQ ZH GHULYH WKH HUURU JUDGLHQWV f§ DV IROORZV G_; ( A A G( Z GrP Gr9 P;PA G[LN: aO¯HP:DPnQHWP:n/ZPLNDW: N P (T ZKHUH DrUf f§A DcWf FDQ EH FRPSXWHG E\ HYDOXDWLRQ RI (T G[LNWf $Q LPSRUWDQW SURSHUW\ LV WKDW WKH EDFNSURSDJDWLRQ SDWK LV VKRUW VLQFH WKH IHHGIRUZDUG QHW LV VWDWLF 7KH HUURUV HVWLPDWHV LQ WKH G\QDPLF LQSXW OD\HU GR QRW GLVSHUVH GXULQJ WUDLQLQJ VLQFH WKH JDPPD PHPRU\ VWUXFWXUHV GR QRW KDYH ODWHUDO FRQQHFWLRQV LQ WKH LQSXW OD\HU 7KLV SURSHUW\ LV FRQILUPHG E\ FRQVLGHULQJ (T ZKLFK SURSDJDWHV WKH HUURUV WKURXJK WLPH 7KXV WKH HUURU HVWLPDWHV GR QRW GLVSHUVH LQ

PAGE 93

WKH IRFXVHG JDPPD QHW ZKLFK H[SODLQV WKH DGMHFWLYH f¯IRFXVHGf° HDPPD QHW DUFKLWHFWXUH 1 XQLWV PHPRU\ RUGHU . 7 WLPH VWHSV 575/ OOLOOLOILOLLO )2&86(' ; M\ OOOOAS£F«OLL 1.f 1.7f 91.f Hr R _ OLWLUULHILL 1.7f .AL37f 2LUIt7f !! r ,_L_L6S£&H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f WKH LQSXW SDWWHUQ
PAGE 94

IRFXVHG JDPPD QHW 1RWH WKDW D OLQHDU RQHOD\HU IRFXVHG JDPPD QHW JHQHUDOL]HV :LGURZf¬V DGDOLQH VWUXFWXUH

PAGE 95

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

PAGE 96

DGDOLQHMLf 0RUH FRPSOH[ DUFKLWHFWXUHV DUH FHUWDLQO\ SRVVLEOH EXW LQ WKLV ZRUN ZH DUH PDLQO\ LQWHUHVWHG LQ D FRPSDUDWLYH HYDOXDWLRQ RI WKH JDPPD PHPRU\ VWUXFWXUH SHU VH 7KH WRSLF RI GHVLJQLQJ FRPSOH[ JOREDOO\ UHFXUUHQW QHXUDO QHW DUFKLWHFWXUHV ZLWK JDPPD PHPRU\ LV QRW DGGUHVVHG KHUH $OVR H[SHULPHQWDO HYDOXDWLRQ RI QHXUDO QHWZRUNV LQ UHODWLRQ WR DOWHUQDWLYH QRQQHXUDO SURFHVVLQJ WHFKQLTXHV LV QRW SUHVHQWHG KHUH ZLWK WKH H[FHSWLRQ RI WKH QRLVH UHGXFWLRQ H[SHULPHQWVf 7KH ODWWHU WRSLF KDV EHHQ VWXGLHG E\ D VSHFLDO '$53$ FRPPLWWHH '$53$ f %HIRUH WKH H[SHULPHQWDO UHVXOWV DUH SUHVHQWHG VRPH JHQHUDO SUDFWLFDO LVVXHV FRQFHUQLQJ JDPPD QHW VLPXODWLRQ DQG DGDSWDWLRQ DUH GLVFXVVHG *DPPD 1HW 6LPXODWLRQ DQG 7UDLQLQJ ,VVXHV 7KH V\VWHP DUFKLWHFWXUH WKDW LV XVHG LQ WKH H[SHULPHQWV LV VKRZQ LQ )LJXUH 7KH VLJQDO WR EH SURFHVVHG VRXUFH VLJQDO LV GHQRWHG E\ VWf %RWK WKH QHXUDO QHW LQSXW VLJQDO DQG WKH GHVLUHG VLJQDO GWf DUH GHULYHG IURP VWf 7KH SDUWLFXODU IRUP RI WKLV WUDQVIRUPDWLRQ GHSHQGV RQ WKH SURFHVVLQJ JRDO $ VXEVHW RI WKH QHXUDO QHW VWDWHV [Wf WKH RXWSXWV DUH PHDVXUHG DQG FRPSDUHG WR WKH GHVLUHG VLJQDOV 7KH GLIIHUHQFH VLJQDO HWf GWf [Wf LV FDOOHG WKH LQVWDQWDQHRXV HUURU VLJQDO DQG LW LV XVHG DV WKH LQSXW WR WKH VWHHSHVW GHVFHQW WUDLQLQJ SURFHGXUH VWR, P IRFXVHG VWf JDPPD SIHW GWf HWf )LJXUH ([SHULPHQWDO DUFKLWHFWXUH

PAGE 97

*DPPD 1HW $GDSWDWLRQ 7KH WUDLQLQJ VWUDWHJ\ RI WKH JDPPD QHXUDO QHW GHVHUYHV PRUH DWWHQWLRQ ,Q DOO FDVHV WKH QHWZRUN SDUDPHWHUV Z DQG S ZHUH DGDSWHG XVLQJ WKH IRFXVHG EDFNSURSDJDWLRQ PHWKRG DV GHULYHG LQ FKDSWHU :H XVHG WKH VLPSOH VWHHSHVW GHVFHQW XSGDWH PHWKRG G( WKDW LV $Z 7_A ,Q DOO H[SHULPHQWV ZH XVHG UHDOWLPH XSGDWLQJ 7KXV WKH ZHLJKWV ZHUH DGDSWHG DIWHU HDFK QHZ VDPSOH 7KH VWHSVL]H OHDUQLQJ UDWHf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f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f ,Q SUDFWLFH LW KDV EHHQ IRXQG WKDW JUDGLHQW GHVFHQW SURFHGXUHV ILUVW DGDSW WR WKH JURVV IHDWXUHV RI WKH WUDLQLQJ

PAGE 98

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f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

PAGE 99

WLPH 7KH DGDSWDWLRQ WLPH LV WKH QXPEHU RI SDWWHUQV RU HSRFKV IRU EDWFK OHDUQLQJf WKDW KDYH WR EH SUHVHQWHG WR WKH QHXUDO QHW EHIRUH WKH ZHLJKWV FRQYHUJH 7KHUH LV QRW PXFK WKHRU\ DERXW WKH DGDSWDWLRQ WLPH RI EDFNSURSDJDWLRQ DOJRULWKPV +RZHYHU WKH TXHVWLRQ ZKHWKHU DQG KRZ WKH YDOXH RI IL DIIHFWV WKH DGDSWDWLRQ WLPH FDQ EH H[SHULPHQWDOO\ WDFNOHG 7KLV SUREOHP ZDV VWXGLHG IRU D WKLUGRUGHU HOOLSWLF ILOWHU PRGHOLQJ SUREOHP ZKLFK LV FRYHUHG LQ PRUH GHWDLO LQ VHFWLRQ 7KH DUFKLWHFWXUH ZDV DQ DGDOLQHM[f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

PAGE 100

1RQ0LQHDU 3UHGLFWLRQ RI D &RPSOH[ 7LPH 6HULHV 3UHGLFWLRQ1RLVH 5HPRYDO RI 6LQXVRLGDOV FRQWDPLQDWHG EY *DXVVLDQ 1RLVH :H FRQVWUXFWHG DQ LQSXW VLJQDO FRQVLVWLQJ RI D VXP RI VLQXVRLGV FRQWDPLQDWHG E\ DGGLWLYH ZKLWH JDXVVLDQ QRLVH $:*1f 6SHFLILFDOO\ f ZDV GHVFULEHG E\ VLQ LW ff P W f f VLQ LF ff VLQ Q ff $:*1 (R 7KH VLJQDOWRQRLVH UDWLR LV G% 7KLV VLJQDO LV VKRZQ LQ )LJXUH 7KH SURFHVVLQJ JRDO ZDV WR SUHGLFW WKH QH[W VDPSOH RI WKH VXP RI VLQXVRLGDOV +HQFH WKH SURFHVVLQJ SUREOHP LQYROYHV D FRPELQDWLRQ RI SUHGLFWLRQ DQG QRLVH FDQFHODWLRQ 7KH SURFHVVLQJ V\VWHP ZDV DGDOLQHSf 7KH JRDOV RI WKLV H[SHULPHQW DUH WKH IROORZLQJ 'HWHUPLQH WKH RSWLPDO V\VWHP SHUIRUPDQFH DV D IXQFWLRQ RI S IRU S DQG . 1RWH WKDW WKLV LPSOLHV D FRPSDULVRQ RI WKH JDPPD PHPRU\ VWUXFWXUH YHUVXV WKH WDSSHG GHOD\ OLQH IRU S Of DQG WKH FRQWH[WXQLW PHPRULHV IRU . Of &DQ WKH V\VWHP SDUDPHWHUV ZN DQG S EH DGDSWHG WR FRQYHUJH WR WKH RSWLPDO YDOXHV"

PAGE 101

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f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

PAGE 102

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ƒ N ?? ? ? ? ? ? . 9 AA Af§. 7f¯ « HSRFK QR )LJSUW" 7KH WUDFNV RI S YHUVXV HSRFK QXPEHU GXULQJ DGDSWDWLRQ IRU WKH VLQXVRLGDO SUHGLFWLRQ H[SHULPHQW 7KH UHVXOWV IRU WKLV H[SHULPHQW DUH YHU\ SRVLWLYH 7KH JDPPD PHPRU\ VWUXFWXUH SURYLGHV DQ DGDSWLYH PHGLXP EHWZHHQ WKH FRQWH[WXQLW PHPRU\ DQG WKH WDSSHG GHOD\ OLQH ,Q JHQHUDO WKH RSWLPDO YDOXHV IRU S DQG . GR QRW FRLQFLGH ZLWK HLWKHU RI WKHVH WZR H[WUHPHV $Q H[DPSOH ZKHUH WKH RSWLPDO PHPRU\ VWUXFWXUH LV QHLWKHU D FRQWH[WXQLW QRU D WDSSHG GHOD\ OLQH LV SUHVHQWHG QH[W

