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krut_c_Page_79.jpg 889cbd706d50bc5942f7f977d12677ad 4d971a0c644af2295e8f17144c0f1a5299a2128a ON A CONJECTURE OF WILLIAM HERSCHEL By CHRISTOPHER C. KRUT A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 ( 2008 Christopher C. Krut TABLE OF CONTENTS LIST OF TABLES ............. LIST OF FIGURES ............ ABSTRACT ................ CHAPTER 1 INTRODUCTION ........ ................... 2 FINITE STATE MARKOV CHAINS ................. 2.1 Introduction . . . . . . . 2.2 Basic Definitions and Notation .................. 2.3 Some Standard Results for Markov Chains . . . 2.4 Simulating From a Markov Chain . . . . 3 BASIC HIDDEN MARKOV MODELS . . . . 3.1 Introduction . . . . . . . 3.2 Classical Definitions .. .. .. .. .. ... ... .. 3.3 The Scoring Problem ....................... 3.4 The Decoding Problem ...................... 3.5 Sim ulation . . . .. . . . 4 SELECTED APPLICATIONS OF HIDDEN MARKOV MODELS. 4.1 Isolated W ord Recognition . . . . . 4.2 The HMM Applied to Traffic Monitoring . . . 4.3 A Hidden Markov Model as a Generator for Rainfall Data . 5 ON WILLIAM HERSCHEL'S CONJECTURE ........... 5.1 William Herschel's Conjecture ......... 5.2 Interest in Herschel's Claim .......... 5.3 Analysis Using Modern Data .......... 5.4 Conclusions . . . . . 6 MARKOV CHAIN MODEL FOR THE 11 YEAR SUNSPOT CYCLE 6.1 Introduction .. . . . . 6.2 Description of The States ........... 6.3 Starting State Distribution . . . 6.4 OneStep Transition Distribution . . 6.5 Results From Simulations . . . 5 6 7 . 8 . 45 APPENDIX: THE R CODE USED IN CONSTRUCTING AND SIMULATING FROM THE MARKOV CHAIN MODEL CONSTRUCTED IN CHAPTER 6 ...... 70 REFERENCES ....................................... 81 BIOGRAPHICAL SKETCH ................................ 84 LIST OF TABLES Table page 51 Years of relevant solar minima and average wheat prices reported by William Herschel over periods of no reported sunspot activity . . 59 52 Years of relevant solar minima and average wheat prices reported by William Herschel over periods directly before periods of no reported sunspot activity. 59 53 Years of relavent solar minima and average wheat prices reported by William Herschel over periods directly after periods of no reported sunspot activity. 60 54 Mean and variance of the price of wheat in the United States during periods of low sunspot activity and time periods directly preceding and following time periods of low sunspot activity . . . . . . 60 55 Mean and variance of wheat yields during periods of low sunspot activity and periods directly preceding and directly following time periods of low sunspot activity . . . . . . . . . . 60 LIST OF FIGURES Figure page 61 Average sunspot number over the course of the sunspot cycle from simulated sunspot cycles and the average sunspot number over the course of the sunspot cycle from actual data (18561996) . . . . . . 68 62 Average sunspot number over the course of the sunspot cycle from simulated sunspot cycles and the average sunspot number over the course of the sunspot cycle from actual data rounded down in groups of 20 (18561996). . . 68 63 Average sunspot number over the course of the sunspot cycle from simulated sunspot cycles and the average sunspot number over the course of the sunspot cycle from actual data not used to train the model (19641996). . .... 69 64 Average sunspot number over the course of the sunspot cycle from 18561996 and the average sunspot number over the course of the sunspot cycle from 19641996. 69 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ON A CONJECTURE OF WILLIAM HERSCHEL By Christopher C. Krut August 2008 Chair: James Keesling Major: Mathematics In 1801 William Herschel conjectured the existence of a relationship between the price wheat and the appearance or absence of sunspots. Initially with intent of constructing a Hidden Markov Model representing this relationship, a Markov Chain was constructed modeling the 11 year sunspot cycle. While it was observed that there was not a significant relationship between the appearance of sunspots and the price of wheat, the Markov Chain Model for the 11 year sunspot cycle produced reasonable results. Beginning with a brief introduction to Markov Chains and Hidden Markov Models as well as a few selected applications of Hidden Markov Models, an analysis of Herschel's claim using modern data is presented as well as a Markov Chain Model for the 11 year sunspot cycle. CHAPTER 1 INTRODUCTION In 1801 William Herschel conjectured on the possibility of a relationship between the appearance of sunspots and the price of wheat. "It seems probable that some temporary scarcity or defect in vegetation has generally taken when the sun has been with out those appearances which we surmise to be symptoms of a copius emission of light and heat." [8] The laws of supply and demand dictate that a shortage in supply without a corresponding decrease in demand will cause prices to increase. A defect in vegetation, caused by a lack of sunspots, or otherwise, could affect supply and indirectly sunspot activity could affect the price of wheat. In support of his claim Herschel presented 5 examples where the price of wheat was higher during a time period when records indicated an absence of sunspots than in a time period of equal length directly following and in 2 cases preceding this time period. A seemingly reasonable conjecture as changes in the sun could have some impact on the health of crops whose growth is dependent on solar energy. With this in mind the question arose; could this relationship be modeled using a Hidden Markov Model. Investigating this question required research into Hidden Markov models in general. Beginning with an introduction to finite state, discrete time Markov Chains, basic definitions as well as basic solutions to classic problems concerning Markov Chains are discussed. The fundamentals of Hidden Markov Models as well as a few modern applications are then examined. To examine Herschel's claim wheat prices from the twentieth century were used to analyze whether sufficient evidence exists to support the claim that there is significant difference in the price of wheat during periods of low sunspot activity and the surrounding time periods. To further examine Herschel's claim the average wheat yield, measure in bushels per acre, during time periods of low sunspot activity was compared to the average wheat yield during time periods of equal length directly preceding and following this time period. Though results indicate that there is not sufficient evidence to support Herschel's claim, examining the issue led to the discovery that the sunspot cycle itself can be roughly modeled by a Markov Chain. Sunspots are the result of changes in the sun's magnetic field. These highly magnetic areas in the sun appear and vanish following a regular cycle of approximately 11 years. Chapter 6 presents a simple Markov Chain model for the sunspot cycle. The model presented is only a preliminary model and nothing conclusive can be determined from results obtained. It does however suggest that Markov Chains and/or Hidden Markov Models may prove a useful tool in future models. CHAPTER 2 FINITE STATE MARKOV CHAINS 2.1 Introduction Markov Chains are a simple, very useful class of probability models which have been applied to a variety of problems both as a method of modeling a system directly and as part of larger hierarchical models, namely Hidden Markov Models. The dependence structure of a Markov Chain is such that all information about the likelyhood of the next state of a given system is contained in the current state of the system. This means that the entire history of the system prior to some time t provides no more information regarding the state of the system at time t than simply knowing the state of the system at time t1. This simple framework provides a reasonable method for modeling many real life systems, especially over relatively short periods of time. With the aid of examples, the following sections will present a brief introduction to this classic probability model. 2.2 Basic Definitions and Notation A Markov Chain is a type of stochastic process, a family of random variables {Xt}It > o}, where the probability that Xt takes on a certain value is dependent only on the value of Xt 1. The events of the system which correspond to the values which the random variables can assume are called the states of the Markov Chain and the set of all possible states is referred to as the statespace of the Markov Chain. A Markov Chain can have at most a countably infinite number of states, though in this paper only Markov Chains with a finite statespace will be considered. A stochastic process with at most a countably infinite number of states is said to have a discrete statespace [21],[1]. Additionally the parameter t which indexes the Markov Chain often referred to as the time parameter though it need not represent time explicitly, takes on at most a countably infinite number values. Since the time parameter of a Markov Chain takes on at most a countably infinite number of values a Markov Chain is said to have a discrete time parameter as well as a discrete statespace [21],[1]. Consider the following example. Let's simplify the political party system in the United States a little and suppose that voters can register as Republican, Democrat, or Independent. Suppose further the probability of an individual voter being registered as Republican, Democrat, or Independent in the next presidential election is dependent only on their current party affiliation. It is then possible to model the behavior of an individual voter using a stochastic process and in particular using a Markov Chain. The states of the system are a finite set: Republican, Democrat, and Independent. The time parameter is discrete as we are only interested in the party affiliation of the voter at discrete time points, namely each presidential election. If the random variable Xt which is a member of the family of random variables which make up a stochastic process in general or a Markov Chain in particular takes on a certain value in the state space say sj then it is said that the stochastic process/Markov Chain is in state sj at time t. If a stochastic process is in state sj at time t1 and state si at time t the process is said to have transitioned from state sj to state si at time t. The behavior of a stochastic process is then described by the probability of transitioning to a new state given the previous states of the process. Formally, a Markov Chain is a stochastic process {X,}n > o with discrete time parameter(n) and space space(S) such that the following equality holds for all n and s G S [21],[1],[12],[5]. P(X,+l = sn+1 Xn = Sn, Xn1 = Sn1, ... X0 So) P(X, + = s+li X, = Sn) (21) The probability P(Xn+1 = s+l X, = Sn) is called the one step transition probability for the Markov Chain and common notation is to write P(Xn+ = sj Xn = si) as pij where si, sj G S [1],[12],[21]. The collection of transition probabilities pij, sj G S define a one step transition probability vector which gives the conditional probability distribution of transitioning to a new state in the state space at time n+1 given the chain is in state si at time n. The collection of conditional probability vectors defined by the one step transition probabilities defines a stochastic matrix called the one step transition matrix P. P11 P12 . . Pm P21 P22 . . P2m P Pml Pm2 Pmm f The transition probability matrix governs the behavior of the chain once an initial state is known. The initial state of a system which a Markov Chain represents may however, be determined by a different probability distribution. For this reason, a second probability distribution which gives the probability of initially being in each of the respective states in the statespace is necessary to describe the behavior of a Markov Chain. The starting state distribution for a Markov Chain is a probability vector giving the probability of being in a given state at time n=0 [21],[1],[15],[12]. In this paper the starting state distribution will be denoted Po. Under the assumptions made, the voter example mentioned previously can be modeled using a Markov Chain. The statespace of the chain is defined as follows. Suppose state 1 corresponds to a voter being registered a Republican, state 2 represents a voter being registered a Democrat, and state 3 represents the voter being registered as Independent. Suppose through some sort of analysis the transition probabilities have been determined. The transition probabilities are given in the following transition matrix. .70 .20 .1 P = .20 .725 .075 .20 .25 .55 To completely describe the behavior of the system a starting state distribution is required. A voter's initial party affiliation is determined by a number of factors. These include the party affiliation of their parents, socioeconomic status, education, peer influence, as well as other environmental factors. Let's suppose after analysis it has been determined that the starting state distribution for party affiliation is given by the following probability distribution where the first entry is the probability a voter will initially register Republican, the second entry is the probability a voter will initially register Democrat, and the third entry is the probability a voter will initially register as an independent: Po= (.5 .4 .1 In the example the one step transition matrix provides the probability that a voter will vote as a member of a given political party in the next election given their current party affiliation. Also of interest would be the probability that in some future election a voter would be registered with a certain political party given that we know their current party affiliation. As it turns out this is easily computed from the one step transition matrix, and in general is the probability P(Xm+n = sj X, = si). This is the probability of transitioning to state sj in m steps given the chain started in state si at time n. Throughout this paper all Markov Chains are assumed to be time homogeneous, unless otherwise indicated. A Markov Chain is said to be time homogeneous, or just homogeneous, if the transition probability given by P(Xm+n = sj X, = si) does not depend on n [1],[15],[12]. Note that in a homogeneous Markov Chain if m = 1 the probability P(Xm+n = sj X, = si) P= pi for all values of n. This property simplifies the model greatly, however it may limit to some extent the model's ability to approximate more dynamic systems. When a Markov Chain is homogeneous the transition probability matrix P does not change over time producing the equality P(Xm+n = sj X, = si) P(Xm = sjyXo = si) [1],[21],[15]. The probability defined by P(X, = sj Xo si) is called the nstep transition probability [1],[21],[15],[12]. Lacking a single standard notation for this probability, the convention P(' = P(X, = sj Xo = si) will be used throughout this paper to indicate the probability of transitioning from state si to state sj in n steps [1]. Note that rPZ = 1 i Pkj [1],[12],[21],[15]. It should be clear that P, can then be expressed as PZj = Zr=l  .. 2=1 1=lPikjPkk2..Pkj. This is the sum of the probabilities of all all possible sequences of length n which start in state si and end in state sj. However, from the above equality it is seen P) is equal to ifj' entry of the 0nt power of the transition matrix P, i.e. Pi) = Pf [1],[12],[21],[15]. Using the nstep transition matrix along with the starting state distribution it is possible to compute the unconditional probability of being in some state si at time n. The probability of being in state si at time n is given by the ith entry of the probability vector Po Pn [12],[1]. To see this consider that Po gives the probability of being in a given state at time 0. Then for a given Markov Chain with state space S consisting of 3 states, for example, the probability of being in state s, G S at time t = 1 is given by P(X1 = s1) = P(Xo = s1)* P(Xi sXo = s1) + P(Xo = s2)P(Xi = sXo = s2) + P(Xo = S3)P(Xi = SlXo = ss) This corresponds to the 1st entry of the vector given by Po P. Using induction it can be shown that this result holds for all values of n. Returning to the previous example, suppose starting at time n = 0 we want to know the distribution for party affiliation after 3 presidential elections. To find the 3step transition probabilities for the Markov Chain representing voter behavior take the third power of the transition matrix P. Thus we have. ..475 .36975 .15525 P3 = .35 .512578 .137422 .35 .405406 .244594 Then to find the probability distribution after 4 presidential elections multiply the starting state distribution vector by the 3step transition matrix, which gives: PoP3 = .4125 .430447 .157053 ) The above probability distribution shows that after 4 presidential elections the probability a voter will be registered Republican is .4125, the probability a voter will be registered Democrat is .430447, and the probability a voter will be registered Independent is .157053. An alternative interpretation of the results is that after 4 presidential elections approximately 41.25% of the population will be registered Republican, approximately 43.0447% of the population will be registered Democrat, and approximately 15.7053% of the population will be registered Independent. It should be noted that the system in the voter party affiliation example is currently being modeled using a homogeneous Markov Chain. Over any reasonable time period the probabilities concerning party loyalty will certainly change. Thus modeling voter party affiliation over extended periods of time using a homogeneous Markov Chain is inappropriate. Over a short period of time however, the system is approximately time homogeneous. For instance it is highly unlikely the probabilities concerning party loyalty will change over the course of a few weeks, or perhaps even an election or two. Most systems are dynamic and evolve over time making a homogeneous Markov Chain an inappropriate choice. However in many cases, over a short time period it is not unreasonable to assume a given system is homogeneous which allows the use of a homogeneous Markov Chain. In addition a homogeneous Markov Chain is simpler to implement than a nonhomogeneous Markov Chain and can often provide a preliminary model. Lastly, one may use homogeneous Markov Chain determine the probability of future outcomes if the system does not change. Returning to the voting example, one may wish to determine the percentage of voters registered Republican after n elections if current trends continue. It is certainly possible for a voter to be registered as a member of a certain political party for one election and registered as a member of a different political for the next election. This is reflected in the example by the fact that the probability of transitioning from one state to any other state is positive. In a Markov Chain, when it is possible to go from one state to another, say from state si to state sj, in a finite number of steps, which is indicated by the existence of an n such that the ijt' entry of the nstep transition matrix being positive, state sj is said to be reachable from state si [15],[1],[21]. Common notation is to write si sj, indicating sj is reachable from si. If si sj and sj si then the states si and sj are said to communicate, notated si sj [15],[1],[21]. Communication between states defines an equivalence relation on the state space of the Markov Chain and states which communicate form equivalence classes [15],[1],[21]. There is one exception however. If a state si does not communicate with itself then communication between states does not define an equivalence relation on the state space of the Markov Chain as the equivalence classes would not form a partition of the state space [15]. A state which communicates only with itself is called an absorbing state and corresponds to the transition probability P = 1 for an absorbing state si [15],[21],[1]. A Markov Chain in which every state communicates with every other state, i.e. there is only one equivalence class, is said to be irreducible [12],[1],[15],[21]. Note that the voter party affiliation example is an example of an irreducible Markov Chain as the probability of going from one state to any other state in one step is positive. Another method by which states of a Markov Chain can be described is by the probability that given the chain is in a state si it will return to that state. For example, consider the voter party affiliation example again. It seems reasonable that if a voter has only 3 choices of political parties with which to register a voter who votes as a registered Republican should vote again in some future election as a registered Republican. In general terms, states of a Markov Chain for which the probability that given the chain is in a certain state it will return to that state is 1 are said to be recurrent [1],[15],[12]. States for which the probability that given the chain is in a certain state it will return to that state is less than 1 are said to be transient [21],[1],[15],[12]. It is known that the states of the voter party affiliation example are recurrent, namely it is known that a voter who votes as a registered Republican in a given election will vote again as a registered Republican in some future election. Given the knowledge that a voter who has voted as a registered Republican will vote again as a registered Republican it would be interesting to know the average amount of time that will pass between the elections in which the voter votes as a registered republican. Given a recurrent state sj of a Markov Chain the expected amount of time it will take to return to state sj given the chain started in state sj is called the expected time of return [1],[15],[12]. If we know that a voter who votes as registered Republican will vote again as a registered Republican in a finite amount of time the expected time of return is useful information. If the expected time of return is not finite then even though the probability of return to a given state is 1 the amount of time it will take for the chain to return is expected to be infinite and thus the recurrence of the chain doesn't seem to mean quite as much. For this reason recurrent states of a Markov Chain are described by the amount of time it will take the chain, given it started in a certain state to return to that state. A recurrent state whose expected time of return is finite is referred to as a positive recurrent state while a recurrent state whose expected time of return is infinite is called a null recurrent state [1],[15],[12],[21]. We know that all states in the voter party affiliation example communicate. Thus it is possible that a voter who votes in an election as a registered Democrat will vote in a future election registered as an Independent. An interesting question is to find out, on average how long it will take a voter who is voting in the current election as a registered Democrat to vote in a future election registered as an Independent. In general suppose Yij is a random variable defined as follows: Y = n if and only if X, = s ,X0 = Si,Xm sj 1 < m < n. Yij is then the random variable representing the first time of passage from state si to state sj [12],[21],[15]. The first time of passage provides a method of determining whether a given state is recurrent or transient as well as determining if a given recurrent state is positive or null recurrent. The probability distribution of YEj gives the probabilities of returning to state si for the first time at time n for each possible time n. The sum of the probability of transitioning to state si from state si for the first time at time n, over all possible values of n gives the probability of ever returning to state si given the chain started in state si. A standard notation is to let fij(n) be the probability mass function associated with Yi [12],[21],[15],[1]. Using this notation it is possible to determining if a state is recurrent or transient. A state si is then recurrent if 2E 1 fJi(n) = 1, and transient if E ljf((n) < 1 [12],[21],[1],[15]. The expectation of Yij, E[Yij] gives the expected number of steps it will take to transition to state sj given the system started in state si [12],[21],[1],[15]. For a recurrent state si it is possible to determine if si is positive or null recurrent by computing the expectation of Yij. The state si is positive recurrent if the expectation of Yij is finite and null recurrent if the expectation of Yj is infinite. It should be noted that fij(n) and P/n are not the same as PFI gives the probability of being in state sj at time n given the chain started in state si and gives the probability that the chain has transitioned from state si to state sj for the first time at time n. There is however a relationship between Pij and fij = 1 fij(n) given by Doeblin's formula which states given any si and sj: fjj = lim so j [15] (2 2) 1 + Z=l p3 Using this formula it can be shown that if sj is a transient state then E j1 P1i7 converges, si G S [15]. Since E2 1 Pi converges for any transient state sj, limnoo Pi = 0 [15]. Hence the probability of transitioning to a transient state sj from any state si in n steps tends to zero as the number of steps tends to infinity. Recurrence and transience are class properties. That is, if si is a recurrent state and si sj then sj is a recurrent state [12],[15],[1],[21]. Recurrent states can be reached from transient states, transient states however are not reachable from recurrent states. Thus transient states and recurrent states do not communicate. This implies a recurrent state and a transient state cannot be in the same class. In this way classes of states can be classified as either transient or recurrent. This also implies that a Markov Chain cannot transition from a recurrent class to a transient class [15],[1],[21]. In a finite Markov Chain with k states the chain can transition at most k times before it must return to some state. Thus in a finite Markov Chain at least one state must be recurrent. This generalizes to every finite Markov Chain contains at least one recurrent class of states [1],[21],[15]. Given that every finite Markov Chain contains at least 1 recurrent class, and the probability of being in any given transient state at time n tends to 0 as n tends to infinity, it follows that a finite Markov Chain will spend only a finite amount of time in transient classes before transitioning into a recurrent class [1],[21],[15],[12]. Returning to the example regarding voter party affiliation; it is known that the chain is irreducible. Then since every finite Markov Chain contains at least one recurrent class it is known that the chain representing this system consists of a single recurrent class. If we know a voter is initially registered Independent it would be interesting to know if there exist future elections for which the probability of the voter being registered Independent is 0. If PFi > 0, then it is known that it is possible to transition from state si back to state si in n steps. The period of a given state si is the greatest common divisor of all possible times n where it is possible to transition from from state si back to state si, i.e the greatest common divisor of {n PF > 0} [12],[15],[21],[1]. If the period of some given state is 1 then the state is said to be periodic [12],[15],[21],[1]. As with recurrence, periodicity is a class property. That is if si is of period m and si sj then sj is of period m[12][15][21][1]. Therefore, a class of states can be said to have a certain period. This is important since larger Markov Chains will have multiple classes of states and the behavior of the system with respect to these classes will be important. Lastly, consider the voter party affiliation example one more time. Recall that the one step transition matrix for the voter party affiliation example has all positive entries, hence the period of the Markov Chain is going to be 1 for each state in the chain. Thus, each state in the chain is periodic. In the following section some basic questions regarding the behavior of Markov Chains are examined along with classic solutions to these problems. 2.3 Some Standard Results for Markov Chains Quite often, the problem of interest is the long term behavior of the system a Markov Chain is modeling. Returning to the voter party affiliation example of section 2.2 it would be interesting to know the long term probabilities of a voter being registered Republican, Democrat, or Independent, based on current trends. In section 2.2 we were able to find the nstep transition probabilities for the voter party affiliation model. The nstep transition probabilities represented the probability of being registered with a given political party at the n + 1st election given the voter initially registered with a certain political party. In general this was given by P'. Consider lim,,ooP. Does this quantity converge? Does it converge in any special way? Under certain circumstances this quantity does converge? Consider the results of taking successively higher powers of the transition matrix from the voter example in section 2.2. .70 .20 .1 P = .20 .725 .075 .20 .25 .55 .475 .36975 .15525 P3 = .35 .512578 .137422 .35 .405406 .244594 .409375 .428096 .16252 P6 = .39375 .447861 .15838 .39375 .436375 .16987 .400146 .437903 .16195 P12 = .399902 .438127 .16183 .399902 .428127 .16183 Clearly the matrix P' is converging. The matrix is converging to a matrix whose rows are the limiting distribution for the Markov Chain representing voter party affiliation. An interesting fact about the matrix to which P' is converging is that the rows are all the same. This implies that the probability of being in a given state at time n, for a sufficiently large n, is approximately the same for all starting states. In the following paragraphs the convergent behavior of finite Markov Chains is examined further. Given a finite state, irreducible, periodic Markov Chain with transition matrix P it is known that lim,,ooPn converges to matrix whose rows are the vector 7T such that 7 P = 7 [21],[1],[4],[5]. The vector 7 is the limiting distribution for the Markov Chain and gives the long term probabilities of being in each respective state [21],[1][4],[5]. As time tends to infinity the rows of the transition matrix P approach 7 as was observed in the above example. Since the rows of the matrix to which lim,,.ooP' converges are the same it should be clear that the long term probability of being in each respective state does not depend on the initial state[12]. Given the existence of a limiting distribution for a Markov Chain, the next question is how quickly does the chain converge to its limiting distribution? The speed with which the transition matrix converges to its limiting distribution provides an indication of the time necessary for the probability of being in a given state to no longer be dependent on the initial state. It can be shown that for a finite state, periodic, irreducible Markov Chain, P' converges geometrically to its limiting distribution [1],[21]. For a small matrix it is possible to compute the limiting distribution by taking powers of the matrix, as was done in the example. However, for larger matrices and for someone without a computer algebra system taking powers of a matrix may prove unfeasible. To simplify the problem of finding the limiting distribution remember that 7 P = 7. This means that 7rP = along with the condition that the elements of 7 sum to one, produces a set of equations which can be solved, under certain conditions, to find the limiting distribution 7T [12],[1],[15],[21],[12]. Returning to the voter party affiliation example, the following system of equations is obtained from 7 P = . 7i .7 + 72 .2 + 73 .2 = 71 (23) 71 *.2 + 72 .725 + 7T3 .25 = 72 (24) 71 .1 + 72 .075 + 73 .55 = 73 (25) 7T1 + 72 + 73 = 1 (26) Using any 2 of the first 3 equations and the last equation the limiting distribution 7 = (.400090, .438087, .161823) can be computed. From the limiting distribution we can conclude that after a sufficiently long time the probability a voter will be a registered Republican is about .40009, the probability they will be registered a Democrat is about .438087, and the probability the voter will be registered as Independent is about .161823. Note that these probabilities are independent of the party with which the voter originally registered. Analysis like this would be helpful in analyzing the long term "health" of a political party. In addition to giving the fraction of time the chain is expected to spend in each state, the limiting distribution can also be used to find the mean time to recurrence for each state in the chain. It can be shown that the mean time of recurrence for a state si is given by [15]. Thus it is now possible to compute the time it takes a voter who voted as a registered Republican to vote again as a registered republican. The mean recurrence times for the example are given by 1 = (2.49994, 2.28625, 6.17959). One final note the reader should bear in mind is that for a periodic Markov Chain the above method can not be used to compute the limiting distribution. In many cases a Markov Chain is not irreducible, but contains multiple classes of states, at least one of which is recurrent. Suppose a Markov Chain has recurrent classes C1, C2, C3,..., C, and transient classes T1, T2, T3, ... ,Tm. It is possible to rearrange the probabilities in the transition matrix P so that the transition probabilities between states in each of the recurrent classes of a given Markov Chain are contained in stochastic submatrices, the probabilities of transitioning from one transient state in the chain to another transient state in the chain are contained in another submatrix, usually denoted Q, and the probabilities of transitioning from a given transient state to given recurrent state are contained in another submatrix, usually denoted R [15],[21],[1]. It should be noted that the rows of submatrices Q and R will not, in general, sum to 1. Using the above notation the one step transition matrix P for a given Markov Chain can be written as: Pi 0 .. 0 0 0 P2 ... o 0 P = : : [15], [1], [21], o 0 ... P, 0 R1 R2 ... Rn Q The example from section 2.2 will now be expanded as follows. Suppose that it is now possible for someone to remain in a given political party permanently, i.e. they decide never to switch parties. If someone were to remain affiliated with a political party for a sufficiently long time, the probability that they would change affiliation to an opposing political party should decrease. For the purpose of simplification suppose that if someone is registered with a particular party affiliation for 3 consecutive presidential elections then for all practical purposes we assume them to be a permanent member of that political party. This assumption results in a new Markov Chain to model the behavior of voters. In this new chain there are 3 recurrent classes, permanent Republican, permanent Democrat, permanent Independent. Note that these classes are not only recurrent but absorbing as well. The states of the new Markov Chain will be numbered in the following way. 1. A voter is a permanent Republican 2. A voter is a permanent Democrat 3. A voter is a permanent Independent 4. A voter has voted in the last 2 elections as a Republican 5. A voter has voted in the last 2 elections as a Democrat 6. A voter has voted in the last 2 elections as an Independent 7. A voter has voted in the last election as a Republican 8. A voter has voted in the last election as a Democrat 9. A voter has voted in the last election as an Independent In the new Markov Chain representing this process states 1,2, and 3 are absorbing, states 4,5,6,7,8 and 9 are transient. With the assumptions made in constructing this model it is known that a voter will only vote a finite number of times before being absorbed into an absorbing state, becoming a permanent member of a certain political party. Using the transition probabilities from the first example the transition matrix for this Markov Chain has the form. 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 .7 0 0 0 0 0 0 .2 .1 0 .725 0 0 0 0 .2 0 .075 0 0 .55 0 0 0 .2 .25 0 0 0 0 .7 0 0 0 .2 .1 0 0 0 0 .725 0 .2 0 .075 0 0 0 0 0 .55 .2 .25 0 The corresponding Q and R matrices are. 0 0 0 0 .2 .1 .7 0 0 0 0 0 .2 0 .075 0 .725 0 0 0 0 .2 .25 0 0 0 .55 Q R .7 0 0 0 .2 .1 0 0 0 0 .725 0 .2 0 .075 0 0 0 0 0 .55 .2 .25 0 0 0 0 Utilizing the Markov Chain above it is possible to analyze the behavior of voters prior to permanently joining a given political party. The first question of interest is: if someone initially registers with a given political party how many elections will the voter likely be registered with that party prior to becoming a permanent member of one of the political parties? Another interesting question is: if a voter is initially registered with a given political party how many presidential elections will take place prior to the voter becoming a permanent member of one of the political parties? Lastly, one might be interested in the probability of becoming registered permanently as a member of a given political party if they initially register with a certain political party. The question of how many times a voter will vote as a member of a particular party given their current voting behavior is in general the question how many times will a given transient state will be visited prior to the chain entering one of the recurrent classes. If we define a random variable to take on the number of times a state sj is visited prior to the chain entering a recurrent state given an initial starting state si. Then it can be shown that the expectation of this random variable is given by the ij' of: (I Q) 1 = M[15], [1], [12], [21] (27) M is called the fundamental matrix [1],[21],[15]. The ijft entry of M gives the expected number of visits to a transient state sjprior to entering a recurrent class given the chain started in state si [15],[12],[1],[21]. Using the fundamental matrix it is possible to determine the expected number of times a voter will vote as a member of a given political party given their voting history. The fundamental matrix, computed in this manner, for the voter party affiliation Markov Chain is: 1.11429 .232808 .093115 .163267 .321115 .1693 .210187 1.1152 .08063 .300267 .1589 .1466 .269501 .310086 1.06643 .385001 .427704 .120784 M= .894288 .395774 .158295 1.27755 .545895 .287809 .362573 .923724 .13909 .517961 1.2741 .25289 .417726 .480632 .65296 .596751 .662941 1.18721 From the fundamental matrix M we see that the chain progresses through the transient states quickly, averaging at most a little more than one step in a given transient state regardless of the transient state si in which the chain starts. With respect to the model representing voter registration this implies that voters choose their party alliances quickly. Suppose now we want to know the average time it takes for a voter to become a permanent member of a given political party given their voting history. In general this is the mean time spent in transient states prior to entering a recurrent state given the initial state is si and is obtained by summing across the ith row of the fundamental matrix M [1],[21],[15]. The results obtained from the voter party affiliation example are given below. 