PAGE 103

3UHGLFWLRQ RI DQ ((* 6OHHS 6WDJH 7ZR 6HJPHQW $Q HOHFWURHQFHSKDORJUDP ((*f GDWD VHJPHQW IURP VOHHS VWDJH WZR ZDV VHOHFWHG VHH)LJXUH Df 7KH SURFHVVLQJ V\VWHP ZDV D WZROD\HU IRFXVHG OLQHDU JDPPD QHWZRUN ZLWK ILYH KLGGHQ XQLWV 7KH SURFHVVLQJ JRDO ZDV WR SUHGLFW WKH ((* VLJQDO E\ ILYH VDPSOHV DKHDG 7KH SHUIRUPDQFH LQGH[ DIWHU FRQYHUJHQFH LV VKRZQ LQ )LJXUH E $V ZDV WKH FDVH IRU WKH SUHYLRXV H[SHULPHQW WKH SHUIRUPDQFH LQGH[ YHUVXV S LV DQ 8ERZHG FXUYH LQGLFDWLQJ WKDW WKHUH LV RQH RSWLPDO S $SSDUHQWO\ WKH SHUIRUPDQFH RI WKH VHFRQG PHPRU\ RUGHU QHW IRU S LV FRPSDUDEOH WR WKH IRXUWK RUGHU QHW ZLWK D WDSSHG GHOD\ OLQH S f 7KH V\VWHP ZLWK VHFRQG RUGHU PHPRU\ LV WR EH SUHIHUUHG VLQFH WKH QXPEHU RI ZHLJKWV LV OHVV IRU WKLV VWUXFWXUH WKDQ IRU WKH IRXUWK RUGHU PHPRU\ PRGHO ,Q JHQHUDO UHGXFWLRQ RI WKH QXPEHU RI ZHLJKWV SURYLGHV EHWWHU JHQHUDOL]DWLRQ FKDUDFWHULVWLFV DQG IDVWHU DGDSWDWLRQ WLPH )LJXUH ((* SUHGLFWLRQ UHVXOWV IRU D WZROD\HU OLQHDU JDPPD PRGHO ([SHULPHQWDO SDUDPHWHUV LQSXW XQLW . JDPPD PHPRU\ WDSV KLGGHQ XQLWV RXWSXW XQLW SUHGLFWLRQ VWHSV Df WKH ((* VHJPHQW ZDV WDNHQ IURP VOHHS VWDJH WZR Ef QRUPDOL]HG SHUIRUPDQFH LQGH[ DIWHU FRQYHUJHQFH DV D IXQFWLRQ RI S 3UHGLFWLRQ RI 0DFNHY*ODVV FKDRWLF 7LPH VHULHV 7KH 0DFNH\*ODVV V\VWHP LV IUHTXHQWO\ XVHG DV D EHQFKPDUN IRU WKH HYDOXDWLRQ RI YDULRXV SUHGLFWLRQ DOJRULWKPV /DSHGHV DQG )DUEHU 0DFNH\ DQG *ODVV f ,Q WKLV VHFWLRQ WZR JRDOV ZLOO EH HVWDEOLVKHG )LUVW LW ZLOO EH VKRZQ WKDW WKH JDPPD PHPRU\ SHUIRUPV EHWWHU WKDQ WKH WDSSHG GHOD\ OLQH PHPRU\ VWUXFWXUH ZKHQ WKH DGDOLQH

PAGE 104

VWUXFWXUH LV XVHG WR SUHGLFW WKH QH[W VDPSOH RI WKH WLPH VHULHV 6HFRQGO\ DQG PD\EH PRUH LQWHUHVWLQJ LW LV VKRZQ WKDW WKH JDPPD PHPRU\ FDQ EH XVHG DV D V\VWHP RUGHU HVWLPDWRU 7KH 0DFNH\*ODVV V\VWHP LV GHVFULEHG E\ WKH IROORZLQJ GHOD\GLIIHUHQWLDO HTXDWLRQ rIf r U$f [I$f (D 7KH FRPSOH[LW\ RI WKH G\QDPLFV RI WKH V\VWHP LV UHJXODWHG E\ WKH GHOD\ SDUDPHWHU $ 7KH 0DFNH\*ODVV WLPH VHULHV IRU $ LV GLVSOD\HG LQ )LJXUH :H JHQHUDWHG D GLVFUHWH WLPH VHULHV E\ VDPSOLQJ WKH FRQWLQXRXV WLPH VLJQDO E\ SHULRG 7V 1H[W DQ DGDOLQH QHWZRUN ZDV WUDLQHG WR SUHGLFW WKH QH[W VDPSOH RI WKH VDPSOHG WLPH VHULHV 7KH WUDLQLQJ VHW FRQVLVWHG RI VDPSOHV DQG WKH YDOLGDWLRQ VHW RI WKH QH[W VDPSOHV 7KH QRUPDOL]HG WRWDO HUURU DIWHU FRQYHUJHQFH YHUVXV WKH QXPEHU RI GHOD\V ILOWHU RUGHUf LQ WKH DGDOLQH VWUXFWXUH LQ GLVSOD\HG LQ )LJXUH $V H[SHFWHG WKH VHULHV LV KDUGHU WR SUHGLFW ZKHQ WKH GHOD\ SDUDPHWHU $ LQFUHDVHV 1H[W D OLQHDU RQHOD\HU JDPPD QHW DGDOLQHM[ff ZDV WUDLQHG WR SHUIRUP WKH VDPH WDVN 7KH ILOWHU RUGHU . ZDV IL[HG WR . DQG WKH PHPRU\ SDUDPHWHU S ZDV SDUDPHWUL]HG

PAGE 105

)LJXUH $G£OLQH SUHGLFWLRQ UHVXOWV IRU 0DFNH\*ODVV VHULHV EHWZHHQ DQG 5HVXOWV DIWHU WUDLQLQJ FRQYHUJHQFH DUH VKRZQ LQ )LJXUH )LJXUH $GDOLQF_Lf SHUIRUPDQFH YHUVXV _L IRU RQH WLPH VWHS SUHGLFWLRQ RI 0DFNH\*ODVV VHULHV ([SHULPHQWDO SDUDPHWHUV VDPSOLQJ SHULRG 7V JDPPD RUGHU . 7KH SHUIRUPDQFH FXUYHV VKRZ WKH QRZ IDPLOLDU 8ERZ ZKLFK ZH IRXQG WR EH FKDUDFWHULVWLF IRU WKH JDPPD PHPRU\ SHUIRUPDQFH YHUVXV _L 7KHUH LV DQ LQWHUHVWLQJ DSSOLFDWLRQ RI WKH FXUYHV RI WKH IRUP LQ )LJXUH :H KDYH DOUHDG\ GLVFXVVHG WKDW WKH PHDQ PHPRU\ GHSWK RI WKH JDPPD PHPRU\ FDQ EH HVWLPDWHG E\ .?L 7KXV IRU $

PAGE 106

WKH RSWLPDO GHSWK FRPHV RXW WR . .3W DQG IRU $ ZH ILQG . f§f§ f§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f HTXDO WKH V\VWHPf¬V RXWSXWV GWf WKH XQNQRZQ V\VWHP KDV EHHQ LGHQWLILHG E\ PHDQV RI LWV WUDQVIHU IXQFWLRQ $GDOLQHMLf ZDV XVHG WR DSSUR[LPDWH WKH WKLUG RUGHU HOOLSWLF ILOWHU DV GHVFULEHG

PAGE 107

E\ +]f ]a ]a ]a 8 ]a ] (J :KLWH JDXVVLDQ QRLVH ZDV XVHG DV WKH VRXUFH VLJQDO %RWK WKH WUDLQLQJ DQG YDOLGDWLRQ VHW FRQVLVWHG RI VDPSOHV ,Q )LJXUH WKH SHUIRUPDQFH LQGH[ DIWHU FRQYHUJHQFH LV VKRZQ 1RWH WKDW WKH RSWLPDO PHPRU\ GHSWK ' W a LV FRQVWDQW IRU GLIIHUHQW ARSW PHPRU\ RUGHUV :KHQ . DGDOLQHOf SHUIRUPV DV ZHOO DV . IRU DGDOLQHf +RZHYHU ZH VWLOO SUHIHU . VLQFH WKLV VWUXFWXUH KDV IUHH SDUDPHWHUV ZKHUHDV DGDOLQH IRU . f XVHV SDUDPHWHUV 3DUVLPRQ\ LQ WKH QXPEHU RI IUHH SDUDPHWHUV SURYLGHV DGDOLQHf ZLWK EHWWHU PRGHOLQJ JHQHUDOL]DWLRQf FKDUDFWHULVWLFV 7HPSRUDO 3DWWHUQ &ODVVLILFDWLRQ 7UDLQLQJ D &RQFHQWUDWLRQLQ7LPH 1HW ,Q 7DQN DQG +RSILHOG LQWURGXFHG WKH &RQFHQWUDWLRQLQ7LPH &,7f QHXUDO QHW D PRGHO IRU WHPSRUDO SDWWHUQ FODVVLILFDWLRQ 7DQN DQG +RSILHOG f 7KH &,7 QHXUDO QHW KDV EHHQ GLVFXVVHG LQ WKLV WKHVLV LQ VHFWLRQ ,Q WKH &,7 QHW LQSXW

PAGE 108

W D SDWWHUQV DUH GLVSHUVLYHO\ GHOD\HG E\ GHOD\ IXQFWLRQV RI WKH IRUP Af H 7KH GHOD\HG LQSXW VLJQDOV DUH PXOWLSOLHG E\ D ZHLJKW IDFWRU DQG IXOO\ FRQQHFWHG WR D ZLQQHU WDNHDOO :7$f RXWSXW OD\HU $ :7$ OD\HU RI XQLWV LV D VHW RI QHXURQV WKDW ODWHUDOO\ LQKLELW HDFK RWKHU VXFK WKDW DW DQ\ WLPH PD[LPDOO\ RQH XQLW LV DFWLYDWHG 7KH ZHLJKWV IURP WKH GHOD\HG LQSXW VLJQDOV WR WKH RXWSXW OD\HU DUH IL[HG DQG D SULRUL GHWHUPLQHG VXFK WKDW DQ RXWSXW XQLW EHFRPHV DFWLYH DIWHU SUHVHQWDWLRQ RI D VSHFLILF LQSXW SDWWHUQ VHTXHQFH $V ZDV GLVFXVVHG LQ VHFWLRQ LQ 7DQN DQG +RSILHOGf¬V PRGHO WKH GHOD\HG VLJQDOV DUH FRPSXWHG E\ H[SOLFLW FRPSXWDWLRQ RI FRQYROXWLRQ LQWHJUDOV ,Q WKH JDPPD PRGHO D PRUH HIILFLHQW DGGLWLYH QHXUDO QHW PHFKDQLVP JHQHUDWHV WKH PHPRU\ VWDWHV ,W LV LQWHUHVWLQJ WR ILQG RXW ZKHWKHU WKH FRQFHQWUDWLRQLQWLPH PHFKDQLVP FDQ EH LPSOHPHQWHG E\ D JDPPD PHPRU\ VWUXFWXUH 7KXV ZH GHVLJQHG D VLPSOH WHPSRUDO FODVVLILFDWLRQ H[SHULPHQW DQG FRQFHQWUDWHG RQ WKH IROORZLQJ WZR DVSHFWV &DQ WKH &,7 QHXUDO QHW EH LPSOHPHQWHG PDNLQJ XVH RI WKH JDPPD PHPRU\ VWUXFWXUH" &DQ WKH ZHLJKWV FRQQHFWLQJ WKH LQSXW OD\HU WR WKH FODVVLILFDWLRQ OD\HU EH DGDSWHG E\ JUDGLHQW GHVFHQW OHDUQLQJ" 7KH DUFKLWHFWXUH RI WKH f¯JDPPD &,7f° QHXUDO QHW LV GLVSOD\HG LQ )LJXUH 7KLV DUFKLWHFWXUH LV VLPLODU WR 7DQN DQG +RSILHOGf¬V QHW ZLWK WKH GLIIHUHQFH WKDW WKH LQSXW VLJQDO PHPRU\ WUDFHV DUH JHQHUDWHG E\ WKH JDPPD PHPRU\ PHFKDQLVP 7KH LQSXW VLJQDO VSDFH FRQVLVWV RI VL[ UDQGRP SDWWHUQ VHTXHQFHV (DFK SDWWHUQ LQ D VHTXHQFH FRQVLVWV RI DQ HLJKW GLPHQVLRQDO ELQDU\ f SDWWHUQ (DFK SDWWHUQ ODVWV IRU VHFRQG DQG D VHTXHQFH FRQVLVWV RI HLJKW SDWWHUQV 7KXV DQ LQSXW VHTXHQFH ODVWV VHFRQGV 7KH JRDO LV WR WUDLQ WKH QHWZRUN WR FODVVLI\ WKH VL[ VHTXHQFHV 7KH LQSXW OD\HU FRQVLVWV RI HLJKW LQSXW QRGHV WKDW VWRUH WKH SDVW LQ DQ HLJKW GLPHQVLRQDO JDPPD PHPRU\ . f $V LQ 7DQN DQG +RSILHOGf¬V PRGHO ZH XVHG WKH FRQWLQXRXVWLPH JDPPD PRGHO DQG XVHG D