2.0939 2.01178 2.57951 Steps to Recurrence = 3.55961 3.47034 3.99822 Given this information it can be seen that a voter who was registered with their respective party for only the previous election is likely to vote between 3 and 4 times prior to becoming permanently affiliated with a political party. This corresponds to permanently joining a political party the 4th or 5th time a voter votes. Next, suppose we want to know the probability of a voter becoming a permanent member of a given political party given their party registration in previous elections. In general this is the probability of being absorbed into a given recurrent class given some initial state. Using the fundamental matrix M and submatrix R it can be shown that the matrix F whose ifjL entry is the probability of being absorbed in to a given absorbing state is given by the equation: F = MR[12],[15],[1],[21] (28) Returning to the example we can now find the respective probability of a voter eventually becoming a permanent Republican, Democrat, or Independent given their party registration in previous elections. For the voter party affiliation example, F is approximately given by: .7800 .1688 .0512 .1471 .8085 .0444 .1887 .2248 .5865 F= .6260 .2869 .0871 .2538 .6697 .0765 .2924 .3485 .3519 From this result if a voter is registered with a given political party for two presidential elections then the probability is fairly high that they will become a permanent member of that political party. The probabilities of becoming a affiliated permanently with a political party given a voter was registered with the party for one presidential election are slightly lower. Once the system has entered a recurrent class the recurrent behavior of the class controls the behavior of the system. Thus the long term behavior of the system will be determined by the limiting distribution of the recurrent class, if it exists, into which the chain is absorbed. It should be noted however that in a finite Markov Chain with both recurrent and transient states the limiting distributions of the chain, provided they exist, will be dependent on the initial state [15]. Lastly, the example used in this section contained 3 absorbing states. The method for computing F can also be used when the recurrent classes of the Markov Chain contain multiple states. The matrix F is simply the probability of ever transitioning to states in recurrent classes. Thus the probability of being absorbed into a certain recurrent class given an initial state is just the sum of the probabilities of of entering each recurrent state given an initial state [15]. Thus, larger recurrent classes of states do not present any real difficulty, aside from additional computation. The simplified example used in this chapter is fictional and was only intended for illustrative purposes. However after discovering this example the question arose; has this issue been examined? Can a homogeneous Markov Chain be used to model voter loyalty to a political party? Researching this issue I came across a paper by Douglas Dobson and Duane A. Meeter which briefly examines the question [6]. The following is a summary of their findings. Dobson and Meeter begin by classifying voters into one of 5 categories: "Strong Democrat, Weak Democrat, Independent, Weak Republican, and Strong Republican" [6]. The authors define a model which describes the change in voter party affiliation in two parts. The first part describes the probability that the voter will change affiliation the second describes the probability that if it is known the voter is changing party affiliation he/she will change to a particular affiliation [6]. In this way it is possible to describe changes party affiliation in more detail as well as examine whether either part of the model should be modeled using a time varying, i.e. nonstationary, model. In particular, the model used by the Dobson and Meter is given by the transition matrix "P(t) = D(t) + (1D(t))M(t)" where D(t) gives the probabilities of remaining in the current state and M(t) is the probability of transitioning to a state other than the current state [6]. The probability matrices M(t) and D(t) were constructed based on data from 19561958. A second pair of probability matrices M(t) and D(t) are constructed based on data from 19581960. Using a chisquared test for stationarity it was found that neither M(t) or D(t) should be stationary [6]. Thus it seems a nonhomogeneous Markov Chain would be the best choice for modeling voter loyalty. Reasoning that the transition probabilities would vary across the population based on characteristics such as gender, race, etc. Dobson and Meeter partitioned their data into three groups based on the level of interest in the electoral system [6]. Defining 3 categories, "high interest, moderate interest, low interest", they found a homogeneous Markov Chain to be an appropriate model for both the high interest and low interest categories with only the moderate interest category requiring a nonhomogeneous Markov Chain [6]. Another example in [1] attempts to use a homogeneous Markov Chain to model changes in voter preference for a particular presidential candidate. However, it was again determined that a homogeneous Markov Chain was not an appropriate model [1]. Thus it seems that voting behavior, especially over any extended period of time, is likely too dynamic to be modeled well using a homogeneous Markov Chain. 2.4 Simulating From a Markov Chain Simulation from a Markov Chain is a useful tool when analytic methods, such as those presented in section 2.3, are not available. For instance a simulation could be used to evaluate how well a model represents the system it is modeling. Additionally a Markov Chain could be used as part of a larger simulation. In either case simulating from a Markov Chain is a relatively simple task. Most mathematical/statistical software packages have a good, at worst reasonable, function for generating random numbers from a uniform distribution. It is then possible to transform the pseudo random numbers generated from a uniform [0,1] distribution to pseudo random numbers generated from the initial and one step transition probability distributions which characterize a Markov Chain. This is done in the following way [24],[5]. 1. Generate a number from uniform[0,1] distribution, call it p. 2. Sum over the initial probability distribution until the sum > p keeping track of the current state whose probability is being added to the sum, call it k. 3. State k1 is the current state. 4. Generate another number from uniform[0,1] distribution. 5. Using the one step transition distribution with current state k1 in place of the initial probability distribution repeat steps 2,3. Steps 3,4,5 can be repeated to generate a sequence of random numbers from the Markov Chain. The code used to simulate sunspot numbers from the Markov Chain discussed in Chapter 6 is available in the Appendix. Generating multiple sequences gives an idea of how the Markov Chain representing the system being modeled is behaving. If the Markov Chain is based on observations, the sequences generated from the model can be compared to the data to evaluate the fit of the model. Sequences generated from a purely theoretical Markov Chain can be used to gain insight into the system being modeled which might have otherwise been unavailable. CHAPTER 3 BASIC HIDDEN MARKOV MODELS 3.1 Introduction Hidden Markov Models, hereafter referred to as HMM's, are a family of probability models composed of two stochastic processes. In a Hidden Markov Model one of the stochastic processes represents an observable process which is dependent on another stochastic process about which little or no information is available [25],[17],[9]. Consider the following example. Suppose a certain highway is being observed. The traffic flow on the highway can be classified as light, moderate, or heavy. Suppose further that no information is known about any other part of the highway except the observed traffic intensity. It is known that highway traffic is dependent on the time of day, any construction which may be taking place on the highway, and whether or not an accident has occurred. In this way it is possible, under some simplifying assumptions, to model the intensity of traffic as an observable result of the previously mentioned factors. In every Hidden Markov Model there is a set of observations, and a set of underlying states. The behavior of the set of underlying states can be approximated by a Markov Chain thus there is a Markov Chain which is "hidden" and only visible through a set of observations which depend on the underlying states. Hence the name Hidden Markov Model. There are 3 applications for which Hidden Markov Models are particularly well suited. First, it is possible to simulate a sequence observations from a Hidden Markov Model. These observations can be used as part of a larger simulation or to obtain statistical information about the system which may not have been available through analytic methods. Second, Hidden Markov Models have been used in identification applications. Specifically Hidden Markov Models have been used with success in speech recognition and gene identification. Third, Hidden Markov Models can be used to obtain information about the state sequence which generates a given observation sequence. Each of these applications will be discussed in the following sections. Section 3.2 will introduce the basic terminology of Hidden Markov Models along with a simple example to help clarify the ideas presented. Section 3.3 will then examine the classic scoring problem for HMM's. Next section 3.4 will present the classic method of decoding the most likely underlying state sequence which produced a given observation sequence. Then, in section 3.5 an example of a simulation using Hidden Markov Models will be presented. 3.2 Classical Definitions As stated in section 3.1 a Hidden Markov Model is composed of two stochastic processes, call them Ot and Xt. The process Ot is a probabilistic function of the process Xt, taking the values in the set 0 {oi, 02, .... o,}. The process Xt represents a finite state, discrete time Markov chain with transition probabilities pij, taking on values in the set S = {S1, S2, ... s,}. Though there are no explicit restrictions on O6, the ability to estimate the parameters of the distribution often limits potential choices [25]. In this chapter we are only interested in the simplest case when Ot is a finite state, discrete time process, and Ot and Ot+1 are independent given Xt+,. The notation bi(k) = P(Ot = k Xt = si) will be used throughout the chapter to represent the probability of a certain observation k given a current state s [25],[17],[9]. The state space, observation set, observation distribution, underlying state transition probabilities, and starting state distribution completely define a hidden Markov model [25],[22],[17]. Common notation is to write A = {{Ppij}sss, bj(k), 0, S, Po} to represent a Hidden Markov Model [25],[9],[17]. The forwardbackward variables are fundamental in the classical algorithms for Hidden Markov Models. Suppose we have an observation sequence 0 {oi, 02, ... on}. The forward variable is defined to be at(i) = P(O0 = oo, 01 = 01, ..., Ot = Or, Xt = si) [17],[25],[9]. This is the probability of observing the first t observations of the observation sequence and ending in state s at time t. The backward variable is defined P(Ot+l = Ot+l, Ot+2 = Ot+2, , On o.Xt = si} [17],[25],[9]. This is the probability of observing a given sequence starting at time t+1 to time n, starting in state s. The major advantage of using the forwardbackward variables for problems involving hidden Markov models is that they can be calculated relatively easily and there exist algorithms for solving the 3 problems mentioned in section 3.1 which utilize the forwardbackward variables [25],[17],[9]. The forward variable a (i) can be computed as follows. First note that ao(i) is simply the probability of observing oo and starting in state si. Using the law of total probability we can write P(Oo = oo,Xo so)P(Xo s= o), where is P(Xo so) as P(Oo Po(so). Thus for each sj in the state space of Xt we have: ao(i) b= (oo) Po(i)[25], [17], [9] In the second step a1(i) P(Oo = O,1 = O1, Xi = si). A little computation shows that a (i) = (Esesao(si)pss,)b.J(o). In general we can compute aj(i) using t 1(i) and the transition and observation probabilities. at(i) Z= (at (s) ps)b(ot)[17], [25], [9] It is possible to define the backward variables in a similar manner. To begin computing the backwards variables begin with: /3,(i) =1 for all si S[17], [25], [9] The remaining /t(i) can be calculated using: /3t(i) = pssbs(Ot+i)Ai/3t(s)[17], [25], [9] These two variables provide methods for solving the problems above as well as estimating the model parameters. The classical solutions to the problems mentioned in chapter 1 as to be f/t(i) well as illustrative examples will be presented in sections 3.3 and 3.4 respectively. For now we consider only the problem of estimating the parameters of the model. The classic solution to estimating, or reestimating to be more correct, the model's probability distributions is the BaumWelch algorithm. The BaumWelch algorithm can be applied when the only information regarding Xt is contained in Or. If any explicit information about Xt is known this can be reflected in the initial estimates of the model parameters. To apply the BaumWelch algorithm initial values are chosen for Po(s), {pij}j,ssEs, and bs(Ot) 0 < t < n [25],[17],[9]. Next, given an observation sequence 0 (o1,02,...,on), it is possible to reestimate Po(s), {pij}js,sEs, and b,(Ot) 0 as follows. To begin we define: Tt(i,j) P(Xt = si,Xt+1 = s 0O,A) [25],[17],[9]. Then by the definition of conditional probability we have: P(Xt P(xt=s, Xos', oA). First note P(Xt Si, Xt+1 = Sj, 0 A) is the probability of observing the first t observations in the sequence 0 and being in in state xi at time t, at(i), then transitioning to state sj at time t+1, pij, and observing Or+1 given state sj, by(ot+1), then observing the remaining observations ot+2, ..., o, given underlying Markov Chain is in state j at time t+1, /1t+,(j). Since each of the above events is independent P(Xt = si, Xt+1 = sj, 0A) that summing P(Xt = si, Xt = at(i) pi bj(ot+1) A/13t(j) [25],[17],[9]. Next note +1 = Sj, 0 A) over all possible states, si, sj gives P(0 A) [25],[17],[9]. Combining these two observations provides a reasonable method of computing (,j) t() (t+) t+ (j) [25], [17] (3 5) E.Es E.sEs 0t(i) ai bj(ot+1) At+(j) It is now possible to obtain P(Xt states sj [25],[17],[9]. Let Ct(i) = si 0, A) by simply summing Tt(i,j) over all possible P(Xt = si0, model) EZsesn(i,j) [25],[17],[9]. Summing Tt(i,j) over t for 1 < t < n 1 gives the expected number of times the underlying Markov Chain transitions from state si to state sj [25],[17],[9], Similarly < t < n si,Xt+1 = sjO,A) summing 7t(i) over t for 1 < t < n 1 gives the expected number of times the system will be in state si [25],[17],[9]. Thus the new estimates of the parameters can be computed using t(i) and Ft(i,j). Po(s) = 1(s) [25], [17], [9] T1 , o di 1 n) [25], [17], [9]ri To define the new estimate for bjfot) the indicator variable, x(dk) 1 if Ot = k and 0 otherwise, will be used. b/k j) = = (k) (j) bji(k) Enl )t() 2t=l ^U^ [25], [17], [9] The BaumWelch algorithm described above is an iterative procedure in which the iterates, the parameters of the model, converge to the values which maximize P(0 A) [25],[17],[9]. Thus repeating the algorithm gives improved estimates of the parameters until optimal values for the parameters has been obtained. 3.3 The Scoring Problem If the underlying state sequence which produced a sequence of observations is known then computing the probability of the observation sequence given a Hidden Markov Model is trivial. However, if the underlying state sequence producing a sequence of observations is not known the problem of computing the probability of an observation sequence given a model becomes more complicated. A well known method of computing the probability of a given observation sequence utilizes the forward variable defined in the previous section. Recall that a (i) = P(Oo = oo, 01 = O1, ..., Ot = Ot, Xt = si). Thus for an observation sequence 0 {01,02, ...,0n} we Sa,(s) [25], [17], [9] Currently hidden Markov models are applied to identification problems. This is often done by constructing a Hidden Markov Model for each item which is to be identified. P(O1 = O1, 02 = 02, ..., On = On} Then the probability of the observation sequence conditioned on each of the models is computed. The object represented by the model for which the observation sequence has the highest probability is taken to be the identified object. Consider the following example. It would be of interest in any election to be able to identify how an individual will vote on based on an observable set of characteristics. Suppose a survey of potential voters is taken approximately twice a month in the 3 months leading up to an election. Each voter is surveyed multiple times and their responses are recorded. In this example we suppose that there are only two political parties. For this example suppose the questions asked are as follows. First a voter is asked their party affiliation, which will be classified as Republican, Democratic. Next a voter is asked if they are planning to vote for a particular candidate. The potential responses are the Republican candidate, the Democratic candidate, or undecided. The last question asked is if the individual is watching the election coverage. The responses to this question are simply yes or no. Thus we have 12 possible combinations of responses to these questions which make up the set of possible observations. For each candidate a hidden Markov model will be constructed. The underlying states will take on the month in which the survey was taken. In this example we suppose the surveys begin being taken during the 3 months prior to month in which the election will occur. The state space will therefore consist of 3 values 1,2,3. If Xt is in state 1, this indicates the survey is being taken 3 months prior to the month in which the election will occur. If Xt is in state 2 this indicates the survey is being taken 2 months prior to month in which the election will occur. If Xt is in state 3 this indicates the survey is being taken in the month preceding the month in which the election will occur. It is assumed that Xt is a Markov chain, with transition matrix P. .5 .5 0 P = 0 .5 .5 0 0 1 The example presented here is an example of a leftright hidden Markov model. A leftright hidden Markov model is a hidden Markov model in which Xt can not make backwards transitions [25],[17],[9]. That is once Xt has entered state 2 it will never return to state 1. A hidden Markov model is then constructed for the Republican candidate, and the Democratic candidate. Both models will share the same underlying state transition matrix. The observation distribution will however be different for each candidate. Then given a sequence of observation vectors the probability of the sequence will be computed conditioned on the HMM for the Republican Candidate, and the HMM for the Democratic candidate respectively. If the probability of the observation vector is greater using the using the HMM for the Republican candidate we conclude the voter will probably vote for the Republican candidate. If the probability of the observation vector is greater using the HMM for the Democratic candidate then we can conclude the voter will probably vote for the Democratic candidate. Through this simple example the concept of identification using Hidden Markov Models has be presented. In Chapter 4 some examples of current applications of Hidden Markov Models to identification systems will presented. 