PAGE 109

IH IL[HG ; 7KXV WKH NWK PHPRU\ WUDFH RI VWf LV WKH VLJQDO VN Wf VWf fµ B 7KH V\VWHP ZDV VLPXODWHG E\ D IRXUWKRUGHU 5XQJH.XWWD PHWKRG 7KH GHVLUHG RXWSXW VLJQDOV DUH DV VKRZQ LQ )LJXUH 'XULQJ SUHVHQWDWLRQ RI WKH ODVW SDWWHUQ RI WKH VHTXHQFH ZH ZDQW RQH GHVLJQDWHG XQLWf¬V DFWLYDWLRQ WR EH $W DOO RWKHU WLPHV WKH DFWLYLW\ LQ WKH RXWSXW OD\HU VKRXOG EH $V IRU WKH RWKHU H[SHULPHQWV LQ WKLV FKDSWHU WKH V\VWHP SDUDPHWHUV ZN ZHUH RQOLQH DGDSWHG ZLWK WKH EDFNSURSDJDWLRQ PHWKRG 7KH OHDUQLQJ UDWH ZDV VHW DW W_ 7KH GHVLUHG DQG DFWXDO DFWLYDWLRQV RI WKH XQLWV LQ WKH FODVVLILFDWLRQ OD\HU DUH VKRZQ LQ )LJXUH 7KH WRWDO HUURU GHILQHG DV WKH LQWHJUDO RYHU WLPH RQH SUHVHQWDWLRQ RI DOO VHTXHQFHVf RI WKH VTXDUHG GLIIHUHQFH RI WKH GHVLUHG DQG DFWXDO DFWLYDWLRQV LV VKRZQ LQ )LJXUH )LJXUH VKRZV WKDW WKH JDPPD &,7 V\VWHP FDQ EH WUDLQHG YHU\ ZHOO DV D

PAGE 110

GHVLUHG RXWSXW DFWXDO RXWSXW RXWSXW XQLW , RXWSXW XQLW UAK $ ¾ :OnOn\ IW LUWW ‘fµ ? ? M ‘ ‘‘‘‘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f DQG FRQWDPLQDWHG LW ZLWK DGGLWLYH JDXVVLDQ QRLVH 7KH FRQYHQWLRQDO PHWKRG WR UHPRYH DGGLWLYHf QRLVH IURP D VLJQDO LQYROYHV DSSOLFDWLRQ RI D OLQHDU ILOWHU D PHWKRG ZKLFK UHTXLUHV SULRU NQRZOHGJH RI WKH IUHTXHQF\ VSHFWUD RI WKH VLJQDO DQG QRLVH ,I VXFK LQIRUPDWLRQ LV QRW DYDLODEOH RU LI WKH VLJQDO DQG QRLVH VSHFWUD RYHUODS WKH OLQHDU ILOWHULQJ

PAGE 111

)LJXUH 7KH WRWDO HUURU DV £ IXQFWLRQ RI HSRFK QXPEHU IRU WKH! ELQDU\ VHTXHQFH FODVVLILFDWLRQ H[SHULPHQW WHFKQLTXH LV LQDGHTXDWH $GDSWLYH QRLVH FDQFHODWLRQ DV LQWURGXFHG E\ :LGURZ UHTXLUHV WKH DYDLODELOLW\ RI D QRLVH VRXUFH ZKLFK LV FRUUHODWHG WR WKH QRLVH WKDW FRQWDPLQDWHG WKH VRXUFH VLJQDO :LGURZ DQG 6WHDPV f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
PAGE 112

H[SHULPHQWV 7KH DUJXPHQW LV WKDW WKH G\QDPLFV RI WKH ORZGLPHQVLRQDO VRXUFH VLJQDO FDQ EH OHDUQHG E\ DGDSWDWLRQ EXW WKH KLJKGLPHQVLRQDO QRLVH G\QDPLFV FDQQRW EH OHDUQHG E\ D ORZGLPHQVLRQDO QHXUDO QHW

PAGE 113

9DULRXV PHWKRGV IRU UHPRYLQJ DGGLWLYH QRLVH IURP VLJQDOV WKDW DUH JHQHUDWHG E\ FKDRWLF G\QDPLFDO V\VWHPV ZHUH FRPSDUHG +HUH ZH VKRZ VRPH UHVXOWV IRU WKH /RUHQW] V\VWHP FRQWDPLQDWHG E\ DGGLWLYH ZKLWH JDXVVLDQ QRLVH $:*1f 7KH /RUHQW] V\VWHP ZDV GLVFRYHUHG E\ (1 /RUHQW] ZKHQ KH ZRUNHG RQ D PRGHO IRU ZHDWKHU IRUHFDVWLQJ /RUHQW] f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

PAGE 114

JDPPD PHPRU\ 7KH PHPRU\ SDUDPHWHU _L LV DGDSWLYH IRU HDFK XQLW $OO V\VWHP SDUDPHWHUV ZHUH DGDSWHG RQOLQH E\ WKH IRFXVHG EDFNSURSDJDWLRQ PHWKRG 7KH LQSXW OD\HU IHHGV WR D WZROD\HU VWDWLF QHXUDO QHW RQH KLGGHQ OD\HU WZHQW\ XQLWVf DQG DQ RXWSXW OD\HU IRXU XQLWVff 2QO\ WKH XQLWV LQ WKH KLGGHQ OD\HU KDYH D QRQOLQHDU WUDQVIHU IXQFWLRQ DOO RWKHU XQLWV DUH OLQHDU 7KH WDUJHW SDWWHUQV ZHUH FRSLHG IURP WKH LQSXW OD\HU DFWLYDWLRQV :KHQ WKH WLPH VHULHV ZDV UHFRQVWUXFWHG IURP WKH DFWLYDWLRQV RI WKH RXWSXW XQLWV D VLJQDOWRQRLVH UDWLR RI G% LV REWDLQHG :KHQ D WKUHH GLPHQVLRQDO EDVLV LV XVHG DQG KHQFH WKUHH LQSXW DQG RXWSXW XQLWVf WKH SHUIRUPDQFH RI WKH V\VWHP LV VWLOO G% 7KH UHVXOWV DUH VKRZQ LQ )LJXUH 1RWH WKDW WKH VSHFWUD RI WKH VLJQDOV WKDW ZHUH SURFHVVHG E\ WKH JDPPD QHW PDWFK WKH RULJLQDO VLJQDO VSHFWUXP ZHOO LQ FRPSDULVRQ WR WKH ORZ SDVV ILOWHULQJ WHFKQLTXH ZKLFK DWWHQXDWHG WKH /RUHQW] VLJQDO WRR PXFK LQ WKH KLJK IUHTXHQF\ UHJLRQ $QRWKHU LQWHUHVWLQJ REVHUYDWLRQ ZDV PDGH FRQFHUQLQJ WKH RSWLPDO YDOXHV RI WKH JDPPD PHPRU\ SDUDPHWHUV M[ )RU WKH IRXU GLPHQVLRQDO EDVLV WKH YDOXHV RI M[ DIWHU WUDLQLQJ ZHUH DQG )RU WKH WKUHH GLPHQVLRQDO EDVLV WKH Sf¬V

PAGE 115

FRQYHUJHG WR DQG 7KH SDWWHUQ RI PRQRWRQRXV GHFUHDVLQJ YDOXHV ZDV H[SODLQHG DV IROORZV 5HFDOO WKDW WKH FRQWDPLQDWHG VLJQDO ZDV SURMHFWHG RQ WKH ILUVW IRXU RU WKUHHf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

PAGE 116

PHPRU\" ,Q RUGHU WR DQVZHU WKLV TXHVWLRQ WKH QHXUDO QHW H[SHULPHQWV ZHUH UHSHDWHG ZLWK _[ IL[HG WR O)LJXUH VKRZV WKH UHVXOWV 7KLV UHGXFHV WKH JDPPD PHPRU\ WR D WDSSHG GHOD\ OLQH $ IRXU GLPHQVLRQDO EDVLV YHFWRU VSDFH ZDV XVHG DQG WKH QXPEHU RI GHOD\V ZHUH VHW WR IRXU WKDW LV . FRPSDUH WR RQO\ . IRU WKH JDPPD QHWf $ KLJKHU RUGHU PHPRU\ ZDV VHOHFWHG LQ RUGHU WR FRPSHQVDWH IRU WKH ORVV RI GHSWK RI PHPRU\ ZKHQ S LV VHW WR 7KLV V\VWHP ZDV DEOH WR HQKDQFH WR 615 RQO\ WR G% 7KXV WKH DGDSWLYH JDPPD PHPRU\ SHUIRUPHG PXFK EHWWHU WKDQ WKH WDSSHG GHOD\ OLQH 7KH UHVXOWV RI WKH QRLVH UHGXFWLRQ H[SHULPHQWV DUH VXPPDUL]HG LQ 7DEOH SURFHVVLQJ WHFKQLTXH JDLQ LQ G% ORZ SDVV ILOWHULQJ f± GLP EDVLV JDPPD QHW GLP EDVLV QHXUDO QHW ZLWK S 7DEOH 5HVXOWV RI $:*1 UHGXFWLRQ IRU SURFHVVLQJ WHFKQLTXHV YDULRXV ,Q FRQFOXVLRQ WKH SUHGLFWLRQ SURWRFRO ZRUNV ZHOO IRU D QRLVH UHGXFWLRQ WDVN $OVR WKH DGDSWLYH JDPPD PRGHO SHUIRUPV EHWWHU WKDQ ERWK WKH OLQHDU ILOWHULQJ WHFKQLTXH RU WKH DGDOLQH VWUXFWXUH

PAGE 117

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f RU S O WKH WDSSHG GHOD\ OLQHf ,W ZDV SRVVLEOH WR DGDSW WKH SDUDPHWHU S WR FRQYHUJH WR WKH RSWLPDO YDOXH XVLQJ WKH JUDGLHQW GHVFHQW SURFHGXUH DV GHULYHG LQ FKDSWHU $ V\VWHP LGHQWLILFDWLRQ PRGHOLQJf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

PAGE 118

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

PAGE 119

&+$37(5 7+( /,1($5 ),/7(5,1* 3(563(&7,9( ,QWURGXFWLRQ 6RIDU WKH JDPPD PRGHO KDV EHHQ WUHDWHG DV D QHXUDO QHW PRGHO $V DQ DUFKLWHFWXUDO H[DPSOH WKH IRFXVHG JDPPD QHW ZDV LQWURGXFHG EXW VHYHUDO RWKHU QHXUDO QHW DUFKLWHFWXUHV WKDW PDNH XVH RI JDPPD PHPRU\ DUH SRVVLEOH ,Q WKLV FKDSWHU ZH PDNH WKH OLQN WR WKH VLPSOHVW QHXUDO QHW WKH OLQHDU DGDSWLYH FRPELQHU RU DGDOLQH ,Q FKDSWHU LW ZDV DOUHDG\ PHQWLRQHG WKDW DGDOLQH LV WKH EDVLF FRPSRQHQW RI PRVW QHXUDO QHW PRGHOV %XW DGDOLQH KDV IRXQG ZLGH DSSOLFDWLRQ E\ LWVHOI DV DQ DGDSWLYH ILOWHU LQ DUHDV VXFK DV FRPPXQLFDWLRQ DGDSWLYH FRQWURO DQG DGDSWLYH DQWHQQD SRVLWLRQLQJ :LGURZ DQG 6WHDPV f ,I WKH LQSXW VLJQDO LV JHQHUDWHG E\ D WDSSHG GHOD\ OLQH WKHQ WKH DGDOLQH VWUXFWXUH LV VLPLODU WR D UHJXODU ILQLWH LPSXOVH UHVSRQVH ),5f ILOWHU 1RZ DVVXPH WKDW WKH LQSXW VLJQDO WR DGDOLQH LV JHQHUDWHG LQVWHDG E\ D WDSSHG JDPPD GHOD\ OLQH 7KLV VWUXFWXUH LV QR ORQJHU VWULFWO\ IHHGIRUZDUG VLQFH WKH LQSXW GHOD\ OLQH LV LQKHUHQWO\ UHFXUVLYH $V ZDV GLVFXVVHG LQ FKDSWHU ZH FDOO WKLV DUFKLWHFWXUH DGDOLQHMLf $GDOLQHSf LV D YHU\ LQWHUHVWLQJ DGDSWLYH ILOWHU 7KLV FKDSWHU LV GHGLFDWHG WR WKH DQDO\VLV RI DGDOLQHQf ,Q RUGHU WR HQKDQFH WKH FRPSOHWHQHVV RI WKLV FKDSWHU ZH UHLQWURGXFH D IHZ WHUPV DQG FRQFHSWV WKDW KDYH EHHQ WUHDWHG HOVHZKHUH LQ WKLV GLVVHUWDWLRQ 7KH JRDO LV WKDW WKLV FKDSWHU LV UHDGDEOH ZLWKRXW NQRZOHGJH RI WKH SUHYLRXV FKDSWHUV IRU WKRVH ZKR DUH RQO\ LQWHUHVWHG LQ WKH OLQHDU ILOWHULQJ YLHZSRLQW DQG DSSOLFDWLRQV RI WKH JDPPD PRGHO ,OO

PAGE 120

$ 5HFDSLWXODWLRQ RI /LQHDU 'LJLWDO )LOWHU $UFKLWHFWXUHV 'LJLWDO ILOWHUV FDQ EH FDWHJRUL]HG LQWR WZR PDLQ JURXSV ,QILQLWH ,PSXOVH 5HVSRQVH +5f ILOWHUV DQG )LQLWH ,PSXOVH 5HVSRQVH ),5f ILOWHUV ),5 ILOWHUV DUH FKDUDFWHUL]HG E\ WKH IDFW WKDW WKH ILOWHU RXWSXW \Wf LV D OLQHDU FRPELQDWLRQ RI WKH ILOWHU LQSXW [Wf DQG LWV SDVW YDOXHV 7KXV D .WK RUGHU ),5 ILOWHU VWUXFWXUHV FDQ EH GHVFULEHG E\ . \Wf ; EALWNf (TL MIH +5 ILOWHUV JHQHUDOL]H WKH ),5 VWUXFWXUH E\ DOORZLQJ SDVW RXWSXW YDOXHV WR DIIHFW WKH FXUUHQW LQSXW +HQFH +5 ILOWHUV FDQ EH ZULWWHQ DV / . \Wf ;D]\Wf (T MIF R 7KH UHFXUUHQW OLQNV LQ WKH +5 VWUXFWXUH VXEVWDQWLDOO\ DIIHFWV WKH SURSHUWLHV RI +5 ILOWHUV LQ FRPSDULVRQ WR ),5 ILOWHUV )LUVW LW LV HDV\ WR VHH WKDW DQ ),5 ILOWHU LV DOZD\V ERXQGLQSXWERXQGHGRXWSXW %,%2f VWDEOH 7KLV PHDQV WKDW IRU ILQLWH ILOWHU FRHIILFLHQW YDOXHV D ERXQGHG LQSXW VLJQDO ?[Wf?0 0 D FRQVWDQWf DOZD\V OHDGV WR D ERXQGHG RXWSXW VLJQDO 6WDELOLW\ LV QRW JXDUDQWHHG IRU +5 ILOWHUV DV WKH IHHGEDFN IURP SDVW RXWSXW YDOXHV PD\ FDXVH WKH RXWSXW WR GLYHUJH HYHQ IRU D ERXQGHG LQSXW VLJQDO 7KLV SUREOHP SDUWLFXODUO\ OLPLWV WKH DSSOLFDWLRQ RI DGDSWLYH +5 ILOWHUV VLQFH SDUDPHWHU XSGDWLQJ PD\ OHDG WR LQVWDELOLW\ $Q DGGLWLRQDO SUREOHP DVVRFLDWHG ZLWK ,,5 V\VWHPV LV WKDW JUDGLHQW GHVFHQW DGDSWLYH SURFHGXUHV DUH QRW JXDUDQWHHG WR ILQG JOREDO RSWLPD LQ WKH QRQFRQYH[ HUURU VXUIDFHV RI ,,5 V\VWHPV 6K\QN f $V D UHVXOW LQ DGDSWLYH VLJQDO SURFHVVLQJ ),5 V\VWHPV DUH DOPRVW H[FOXVLYHO\ XVHG +D\NLQ :LGURZ DQG 6WHDUQV f
PAGE 121

DGDSWLYH SDUDPHWHUV HTXDO . )RU DQ ,,5 V\VWHP WKH OHQJWK RI WKH LPSXOVH UHVSRQVH LV XQFRXSOHG IURP WKH RUGHU DQG QXPEHU RI SDUDPHWHUVf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f VWUXFWXUH RU DGDSWLYH JDPPD ILOWHU LV VKRZQ WR EH D SDUWLFXODU LQVWDQFH RI WKH JHQHUDOL]HG IHHGIRUZDUG ILOWHU 7KH JDPPD ILOWHU LV DQDO\]HG LQ GHWDLO DQG ZH VKRZ WKDW LW ERUURZV GHVLUDEOH IHDWXUHV IURP ERWK ,,5 DQG ),5 V\VWHPV WULYLDO VWDELOLW\ HDV\ DGDSWDWLRQ DQG \HW WKH XQFRXSOLQJ RI WKH UHJLRQ RI VXSSRUW RI WKH LPSXOVH UHVSRQVH DQG WKH ILOWHU RUGHU *HQHUDOL]HG )HHGIRUZDUG )LOWHUV 'HILQLWLRQV &RQVLGHU WKH FRQYROXWLRQ ILOWHU DV GHVFULEHG E\ \Wf ZWf fµ[Wf (T ZKHUH [Wf LV WKH LQSXW VLJQDO \Wf WKH ILOWHU RXWSXW DQG ZWf D WLPH GHSHQGHQW ZHLJKW IXQFWLRQ 7KH V\PERO fµ LV WKH FRQYROXWLRQ RSHUDWRU ,I ZH DVVXPH WKDW WKH ZHLJKW IXQFWLRQ KDV . GHJUHHV RI IUHHGRP LW LV SRVVLEOH WR ZULWH ZWf DV . :Wf AZNJNWf )T N ZKHUH JNWf DUH QRUPDOL]HGf GHOD\ NHUQHOV 6XEVWLWXWLRQ RI (T LQWR (T OHDGV WR WKH SDUDOOHO FRQYROXWLRQ ILOWHU ZKLFK KDV . SDUDPHWHUV 7KLV VWUXFWXUH LV VKRZQ LQ )LJXUH 7KH SDUDOOHO FRQYROXWLRQ ILOWHU LV GHVFULEHG E\

PAGE 122

. \Wf eZrrr (J N N 7KH VLJQDOV [NWf DUH WKH VWDWH YDULDEOHV RU WDS YDULDEOHV ,I WKH GHOD\ NHUQHOV JNWf FDQ EH UHFXUVLYHO\ FRPSXWHG E\ JNWf JWf fµJNBOWf (DWKHQ WKH ILOWHU WDSV [NWf DUH DOVR UHODWHG E\ [NWf J Wf fµ [NB M Wf 7KLV ILOWHU VWUXFWXUH UHVHPEOHV D WDSSHG GHOD\ OLQH IHHGIRUZDUG ILOWHU DQG LV GHVFULEHG LQ WKH ] GRPDLQ E\ . )]f e :W[W]f (R N 4 ;N]f *]f;NBO]fN O. (T ZKHUH ;T ]f ; ]f LV WKH LQSXW VLJQDO DQG <]f WKH ILOWHU RXWSXW :H UHIHU WR WKLV VWUXFWXUH DV WKH JHQHUDOL]HG IHHGIRUZDUG ILOWHU 7KH JHQHUDOL]HG IHHGIRUZDUG LV VKRZQ LQ )LJXUH 7KH WDSWRWDS WUDQVIHU IXQFWLRQ *]f LV FDOOHG WKH JHQHUDOL]HGf GHOD\ RSHUDWRU DQG LW FDQ HLWKHU EH UHFXUVLYH RU QRQUHFXUVLYH :KHQ