3.4 The Decoding Problem Consider the following example. Suppose a local hospital has seen a recent increase in the number of patients being a treated for certain contagious disease. The department of health is concerned about the spread of this disease and wants to know the likely number of current people who are infected and possibly contagious. The number of people being treated for the disease at the hospital is an observable result of the number of people infected with the disease. The question is then how does one obtain information about the number of people infected with the disease in the general population which the hospital serves from the number of people treated. Given an observation sequence the decoding problem for hidden Markov models attempts to uncover the most likely state sequence which would have generated the observations. With respect to the above example this equates to finding the number of infected and possibly contagious individuals in the area. Given an observation sequence of length n, common practice is to find the state sequence of length n which maximizes P(0o = o0,1 01, ...,0, = oX0 = ,Xi si,,...,X = si A) [25],[17],[9]. The Viterbi algorithm provides an efficient method of finding such a sequence. The Viterbi Algorithm begins by defining two variables the first of which is Jt(s) = rtaxgs ,, tesP(Oo = ,01 = oi,...,Ot = otXo = si',X = Si,...,X t = s) [25],[17],[9]. This is the maximum probability of observing the first t observations and being in state s at time t. The states which maximize J6(s) are retained using the variable dt(s) [25],[17][9]. This entry gives the state at time t1 which maximizes the probability of observing the first t observations and being in state s at time t. The algorithm begins by initializing Jo(s) and po(s) as follows: Jo(s) = ro(s)b,(oo) for all s C S[25], [17], [9] (310) o0(s) = 0[25], [17], [9] (311) The remaining values of J6(s) and %t(s) are computed as follows: Jt(si) = maxss(Jt (s) as,)bs,(ot) 0 < t < n and s, E S[25], [17], [9] (312) t(si) = argmaxses(Jt (s) ass,) 0 < t < n and s1 E S[25], [17], [9] (313) Once Jt(sl) and pt(sl) have been computed for all values of s1 and t, the final state in the state sequence is chosen such that st = argrmaxss(Jt(s)) [25],[17],[9]. This is the state at time which which maximizes the probability of observing the entire observation sequence and being this state at the last observation. Using this value, the remaining states in the sequence are chosen as follows: st = t+1(st+1)[25],[17], [9] (314) States at time t1 are chosen to maximize the probability of observing the sequence through time t and being in the state which maximized the probability of observing the sequence through time t+1 and being in the appropriate state. In the beginning of this section a situation was discussed in which local officials are worried about the sudden increase in the number of cases of a particular infectious disease at a local hospital and want to determine how widespread the disease may be among the general public. There is a classic example in which the spread of an infectious disease in a small population is modeled using a Markov chain [15]. The number of patients treated at the hospital is an observable result of the number of people infected with the disease in the general population. Thus a hidden Markov model may prove a suitable tool for analyzing the situation. Suppose the hospital serves a community of 10,000 and that the based on previous experience local officials have developed a various methods for stopping/controlling the spread of the disease based on how widespread the disease is at any given time. As part of this plan officials have developed a scale ranging from 1 to 10 describing the extent to which the disease has spread. Level 1 corresponds to normal circumstances, while level 10 corresponds to a serious epidemic. Let Xt be a Markov Chain with 10 states corresponding to the level of infection at time t. The time parameter is assumed to be discrete and represents the time at which observations are taken. Suppose based on previous experience the transition probabilities are known and defined by matrix P. Note that these probabilities correspond to the changes which occur prior to any action taken to stop the spread of the disease. .85 .13 .02 0 0 0 0 0 0 0 .05 .50 .4 .05 0 0 0 0 0 0 .02 .08 .4 .35 .15 0 0 0 0 0 0 .03 .07 .38 .42 .1 0 0 0 0 0 0 .05 .15 .40 .35 .05 0 0 0 P 0 0 0 .05 .15 .40 .35 .05 0 0 0 0 0 0 .05 .22 .38 .3 .05 0 0 0 0 0 0 .08 .22 .30 .35 .05 0 0 0 0 0 0 .1 .2 .5 .3 0 0 0 0 0 0 0 .2 .3 .5 Let Ot be a probabilistic function of Xt. For this example suppose the number of individuals who have been treated for the disease can represented on a scale from 1 to 10. Under this scale the 1 represents a low number of people treated, 10 represents and extremely high number of people treated. The distribution bs(k) with entries bsk gives the probability of observing k given the underlying state s. .9 .09 .01 0 0 0 0 0 0 0 .65 .25 .1 0 0 0 0 0 0 0 .35 .45 .1 .09 .01 0 0 0 0 0 .1 .55 .2 .1 .05 0 0 0 0 0 0 .25 .35 .15 .1 .05 0 0 0 0 bs(k) = 0 .05 .1 .3 .4 .1 .05 0 0 0 0 0 .01 .09 .37 .38 .1 .05 0 0 0 0 0 0 .15 .30 .30 .20 .05 0 0 0 0 0 0 0 .15 .25 .3 .3 0 0 0 0 0 0 .1 .2 .2 .50 Suppose Xt starts in state 1 with probability 1. Thus Po 1 if so = 1 and 0 otherwise. Suppose the following sequence of observations has taken place: 0 = 1, 2, 3, 3, 4. Using the Viterbi algorithm it is possible to find both how the level of infection is changing and the current level of infection. The Viterbi algorithm suggests the most likely state sequence, given the model, is: 1,3,5,6,8. Thus indicating the disease is spreading very quickly through the population and that immediate action should be taken to stop the epidemic. It is clear that the ability to obtain information regarding the state sequence which produces a given observation sequence can be very useful in a wide variety of applications. 3.5 Simulation As stated in section 3.1 a hidden Markov model can be used for generating observations. The generated observations can be used as a component in a larger simulation, or as an independent simulation. A state/observation sequence can be generated from a hidden Markov model in the following way. 1. Generate an initial state using the initial distribution. 2. Conditioned on the initial state generate the first transition from Xt. 3. Generate an observation from bs(k) where s is the state generated in step 2. 4. Repeat steps 2 and 3 to generate the desired number of observations/states. [25],[17],[9] The state sequence can be generated using the method presented in Chapter 2. Note that the method used to generate the state sequence also works to generate the observation sequence provided the observation distribution is discrete. Consider the following example. Suppose a town has recently observed an increase in the amount of traffic on the main highway through town and now frequently observes heavy or stopped traffic. To alleviate the traffic problems it is suggested that an additional lane be constructed in each direction. The town wants to see if this would change the observed traffic intensity. A simplified, fictional, Hidden Markov Model is presented to analyze this problem. Let Xt be a discrete Markov Chain describing the underlying causes of the observed traffic intensity. Let Ot be a probabilistic function of Xt Suppose traffic intensity is observed every hour and at each observation we can classify the observed traffic intensity in one of four ways light, moderate, heavy, and stopped. Thus we have the following observation set 0. 1. O = light 2. 02 = moderate 3. 03 = heavy 4. 04 = stopped Suppose further that we consider only two causes for the observed traffic intensity. First, is the time of day which will be classified as either rush hour or nonrush hour. The second is whether an accident has occurred. With these assumptions we have the following underlying statespace for Xt. 1. s, = nonrush hour, no accident 2. s2 = nonrush hour, accident 3. s3 = rush hour, no accident 4. s4 = rush hour, accident Though we make the assumption that Xt is Markovian, the system which Xt describes may not be Markovian in the strict sense. Since the current state is certainly going to be dependent on the previous state it is not unreasonable to say that the system is approximately Markovian. In making this assumption we are simply saying that there only exists a first order dependence among the states. Over any extended period of time, months or years, the transition probabilities describing Xt are certainly not going to be stationary. Simply the accident rates and the times during which a "rush hour" is observed will, in all likelyhood, change. However over a short period of time say a single month, or perhaps a single year, we can say that the Xt is approximately stationary and thus under our assumptions Xt is a finite state, discrete time homogeneous Markov chain. Suppose that currently the town observes that 20% of the time traffic intensity can be classified as light, 30% of the time traffic intensity can be classified as moderate, 40% of the time traffic intensity can be classified as heavy, and 10% of the time traffic intensity can be classified a stopped. With the new lanes added it is possible that the number of accidents could increase, however the impact of a single accident on traffic flow will be decreased. In addition, during rush hour periods traffic will flow more quickly thus decreasing the amount of time the spent in a rush hour state. Suppose then the following distributions were obtained using information regarding the current system and the expected impact of the new lanes. .70 .10 Po .15 .05 .8 .10 .0775 .025 .5 .3 .15 .05 P .30 .01 .5 .19 .2 .05 .45 .3 .60 .20 .15 .05 .25 .55 .10 .10 bs(k) = .05 .45 .45 .05 .02 .30 .53 .15 Then with the current distributions we simulate a 24 hour period. From 30 simulations we expect, based on the model, the town to observe light traffic about 40% of the time, moderate traffic about 29% of time, heavy traffic about 24% of the time, and stopped traffic about 7% of the time. Based on the simulations we get the impression that adding another lane would decrease the amount of time the town observes undesirable traffic intensities. Though this is a simple, fictional, example it does illustrate the potential for hidden Markov models to be used in conducting simulations. CHAPTER 4 SELECTED APPLICATIONS OF HIDDEN MARKOV MODELS 4.1 Isolated Word Recognition Automatic speech recognition is one of the first areas to which Hidden Markov Markov Models were applied. It seems appropriate then to begin a chapter on applications of Hidden Markov Models with this topic. The following paragraphs present a brief synopsis of how Hidden Markov Models can be applied to the well known isolated word recognition problem and is based on [25],[11],[9]. A spoken word or word fragment, can be characterized as a speech signal. This signal is a sequence of vectors which characterize the properties of the spoken word or word fragment over time [11],[9]. For any single word there are a variety of possible speech signals. This variation can be attributed to differences in the source of signal, i.e. the speaker. Dialect, gender, as well as environmental factors all influence a given speech signal. Automatic speech recognition systems attempt to match the signal of a spoken word to a standard representation of that word. A speech signal will change over time making it is necessary to partition a given speech signal into regions on which the properties of the signal are approximately stationary [11],[9]. The partitions of the signal are sometimes called frames and the length of partitions the frame length. If the signal is framed in terms of the individual sounds in a word then the signal produced can be seen as a probabilistic function of the intended sound. A Hidden Markov Model then seems particularly well suited to the problem of isolated word recognition. In practice a Hidden Markov Model is constructed for each word in a predetermined vocabulary. Given a speech signal, the probability that the signal represents a given word is determined by computing the probability of the observation signal given the model for the word. The word represented by the model for which the signal has the highest probability is taken to be the identified word. The underlying states for a hidden Markov Model representing a specific word can be interpreted as the sounds which compose the word [25]. Defining the states this way implies that the model is a left right model in that the sounds which compose a word follow a certain order. The underlying Markov Chain for a word model need not be strictly left right as different pronunciations of a word may utilize different sounds [25],[11]. This corresponds to a left right model with some variation in the progression of states. The observations associated with each state could be seen as a particular frame of the signal. Implementing a discrete Hidden Markov Model requires the set of possible observation vectors be reduced to a finite set. This is accomplished using a process called vector quantization which partitions the set of all possible vectors into a finite number of subsets [9],[25],[11]. The centroid of the set is then chosen as the representative of the that particular set of vectors [11],[9]. Classifying a new vector is then merely the process of finding the set whose representative is the closest in distance to the new vector [11],[9]. The set of representatives of each class form a codebook. Once a codebook has been designed a Hidden Markov Model for each word in the vocabulary is constructed from a training set consisting of spoken versions of the word [11],[9]. Vector quantization allows for discrete probability distributions to be used and the methods of Chapter 3 are directly applicable. The major disadvantage to vector quantization is distortion caused by partitioning the set of all possible vectors to a finite set where each set has a chosen representative [11],[9]. An alternative to vector quantization is to apply Hidden Markov Models using continuous observation densities. One of the most common distributions, for all continuous HMMs, is a mixture of multivariate normal distributions where bi(o) = Em cikN(o, pik, (ik) [25],[9]. Here o is an observation vector, cik are the mixing coefficients for the distributions, [ik is a mean vector in state i for mixture k, and aik is the covariance matrix for mixture k in state i [9],[11]. Following parameter estimation the identification process is similar to that in the discrete case. The preceding paragraphs hopefully gave an idea as to how Hidden Markov Models can be applied to isolated speech recognition. An interested reader can consult books by Huang, Arriki and Jack [9] or by Rabiner and Juang [11] for further information on the topic. 4.2 The HMM Applied to Traffic Monitoring Another area to which Hidden Markov Models have recently been applied is image processing. In particular, I have found 2 specific examples where Hidden Markov Models have been applied to image processing. The first example is in an article by J. Kato, et al. and describes a Hidden Markov Model for distinguishing the background, from the foreground, from the shadows caused by the foreground in a sequence of images [16]. Presented in a paper by Li and Porikli, the second example is a Hidden Markov Model used in an automated method of identifying highway traffic patterns using video images of the highway [19]. Both of these models will be discussed in the following paragraphs. Consider a small block of pixels of an image. The purpose of the first example, [16], is to determine if the block of pixels is showing the background of an image, the foreground of an image, or a shadow. This was accomplished with moderate success by implementing a 3 state Hidden Markov Model. The states of the model are "Background, shadow, and foreground" [16]. The observations are 2 dimensional vectors characterizing the image in the block of pixels. The state transition probabilities are characterized by a 3 by 3 transition matrix and a starting state distribution [16]. The observation probabilities for the background state and the shadow state are approximated by multivariate normal mixture densities of the type described in the previous section while the observation density for the foreground state is approximated by a uniform distribution [16]. To segment the image, an HMM of the type described in the preceding paragraph is constructed for each block of pixels which make up the image. After training the model determining whether the pixels represent a foreground, background or shadow of the image at a given time t is taken to be the state which maximizes the probability of observing the first t1 observations and being in that state at time t [16]. The results presented in the paper were fairly impressive. It was seen that the model identified light colored cars easily but had more difficulty with dark and gray colored cars though incorporating features of the surrounding region did improve the results obtained from dark and gray cars [16]. The second application of Hidden Markov Models to traffic modeling involved identifying highway traffic patterns by analyzing video images. Different than most identification systems implemented using Hidden Markov Models the detection system described in [19] is implemented using a single 6 state hidden Markov Model. The states of the model correspond to number of vehicles on the highway and the speed at which they are traveling [19]. States are classified as "heavy congestion, high density with low speed, high density with high speed, low density with low speed, low density with high speed, and vacancy" [19]. The observations are defined by a 4 dimensional vector which describes a piece of the image in a frame of video at a given instant [19]. Transitions between states are defined by a transition matrix and the observation probabilities are either defined by a single normal distribution or a Gaussian mixture, with training accomplished using the BaumWelch algorithm [19]. Following training, the Viterbi algorithm was used to identify the most likely state sequence given an observation sequence and identify a given traffic pattern [19]. 4.3 A Hidden Markov Model as a Generator for Rainfall Data The next application considered is hidden Markov Models applied to weather, in particular rainfall, simulation. Two applications of hidden Markov Models seemed particularly interesting and will be discussed in the following paragraphs. First the model suggested by Hughes and Guttorp for modeling rainfall occurrence at a set of sites in a region will be discussed. Second a variation of the general model described by Hughes and Guttorp which was suggested by Robertson, Ines, and Hansen [26] for modeling rainfall at different sites throughout a region using regional rainfall statistics for the region will be discussed. Rainfall occurrence in a given area can be viewed as a result of an underlying weather pattern in that area. Building on this idea Hughes and Guttorp suggest a Hidden Markov Model which views the occurrence of rainfall at a network of detection stations as an observable result of the current state of weather in that area. Hughes and Guttorp, [14],[13]. view changes in weather states as a result of the previous weather state and current atmospheric conditions [14]. Thus, state transitions can not be modeled as a simple Markov Chain as the Markov property is not satisfied. Instead Hughes and Guttorp suggest a nonhomogeneous Markov Chain to model the transitions between weather states [14]. Thus the model is referred to as a nonhomogeneous Hidden Markov Model. An observation consists of an ndimensional vector, RP which corresponds to a network of sites where rain fall at time t at site i is indicated by a 1 in position i of the observation vector and a lack of rainfall at time t at site i is indicated by a 0 in position i of the observation vector [14]. The weather state at time t is notated St [14]. Atmospheric conditions at time t are represented by Xt [14]. Two assumptions are made regarding the model: first it is assumed that observations are dependent only on the current weather state, second it is assumed that the probability of being in a given weather state at time t+1 is dependent only on the weather state at time t and atmospheric conditions at time t+1 [14]. Their model is then defined by fitting two probability distributions [14]. The first distribution which must be determined is the probability of a certain rainfall observation given a certain weather state[14]. The second distribution which must be determined is the probability of a certain weather state given the previous weather state and current atmospheric conditions[14]. The authors provide suggestions for distributions, under various assumptions, to parameterize the model. An interested reader can consult [13] and [14] for further information on the technical aspects of implementing this model. Applying a variation of the model suggested by Hughes and Guttorp, Robertson, Ines and Hansen have shown that a nonhomgeneous Hidden Markov Model can be used to model rainfall occurrences at a set of sites in a region based on average rainfall statistics for the region [26]. In addition to modeling rainfall occurrence at a number of sites throughout a region the model also models the amount of rainfall at a particular site [26]. For their model the authors assume that rainfall on a particular day is independent of all other states and rainfall on any previous days [26]. In addition the authors assume that rainfall in different locations is independent given the current weather state [26]. Transition between states is a function of the average observed daily rainfall [26]. Following construction of the model it was applied to crop simulation and decoding state sequences corresponding to historical rainfall data [26]. CHAPTER 5 ON WILLIAM HERSCHEL'S CONJECTURE 5.1 William Herschel's Conjecture In 1801 William Herschel published in the Phzlosophzcal Transactzons of the Royal Society of London a paper entitled "Observations Tending to Investigate the Nature of the Sun in Order to Find the Causes or Symptoms of Its Variable Emission of Light and Heat; With Remarks on the Use that May Possibly Be Drawn from Solar Observations". Here he suggests a lack of observable solar activity, sunspots, may be associated with a decrease in the amount of light and heat generated by the sun [8]. Having no direct method for testing this hypothesis, Herschel suggested an indirect method. Reasoning that the growth of vegetation is affected by heat and light from the sun, Herschel argued that if it could be shown that a decrease in the health of a crop corresponded to times when there are no visible sunspots, it might suggest that the sun is operating differently during these time periods than during time periods when sunspots are present [8]. Using the price of wheat as an indicator of the health of the wheat crop, Herschel compared the average price of wheat during time periods which records indicate that no sunspots were visible with the average price of wheat over the time period of equal length directly following this time period and in 2 cases preceding it [8]. Herschel's observations as well as the years of relevant solar minimums, the time period during which the fewest sunspots are visible are given in the tables 51, 52, 53 [23],[8]. In each case the price of wheat is higher during time periods when sunspots were not present than during time periods of equal length directly following these time periods. Additionally in 2 cases the price of wheat is lower in the time period directly preceding the period of no sunspot activity than during the period of no sunspot activity. This suggests there might be a relationship between the appearance of sunspots and the health of wheat crops since a decrease in supply without a corresponding decrease in demand should cause an increase in price. The evidence presented by Herschel was not intended to be conclusive, only a suggestion of something to consider. Herschel acknowledged that there are many other factors which can influence the growth of vegetation, but conjectured that these factors may be influenced by changes in the amount of heat and light generated by the sun [8]. In the 1970's was discovered that there may be a relationship between specific time periods in the sunspot cycle and drought periods in certain regions [3]. It was also found that drought periods tend to be associated with lower corn yields, a fact which may easily carry over to wheat yields [2]. If it is known that drought periods tend occur during specific time periods in the solar cycle and that during these drought periods yields tend to be lower it is quite possible that sunspots could provide an indication of processes taking place in the sun which will impact vegetation. Considering the evidence presented by William Herschel as well as more modern investigations it seems possible that there may be a relationship between sunspot activity and the price/health of wheat crops. 5.2 Interest in Herschel's Claim Investigation into William Herschel's claim began with the intent of determining whether the relationship, if it existed, between sunspots and the price of wheat could be modeled using a Hidden Markov Model where the price of wheat is modeled as an observable result of sunspot activity. The evidence presented by Herschel seemed to indicate that it might be reasonable to model the price of wheat using a Hidden Markov Model. There is however a problem with Herschel's observation which is described in the following paragraph. The average sunspot cycle lasts approximately 11 years from solar minimum to solar minimum. In any given cycle a solar maximum should be observed during which many sunspots should be visible. Some of the periods in Herschel's data which records indicated no sunspots were observed seem too long to be accurate. It is possible that sunspots which were present went unobserved due to poor technology. Thus the observations on which Herschel based his suggestion may not be accurate. There is evidence however that during this time period the sunspot cycle was not as regular as it is now. For example the Maunder Minimum was the period from 16381715 during which few sunspots were observed [18]. Sunspots which were observed did not appear to follow the current approximately 11 year cycle [18]. The Maunder Minimum spans Herschel's entire observation set and therefore it is possible that the observations on which Herschel based his claim are valid. Further investigation is needed before any conclusions can be reached. 5.3 Analysis Using Modern Data The following section examines William Herschel's suggestion using data from the 19th and 20th centuries in the United States. William Herschel's claim was really about a possible relationship between the health of wheat crops and the appearance of sunspots with price used only as an indicator of the health of the crop. For this reason both the price and the yield of wheat crops will be examined during time periods of low sunspot activity and time periods of equal length directly preceding and following periods of low sunspot activity to see if it can be concluded that such a relationship exists. Sunspot data, in the form of the daily international sunspot number, was obtained from the National Geophysical Data Center/NOAA Satellite and Information Service website [22]. The average annual wheat price and yield for the United States were obtained from the United States Department of Agriculture Wheat Data Yearbook [30]. Initially, it was necessary to identify the time periods which had the least solar activity as indicated by sunspot activity. It was found that the average annual sunspot number tended to be the lowest in the year in which a solar minimum occurred, and the years preceding and following that year. So, the 3 year period surrounding the year in which a solar minimum occurs will be used as the period of low sunspot activity. Having found the period which, on average, has the least sunspot activity, the average price of wheat during the period of low sunspot activity will be compared to the average price of wheat in the 3 years preceding and 3 years following this time period. Similarly, the average yield during the 3 year period of low sunspot activity will be compared to the average yield during the 3 years directly preceding and the 3 years directly following this time period. Evaluation of Herschel's suggestion using the price of wheat will be done first. Over the course of 80 years inflation will certainly have affected the price of wheat considerably. For this reason prior to analysis it was necessary to adjust the price of wheat to a single year. The consumer price index was used to adjust all prices to be measured in dollars in 19821984. This was done by multiplying the 100ear price in year x. The average adjusted price was then computed over each time period. The consumer price index from 1913 to 2007 was obtained from the Federal Reserve Bank of Minneapolis website [7]. A summary of the results obtained, using prices from 1922 to 2000, is given in table 54. For notational convenience the sample mean for the 3 year period containing a solar minimum will be called l = $ 5.22 with corresponding sample variance S1 = $ 5.51. The sample mean for the 3 year period before the period containing a solar minimum will be called P2 = $ 5.81 with corresponding sample variance S2 = $ 5.51. The sample mean for the 3 year period after the period containing a solar minimum will be called 3 = $ 5.43 with corresponding sample variance S3 = $ 6.67. From these results it is seen that P2 > u1 and p3 > p/. Thus it seems that the average price of wheat is lower during the period of low sunspot activity than during time periods directly before and after this time period. It remains however to be determined if the differences in the sample are significant enough to infer that the samples represent different populations. Independence of the samples can not be assumed as the prices in consecutive 3 year periods have some degree of dependence. Because of this dependence the differences between pairs of observations will be examined to determine whether the prices are significantly different. Lacking conclusive evidence that the differences in the pairs of observations come from a normal distribution, a nonparametric test must be used. The test that will be used is the Binomial Sign Test. The Binomial Sign Test attempts to determine whether two samples likely come from two different populations by examining the number of positive and negative differences. The number of positive differences in a sample of n signed differences can be approximated by a binomial distribution. The test statistic is then the number of positive differences [27]. The null hypothesis for the test is that the samples come from the same population which implies that the probability of observing a positive difference should be .5 [27]. The alternative hypothesis is that the samples do not come from the same population which, in this case, implies the probability of observing a positive difference is greater than .5 [27]. To determine whether the the test statistic lies in the rejection region, i.e. the probability of observing a given number of positive differences, call it x, in a sample of n signed differences is significantly small so as to imply the null hypothesis is likely false, one computes the probability of observing at least x positive differences in a sample of n signed differences, assuming the probability of positive difference is .5, using the binomial distribution [27]. If the probability observing at least x positive differences in a sample of n signed differences is significantly small, less than .05, the null hypothesis can be rejected in favor of the alternative hypothesis [27]. When the Binomial Sign Test is applied to the differences in the price of wheat in the 3 year period preceding the period of low sunspot activity and the period of low sunspot activity the null hypothesis is that the samples represent different populations, i.e. the average price in the period of low sunspot activity is not significantly different than the average price in the preceding time period. The alternative hypothesis is that the samples represent different populations, i.e. the average price in the preceding 3 year time period is higher than the price in the 3 year time period around a minimum. Applying the Binomial Sign Test it was found that there were 5 positive differences out of 8 signed differences. The probability of observing 5,6,7, or 8 positive differences is .3632 and thus the null hypothesis must be retained. Thus it can not be concluded that the average price over the respective time periods is significantly different. If the Binomial Sign Test is applied to the differences in the price of wheat in the 3 year period following the period of low sunspot activity and period of low sunspot activity the null hypothesis is that the samples do not represent different populations. The alternative hypothesis is that the samples represent different populations, i.e. the average price of wheat in the 3 year period following a period of low sunspot activity is higher than the average price of wheat during the 3 year period of low sunspot activity. Applying the Binomial Sign Test it was found that there were 4 positive differences out of 8 signed differences. The probability of observing 4,5,6,7 or 8 positive differences is .64 and thus the null hypothesis must be retained. So, it cannot be concluded that the average price of wheat over the respective time periods is significantly different. William Herschel's suggestion was that a lack of sunspots may cause a decrease in the health of vegetation. The price of wheat was used simply as an indicator of the health of the crop. In the following paragraphs the average yield in the 3 year period containing a solar minimum is compared to the average yield in the 3 year period preceding and the 3 year period following this time period. This will provide a more direct evaluation of Herschel's suggestion. Using data from 18782000 the average yield, bushels/acre, was computed over the 3 year period of low sunspot activity, the 3 year period preceding this time period, and the 3 year period following this time period. The results are summarized in table 25. It is seen that the average yield is higher during the 3 year period containing a solar minimum, 20.61 bushels per acre, than in the preceding 3 year period, just 20.52 bushels per acre. It is also seen that the average yield during the 3 year period containing a solar minimum is less than the average yield over the following 3 year period, 21.87 bushels per acre. It seems that there is a difference in the average yield over the respective time periods. Again, it remains to be determined if these differences are significant enough to infer that there exists a difference in the population. Independence of the samples can not be assumed as yields in consecutive 3 year periods will have some degree of dependence. So the differences in the yields will be tested to determine if the samples likely come from two different populations. The Binomial Sign Test was again used to test if the differences in the average yield in the period of low sunspot activity and 3 years preceding this time period. The null hypothesis is that the samples represent the same population and the alternative hypothesis is that the samples do not represent the same population, i.e. the average yield is higher in the 3 year period of low sunspot activity than in the preceding 3 year period. It was found that there were 6 positive differences out of 11 signed differences. The probability of 6, 7, 8, 9, 10, or 11 positive differences is .5 and thus the null hypothesis must be retained. Hence, it cannot be concluded that the average wheat yield is different during the period of low sunspot activity than during the 3 year period directly preceding this time period. Again, the Binomial Sign Test was used to determine if the differences in average yield in the 3 year period of low sunspot activity and the 3 year period directly following this period are significant. The null hypothesis is that the samples represent the same population, i.e. there is no difference in the yields over the respective time periods. The alternative hypothesis is that the samples represent different populations, i.e. the average yield is higher during the 3 year period following a period of low sunspot activity than in the 3 year period of low sunspot activity. It was found that there were 7 positive differences out of 12 signed differences. The probability of observing 7, 8, 9, 10, 11, or 12 positive differences is .38 and thus the null hypothesis must be retained. Hence, it cannot be concluded that there is a significant difference in the average wheat yield during the period of low sunspot activity and the 3 year period directly following this time period. 