PAGE 123

* ]f ] WKLV ILOWHU VWUXFWXUH UHGXFHV WR DQ ),5 ILOWHU 7KH PHPRU\ VWUXFWXUH RI DQ ),5 ILOWHU LV VLPSO\ D WDSSHG GHOD\ OLQH %\ LWHUDWLRQ RI (T ZH FDQ ZULWH <]f DV D IXQFWLRQ RI WKH LQSXW ;]f DV . <]f = ZN>*]f@N;]f (TN :H ZLOO DOVR ZULWH *N ]f >* ]f @N IRU WKH LQSXWWRWDSN WUDQVIHU IXQFWLRQ 7KXV WKH WUDQVIHU IXQFWLRQ RI WKH JHQHUDOL]HG IHHGIRUZDUG ILOWHU LV \}W>6Ef@f« (J r]f r7 ,W IROORZV IURP (T WKDW +I]O LV VWDEOH ZKHQHYHU *]f LV VWDEOH 7KH SDVW RI [Wf LV UHSUHVHQWHG LQ WKH WDS YDULDEOHV [NWf VKDGHG DUHD LQ )LJXUH f ,Q JHQHUDO WKH RSWLPDO PHPRU\ VWUXFWXUH *]f LV D IXQFWLRQ RI WKH LQSXW VLJQDO FKDUDFWHULVWLFV DV ZHOO DV WKH JRDO RI WKH ILOWHU RSHUDWLRQ 7KLV REVHUYDWLRQ KDV OHG XV WR FRQVLGHU DGDSWLYH GHOD\ RSHUDWRUV *]f 7KH DGDSWLYH FRHIILFLHQWV RI *]f ZLOO EH FDOOHG PHPRU\ SDUDPHWHUV ,Q FRPSDULVRQ WR D JHQHUDO +5 ILOWHU WKH JHQHUDOL]HG IHHGIRUZDUG ILOWHU DUFKLWHFWXUH LV UHVWULFWHG E\ WZR FRQGLWLRQV

PAGE 124

&Of UHFXUUHQW OLQNV LQ WKH JHQHUDOL]HG IHHGIRUZDUG ILOWHU DUH UHVWULFWHG WR ZLWKLQ WKH GHOD\ RSHUDWRUV *]f 7KXV IHHGEDFN LQ WKH JHQHUDOL]HG IHHGIRUZDUG QHW LV RI ORFDO QDWXUH &f 1RWH WKDW VLQFH *]f LV LQVHUWHG DW HYHU\ WDS LQ WKH PHPRU\ WKH PHPRU\ SDUDPHWHUV DUH UHSHDWHG WKURXJKRXW WKH PHPRU\ 7KXV WKH PHPRU\ SDUDPHWHUV DUH JOREDO ZLWK UHVSHFW WR WKH PHPRU\ VWUXFWXUH 7KLV FKDSWHU DQDO\]HV LQ GHWDLO WKH FDVH * ]f r WKH JDPPD GHOD\ ]GOLf RSHUDWRU 7KH JDPPD GHOD\ RSHUDWRU FDQ EH LQWHUSUHWHG DV D OHDN\ LQWHJUDWRU ZKHUH ; LV WKH JDLQ LQ WKH LQWHJUDWLRQ IHHGEDFNf ORRS 7KH $GDSWLYH *DPPD )LOWHU 'HILQLWLRQV 7KH JDPPD ILOWHU LV GHILQHG LQ WKH WLPH GRPDLQ DV \Wf . ( ZN[NA N (Tf±4 [NWf ^fre f >X$B W f N (T ZKHUH MFT Lf [Wf LV WKH LQSXW VLJQDO DQG \Wf WKH ILOWHU RXWSXW ,I ZZZN DQG ML DUH DGDSWLYH WKLV VWUXFWXUH LV FDOOHG WKH DGDSWLYH JDPPD ILOWHU RU DGDOLQH_Qf 7KH DGDSWLYHf JDPPD ILOWHU VWUXFWXUH LV GLVSOD\HG LQ )LJXUH 7KH JDPPD LQSXWWRWDSN WUDQVIHU IXQFWLRQ *N]f LV JLYHQ E\ *$]f ] M[f f (J ,QYHUVH ]WUDQVIRUPDWLRQ \LHOGV WKH LPSXOVH UHVSRQVH IRU WDS N

PAGE 125

rW:m=^&]f` ?c7B?fYN99fQN89Nf ST ZKHUH 8Wf LV WKH XQLW VWHS IXQFWLRQ :KHQ L WKH DGDSWLYH JDPPD ILOWHU UHGXFHV WR :LGURZf¬V DGDOLQH VWUXFWXUH :LGURZ DQG 6WHDUQV f )RU [ r WKH JDPPD ILOWHU WUDQVIHU IXQFWLRQ LV RI +5 W\SH GXH WR WKH UHFXUVLRQ LQ (T DQG *]f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f LV VWDEOH ,Q WKH JDPPD ILOWHU WKLV WUDQVODWHV WR WKH IROORZLQJ UHTXLUHPHQW }[_ (T +HQFH WKH JDPPD ILOWHUf¬V VWDELOLW\ LV JXDUDQWHHG IRU S

PAGE 126

0HPRU\ 'HSWK YHUVXV )LOWHU 2UGHU 7KH WLJKW FRXSOLQJ RI WKH PHPRU\ GHSWK WR WKH QXPEHU RI IUHH SDUDPHWHUV LQ WKH DGDSWLYH ),5 ILOWHU VWUXFWXUH KDV EHHQ GLVFXVVHG DQG LW ZDV DUJXHG WKDW WKLV SURSHUW\ OHDGV WR SRRU PRGHOLQJ RI ORZ SDVV IUHTXHQF\ ERXQGHG VLJQDOV +5 ILOWHUV RQ WKH RWKHU KDQG KDYH IHHGEDFN FRQQHFWLRQV DQG FRQVHTXHQWO\ WKH PHPRU\ GHSWK LV QRW FRXSOHG WR WKH QXPEHU RI ILOWHU SDUDPHWHUV ,Q VHFWLRQ ZH GHULYHG IRU D eWK RUGHU JDPPD ILOWHU WKH IROORZLQJ UHODWLRQ . '[5 (J . ZKHUH ' f§ LV WKH PHPRU\ GHSWK DQG 5 S LV WKH UHVROXWLRQ RI WKH ILOWHU 7KXV WKH SDUDPHWHU IL SURYLGHV D PHFKDQLVP WR DGDSW WKH PHPRU\ GHSWK IRU FRQVWDQW ILOWHU RUGHU . /06 $GDSWDWLRQ ,Q WKLV VHFWLRQ ZH GHULYH WKH OHDVW PHDQ VTXDUH /06f DGDSWDWLRQ XSGDWH UXOHV IRU WKH JHQHUDOL]HG IHHGIRUZDUG QHW ,Q WKH WLPH GRPDLQ WKLV VWUXFWXUH LV GHVFULEHG E\ \Wf
PAGE 127

O ( e Hm nW8GOf\Off (R I W W ZKHUH GWf LV D WDUJHW VLJQDO 7KH /06 DOJRULWKP FRUUHFWV WKH ILOWHU FRHIILFLHQWV SURSRUWLRQDOO\ WR WKH QHJDWLYH RI WKH ORFDO JUDGLHQW WKDW LV WKH FRHIILFLHQW XSGDWH HTXDWLRQV DUH LQ WKH GLUHFWLRQ RI WKH QHJDWLYH JUDGLHQWV G( $Z f§7_ GZ (T $ @/ f§7_ G( Sf¬ ZKHUH 7_ LV D VWHS VL]H SDUDPHWHU :H ILUVW H[SDQG IRU ZN \LHOGLQJ 7 7 $ZN B7OOA ‹ HfA7  r L r W 6LPLODUO\ WKH XSGDWH HTXDWLRQ IRU _L HYDOXDWHV WR ZKHUH D $Wf G[Wf (R (R RF r r G[NQf r BU!A ; HQf ; ,f§ 7O; ; (D-/LO A m IF A Q N 7KH JUDGLHQW VLJQDO DNWf FDQ EH FRPSXWHG RQOLQH E\ GLIIHUHQWLDWLQJ (T 6K\QN :LOOLDPV DQG =LSVHU f OHDGLQJ WR 02 02 _[ D nAMWBL DrBL A (Tf± )RU WKH JDPPD ILOWHU (T HYDOXDWHV WR DNWf f§ 0f RWA f§ f SDIFB M f >reB M W f f@ (TMMM 7KH VHW RI HTXDWLRQV (T (T DQG (T FRQVWLWXWH WKH XSGDWH DOJRULWKP LQ

PAGE 128

HSRFKZLVH DGDSWDWLRQ ,Q SUDFWLFH D ORFDO LQ WLPH DSSUR[LPDWLRQ LH VDPSOH E\ VDPSOHf RI WKH IRUP $ZNWf [?HWf[NWf N . (J . $SWf U_HWf ; ZNDNWf (J N ZRUNV ZHOO LI 7_ LV VXIILFLHQWO\ VPDOO 7KH XSGDWH V\VWHP (T FDQ EH UHFRJQL]HG DV WKH /06 DOJRULWKP 1RWLFH WKH QXPEHU RI RSHUDWLRQV SHU WLPH VWHS IRU DQG (T VFDOH ERWK DV .f ZKHUHDV (T LV f 7KXV WKH HQWLUH /06 DOJRULWKP VFDOHV DV 2.f ZKLFK FRLQFLGHV ZLWK WKH FRPSOH[LW\ IRU :LGURZf¬V DGDOLQH 7KH JDLQ ZLWK UHVSHFW WR D JHQHUDO +5 /06 URXWLQH VFDOHV DV 2L.ff LV GXH WR WKH UHVWULFWHG QDWXUH RI WKH IHHGEDFN ORRSV LQ WKH JDPPD ILOWHU ,W LV HDVLO\ YHULILHG WKDW WKH /06 DOJRULWKP KHUH LV FRQVLVWHQW ZLWK WKH OHDUQLQJ DOJRULWKP DV GHULYHG LQ VHFWLRQ IRU WKH IRFXVHG JDPPD PRGHO 7KH UHVXOWV RI WKH ODVW WKUHH VHFWLRQV DUH VXPPDUL]HG LQ 7DEOH &OHDUO\ WKH JDPPD ILOWHU VKDUHV GHVLUDEOH IHDWXUHV IURP ERWK ),5 DQG +5 ILOWHUV :LHQHU+RSI (TXDWLRQV IRU WKH $GDSWLYH *DPPD )LOWHU 7KH RSWLPDO ZHLJKWV IRU DQ DGDOLQH VWUXFWXUH LQ D JLYHQ VWDWLRQDU\ HQYLURQPHQW FDQ EH DQDO\WLFDOO\ H[SUHVVHG E\ WKH :LHQHU+RSI RU QRUPDO HTXDWLRQV +D\NLQ f :H GHULYHG WKH QRUPDO HTXDWLRQV IRU WKH DGDOLQH VWUXFWXUH LQ VHFWLRQ +HUH WKHVH H[SUHVVLRQV DUH H[WHQGHG WR WKH JHQHUDOL]HG IHHGIRUZDUG ILOWHU ,W ZLOO EH VKRZQ WKDW WKH JHQHUDOL]HG QRUPDO HTXDWLRQV JHQHUDOL]H :LHQHUf¬V IRUPXODWLRQ IRU VWULFW IHHGIRUZDUG ILOWHUV &RQVLGHU WKH JHQHUDOL]HG IHHGIRUZDUG VWUXFWXUH DV GHVFULEHG E\ (T DQG (T 7KH SHUIRUPDQFH LQGH[ LV GHILQHG DV b (>H Wf@ ZKHUH

PAGE 129

.WK RUGHU ILOWHU ),5 *$00$ ,,5 67$%,/,7< DOZD\V VWDEOH WULYLDO VWDELOLW\ _; QRQWULYLDO VWDELOLW\ 0(025< '(37+ YV 25'(5 FRXSOHG . XQFRXSOHG .S XQFRXSOHG &203/(;,7< RI $'$37$7,21 2.f 2.f .f 7DEOH FRPSDULVRQ RI ),5 ,,5 DQG *DPPD ILOWHU SURSHUWLHV HWf GWf \Wf LV DQ HUURU VLJQDO DQG e>@ WKH H[SHFWDWLRQ RSHUDWRU ,Q RUGHU WR PDLQWDLQ D FRQVLVWHQW QRWDWLRQ ZLWK UHVSHFW WR WKH DGDSWLYH VLJQDO SURFHVVLQJ OLWHUDWXUH ZH LQWURGXFH WKH YHFWRUV ;I >rR }rL [.Wf @ 7 DQG : >Z :MA@ 7 1RWH WKDW ; KROGV WKH WDS YDULDEOHV DQG QRW WKH LQSXW VLJQDO VDPSOHV (YDOXDWLQJ e OHDGV WR O (>GWff :U5:37: (R ZKHUH 5 ( >;W;^ @ DQG 3 (>GWf;^@ 7KH JRDO RI DGDSWDWLRQ LV WR PLQLPL]H A LQ ‹ WKH VSDFH RI A7O ZHLJKWV DQG ; :KHQ A LV PLQLPDO WKH FRQGLWLRQV Z DQG QHFHVVDULO\ KROG 3DUWLDO GLIIHUHQWLDWLRQ RI (T ZLWK UHVSHFW WR WKH V\VWHP _L SDUDPHWHUV \LHOGV 5: 3 DQG (R

PAGE 130

:n >5A:,3A@ (J ZKHUH 5 A O RML U G[7? GS U ( DQG 3 Yf§ A G_L ( r _UR f§1 1RWH WKDW (T LV WKH VDPH H[SUHVVLRQ DV WKH :LHQHU+RSI HTXDWLRQ IRU WKH DGDOLQH QHWZRUN 7KH GLIIHUHQFH OLHV LQ WKH IDFW WKDW WKH YHFWRU ;W KROGV WKH WDS YDULDEOHV [NWf DQG QRW WKH VDPSOHV [^WNf 7KH H[WUD VFDODU FRQGLWLRQ (T LV D UHVXOW RI GW UHTXLULQJ 7KXV (T SURYLGHV DQ DQDO\WLFDO H[SUHVVLRQ IRU WKH RSWLPDO G[NWf PHPRU\ GHSWK 7KLV H[SUHVVLRQ DOVR UHYHDOV WKDW WKH VLJQDO DNWf f§Af§ LV QHHGHG LQ RUGHU WR FRPSXWH WKH RSWLPDO PHPRU\ VWUXFWXUH WKDW LV WKH RSWLPDO YDOXH RI Sf 7KLV REVHUYDWLRQ LV FRQILUPHG LQ WKH H[SUHVVLRQV IRU WKH /06 DOJRULWKP ,W LV LQVLJKWIXO WR UHZULWH WKH :LHQHU+RSI HTXDWLRQV LQ WHUPV RI WKH LQSXW VLJQDO [ Wf /HW XV GHILQH WKH GHOD\ NHUQHO YHFWRU * Wf >J Wf J WfJ. Wf@7 7KHQ (T DQG (T HYDOXDWH WR (>*Wf rrf *If fµ[Wff7@ : (>GWf *Lf fµrrff@ (J :7( G*Wf A *If fµrrff Arrf : :7( G*Wf GWf J rrf (D (T DQG (T H[WHQG WKH :LHQHU+RSI HTXDWLRQV WR JHQHUDOL]HG IHHGIRUZDUG VWUXFWXUHV 1RWH WKDW WKHVH HTXDWLRQV LQ WKH WLPH GRPDLQ LQFOXGH LQILQLWH VXPPDWLRQV JWf PD\ EH RI LQILQLWH OHQJWKf EXW LQ WKH ]GRPDLQ WKH\ FDQ EH FRPSXWHG H[DFWO\ E\ FRQWRXU LQWHJUDWLRQ ([SHULPHQWDO 5HVXOWV 7ZR IUDPHZRUNV ZHUH SUHVHQWHG WR REWDLQ DQ RSWLPDO ILOWHU DUFKLWHFWXUH

PAGE 131

DGDOLQH4Lf ,Q VHFWLRQ WKH /06 DGDSWDWLRQ DOJRULWKP ZDV GHULYHG DQG VHFWLRQ ZDV GHYRWHG WR WKH :LHQHU+RSI HTXDWLRQV IRU WKH JHQHUDOL]HG IHHGIRUZDUG ILOWHU ,Q WKLV VHFWLRQ QXPHULFDO VLPXODWLRQ UHVXOWV DUH SUHVHQWHG IRU ERWK RSWLPL]DWLRQ PRGHOV ZKHQ DGDOLQHM[f LV XVHG LQ D V\VWHP LGHQWLILFDWLRQ FRQILJXUDWLRQ 7KH JRDO RI WKLV VHFWLRQ LV WZRIROG )LUVW , ZLOO VKRZ WKDW WKH RSWLPDO ILOWHU DUFKLWHFWXUH LQGHHG RXWSHUIRUPV :LGURZf¬V DGDOLQHOf $OVR LW ZLOO EH VKRZQ WKDW WKH ILOWHU FRHIILFLHQWV FRQYHUJH WR WKH RSWLPDO YDOXHV LI WKH /06 XSGDWH UXOHV RI VHFWLRQ DUH XVHG 7KLV H[SHULPHQW KDV LQ SDUW EHHQ GLVFXVVHG LQ VHFWLRQ 7KH V\VWHP WR EH LGHQWLILHG LV WKH UG RUGHU HOOLSWLF ORZ SDVV ILOWHU GHVFULEHG E\ +]f ]ar ]a ]a ]B ]a ]f¯ (R 7KH SHUIRUPDQFH LQGH[ A DV D IXQFWLRQ RI _L ZDV FRPSXWHG E\ HYDOXDWLQJ (T LQ WKH ]GRPDLQ UHVLGXH WKHRUHPf 7KH RSWLPDO ZHLJKW YHFWRU Zr LV FRPSXWHG E\ VROYLQJ WKH :LHQHU+RSI HTXDWLRQ (T :H DVVXPHG D QRUPDO OfGLVWULEXWHG ZKLWH QRLVH LQSXW ZKLFK WUDQVODWHV WR D FRQVWDQW VSHFWUXP LQ WKH ]GRPDLQ IL ZDV SDUDPHWUL]HG RYHU WKH UHDO GRPDLQ >@ 7KH VLPXODWLRQV SORWWHG LQ )LJXUH Df ZHUH SHUIRUPHG ZLWK 0DWKHPDWLFD :ROIUDP f RQ D 1H;7 FRPSXWHU 7KH WLPH GXUDWLRQ RI WKH VLPXODWLRQV UHVWULFWHG WKH HYDOXDWLRQ WR . 2EVHUYH WKDW IRU DOO PHPRU\ RUGHUV . WKH RSWLPDO SHUIRUPDQFH LV REWDLQHG IRU MM +HQFH WKH RSWLPDO JDPPD QHW RXWSHUIRUPV WKH FRQYHQWLRQDO DGDOLQH E\ D ODUJH PDUJLQ7KHUH LV D ORW RI VWUXFWXUH LQ WKH FXUYHV . 1RWH WKDW WKH RSWLPDO PHPRU\ GHSWK ' S ARSW LV FRQVWDQW IRU GLIIHUHQW PHPRU\ RUGHUV ,Q )LJXUH Ef WKH UHODWLYH WRWDO HUURU H D LV VKRZQ DIWHU FRQYHUJHQFH 7KLV ILOWHU KDV EHHQ GHVFULEHG LQ 2SSHQKHLP DQG 6FKDIHU SJ

PAGE 132

)LJXUH 2SWLPDO SHUIRUPDQFH LQGH[ DV D IXQFWLRQ RI S IRU LGHQWLILFDWLRQ RI HOOLSWLF ILOWHU +]f Df e LV FRPSXWHG XVLQJ IURP WKH :LHQHU+RSI HTXDWLRQV IRU DGDOLQHSf Ef -PP YDU¯HWf@YDU>GWf@ LV FRPSXWHG DIWHU DGDSWDWLRQ XVLQJ WKH /06 XSGDWH UXOH XVLQJ WKH UHDOWLPH /06 XSGDWH UXOH (T S ZDV SDUDPHWUL]HG RYHU WKH GRPDLQ >@ XVLQJ D VWHS VL]H $S 7KH UHVXOWV PDWFK WKH WKHRUHWLFDO RSWLPDO SHUIRUPDQFH )LJXUH Dff YHU\ ZHOO 7KLV H[SHULPHQW VKRZV WKDW WKH ILOWHU ZHLJKWV ^Zr` FDQ LQGHHG EH OHDUQHG E\ RQOLQH /06 OHDUQLQJ :KHQ . DGDOLQHOf SHUIRUPV DV ZHOO DV . IRU DGDOLQHf +RZHYHU DGDOLQHf ZLWK . VKRXOG EH SUHIHUUHG VLQFH WKLV VWUXFWXUH KDV IUHH SDUDPHWHUV ZKHUHDV DGDOLQH XVHV SDUDPHWHUV 3DUVLPRQ\ LQ WKH QXPEHU RI IUHH SDUDPHWHUV SURYLGHV DGDOLQH2µf ZLWK EHWWHU PRGHOLQJ JHQHUDOL]DWLRQf FKDUDFWHULVWLFV 7KH HIIHFW RI WKH PHPRU\ SDUDPHWHU S RQ WKH ILOWHU SHUIRUPDQFH LQFUHDVHV ZKHQ ZH PRGHO D V\VWHP ZLWK VPDOOHU FXWRII IUHTXHQF\ EXW WKH VDPH QXPEHU RI SDUDPHWHUV ,Q )LJXUH WKH SHUIRUPDQFH LQGH[ e YHUVXV S LV SORWWHG IRU D WKLUGRUGHU HOOLSWLF ORZ SDVV ILOWHU +]f ZLWK VPDOOHU FXWRII IUHTXHQF\ ZFR 22•7W UDGf f± O] ]a ]a A f§ f§ f§ O] ] ] ,W LV FOHDU WKDW WKH WKLUGRUGHU DGDOLQH VWUXFWXUH SHUIRUPV YHU\ SRRUO\ e m f ZKHUHDV WKH WKLUGRUGHU DGDOLQHf DFKLHYHV D SHUIRUPDQFH LQGH[ RI e