5.4 Conclusions There is a difference in the price of wheat in the sample data. However, the differences found in the sample do not provide sufficient evidence to conclude the samples come from different populations. Thus it can not be concluded that the price of wheat during the 3 year period of low sunspot activity is less than the price of wheat during the 3 year periods preceding and following the 3 year period containing a solar minimum. It seems that the level solar activity, as indicated by the appearance or absence of sunspots, has a minimal, if any, impact on the price of wheat. Based on the sample data it cannot be concluded that, in the United States, the average yield is different in the 3 year period containing a solar minimum than in the preceding 3 year period. In addition it cannot be concluded that the average yield in the 3 year period following a period of low sunspot activity is different than the yield in the 3 year period of low sunspot activity. Thus it does not seem that the appearance or absence of sunspots is related to higher or lower wheat yields. Based on the results obtained in this chapter it does not seem reasonable to model the price of wheat or the yield using a Hidden Markov Model. However, investigating this problem did lead to a simple Markov Chain Model for the 11 year sunspot cycle which will be presented in the following Chapter. Table 51. Years of relevant solar minima and average wheat prices reported by William Herschel over periods of no reported sunspot activity. Year(s) of solar minima Year(s) of no observed Average price sunspots 1655,1666 16501670 2. 10s. 5'9d. 1679 16771684 2. 7. 7d. 1689 16861688 1. 15s. 2d. 1698 16951670 3. 3s. 3 d. 1712 17101713 2. 17s. 4d. W. Herschel, "Observations Tending to Investigate the Nature of the Sun, in Order to Find the Causes or Symptoms of its Variable of Light and Heat; With Remarks on the Use That May Possibly be Drawn from Solar Observations," Phzlosophical Transactions of the Royal Soczety of London, vol. 91, pp. 265318, 1801. National Atmospheric and Oceanographic Administration/National Geophysical Data Center, "Minima and Maxima of Sunspot Number Cycles," NGDC.NOAA.gov, 2008, 7 Jan. 2008, ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SUNSPOT NUMBERS/maxmin.new. Table 52. Years of relevant solar minima and average wheat prices reported by William Herschel over periods directly before periods of no reported sunspot activity. Year(s) of solar minima Time prior to years) of no Average price observed sunspots 1689 16901694 2. 9s. 4d. 1698 17061709 2. 3s. 71d. W. Herschel, "Observations Tending to Investigate the Nature of the Sun, in Order to Find the Causes or Symptoms of its Variable of Light and Heat; With Remarks on the Use That May Possibly be Drawn from Solar Observations," Phzlosophical Transactions of the Royal Soczety of London, vol. 91, pp. 265318, 1801. National Atmospheric and Oceanographic Administration/National Geophysical Data Center, "Minima and Maxima of Sunspot Number Cycles," NGDC.NOAA.gov, 2008, 7 Jan. 2008, ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SUNSPOT NUMBERS/maxmin.new. Table 53. Years of relavent solar minima and average wheat prices reported by William Herschel over periods directly after periods of no reported sunspot activity. Year(s) of solar minima Time after years) of No Average price observed sunspots 1679,1689 16711691 2. 4s. 4 d. 1689 16851691 1. 17s. 1 d. 1689 16891692 1. 12s. 102d. 1698 17001704 1. 17s. 111d. 1712 17141717 2. 6s. 9d. W. Herschel, "Observations Tending to Investigate the Nature of the Sun, in Order to Find the Causes or Symptoms of its Variable of Light and Heat; With Remarks on the Use That May Possibly be Drawn from Solar Observations," Phzlosophical Transactons of the Royal Soczety of London, vol. 91, pp. 265318, 1801. National Atmospheric and Oceanographic Administration/National Geophysical Data Center, "Minima and Maxima of Sunspot Number Cycles," NGDC.NOAA.gov, 2008, 7 Jan. 2008, ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SUNSPOT NUMBERS/maxmin.new. Table 54. Mean and variance of the price of wheat in the United States during periods of low sunspot activity and time periods directly preceding and following time periods of low sunspot activity. Preceding 3 year 3 Year period Following 3 year period surrounding a period. minimum Mean $ 5.81 $ 5.22 $ 5.43 Variance $ 5.51 $ 4.18 $ 6.67 Table 55. Mean and variance of wheat yields during periods of low sunspot activity and periods directly preceding and directly following time periods of low sunspot activity. Preceding 3 year 3 Year period Following 3 year period surrounding a period. minimum Mean 20.52 20.61 21.87 Variance 97.68 88.32 109.73 CHAPTER 6 MARKOV CHAIN MODEL FOR THE 11 YEAR SUNSPOT CYCLE 6.1 Introduction Sunspots are magnetic regions of the sun which are relatively cool when compared to the surrounding regions causing these regions to appear as dark spots. The number of sunspots and sunspot groups obeys a fairly regular cycle of approximately 11 years resulting from periodic changes in the magnetic field of the sun. Beginning in a period of low sunspot activity called a solar minimum, the number of visible sunspots and sunspot groups increases until a solar maximum is reached. This is the period of time when most the sunspots and sunspot groups are visible. Following a solar maximum the number of visible sunspots and sunspot groups decreases until a solar minimum is reached beginning the cycle again. Because of the nature of the sunspot cycle it is necessary to take into account both the current sunspot number and the position in the sunspot cycle in building a model. The International Sunspot Number is weighted average of the Wolf sunspot number as reported by reliable observatories throughout the world. The Wolf Sunspot Number, devised by Johann Rudolf Wolf in 1848, measures overall sunspot activity by taking into account the number of individual sunspots and the number of sunspot groups [22]. The sunspot number is given by R = k(10*g + s) where g is the number of sunspot groups, s is the number of individual sunspots and k is the observation constant which takes into account the conditions under which the observations were made [22]. A combination of the sunspot number and the position in the solar cycle will be used in constructing the Markov chain representing the system. A few assumptions were made in constructing the Markov chain. The first assumption is a uniform, discrete time step of 1 day. The second assumption that was made is one step dependence. Namely it is assumed that the probability of transitioning to a new state on the next day is dependent only on the current state. Since sunspot groups tend to last for multiple days, or even weeks, a higher order Markov Chain may provide a more accurate model, but for this model one step dependence is assumed. The next assumption is that the system can be modeled using a time homogeneous Markov Chain. The sun is evolving and therefore a nonhomogeneous model might give a more accurate model. However, over a relatively short period of time it is not unreasonable to assume that the system is approximately stationary and for a preliminary model a time homogeneous Markov Chain is used. Throughout the data, which ranges from 1818 to 2007, the sunspot number has never risen above 400. For this reason the maximum sunspot number for this Markov chain will be 400. With these assumptions the following sections describe how the Markov chain was constructed and some results obtained from simulations using the model. 6.2 Description of The States Since the sunspot number varies with the sunspot cycle it is necessary to consider the location in the sunspot cycle and the sunspot number itself when defining the states of the Markov Chain. The position in the sunspot cycle is represented in the following way. A total of 10 sunspot cycles were used in training the Markov chain, the first sunspot cycle starting in January of 1856 and the last sunspot cycle starting in March of 1954, using data obtained from the National Oceanographic and Atmospheric Administration website [22]. It should be noted that the data is reported by the NOAA website but was compiled by the Solar Influences Data Center, World Data Center for The Sunspot Index, at the Royal Observatory of Belgium [22]. Each sunspot cycle was divided into 100 approximately equal partitions. The code used to partition the cycles can be found in the appendix. The position in the sunspot cycle is then described as hundredths of the way through a sunspot cycle. The second component of the states of the Markov Chain is the sunspot number. Due to limited data it is necessary to group sunspot numbers to decrease the number of states in the Markov Chain. For this reason the sunspot numbers are grouped in sets of 20. The first group consists of the sunspot numbers from 0 to 19, the second group consists of sunspot numbers from 20 to 39, sunspot numbers are grouped in this way with the last group containing sunspot numbers from 380 to 400. The state space of the Markov Chain is the set of all ordered pairs s=(x,y), where x is the sunspot number group as defined above and y is the position in the sunspot cycle as defined earlier. Since there are 20 sunspot groups and 100 possible positions in the sunspot cycle the state space consists of 2000 states. The states are numbered from 1 to 2000 in the following way; state 1 corresponds to the ordered pair (0,1) this represents observing a sunspot number between 0 and 19 in the first position in the sunspot cycle. State 2 corresponds to the ordered pair (20,1) which represents observing a sunspot number between 20 and 39 in the first position of the sunspot cycle. The states 1 through 20 are numbered in this way. State 21 corresponds to the ordered pair (0,2) which represents observing a sunspot number between 0 and 19 in the second position of the sunspot cycle. States are numbered in order by indexing the x coordinate by 1 until the max x value of 380 is reached, the next state then indexes the y coordinate and resets the x coordinate to 0. In this way the states are numbered from 1 to 2000. The states are numbered this way initially to simplify the programming when constructing and simulating from the distribution. However due to the fact that some states never occur, for instance according to the data the sunspot number has never been 360 in during a minimum, some rows in the initial construction of the Markov chain are a row of zeros. With these rows the matrix it is no longer a stochastic matrix, since all rows in the matrix do not sum to one for this reason the rows which sum to zero should be deleted in the final matrix and the rows and columns of the matrix labeled with the (x,y) pair representing the state. 6.3 Starting State Distribution Let X(t) be a random variable taking on the values of the x coordinate of the ordered pair defining the states of the Markov chain. Let Y(t) be a random variable taking on the values of the y coordinate of the ordered pair defining the states of the Markov chain. The staring state probabilities will be estimated as: Number of Times a Sunspot Cycle Started in State i Number of Sunspot Cycles It should be noted that the in the training data only one sunspot cycle did not start in state 1; it started in state 2. Thus there are only two possible starting states. 6.4 OneStep Transition Distribution Defining X(t) and Y(t) as in the previous section, it is possible to estimate the probability that if system is in a given state s1 (x2,y2) at time tl + (xl,yl) at time tl it will transition 1. The transition probability is formally defined as P(X(t) = x, Y(t) y= X(t 1) = xi, Y(t 1) = y). Consider the transition probability Ps S2 P(St 2= s2St = si). Then using the definition of conditional probability we know P(St estimated as P(St S2, St1 ) s s2 Ptl = s1) = The probability P(St_1 PFi 1 sl) number of times in state s1 number of total sample points can be estimated as . Next the probability P(St = s) can be 2 St = sl) number of transitions from state s1 to s2 total number of transitions from s Since since every point total number of transitions from sl in the training data transitions to another point, the number of transitions from state s2 equals the number of times state s2 occurs in the training data. Then P(St s2St 1= l) can be estimated as: the number of transitions from state s1 to s2 the number of times in s2 It is interesting to note that the y coordinate of the state must either stay the same or increase by one, except when the system is in a state with a y coordinate of 100 in which case the y coordinate can stay the same or return to one. Thus if the current state is given by the pair (x,y) the probability of transitioning to a state other than those of the form (*,y), (*,y+l) will be 0. There is one exception, namely when the y coordinate is 100 the system can transition into a state with y to state s2 S The probability of observing a certain sunspot number is certainly affected by the y coordinate representing the state in the system. Additionally the transition probability between partitions of the sunspot cycle seems to be affected by what sunspot numbers are observed. This supports the idea that sunspots are a visible indicator of the underlying changes in the sun's magnetic field and is noticeable from differences in the transition probabilities for states which share a y component. 6.5 Results From Simulations Following the construction of the distribution, 100 simulations were generated according to the starting distribution. This means that each simulated sunspot cycle began in a solar minimum, allowing the sunspot cycles generated by the Markov chain to be compared to the actual sunspot data. The method used for simulating is the same as that presented in chapter 2. The generated sunspot cycles on average were similar to the actual sunspot cycles. First the average length of the simulated sunspot cycles was 3955.51 days with a standard deviation of 340.32 days. The average length of the actual sunspot cycles, based on the entire data set ranging from 1856 to 1996, was 3942.154 days with a standard deviation of 280.9359. The average length of the sunspot cycle for the data not used to train the model, 1964 to 1996, was 3834.667 days with a standard deviation of 297.342 days. The results indicate that, as a preliminary model, the Markov chain constructed in this chapter does a reasonable job of modeling the length of a sunspot cycle. Next, the average sunspot number from the simulations was plotted against the average sunspot number from the data set over the course of the sunspot cycle. The plots of the average sunspot number over the sunspot cycle, and the average simulated sunspot number over the sunspot cycle provide an indication of how the simulated sunspot numbers compare to the actual sunspot numbers. Figure 61 shows a plot of the average simulated sunspot number versus the average sunspot number from the entire data set. The plot of the average sunspot number, 18561996, is quite similar to the plot of the simulated sunspot number, however the plot of the average sunspot number is slightly higher than the plot of the simulated sunspot number. However, the simulated sunspot numbers are rounded down in groups of twenty. If the average sunspot numbers are grouped the same way as the simulated sunspot number the plots are more similar as is seen in figure 62. It seems that overall, the model does a reasonable job of modeling the sunspot number. However it remains to be seen how the simulated sunspot cycles compare to the actual sunspot cycles which were not used to train the model. Figure 63 is a plot of the average simulated sunspot number versus the average sunspot number from the nontraining data which ranges from 19641996. Compared to the nontraining data, the average sunspot number tends to be larger than the average simulated sunspot number. Since the simulated sunspot cycles represented the overall data so well it would be interesting to see how the recent average sunspot numbers compare to the overall average sunspot numbers. A plot of the average sunspot number from 1856 to 1996 compared with the plot of the average sunspot number from 19641996, given in figure 64, reveals that the average sunspot number over the last 3 solar cycles tends to be higher than the average sunspot number over the previous 150 years. A partial cause for this difference could be improved methods of observation which allow smaller changes in the sun, which would have previously gone unobserved, to be observed. A second potential cause for differences in observations could be changes in the sun which produce a greater number of sunspots. The Markov chain presented is only a preliminary model and the similarities in the average behavior of the model and the sunspot cycle indicate that a Markov Chain/Hidden Markov Model may prove a useful as a model of the sunspot cycle. Improvements in the model would need to be made before anything conclusive can be said. Future work on this model would involve attempting to model not only the sunspot number but other solar observations as well using a Hidden Markov Model. A Hidden Markov Model would provide many advantages over the homogeneous Markov Chain presented in this chapter. First, the system being modeled is likely nonhomogeneous. Modeling a nonhomogeneous system with a homogeneous model produces only moderate results as was seen in this chapter. A Hidden Markov Model would allow for the underlying state transitions to be modeled using a nonhomogeneous Markov Chain. Second, a Hidden Markov Model would allow for state durations to be modeled explicitly. This would almost certainly be an advantage over the geometric distribution which models state duration in a Markov Chain. Lastly, a Hidden Markov Model would allow for more complex distributions to be used in modeling observations. These would hopefully allow for more accurate modeling of not only the sunspot number but other solar observations as well. * Actual Data * Simulated Numbers 10o 2000 3000 400C 5000 Figure 61. Average sunspot number over the course of the sunspot cycle from simulated sunspot cycles and the average sunspot number over the course of the sunspot cycle from actual data (18561996). 150 140 1 2 0 T  SSimulated Data I000 2000 5000 4000 5000 Day. Average sunspot number over the course of the sunspot cycle from simulated sunspot cycles and the average sunspot number over the course of the sunspot cycle from actual data rounded down in groups of 20 (18561996). Figure 62. * Actual Data S Simulated Data 1000 2000 3000 4000 5C00 Average sunspot number over the course of the sunspot cycle from simulated sunspot cycles and the average sunspot number over the course of the sunspot cycle from actual data not used to train the model (19641996). * 1~E4 1~E 0 1000 2000 3000 4000 5000 Average sunspot number over the course of the sunspot cycle from 18561996 and the average sunspot number over the course of the sunspot cycle from 19641996. Figure 63. Figure 64. APPENDIX: THE R CODE USED IN CONSTRUCTING AND SIMULATING FROM THE MARKOV CHAIN MODEL CONSTRUCTED IN CHAPTER 6 #this program formats the data prior to constructing #the transition distribution #This code partitions the first sunspot cycle in the data k<l m<l 1<1 for(s in c(1:100)) { length< 40\ if(m <= 59 ) { if(1 == 2   1 == 3) { length < 41] m<m+l } } j< k+length1 for(t in c(k:j)) { ssndata[t,2]<s } k<k+length 1<1+1 if ( 1 == 4) { 1 <1 } #This code partitions The second sunspot cycle in the data k<l m<l 1<1 for(s in c(1:100)) { length< 42 if(m <= 74) { if(l == 2 I 1 == 3 I 1 == 4 I 1 == 6 II 1 == 7 II 1 == 8 II 1 == 9 II 1 == 10) { length < 43 } } 1<1+1 j<k+length1 for(t in c(k:j)) { ssndata[t,4]<s } k<k+length if(l >= 11) 1<1 } #This code partitions the third Sunspot cycle in the data set k<1 for(s in c(1:100)) { length< 39 if(s == 10 I s == 30 I s == 50 I s == 60 II s ==70 II s == 90 II s == 100) { length < 40 } j<k+length1 for(t in c(k:j)) { ssndata[t,6]<s } k<k+length #This code partitions The fourth sunspot cycle k<l m<l 1<1 for(s in c(1:100)) { length< 44 if( m <= 19 ) { if(l == 5) { length < 45 m<m+1 1<1 71 } } j<k+length1 for(t in c(k:j)) { ssndata[t,8]<s } k<k+length 1<1+1 } #This code partitions the fifth sunspot cycle in the data k<l m<l 1<1 for(s in c(1:100)) { length< 43 if(s >= 20) { if( m <= 46) { if(l == 2) { length < 44 m<m+l } } } j<k+length1 for(t in c(k:j)) { ssndata[t,10]<s } k<k+length 1< 1+1 if(l >= 3) { 1<1 } if(s == 19) { 1<1 } } 72 #This Code Partitions the sixth sunspot cycle sunspot cycle in the data k<l m<l 1<1 for(s in c(1:100)) { length< 36 if(m <= 52 ) { if(l == 2   1 == 3) { length < 37 m<m+l } } j< k+length1 for(t in c(k:j)) { ssndata[t,12]<s } k<k+length 1<1+1 if( 1 == 5) { 1 <1 } } #This code partitions the seventh sunspot cycle in the data k<l m<l 1<1 for(s in c(1:100)) { length< 37 if( m <= 27 ) { if(l == 3) { length < 38 m<m+l 1<1 } } j<k+length1 for(t in c(k:j)) { ssndata[t,14]<s } k<k+length 1<1+1 } #This code partitions the eight sunspot cycle in the data k<l for(s in c(1:100)) { length < 37 if(s != 1 && s != 2) { length < 38 } j< k + length 1 for(t in c(k:j)) { ssndata[t,16]<s } k< k + length } #This code partitions the 9th sunspot cycle in the data k<l m<l 1<1 for(s in c(1:100)) { length< 36 if( m <= 89) { if(l == 1 II 1 == 2 II 1 == 3 II 1 == 4 II 1 == 5 II 1 == 61 1 == 7 Ill == 8 II 1 == 9) { length < 39 m<m+l } } j<k+length1 for(t in c(k:j)) { ssndata[t,18]<s } k<k+length 1<1+1 if( 1 == 11) { 1 <1 } } This code partitions the 10th sunspot cycle in the data k<l m<l 1<1 for(s in c(1:100)) { length< 38 if( m <= 73) { if(l == 1 II 1 == 7 II 1 == 3 II 1 == 4 II 1 ==5 1 == 8 II 1 == 9 II 1 == 10 ) { length < 39 m<m+l } } j<k+length1 for(t in c(k:j)) { ssndata[t,20]<s } k<k+length 1<1+1 if( 1 == 11) { 1 <1 #This program constructs the sunspot transition matrix m<1 place < array(0, c(1,19)) tran < array(0, c(2000,2000)) #Constructs the matrix for the first 99 y values for(y in c(1:99)) { place < place+1 for(x in c(0,20,40,60,80,100,120,140,160,180,200,220,240, 260,280,300,320,340,360,380)) {n<(((y1)*20)+1) z<y+l for(ynew in c(y,z)) { for(xnew in c(0,20,40,60,80,100,120,140,160,180,200,220,240,260, 280,300,320,340,360,380)) {k<0 1<0 for( j in c(1,3,5,7,9,11,13,15,17,19)) {i<place[l,j] while( ssndata[i,j+l] == y II ssndata[i,j+l] == y+1) if(ssndata[i,j+l] == y && ssndata[i,j] >= x && ssndata[i,j] k < k+1 if(ssndata[i+1,j+l] == ynew && ssndata[i+l1, j] && ssndata[i+l,j] { 1 < 1+1 } } i< i+1 if(ssndata[i,j+l] < 20 + xnew) == y && x == 380 && xnew == 380) place[1,j]< place[l1,j]+l1 } } } if(k != 0) {tran[m,n]<l/k} n<n+l } } m < m+1 } } #Constructs Matrix For y value 100 y < 100 ynew <100 m<1981 place< array(l, c(1,19)) for(j in c(1,3,5,7,9,11,13,15,17,19)) while(ssndata[i,j+l] { != 100) place[1,j] i < i+1 < place[1,j] < x+20) >= xnew } for(x in c(0,20,40,60,80 220,240,260,280,300,320, {n<1981 ,100,120,140,160,180,200, 340,360,380)) for(xnew in c(0,20,40,60,80 220,240,260,280,300,320,340 { 1<0 k<0 ,100,120,140 ,360,380)) ,160,180,200, for(j in c(1,3,5,7,9,11,13,15,17,19)) i < place[l,j] while( ssndata[i,j] { != 9999) if( ssndata[i,j+1] == y && ssndata[i,j] { k<k+1 if( ssndata[i+l, >=xnew && ssndata[i+l,j] { 1 <1+} } i < i+1 } } if(k!=0) { tran[m,n]< 1/k } n<n+l } m<m+l I >= x && ssndata[i,j] j+1] == ynew && ssndata[i+l,j] < xnew+20) #Constructs the part of the distribution which returns the start of a solarcycle m<1981 for(x in c(0,20,40,60,80,100,120,140,160,180,200, 220,240,260,280,300,320,340,360,380)) { n<l for(xnew in c(0,20,40,60,80,100,120,140,160, 180,200,220,240,260,280,300,320,340,360,380)) { 1<0 k<0 for(j in c(1,3,5,7,9,11,13,15,17,19)) < x+20) { i < place[1,j] while(ssndata[i,j] != 9999) { if(ssndata[i,j+1] == y && ssndata[i,j]>=x && ssndata[i,j] < x+20) { k < k+1 if(ssndata[i+1,j] == 9999) { if(ssndata[1,j+2] >= xnew && ssndata[1,j+2] < xnew + 20) i{1 < 1 + 1} } i < i+1 if(k != 0) { tran[m,n]< 1/k n<n+l m<m+l } #This program simulates an observation sequence from the Markov #chain model #This program produces a simulated sunspot cycle transim <array(0,c(100,6000)) initial< array(0,c(10,1)) #initial conditions initial[1,1]<1 initial[2,1]<l initial [3,1]<2 initial [4,1]<2 initial[5,1]<2 initial[6,1]<2 initial [7,1]<i initial[8,1]<l initial[9,1]<l initial[10,1]<2 1<1 #simulation code for(n in c(1:7)) { m<1 for(j in c(l:m)) { transim[j,1]<initial[n,1] #simulates the remaining days from transition matrix for(i in c(2:6000)) { sum<O point<runif(l, min = 0, max = 1) k<l while( sum < point && k < 2000) { sum< sum+ tran[transim[j,il],k] k<k+l } transim[j,i]< k1 if(transim[j,i1] > 400 && floor(transim[j,i]/20) <= 1) { break } } } 1<1+1 } #This program extracts the sunspot group from the simulated state sunspots < array(0,c(40,5000)) for(i in c(1:7)) { for(j in c(1:5000)) { sunspots[i,j] if(j > 2) <((transim[i,j]1)%%20)*20 if(transim[i,j1] > 400 && floor(transim[i,j]/20) <= 1) { break } } } } #This program determines the length of the sunspots cycle #I         #generated by the model sum<array(O,c(2000,1)) for( i in c(1:33)) { k<l for( j in c(1:5000)) { if(k > 400 && floor(transim[i,j]/20) { break } k<k+l } sum[i, ] <k } <= 1) REFERENCES [1] N.U. Bhat, Elements of Applzed Stochastic Processes 2nd ed. New York: John Wiley and Sons, 1984. [2] R. Black, "Weather Variation as a CostPrice Uncertainty Factor as it Affects Corn and Soybean Production," Jounal of Agrzcultural Economzcs, vol. 57, no 5, pp. 940944, 1975. [3] R.J. Black, and S. R. Thompson, "Evidence on WeatherCropYield Interpretation," American Jounal of Agrzcultural Economzcs, vol. 60, no. 3, pp. 540543, 1978. [4] E. Cinclair, Introductzon to Stochastic Processes. Englewood Cliffs: Prentice Hall, 1975. [5] W.K. Ching, and M.K. Ng, Markov Chains, Models Algorzthms and Applzcatzon. New York: Springer Science + Buisness Media, 2006. [6] D. Dobson, and D.A. Meeter, "Alternative Models for Describing Change in Party Identification," American Journal of Polztzcal Sczence, vol. 18, no. 3, pp. 487500, 1974. [7] Federal Reserve Bank of Minneapolis, "Consumer Price Index, 1913", woodrow.mpls.frb.fed.us, 2008, 24 April 2008, http://woodrow.mpls.frb.fed.us/research/data/us/calc/histl913.cfm. [8] W. Herschel, "Observations Tending to Investigate the Nature of the Sun, in Order to Find the Causes or Symptoms of its Variable of Light and Heat; With Remarks on the Use That May Possibly be Drawn from Solar Observations," Phzlosophzcal Transactions of the Royal Soczety of London, vol. 91, pp. 265318, 1801. [9] X.D Huang, Y. Ariki, and M.A. Jack Hzdden Markov Models for Speech Recognziton. George Square: Edinburg University Press, 1990. [10] B.H. Juang, and L. Rabiner. "Hidden Markov Models for Speech Recognition," Technometrzcs, vol. 33, pp. 251272, 1991. [11] B.H. Juang, and L. Rabiner. Fundamentals of Speech recognziton. Englewood Cliffs: Prentice Hall Signal Processing Series, 1993. [12] R.A. Howard, Dynamzc Probabzlzstzc Systems Volume 1, Markov Models. Mineola: Dover, 2007. [13] J.P. Hughes, P. Guttorp, and S.P. Charles. "A NonHomogeneous Hidden Markov Model For Precipitation Occurrence," Applzed Statzstzcs, vol. 48, no. 1, pp. 1530, 1999. 82 [14] J.P. Hughes, and P. Guttorp. "Incorporating Spatial Dependence and Atmospheric Data in a Model of Precipitation," American Meteorological Society, December, pp. 15031515, 1994. [15] M. Iosifescu, Fznrte Markov Processes and Their Applzcatzons. Mineola: Dover, 2007. [16] J. Kato, T. Watanabe, S. Joga, J. Rittscher, and A. Blake, "An HMMBased Segmentation Method for Traffic Monitoring Movies," IEEE Transactions on Pattern Analysis and Machine Intellzgence, vol. 24, no. 9, pp. 12911296, December 2002. [17] T. Koski, Hzdden Markov Models for Bzoznformatzcs. Norwell: Kluwer Academic Publishers, 2001. [18] K.R. Lang, Sun, Earth, and Sky 2nd ed., New York: Springer Science+Business Media.LLC, 2006. [19] X. Li, and F.M. Porikli, "A Hidden Markov Model Framework for Traffic Event Detection Using Video Features." 2004 Internatzonal Conference on Image Process rng, IEEE: pp. 29012904, 2004. [20] M. Longnecker, and R.L. Ott, An Introductzon to Statzstzcal Methods and Data Analysis 2nd ed. Pacific Grove: Duxbury, 2001. [21] J. Medhi, Stochastzc Processes 2nd ed. New Delhi: Wiley Eastern Limited, 1994. [22] National Atmospheric and Oceanographic Administration/National Geophysical Data Center, "International Sunspot Numbers," NGDC.NOAA.gov, 2008, 28 Feb. 2008, http://www.ngdc.noaa.gov/stp/SOLAR_ data/sunspot_ numbers/ridaily.plt. [23] National Atmospheric and Oceanographic Administration/National Geophysical Data Center, "Minima and Maxima of Sunspot Number Cycles," NGDC.NOAA.gov, 2008, 7 Jan. 2008, ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SUNSPOT NUMBERS/maxmin.new. [24] B.L. Nelson, Stochastzc Modelzng Analyszs & Szmulatzon, Mineola: Dover Books, 2002. [25] L.R. Rabiner, "A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition," Proceedzngs of the IEEE, vol. 77, no. 2, pp. 257286, 1989. [26] A.W. Robertson, A.V.M. Ines, and J.W. Hansen. "Downscaling of Seasonal Precipitation for Crop Simulation" Journal of Applied Meterology and Climatology, vol. 46, pp. 678693, 2007. [27] D.J. Sheskin, Handbook of Parametrzc and NonParametrzc Statzstical Procedures. Boca Raton: CRC Press LLC, 1997. [28] M. Stix, The Sun, An Introductzon 2nd ed., Berlin, Heidelberg: Springer, 2004. 83 [29] D.D. Wackerly, W. Mendenhall III, and R.L. Scheaffer. Mathematical Statistics wzth Applhcatzons, Pacific Grove: Duxbury, 2002. [30] USDA Economic Research Service, "Wheat Data: Yearbook Tables," ers.usda.gov, 2008, 20 May 2008, http://www.ers.usda.gov/data/wheat. BIOGRAPHICAL SKETCH Christopher C. Krut was born in Beaver, Pennsylvania in 1983. Raised in the Monongahela Valley, White Oak Pennsylvania to be exact, he graduated from McKeesport Area High School in 2002. After receiving his B.S. in applied mathematics from the University of Pittsburgh, Greensburg, in 2006, he accepted a teaching assistantship in the Mathematics Department at the University of Florida which allowed him to pursue an M.S. in applied mathematics. After completing his M.S. in Applied Mathematics Christopher will begin working on an M.S. and PhD. in statistics at North Carolina State University. PAGE 1 1 PAGE 2 2 PAGE 3 page LISTOFTABLES ..................................... 5 LISTOFFIGURES .................................... 6 ABSTRACT ........................................ 7 CHAPTER 1INTRODUCTION .................................. 8 2FINITESTATEMARKOVCHAINS ........................ 10 2.1Introduction ................................... 10 2.2BasicDenitionsandNotation ......................... 10 2.3SomeStandardResultsforMarkovChains .................. 20 2.4SimulatingFromaMarkovChain ....................... 29 3BASICHIDDENMARKOVMODELS ....................... 31 3.1Introduction ................................... 31 3.2ClassicalDenitions .............................. 32 3.3TheScoringProblem .............................. 35 3.4TheDecodingProblem ............................. 37 3.5Simulation .................................... 41 4SELECTEDAPPLICATIONSOFHIDDENMARKOVMODELS ........ 45 4.1IsolatedWordRecognition ........................... 45 4.2TheHMMAppliedtoTracMonitoring ................... 47 4.3AHiddenMarkovModelasaGeneratorforRainfallData ......... 48 5ONWILLIAMHERSCHEL'SCONJECTURE .................. 51 5.1WilliamHerschel'sConjecture ......................... 51 5.2InterestinHerschel'sClaim .......................... 52 5.3AnalysisUsingModernData .......................... 53 5.4Conclusions ................................... 58 6MARKOVCHAINMODELFORTHE11YEARSUNSPOTCYCLE ..... 61 6.1Introduction ................................... 61 6.2DescriptionofTheStates ........................... 62 6.3StartingStateDistribution ........................... 63 6.4OneStepTransitionDistribution ....................... 64 6.5ResultsFromSimulations ........................... 65 3 PAGE 4 ...... 70 REFERENCES ....................................... 81 BIOGRAPHICALSKETCH ................................ 84 4 PAGE 5 Table page 51YearsofrelevantsolarminimaandaveragewheatpricesreportedbyWilliamHerscheloverperiodsofnoreportedsunspotactivity. ............... 59 52YearsofrelevantsolarminimaandaveragewheatpricesreportedbyWilliamHerscheloverperiodsdirectlybeforeperiodsofnoreportedsunspotactivity. .. 59 53YearsofrelaventsolarminimaandaveragewheatpricesreportedbyWilliamHerscheloverperiodsdirectlyafterperiodsofnoreportedsunspotactivity. ... 60 54MeanandvarianceofthepriceofwheatintheUnitedStatesduringperiodsoflowsunspotactivityandtimeperiodsdirectlyprecedingandfollowingtimeperiodsoflowsunspotactivity. ........................... 60 55Meanandvarianceofwheatyieldsduringperiodsoflowsunspotactivityandperiodsdirectlyprecedinganddirectlyfollowingtimeperiodsoflowsunspotactivity. ........................................ 60 5 PAGE 6 Figure page 61Averagesunspotnumberoverthecourseofthesunspotcyclefromsimulatedsunspotcyclesandtheaveragesunspotnumberoverthecourseofthesunspotcyclefromactualdata(18561996). ......................... 68 62Averagesunspotnumberoverthecourseofthesunspotcyclefromsimulatedsunspotcyclesandtheaveragesunspotnumberoverthecourseofthesunspotcyclefromactualdataroundeddowningroupsof20(18561996). ........ 68 63Averagesunspotnumberoverthecourseofthesunspotcyclefromsimulatedsunspotcyclesandtheaveragesunspotnumberoverthecourseofthesunspotcyclefromactualdatanotusedtotrainthemodel(19641996). ......... 69 64Averagesunspotnumberoverthecourseofthesunspotcyclefrom18561996andtheaveragesunspotnumberoverthecourseofthesunspotcyclefrom19641996. 69 6 PAGE 7 7 PAGE 8 8 ]Thelawsofsupplyanddemanddictatethatashortageinsupplywithoutacorrespondingdecreaseindemandwillcausepricestoincrease.Adefectinvegetation,causedbyalackofsunspots,orotherwise,couldaectsupplyandindirectlysunspotactivitycouldaectthepriceofwheat.InsupportofhisclaimHerschelpresented5exampleswherethepriceofwheatwashigherduringatimeperiodwhenrecordsindicatedanabsenceofsunspotsthaninatimeperiodofequallengthdirectlyfollowingandin2casesprecedingthistimeperiod.Aseeminglyreasonableconjectureaschangesinthesuncouldhavesomeimpactonthehealthofcropswhosegrowthisdependentonsolarenergy.Withthisinmindthequestionarose;couldthisrelationshipbemodeledusingaHiddenMarkovModel.InvestigatingthisquestionrequiredresearchintoHiddenMarkovmodelsingeneral.Beginningwithanintroductiontonitestate,discretetimeMarkovChains,basicdenitionsaswellasbasicsolutionstoclassicproblemsconcerningMarkovChainsarediscussed.ThefundamentalsofHiddenMarkovModelsaswellasafewmodernapplicationsarethenexamined.ToexamineHerschel'sclaimwheatpricesfromthetwentiethcenturywereusedtoanalyzewhethersucientevidenceexiststosupporttheclaimthatthereissignicantdierenceinthepriceofwheatduringperiodsoflowsunspotactivityandthesurroundingtimeperiods.TofurtherexamineHerschel'sclaimtheaveragewheatyield,measureinbushelsperacre,duringtimeperiodsoflowsunspotactivitywascomparedtotheaveragewheatyieldduringtimeperiodsofequallengthdirectlyprecedingandfollowingthistime 8 PAGE 9 9 PAGE 10 21 ],[ 1 ].AdditionallytheparametertwhichindexestheMarkovChain,oftenreferredtoasthetimeparameterthoughitneednotrepresenttimeexplicitly,takesonatmostacountablyinnitenumbervalues.SincethetimeparameterofaMarkovChaintakesonatmostacountablyinnitenumberofvaluesaMarkovChainissaidtohaveadiscretetimeparameteraswellasadiscretestatespace[ 21 ],[ 1 ]. 10 PAGE 11 21 ],[ 1 ],[ 12 ],[ 5 ]. 1 ],[ 12 ],[ 21 ].Thecollectionoftransitionprobabilitiespij,sj2Sdeneaonesteptransitionprobabilityvectorwhichgivestheconditionalprobabilitydistributionoftransitioningtoanewstateinthestatespaceattimen+1 11 PAGE 12 21 ],[ 1 ],[ 15 ],[ 12 ].InthispaperthestartingstatedistributionwillbedenotedP0.Undertheassumptionsmade,thevoterexamplementionedpreviouslycanbemodeledusingaMarkovChain.Thestatespaceofthechainisdenedasfollows.Supposestate1correspondstoavoterbeingregisteredaRepublican,state2representsavoterbeingregisteredaDemocrat,andstate3representsthevoterbeingregisteredasIndependent.Supposethroughsomesortofanalysisthetransitionprobabilitieshavebeendetermined.Thetransitionprobabilitiesaregiveninthefollowingtransitionmatrix.P=0BBBB@:70:20:1:20:725:075:20:25:551CCCCATocompletelydescribethebehaviorofthesystemastartingstatedistributionisrequired.Avoter'sinitialpartyaliationisdeterminedbyanumberoffactors.Theseincludethepartyaliationoftheirparents,socioeconomicstatus,education,peer 12 PAGE 13 1 ],[ 15 ],[ 12 ].NotethatinahomogeneousMarkovChainifm=1theprobabilityP(Xm+n=sjjXn=si)=pijforallvaluesofn.Thispropertysimpliesthemodelgreatly,howeveritmaylimittosomeextentthemodel'sabilitytoapproximatemoredynamicsystems.WhenaMarkovChainishomogeneousthetransitionprobabilitymatrixPdoesnotchangeovertimeproducingtheequalityP(Xm+n=sjjXn=si)=P(Xm=sjjX0=si)[ 1 ],[ 21 ],[ 15 ].TheprobabilitydenedbyP(Xn=sjjX0=si)iscalledthensteptransitionprobability[ 1 ],[ 21 ],[ 15 ],[ 12 ].Lackingasinglestandardnotationforthisprobability,theconventionP(n)ij=P(Xn=sjjX0=si)willbeusedthroughoutthispapertoindicatetheprobabilityoftransitioningfromstatesitostatesjinnsteps 13 PAGE 14 1 ].NotethatP(n)ij=PNk=1P(n1)ikpkj[ 1 ],[ 12 ],[ 21 ],[ 15 ].ItshouldbeclearthatP(n)ijcanthenbeexpressedasP(n)ij=PNkn=1:::PNk2=1PNk1=1pik1pk1k2:::pknj.Thisisthesumoftheprobabilitiesofallallpossiblesequencesoflengthnwhichstartinstatesiandendinstatesj.However,fromtheaboveequalityitisseenP(n)ijisequaltoijthentryofthenthpowerofthetransitionmatrixP,i.e.P(n)ij=Pnij[ 1 ],[ 12 ],[ 21 ],[ 15 ].Usingthensteptransitionmatrixalongwiththestartingstatedistributionitispossibletocomputetheunconditionalprobabilityofbeinginsomestatesiattimen.TheprobabilityofbeinginstatesiattimenisgivenbytheithentryoftheprobabilityvectorP0Pn[ 12 ],[ 1 ].ToseethisconsiderthatP0givestheprobabilityofbeinginagivenstateattime0.ThenforagivenMarkovChainwithstatespaceSconsistingof3states,forexample,theprobabilityofbeinginstates12Sattimet=1isgivenbyP(X1=s1)=P(X0=s1)P(X1=s1jX0=s1)+P(X0=s2)P(X1=s1jX0=s2)+P(X0=s3)P(X1=s1jX0=s3)Thiscorrespondstothe1stentryofthevectorgivenbyP0P.Usinginductionitcanbeshownthatthisresultholdsforallvaluesofn.Returningtothepreviousexample,supposestartingattimen=0wewanttoknowthedistributionforpartyaliationafter3presidentialelections.Tondthe3steptransitionprobabilitiesfortheMarkovChainrepresentingvoterbehaviortakethethirdpowerofthetransitionmatrixP.Thuswehave.P3=0BBBB@::475:36975:15525:35:512578:137422:35:405406:2445941CCCCAThentondtheprobabilitydistributionafter4presidentialelectionsmultiplythestartingstatedistributionvectorbythe3steptransitionmatrix.whichgives:P0P3=:4125:430447:157053Theaboveprobabilitydistributionshowsthatafter4presidentialelectionstheprobabilityavoterwillberegisteredRepublicanis.4125,theprobabilityavoterwillbe 14 PAGE 15 15 PAGE 16 15 ],[ 1 ],[ 21 ].Commonnotationistowritesi!sj,indicatingsjisreachablefromsi.Ifsi!sjandsj!sithenthestatessiandsjaresaidtocommunicate,notatedsi$sj[ 15 ],[ 1 ],[ 21 ].CommunicationbetweenstatesdenesanequivalencerelationonthestatespaceoftheMarkovChainandstateswhichcommunicateformequivalenceclasses[ 15 ],[ 1 ],[ 21 ].Thereisoneexceptionhowever.IfastatesidoesnotcommunicatewithitselfthencommunicationbetweenstatesdoesnotdeneanequivalencerelationonthestatespaceoftheMarkovChainastheequivalenceclasseswouldnotformapartitionofthestatespace[ 15 ].AstatewhichcommunicatesonlywithitselfiscalledanabsorbingstateandcorrespondstothetransitionprobabilityPii=1foranabsorbingstatesi[ 15 ],[ 