PAGE 133

7KLV H[SHULPHQW FRQILUPV RXU K\SRWKHVLV WKDW WKH JDPPD PHPRU\ SHUIRUPV EHWWHU DV WKH GHPDQGV RQ D GHHS PHPRU\ LQFUHDVHV VHH VHFWLRQ f 7KH *DPPD 7UDQVIRUP $ 'HVLJQ DQG $QDO\VLV 7RRO )RU *DPPD )LOWHUV 6RIDU WKH DGDSWLYHf JDPPD ILOWHU SURSHUWLHV KDYH EHHQ DQDO\]HG XVLQJ WKH WLPH DQG ]GRPDLQ WRROV DQG FRPSDUHG WKH UHVXOWV WR ),5 DQG ,,5 ILOWHUV ,W ZDV VKRZQ WKDW WKH UHVWULFWHG QDWXUH RI WKH IHHGEDFN FRQQHFWLRQV LQ WKH JDPPD ILOWHU KDV UHQGHUHG WKLV PRGHO ZLWK GHVLUDEOH SURSHUWLHV IURP ERWK ILOWHU FODVVHV ,Q IDFW WKH JDPPD ILOWHU FDQ EH YLHZHG DV DQ LQVWDQFH RI D K\EULG ILOWHU FODVV WKH JHQHUDOL]HG IHHGIRUZDUG ILOWHU $Q LQWHUHVWLQJ IHDWXUH RI JDPPD ILOWHUV DOUHDG\ H[SORUHG IRU WKH H[WHQVLRQ RI WKH :LHQHU +RSI VROXWLRQ LV WKDW WKH\ FDQ EH IRUPXODWHG DV ),5 ILOWHUV ZLWK UHVSHFW WR D GHOD\ RSHUDWRU *]f ,Q WKLV VHFWLRQ ZH H[SORUH WKH LPSOLFDWLRQV RI GHVFULELQJ WKH V\VWHP LQ D QHZ WUDQVIRUP GRPDLQ WKH \GRPDLQ ZKLFK ZH GHILQH DV < *]f (TL ,Q WKH \GRPDLQ JHQHUDOL]HG IHHGIRUZDUG ILOWHUV DUH RUGLQDU\ ),5 ILOWHUV GHILQHG DURXQG GHOD\ RSHUDWRUV \ )RU JDPPD ILOWHUV (T HYDOXDWHV WR

PAGE 134

< ]^ +f (T $ VLJQDO [¯f FDQ EH H[SUHVVHG LQ WKH \GRPDLQ E\ VXEVWLWXWLQJ (T LQ WKH ] WUDQVIRUP 7KLV OHDGV WR WKH IROORZLQJ H[SUHVVLRQ IRU WKH \WUDQVIRUP RI D VLJQDO [Qf ;7 ‘Sf W ] 3< Sf RR A ; A ^< f§7A` ¯ A (R 7KXV WKH \WUDQVIRUP LV HTXLYDOHQW WR WKH /DXUHQW VHULHV H[SDQVLRQ RI WKH VLJQDO MM r[ Wf HYDOXDWHG DW WKH SRLQW \4 f§f§ 7KLV LGHD LV GLVSOD\HG LQ )LJXUH S 7KH FRUUHVSRQGLQJ WLPH VHULHV REWDLQHG E\ WKH LQYHUVH \WUDQVIRUP FDQ EH FRPSXWHG DV [f si;]f]nnG]

PAGE 135

WLM _;
PAGE 136

(J 7KH LPSXOVH UHVSRQVH RI D JDPPD ILOWHU FDQ EH H[SUHVVHG DV . KWf !$ N JR Jf WOfSO_[fLBeLf (T ,Q )LJXUH WKH V\VWHPf¬V PDJQLWXGH IUHTXHQF\ DQG LPSXOVH UHVSRQVHV DUH GLVSOD\HG DV D IXQFWLRQ RI S 1RWH WKDW LI S LV FORVH WR WKH JDPPD ILOWHU EHKDYHV DV WKH ),5 V\VWHP +]f ]n :KHQ S JHWV VPDOOHU WKH f¯SHDNf° RI WKH IUHTXHQF\ UHVSRQVH EHFRPHV VKDUSHU ZKLFK LV W\SLFDO IRU +5 ILOWHUV DV FRPSDUHG WR ),5 ILOWHUV RI WKH VDPH RUGHU 7KXV WKH JOREDO ILOWHU SDUDPHWHU S GHWHUPLQHV ZKHWKHU ),5 RU +5 ILOWHU FKDUDFWHULVWLFV DUH REWDLQHG Df Ef R )LHXUH )UHTXHQF\ PDJQLWXGH UHVSRQVH Df DQG LPSXOVH UHVSRQVH Ef RI JDPPD ILOWHU (T DV D IXQFWLRQ RI _L

PAGE 137

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f 7KXV WKH PHPRU\ SDUDPHWHUV Y DQG _L DUH JOREDO DQG WKH IHHGEDFN LV ORFDO EHWZHHQ WDSV $V D UHVXOW /06 IRU WKH JDPPD ,, ILOWHU VFDOHV DOVR E\ .f DQG WKH VWDELOLW\ FRQGLWLRQV DUH WULYLDO 1H[W D IHZ SURSHUWLHV RI WKH JDPPD ,, PHPRU\ HOHPHQW DUH GHULYHG 7KH WUDQVIHU L

PAGE 138

IXQFWLRQ * ]f <]f ;]f RI WKLV VWUXFWXUH HYDOXDWHV WR *^]f Yf*]f Y*]f +OYf >] ,;f@ >] f§ ,;f @ QY (D-/=L 7KXV *]f KDV D ]HUR DW =T _; DQG SROHV DW ]S Sf s \ 9M;Y 7KH IRUZDUG JDLQ IDFWRU Y HQVXUHV WKH QRUPDOL]DWLRQ RI WKH JDPPD ,, GHOD\ HOHPHQW 6LPLODUO\ WR WKH JDPPD , PHPRU\ ZH KDYH AJNWf *N]f?] O >*Of@N O (T W ,Q RUGHU WR GHULYH WKH VWDELOLW\ FRQGLWLRQ ZH DVVXPH ; ! DQG Y ! 7KHQ WKH V\VWHP LV VWDEOH LI Sf _LY (D (T FDQ EH UHGXFHG WR SS Yf DQG WRJHWKHU ZLWK S! LW IROORZV WKDW VXIILFLHQW FRQGLWLRQV IRU VWDELOLW\ DUH JLYHQ E\ _; f D 9f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

PAGE 139

FRMW )LJXUH )UHTXHQF\ PDJQLWXGH UHVSRQVH Df DQG JURXS GHOD\ Ef IRU WKH *DPPD ,, PHPRU\ HOHPHQW

PAGE 140

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f RI WKH VDPH RUGHU ,Q JHQHUDO WKH JDPPD ILOWHU LV SUHIHUDEOH LI WKH SURFHVVLQJ SUREOHP LQYROYHV VLJQDOV ZLWK HQHUJ\ FRQFHQWUDWHG DW ORZ IUHTXHQFLHV DQG UHODWLYHO\ IHZ GHJUHHV RI IUHHGRP $SSOLFDWLRQV LQYROYLQJ ORQJ GHOD\V DV LQ FKDQQHO HTXDOL]DWLRQ URRP DFRXVWLFV RU LGHQWLILFDWLRQ RI V\VWHPV ZLWK ORQJ LPSXOVH UHVSRQVHV VHHP WR EH SDUWLFXODUO\ DSSURSULDWH IRU WKH JDPPD ILOWHU
PAGE 141

SURYLGH DQ H[WHQGHG IUDPHZRUN

PAGE 142

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f KRZ GR ZH HQVXUH VWDELOLW\ RI WKH QHWZRUN DQG LLf KRZ WR HIILFLHQWO\ WUDLQ WKH QHWZRUN 7KXV LQ VHFWLRQ WKH GHYHORSPHQW RI D QHXUDO PRGHO ZLWK DGDSWLYH VKRUW WHUP PHPRU\ IRU WHPSRUDO SURFHVVLQJ ZDV GHWHUPLQHG DV WKH PDLQ UHVHDUFK JRDO RI WKLV ZRUN ,Q UHVSRQVH WKH JDPPD PRGHO ZDV GHYHORSHG 7KH JDPPD QHXUDO PRGHO JHQHUDOL]HV DQG XQLILHV QHXUDO QHWV ZLWK WDSSHG GHOD\ OLQHV DQG QHWV ZLWK ORFDO IHHGEDFN

PAGE 143

LQWR D VLQJOH IUDPHZRUN 7KH KLVWRU\ WUDFHV LQ WKH JDPPD QHW DUH DGDSWLYH DQG DUH JHQHUDWHG E\ DQ DGGLWLYH QHXUDO PRGHO 7KH JDPPD PHPRU\ VWUXFWXUH LV DQ LQVWDQFH RI D VWUXFWXUH ZKLFK ZH FDOO WKH JHQHUDOL]HG WDSSHG GHOD\ OLQH 7KH .WK RUGHU JHQHUDOL]HG WDSSHG GHOD\ OLQH LV GLVSOD\HG LQ )LJXUH ,I WKH GHOD\ RSHUDWRU *]f LV QRUPDOL]HG E\ *Of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

PAGE 144

IRFXVHG JDPPD QHW DSSHDUV DV D YHU\ YHUVDWLOH QHWZRUN VWUXFWXUH IRU YDULRXV SURFHVVLQJ SURWRFROV ,Q FKDSWHU WKH OLQHDU RQHGLPHQVLRQDO JDPPD PRGHO ZDV DQDO\]HG LQ GHWDLO $ QHZ DGDSWLYH ILOWHU WKH DGDOLQHSf VWUXFWXUH ZDV LQWURGXFHG DQG DQDO\]HG $GDOLQHMMf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f¬V VWXGHQW LQ WKH FRPSXWDWLRQDO QHXURHQJLQHHULQJ ODERUDWRU\ LV FXUUHQWO\ ZRUNLQJ RQ WKH GHYHORSPHQW RI D UHDOWLPH LVRODWHG ZRUG

PAGE 145

UHFRJQLWLRQ V\VWHP 7KH SURWRW\SH V\VWHP LV VKRZQ LQ )LJXUH 7KLV V\VWHP LV LPSOHPHQWHG RQ WKH 1H;7r1 FRPSXWHU XWLOL]LQJ WKH '63r1 GLJLWDO VLJQDO SURFHVVLQJ FKLS E\ 0RWRUROD 7KH JRDO RI WKH V\VWHP LV WR OHDUQ WR UHFRJQL]H ZLQGRZ FRPPDQGV IRU WKH 1H;7 V\VWHP VXFK DV f¯SDXVHf° f¯KDOWf° RU f¯JRf° 7KH V\VWHP LV GHVLJQHG WR OHDUQ DQG FODVVLI\ LQFRPLQJ ZRUGV LQ UHDOWLPH $ IUHTXHQF\YHUVXVWLPH UHSUHVHQWDWLRQ RI WKH VSHHFK VLJQDO LV JHQHUDWHG E\ D FRFKOHDU HQHUJ\ PRGHO *UHHQ DQG 6ZHWV f 7KH FRFKOHDU HQHUJ\ PRGHO LV D VLPSOH PRGHO IRU SHULSKHUDO DXGLWRU\ SURFHVVLQJ 7KH RXWSXW RI WKH FRFKOHDU PRGHO LV VWRUHG LQ D JDPPD PHPRU\ OD\HU 7KH UHVROXWLRQ RI WKH PHPRU\ FDQ EH DGDSWHG E\ YDULDWLRQ RI P 7KH VSDWLRWHPSRUDO VLJQDO UHSUHVHQWDWLRQ LQ WKH JDPPD PHPRU\ OD\HU LV VXEVHTXHQWO\ SURFHVVHG E\ D VWDWLF QRQn OLQHDU IHHGIRUZDUG QHW 7KH RXWSXW QRGHV HQFRGH WKH YRFDEXODU\ 7KH REMHFWLYHV RI WKLV SURMHFW DUH WKH IROORZLQJ Lf &DQ D UHDOWLPH ZRUG UHFRJQL]HU EH LPSOHPHQWHG RQ D VPDOO GHVN FRPSXWHU ZLWKRXW H[WUD KDUGZDUH" 7KH 1H;7 FRPSXWHU XWLOL]HV 0RWRURODf¬V &38 DQG '63 SURFHVVRUV LLf +RZ GR ZH WUDLQ WKH JDPPD QHWZRUN SDUDPHWHUV LQ SDUWLFXODU S IRU WKH WHPSRUDO FODVVLILFDWLRQ SUREOHP LLLf 0HDVXUH WKH V\VWHP SHUIRUPDQFH DV D IXQFWLRQ RI S DQG WKH PHPRU\ RUGHU . &DQ ZH WUDLQ WKH PHPRU\ WR FRQYHUJH WR WKH RSWLPDO S $W WKH WLPH RI ZULWLQJ RI WKLV WKHVLV WKH PRGHOV KDYH EHHQ LPSOHPHQWHG DQG WKH ILUVW H[SHULPHQWV DUH XQGHUZD\ ,Q DQRWKHU SURMHFW WKH SHUIRUPDQFH RI WKH JDPPD PRGHO LV WHVWHG IRU FODVVLILFDWLRQ RI WKH SKRQHPHV f¯Ef° f¯Gf° DQG f¯Jf° 7KH GDWD VHW ZDV JHQHUDWHG E\ ,%0 DW WKH -DPHV :DWVRQ 5HVHDUFK &HQWHU LQ
PAGE 146

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f KDYH DOUHDG\ EHHQ GLVFXVVHG 7KH JDPPD PHPRU\ ZDV LQWURGXFHG DV DQ H[DPSOH RI WKH JHQHUDOL]HG WDSSHG GHOD\ OLQH

PAGE 147

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n OLQHDU QHXUDO QHWZRUN KY LWVHOI 7KH ORQJ WHUP JRDO LV WR KDYH D QXPEHU RI ORFDO PHPRU\ VWUXFWXUHV DYDLODEOH WKDW KDYH EHHQ WHVWHG DQG FDWHJRUL]HG IRU YDULRXV VLJQDO VWDWLVWLFV DQG SURFHVVLQJ SUREOHPV 7KH VHFRQG OLQH RI UHVHDUFK ZKLFK , HQFRXUDJH FRQFHUQV JOREDO G\QDPLF QHXUDO QHWZRUN DUFKLWHFWXUHV 7KLV WKHVLV IRFXVHV RQ WKH FRPSXWDWLRQDO IHDWXUHV RI GHOD\V YHUVXV ORFDO IHHGEDFN IRU VWRULQJ RI SDVW LQIRUPDWLRQ ZKLFK FRQFHUQV G\QDPLF QHWZRUN DUFKLWHFWXUH DW WKH XQLW ORFDOf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f RI VLJQLILFDQW QHWZRUN DUFKLWHFWXUHV )RU H[DPSOH WKH WLPH GHOD\ QHXUDO QHW DQG VHTXHQWLDO QHWZRUN DUFKLWHFWXUHV VKRXOG EH UHDFKDEOH

PAGE 148

SDUDPHWUL]DWLRQV RI WKLV PRGHO 7KH VHFRQG SKDVH RI UHVHDUFK GHYHORSV WKH DOJRULWKPV WR DGDSW WKH QHXUDO QHW DUFKLWHFWXUH WR WKH LQSXW VLJQDO VSDFH 5HFHQWO\ FRQVLGHUDEOH LQWHUHVW KDV HPHUJHG LQ QHWZRUN FRQVWUXFWLRQ DOJRULWKPV WKDW DGDSW WKH QHW DUFKLWHFWXUH WR WKH WUDLQLQJ GDWD VHW )DKOPDQ DQG /HELHUH +DQVRQ f 7KHVH DOJRULWKPV KDYH EHHQ GHYHORSHG IRU VWDWLF QHXUDO QHW PRGHOV DQG UHVXOWV LQGLFDWH WKH IROORZLQJ EHQHILWV VPDOOHU WRWDO SHUIRUPDQFH HUURU EHWWHU JHQHUDOL]DWLRQ FKDUDFWHULVWLFV DQG IDVWHU OHDUQLQJ $Q LQWHUHVWLQJ SODQ LV WR FDWHJRUL]H WKH SURSRVHG FRQVWUXFWLRQ DOJRULWKPV IRU VWDWLF PRGHOV DQG HYDOXDWH WKHLU DSSOLFDELOLW\ IRU G\QDPLF QHXUDO QHWV

PAGE 149

5()(5(1&(6 $PLQ 0 6OLGLQJ 6SHFWUD D 1HZ 3HUVSHFWLYH 3URF WK $QQ $663 :RUNVKRS RQ 6SHFWUXP (VWLPDWLRQ SS $PLW '1HXUDO QHWZRUNV FRXQWLQJ FKLPHV LQ 3URF 1DWO $FDG 6FL 86$ YRO SS $SULO $QGHUVRQ 6 0HUULOO -:/ DQG 3RUW 5 '\QDPLF 6SHHFK &DWHJRUL]DWLRQ ZLWK 5HFXUUHQW 1HWZRUNV LQ 3URFHHGLQJV RI WKH &RQQHFWLRQLVW 6XPPHU 6FKRRO 3LWWVEXUJf 7RXUHW]N\ ' +LQWRQ * DQG 6HMQRZVNL 7 HGVf 0RUJDQ .DXIPDQQ 6DQ 0DWHR &$ %DUU $DQG )HLJHQEDXP ($ 7KH +DQGERRN RI $UWLILFLDO ,QWHOOLJHQFH YRO ,,, DQG ,,, :LOOLDP .DXIPDQQ ,QF /RV $OWRV &$ %RGHQKDXVHQ 8 DQG :DLEHO $ /HDUQLQJ WKH $UFKLWHFWXUH RI 1HXUDO 1HWZRUNV IRU 6SHHFK 5HFRJQLWLRQ ,((( 3URF RI WKH ,&$663 0D\ %UDXQ 0 2UGLQDU\ 'LIIHUHQWLDO (TXDWLRQV DQG WKHLU $SSOLFDWLRQV UG HG 6SULQJHU 9HUODJ %HUOLQ %U\VRQ$( DQG +R<& $SSOLHG 2SWLPDO &RQWURO RSWLPL]DWLRQ HVWLPDWLRQ DQG FRQWURO +HPLVSHUH 3XEO FRUS 1HZ
PAGE 150

*RUL 0 %HQJLR < DQG 'H 0RUL 5 %36 $ /HDUQLQJ $OJRULWKP IRU &DSWXULQJ WKH '\QDPLF 1DWXUH RI 6SHHFK LQ 3URF RI WKH ,QW -RLQW &RQI RQ 1HXUDO 1HWZRUNV YRO SS *UHHQ '0 DQG 6ZHWV -$ 6LJQDO 'HWHFWLRQ 7KHRU\ DQG 3VYFKRSKYVLFV :LOH\ ,QF +DQVRQ 60HLRVLV 1HWZRUNV LQ $GYDQFHV LQ 1HXUDO ,QIRUPDWLRQ 3URFHVVLQJ 6\VWHPV 7RXUHW]N\ '6 HGf 0RUJDQ .DXIPDQQ 3XE 6DQ 0DWHR &$ +D\NLQ 6 $GDSWLYH )LOWHU 7KHRU\ QG HG 3UHQWLFH+DOO (QJOHZRRG &OLIIV 1+HFKW1LHOVHQ 5 .ROPRJRURYf¬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
PAGE 151

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
PAGE 152

6RPSROLQVN\ + DQG .DQWHU , 7HPSRUDO DVVRFLDWLRQ LQ DV\PPHWLF QHXUDO QHWZRUNV LQ 3K\VLFDO 5HYLHZ /HWWHUV YRO f 'HFHPEHU 6WRPHWWD :6 +RJJ 7 DQG +XEHUPDQ %$ $ '\QDPLFDO $SSURDFK WR 7HPSRUDO 3DWWHUQ 3URFHVVLQJ LQ 1HXUDO ,QIRUPDWLRQ 3URFHVVLQJ 6\VWHPV $QGHUVRQ '= HGf SS 6]HJR * 2UWKRJRQDO 3RO\QRPLDOV &ROORTXLXP 3XEOLFDWLRQV YRO $PHULFDQ 0DWKHPDWLFDO 6RFLHW\ 3URYLGHQFH 5, 7DNHQV ) 'HWHFWLQJ 6WUDQJH $WWUDFWRUV LQ )OXLG 7XUEXOHQFH /HFWXUH 1RWHV LQ 0DWKHPDWLFV 6SULQJHU9HUODJ 1HZ
PAGE 153

%,2*5$3+,&$/ 6.(7&+ %HUW GH 9ULHV ZDV ERUQ RQ -XQH WK LQ 8WUHFKW WKH 1HWKHUODQGV $OWKRXJK KLV IDWKHU VHFUHWO\ KRSHG %HUW ZRXOG HPHUJH DV D FHOHEUDWHG SURIHVVLRQDO VRFFHU SOD\HU IDWH GLYHUWHG KLP LQWR DQ DFDGHPLF HGXFDWLRQ ,Q 1RYHPEHU DV D VWXGHQW RI HOHFWULFDO HQJLQHHULQJ DW WKH (LQGKRYHQ 8QLYHUVLW\ RI 7HFKQRORJ\ KH VSHQW D \HDU DW WKH GHSDUWPHQW RI DQHVWKHVLRORJ\ DW WKH 8QLYHUVLW\ RI )ORULGD WR FRPSOHWH KLV PDVWHUf¬V WKHVLV ZRUN +H UHFHLYHG WKH f¯LQJHQLHXUf° GHJUHH LQ 'HFHPEHU IURP (LQGKRYHQ 8QLYHUVLW\ ,Q $XJXVW KH HQUROOHG LQ WKH HOHFWULFDO HQJLQHHULQJ GHSDUWPHQW RI WKH 8QLYHUVLW\ RI )ORULGD WR REWDLQ D 3K' GHJUHH 6LQFH WKHQ KH KDV ZRUNHG ZLWK 'U -RVH 3ULQFLSH RQ YDULRXV WRSLFV LQ VLJQDO SURFHVVLQJ DQG QHXUDO QHWZRUNV 'XULQJ IUHH WLPH KH OLNHV WR EH DFWLYH LQ YDULRXV VSRUWV VXFK DV WHQQLV UXQQLQJ RU F\FOLQJ

PAGE 154

, FHUWLI\ WKDW , KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ , FHUWLI\ WKDW , KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ W/\-InG 'RQDOG * &KLOGHUV 3URIHVVRU RI (OHFWULFDO (QJLQHHULQJ , FHUWLI\ WKDW , KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ r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

PAGE 155

7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 'HFHPEHU :LQIUHG 0 3KLOOLSV 'HDQ &ROOHJH RI (QJLQHHULQJ 0DGHO\Q 0 /RFNKDUW 'HDQ *UDGXDWH 6FKRRO