21 ],[ 1 ].AMarkovChaininwhicheverystatecommunicateswitheveryotherstate,i.e.thereisonlyoneequivalenceclass,issaidtobeirreducible[ 12 ],[ 1 ],[ 15 ],[ 21 ].NotethatthevoterpartyaliationexampleisanexampleofanirreducibleMarkovChainastheprobabilityofgoingfromonestatetoanyotherstateinonestepispositive.AnothermethodbywhichstatesofaMarkovChaincanbedescribedisbytheprobabilitythatgiventhechainisinastatesiitwillreturntothatstate.Forexample,considerthevoterpartyaliationexampleagain.Itseemsreasonablethatifavoterhasonly3choicesofpoliticalpartieswithwhichtoregisteravoterwhovotesasaregisteredRepublicanshouldvoteagaininsomefutureelectionasaregisteredRepublican.Ingeneralterms,statesofaMarkovChainforwhichtheprobabilitythatgiventhechainisinacertainstateitwillreturntothatstateis1aresaidtoberecurrent[ 1 ],[ 15 ],[ 12 ].Statesforwhichtheprobabilitythatgiventhechainisinacertainstateitwillreturntothatstateislessthan1aresaidtobetransient[ 21 ],[ 1 ],[ 15 ],[ 12 ].Itisknownthatthestatesofthevoterpartyaliationexamplearerecurrent,namelyitisknownthatavoterwhovotesasaregisteredRepublicaninagivenelectionwillvoteagainasaregisteredRepublicaninsomefutureelection.GiventheknowledgethatavoterwhohasvotedasaregisteredRepublicanwillvoteagainasaregisteredRepublican 16 PAGE 17 1 ],[ 15 ],[ 12 ].IfweknowthatavoterwhovotesasregisteredRepublicanwillvoteagainasaregisteredRepublicaninaniteamountoftimetheexpectedtimeofreturnisusefulinformation.Iftheexpectedtimeofreturnisnotnitetheneventhoughtheprobabilityofreturntoagivenstateis1theamountoftimeitwilltakeforthechaintoreturnisexpectedtobeinniteandthustherecurrenceofthechaindoesn'tseemtomeanquiteasmuch.ForthisreasonrecurrentstatesofaMarkovChainaredescribedbytheamountoftimeitwilltakethechain,givenitstartedinacertainstatetoreturntothatstate.Arecurrentstatewhoseexpectedtimeofreturnisniteisreferredtoasapositiverecurrentstatewhilearecurrentstatewhoseexpectedtimeofreturnisinniteiscalledanullrecurrentstate[ 1 ],[ 15 ],[ 12 ],[ 21 ].Weknowthatallstatesinthevoterpartyaliationexamplecommunicate.ThusitispossiblethatavoterwhovotesinanelectionasaregisteredDemocratwillvoteinafutureelectionregisteredasanIndependent.Aninterestingquestionistondout,onaveragehowlongitwilltakeavoterwhoisvotinginthecurrentelectionasaregisteredDemocrattovoteinafutureelectionregisteredasanIndependent.IngeneralsupposeYijisarandomvariabledenedasfollows:Yij=nifandonlyifXn=sj;X0=si;Xm6=sj1m PAGE 18 12 ],[ 21 ],[ 15 ],[ 1 ].Usingthisnotationitispossibletodeterminingifastateisrecurrentortransient.AstatesiisthenrecurrentifP1n=1fii(n)=1,andtransientifP1n=1fii(n)<1[ 12 ],[ 21 ],[ 1 ],[ 15 ].TheexpectationofYij,E[Yij],givestheexpectednumberofstepsitwilltaketotransitiontostatesjgiventhesystemstartedinstatesi[ 12 ],[ 21 ],[ 1 ],[ 15 ].ForarecurrentstatesiitispossibletodetermineifsiispositiveornullrecurrentbycomputingtheexpectationofYii.ThestatesiispositiverecurrentiftheexpectationofYiiisniteandnullrecurrentiftheexpectationofYiiisinnite.Itshouldbenotedthatfij(n)andPnijarenotthesameasPnijgivestheprobabilityofbeinginstatesjattimengiventhechainstartedinstatesiandgivestheprobabilitythatthechainhastransitionedfromstatesitostatesjforthersttimeattimen.ThereishoweverarelationshipbetweenPijandfij=P1n=1fij(n)givenbyDoeblin'sformulawhichstatesgivenanysiandsj: 15 ](2{2)UsingthisformulaitcanbeshownthatifsjisatransientstatethenP1n=1Pnijconverges,si2S[ 15 ].SinceP1n=1Pnijconvergesforanytransientstatesj,limn!1Pnij=0[ 15 ].Hencetheprobabilityoftransitioningtoatransientstatesjfromanystatesiinnstepstendstozeroasthenumberofstepstendstoinnity.Recurrenceandtransienceareclassproperties.Thatis,ifsiisarecurrentstateandsi$sjthensjisarecurrentstate[ 12 ],[ 15 ],[ 1 ],[ 21 ].Recurrentstatescanbereachedfromtransientstates,transientstateshoweverarenotreachablefromrecurrentstates.Thustransientstatesandrecurrentstatesdonotcommunicate.Thisimpliesarecurrentstateandatransientstatecannotbeinthesameclass.Inthiswayclassesofstatescanbeclassiedaseithertransientorrecurrent.ThisalsoimpliesthataMarkovChaincannottransitionfromarecurrentclasstoatransientclass[ 15 ],[ 1 ],[ 21 ]. 18 PAGE 19 1 ],[ 21 ],[ 15 ].GiventhateveryniteMarkovChaincontainsatleast1recurrentclass,andtheprobabilityofbeinginanygiventransientstateattimentendsto0asntendstoinnity,itfollowsthataniteMarkovChainwillspendonlyaniteamountoftimeintransientclassesbeforetransitioningintoarecurrentclass[ 1 ],[ 21 ],[ 15 ],[ 12 ].Returningtotheexampleregardingvoterpartyaliation;itisknownthatthechainisirreducible.ThensinceeveryniteMarkovChaincontainsatleastonerecurrentclassitisknownthatthechainrepresentingthissystemconsistsofasinglerecurrentclass.IfweknowavoterisinitiallyregisteredIndependentitwouldbeinterestingtoknowifthereexistfutureelectionsforwhichtheprobabilityofthevoterbeingregisteredIndependentis0.IfPnii>0,thenitisknownthatitispossibletotransitionfromstatesibacktostatesiinnsteps.Theperiodofagivenstatesiisthegreatestcommondivisorofallpossibletimesnwhereitispossibletotransitionfromfromstatesibacktostatesi,i.ethegreatestcommondivisoroffnjPnii>0g[ 12 ],[ 15 ],[ 21 ],[ 1 ].Iftheperiodofsomegivenstateis1thenthestateissaidtobeaperiodic[ 12 ],[ 15 ],[ 21 ],[ 1 ].Aswithrecurrence,periodicityisaclassproperty.Thatisifsiisofperiodmandsi$sjthensjisofperiodm[ 12 ][ 15 ][ 21 ][ 1 ].Therefore,aclassofstatescanbesaidtohaveacertainperiod.ThisisimportantsincelargerMarkovChainswillhavemultipleclassesofstatesandthebehaviorofthesystemwithrespecttotheseclasseswillbeimportant.Lastly,considerthevoterpartyaliationexampleonemoretime.Recallthattheonesteptransitionmatrixforthevoterpartyaliationexamplehasallpositiveentries,hencetheperiodoftheMarkovChainisgoingtobe1foreachstateinthechain.Thus,eachstateinthechainisaperiodic. 19 PAGE 21 21 ],[ 1 ],[ 4 ],[ 5 ].ThevectoristhelimitingdistributionfortheMarkovChainandgivesthelongtermprobabilitiesofbeingineachrespectivestate[ 21 ],[ 1 ][ 4 ],[ 5 ].AstimetendstoinnitytherowsofthetransitionmatrixPapproachaswasobservedintheaboveexample.Sincetherowsofthematrixtowhichlimn!1Pnconvergesarethesameitshouldbeclearthatthelongtermprobabilityofbeingineachrespectivestatedoesnotdependontheinitialstate[ 12 ].GiventheexistenceofalimitingdistributionforaMarkovChain,thenextquestionishowquicklydoesthechainconvergetoitslimitingdistribution?Thespeedwithwhichthetransitionmatrixconvergestoitslimitingdistributionprovidesanindicationofthetimenecessaryfortheprobabilityofbeinginagivenstatetonolongerbedependentontheinitialstate.Itcanbeshownthatforanitestate,aperiodic,irreducibleMarkovChain,Pnconvergesgeometricallytoitslimitingdistribution[ 1 ],[ 21 ].Forasmallmatrixitispossibletocomputethelimitingdistributionbytakingpowersofthematrix,aswasdoneintheexample.However,forlargermatricesandforsomeonewithoutacomputeralgebrasystemtakingpowersofamatrixmayproveunfeasible.TosimplifytheproblemofndingthelimitingdistributionrememberthatP=.ThismeansthatP=,alongwiththeconditionthattheelementsofsumtoone,producesasetofequationswhichcanbesolved,undercertainconditions,tond 21 PAGE 22 12 ],[ 1 ],[ 15 ],[ 21 ],[ 12 ].Returningtothevoterpartyaliationexample,thefollowingsystemofequationsisobtainedfromP=. (2{6) Usingany2oftherst3equationsandthelastequationthelimitingdistribution=(:400090;:438087;:161823)canbecomputed.FromthelimitingdistributionwecanconcludethatafterasucientlylongtimetheprobabilityavoterwillbearegisteredRepublicanisabout.40009,theprobabilitytheywillberegisteredaDemocratisabout.438087,andtheprobabilitythevoterwillberegisteredasIndependentisabout.161823.Notethattheseprobabilitiesareindependentofthepartywithwhichthevoteroriginallyregistered.Analysislikethiswouldbehelpfulinanalyzingthelongterm"health"ofapoliticalparty.Inadditiontogivingthefractionoftimethechainisexpectedtospendineachstate,thelimitingdistributioncanalsobeusedtondthemeantimetorecurrenceforeachstateinthechain.Itcanbeshownthatthemeantimeofrecurrenceforastatesiisgivenby1 15 ].ThusitisnowpossibletocomputethetimeittakesavoterwhovotedasaregisteredRepublicantovoteagainasaregisteredrepublican.Themeanrecurrencetimesfortheexamplearegivenby1 22 PAGE 23 15 ],[ 21 ],[ 1 ].ItshouldbenotedthattherowsofsubmatricesQandRwillnot,ingeneral,sumto1.UsingtheabovenotationtheonesteptransitionmatrixPforagivenMarkovChaincanbewrittenas:P=0BBBBBBBBBB@P10:::000P2:::00............00:::Pn0R1R2:::RnQ1CCCCCCCCCCA[ 15 ];[ 1 ];[ 21 ];Theexamplefromsection2.2willnowbeexpandedasfollows.Supposethatitisnowpossibleforsomeonetoremaininagivenpoliticalpartypermanently,i.e.theydecidenevertoswitchparties.Ifsomeoneweretoremainaliatedwithapoliticalpartyforasucientlylongtime,theprobabilitythattheywouldchangealiationtoanopposingpoliticalpartyshoulddecrease.Forthepurposeofsimplicationsupposethatifsomeoneisregisteredwithaparticularpartyaliationfor3consecutivepresidentialelectionsthenforallpracticalpurposesweassumethemtobeapermanentmemberofthatpoliticalparty.ThisassumptionresultsinanewMarkovChaintomodelthebehaviorofvoters.Inthisnewchainthereare3recurrentclasses,permanentRepublican,permanentDemocrat,permanentIndependent.Notethattheseclassesarenotonlyrecurrentbutabsorbingaswell.ThestatesofthenewMarkovChainwillbenumberedinthefollowingway. 1. AvoterisapermanentRepublican 23 PAGE 24 AvoterisapermanentDemocrat 3. AvoterisapermanentIndependent 4. Avoterhasvotedinthelast2electionsasaRepublican 5. Avoterhasvotedinthelast2electionsasaDemocrat 6. Avoterhasvotedinthelast2electionsasanIndependent 7. AvoterhasvotedinthelastelectionasaRepublican 8. AvoterhasvotedinthelastelectionasaDemocrat 9. AvoterhasvotedinthelastelectionasanIndependentInthenewMarkovChainrepresentingthisprocessstates1,2,and3areabsorbing,states4,5,6,7,8and9aretransient.Withtheassumptionsmadeinconstructingthismodelitisknownthatavoterwillonlyvoteanitenumberoftimesbeforebeingabsorbedintoanabsorbingstate,becomingapermanentmemberofacertainpoliticalparty.UsingthetransitionprobabilitiesfromtherstexamplethetransitionmatrixforthisMarkovChainhastheform.P=0BBBBBBBBBBBBBBBBBBBBBBBB@100000000010000000001000000:7000000:2:10:7250000:20:07500:55000:2:250000:7000:2:10000:7250:20:07500000:55:2:2501CCCCCCCCCCCCCCCCCCCCCCCCA PAGE 25 (IQ)1=M[ 15 ];[ 1 ];[ 12 ];[ 21 ](2{7)Miscalledthefundamentalmatrix[ 1 ],[ 21 ],[ 15 ].TheijthentryofMgivestheexpectednumberofvisitstoatransientstatesjpriortoenteringarecurrentclassgiventhechainstartedinstatesi[ 15 ],[ 12 ],[ 1 ],[ 21 ].Usingthefundamentalmatrixitispossibleto 25 PAGE 26 1 ],[ 21 ],[ 15 ].Theresultsobtainedfromthevoterpartyaliationexamplearegivenbelow.StepstoRecurrence=0BBBBBBBBBBBBBB@2:09392:011782:579513:559613:470343:998221CCCCCCCCCCCCCCAGiventhisinformationitcanbeseenthatavoterwhowasregisteredwiththeirrespectivepartyforonlythepreviouselectionislikelytovotebetween3and4times 26 PAGE 27 12 ];[ 15 ];[ 1 ];[ 21 ](2{8)ReturningtotheexamplewecannowndtherespectiveprobabilityofavotereventuallybecomingapermanentRepublican,Democrat,orIndependentgiventheirpartyregistrationinpreviouselections.Forthevoterpartyaliationexample,Fisapproximatelygivenby:F=0BBBBBBBBBBBBBB@:7800:1688:0512:1471:8085:0444:1887:2248:5865:6260:2869:0871:2538:6697:0765:2924:3485:35191CCCCCCCCCCCCCCAFromthisresultifavoterisregisteredwithagivenpoliticalpartyfortwopresidentialelectionsthentheprobabilityisfairlyhighthattheywillbecomeapermanentmemberofthatpoliticalparty.Theprobabilitiesofbecomingaaliatedpermanentlywithapoliticalpartygivenavoterwasregisteredwiththepartyforonepresidentialelectionareslightlylower.Oncethesystemhasenteredarecurrentclasstherecurrentbehavioroftheclasscontrolsthebehaviorofthesystem.Thusthelongtermbehaviorofthesystemwillbe 27 PAGE 28 15 ].Lastly,theexampleusedinthissectioncontained3absorbingstates.ThemethodforcomputingFcanalsobeusedwhentherecurrentclassesoftheMarkovChaincontainmultiplestates.ThematrixFissimplytheprobabilityofevertransitioningtostatesinrecurrentclasses.Thustheprobabilityofbeingabsorbedintoacertainrecurrentclassgivenaninitialstateisjustthesumoftheprobabilitiesofofenteringeachrecurrentstategivenaninitialstate[ 15 ].Thus,largerrecurrentclassesofstatesdonotpresentanyrealdiculty,asidefromadditionalcomputation.Thesimpliedexampleusedinthischapterisctionalandwasonlyintendedforillustrativepurposes.Howeverafterdiscoveringthisexamplethequestionarose;hasthisissuebeenexamined?CanahomogeneousMarkovChainbeusedtomodelvoterloyaltytoapoliticalparty?ResearchingthisissueIcameacrossapaperbyDouglasDobsonandDuaneA.Meeterwhichbrieyexaminesthequestion[ 6 ].Thefollowingisasummaryoftheirndings.DobsonandMeeterbeginbyclassifyingvotersintooneof5categories:"StrongDemocrat,WeakDemocrat,Independent,WeakRepublican,andStrongRepublican"[ 6 ].Theauthorsdeneamodelwhichdescribesthechangeinvoterpartyaliationintwoparts.Therstpartdescribestheprobabilitythatthevoterwillchangealiationtheseconddescribestheprobabilitythatifitisknownthevoterischangingpartyaliationhe/shewillchangetoaparticularaliation[ 6 ].Inthiswayitispossibletodescribechangespartyaliationinmoredetailaswellasexaminewhethereitherpartofthemodelshouldbemodeledusingatimevarying,i.e.nonstationary,model.Inparticular,themodelusedbytheDobsonandMeterisgivenbythetransitionmatrix"P(t)=D(t)+(1D(t))M(t)"whereD(t)givestheprobabilitiesofremaininginthecurrentstateand 28 PAGE 29 6 ].TheprobabilitymatricesM(t)andD(t)wereconstructedbasedondatafrom19561958.AsecondpairofprobabilitymatricesM(t)andD(t)areconstructedbasedondatafrom19581960.UsingachisquaredtestforstationarityitwasfoundthatneitherM(t)orD(t)shouldbestationary[ 6 ].ThusitseemsanonhomogeneousMarkovChainwouldbethebestchoiceformodelingvoterloyalty.Reasoningthatthetransitionprobabilitieswouldvaryacrossthepopulationbasedoncharacteristicssuchasgender,race,etc.DobsonandMeeterpartitionedtheirdataintothreegroupsbasedonthelevelofinterestintheelectoralsystem[ 6 ].Dening3categories,"highinterest,moderateinterest,lowinterest",theyfoundahomogeneousMarkovChaintobeanappropriatemodelforboththehighinterestandlowinterestcategorieswithonlythemoderateinterestcategoryrequiringanonhomogeneousMarkovChain[ 6 ].Anotherexamplein[ 1 ]attemptstouseahomogeneousMarkovChaintomodelchangesinvoterpreferenceforaparticularpresidentialcandidate.However,itwasagaindeterminedthatahomogeneousMarkovChainwasnotanappropriatemodel[ 1 ].Thusitseemsthatvotingbehavior,especiallyoveranyextendedperiodoftime,islikelytoodynamictobemodeledwellusingahomogeneousMarkovChain. 29 PAGE 30 24 ],[ 5 ]. 1. Generateanumberfromuniform[0,1]distribution,callitp. 2. Sumovertheinitialprobabilitydistributionuntilthesumpkeepingtrackofthecurrentstatewhoseprobabilityisbeingaddedtothesum,callitk. 3. Statek1isthecurrentstate. 4. Generateanothernumberfromuniform[0,1]distribution. 5. Usingtheonesteptransitiondistributionwithcurrentstatek1inplaceoftheinitialprobabilitydistributionrepeatsteps2,3.Steps3,4,5canberepeatedtogenerateasequenceofrandomnumbersfromtheMarkovChain.ThecodeusedtosimulatesunspotnumbersfromtheMarkovChaindiscussedinChapter6isavailableintheAppendix.GeneratingmultiplesequencesgivesanideaofhowtheMarkovChainrepresentingthesystembeingmodeledisbehaving.IftheMarkovChainisbasedonobservations,thesequencesgeneratedfromthemodelcanbecomparedtothedatatoevaluatethetofthemodel.SequencesgeneratedfromapurelytheoreticalMarkovChaincanbeusedtogaininsightintothesystembeingmodeledwhichmighthaveotherwisebeenunavailable. 30 PAGE 31 25 ],[ 17 ],[ 9 ].Considerthefollowingexample.Supposeacertainhighwayisbeingobserved.Thetracowonthehighwaycanbeclassiedaslight,moderate,orheavy.Supposefurtherthatnoinformationisknownaboutanyotherpartofthehighwayexcepttheobservedtracintensity.Itisknownthathighwaytracisdependentonthetimeofday,anyconstructionwhichmaybetakingplaceonthehighway,andwhetherornotanaccidenthasoccurred.Inthiswayitispossible,undersomesimplifyingassumptions,tomodeltheintensityoftracasanobservableresultofthepreviouslymentionedfactors.IneveryHiddenMarkovModelthereisasetofobservations,andasetofunderlyingstates.ThebehaviorofthesetofunderlyingstatescanbeapproximatedbyaMarkovChainthusthereisaMarkovChainwhichis"hidden"andonlyvisiblethroughasetofobservationswhichdependontheunderlyingstates.HencethenameHiddenMarkovModel.Thereare3applicationsforwhichHiddenMarkovModelsareparticularlywellsuited.First,itispossibletosimulateasequenceobservationsfromaHiddenMarkovModel.Theseobservationscanbeusedaspartofalargersimulationortoobtainstatisticalinformationaboutthesystemwhichmaynothavebeenavailablethroughanalyticmethods.Second,HiddenMarkovModelshavebeenusedinidenticationapplications.SpecicallyHiddenMarkovModelshavebeenusedwithsuccessinspeechrecognitionandgeneidentication.Third,HiddenMarkovModelscanbeusedtoobtaininformation 31 PAGE 32 25 ].InthischapterweareonlyinterestedinthesimplestcasewhenOtisanitestate,discretetimeprocess,andOtandOt+1areindependentgivenXt+1.Thenotationbi(k)=P(Ot=kjXt=si)willbeusedthroughoutthechaptertorepresenttheprobabilityofacertainobservationkgivenacurrentstates[ 25 ],[ 17 ],[ 9 ].Thestatespace,observationset,observationdistribution,underlyingstatetransitionprobabilities,andstartingstatedistributioncompletelydeneahiddenMarkovmodel[ 25 ],[ 22 ],[ 17 ].Commonnotationistowrite=ffpijgsi;sj2S;bj(k);O;S;P0gtorepresentaHiddenMarkovModel[ 25 ],[ 9 ],[ 17 ].TheforwardbackwardvariablesarefundamentalintheclassicalalgorithmsforHiddenMarkovModels.SupposewehaveanobservationsequenceO=fo1;o2;:::ong.Theforwardvariableisdenedtobet(i)=P(O0=o0;O1=o1;:::;Ot=ot;Xt=si)[ 17 ],[ 25 ],[ 9 ].Thisistheprobabilityofobservingthersttobservationsof 32 PAGE 33 17 ],[ 25 ],[ 9 ].Thisistheprobabilityofobservingagivensequencestartingattimet+1totimen,startinginstates.ThemajoradvantageofusingtheforwardbackwardvariablesforproblemsinvolvinghiddenMarkovmodelsisthattheycanbecalculatedrelativelyeasilyandthereexistalgorithmsforsolvingthe3problemsmentionedinsection3.1whichutilizetheforwardbackwardvariables[ 25 ],[ 17 ],[ 9 ].Theforwardvariablet(i)canbecomputedasfollows.Firstnotethat0(i)issimplytheprobabilityofobservingo0andstartinginstatesi.UsingthelawoftotalprobabilitywecanwriteP(O0=o0;X0=s0)asP(O0=o0jX0=s0)P(X0=s0),whereisP(X0=s0)=P0(s0).ThusforeachsjinthestatespaceofXtwehave: 25 ];[ 17 ];[ 9 ](3{1)Inthesecondstep1(i)=P(O0=o0;O1=o1;X1=si).Alittlecomputationshowsthat1(i)=(Ps2S0(si)pssi)bsi(o1).Ingeneralwecancomputet(i)usingt1(i)andthetransitionandobservationprobabilities. 17 ];[ 25 ];[ 9 ](3{2)Itispossibletodenethebackwardvariablesinasimilarmanner.Tobegincomputingthebackwardsvariablesbeginwith: 17 ];[ 25 ];[ 9 ](3{3)Theremainingt(i)canbecalculatedusing: 17 ];[ 25 ];[ 9 ](3{4)Thesetwovariablesprovidemethodsforsolvingtheproblemsaboveaswellasestimatingthemodelparameters.Theclassicalsolutionstotheproblemsmentionedinchapter1as 33 PAGE 34 25 ],[ 17 ],[ 9 ].Next,givenanobservationsequenceO=(o1;o2;:::;on),itispossibletoreestimateP0(s),fpijgsi;sj2S,andbs(Ot)0tnasfollows.Tobeginwedene:t(i;j)=P(Xt=si;Xt+1=sjjO;)[ 25 ],[ 17 ],[ 9 ].Thenbythedenitionofconditionalprobabilitywehave:P(Xt=si;Xt+1=sjjO;)=P(Xt=si;Xt+1=sj;Oj) 25 ],[ 17 ],[ 9 ].NextnotethatsummingP(Xt=si;Xt+1=sj;Oj)overallpossiblestates,si;sjgivesP(Oj)[ 25 ],[ 17 ],[ 9 ].Combiningthesetwoobservationsprovidesareasonablemethodofcomputingt(i;j). 25 ];[ 17 ](3{5)ItisnowpossibletoobtainP(Xt=sijO;)bysimplysummingt(i;j)overallpossiblestatessj[ 25 ],[ 17 ],[ 9 ].Lett(i)=P(Xt=sijO;model)=Psj2St(i;j)[ 25 ],[ 17 ],[ 9 ].Summingt(i;j)overtfor1tn1givestheexpectednumberoftimestheunderlyingMarkovChaintransitionsfromstatesitostatesj[ 25 ],[ 17 ],[ 9 ],Similarly 34 PAGE 35 25 ],[ 17 ],[ 9 ].Thusthenewestimatesoftheparameterscanbecomputedusingt(i)andt(i;j). ^P0(s)=1(s)[ 25 ];[ 17 ];[ 9 ](3{6) ^pij=Pn1t=1t(i;j) 25 ];[ 17 ];[ 9 ](3{7)Todenethenewestimateforbj(ot)theindicatorvariable,t(k)=1ifOt=kand0otherwise,willbeused. ^bj(k)=Pnt=1t(k)t(j) 25 ];[ 17 ];[ 9 ](3{8)TheBaumWelchalgorithmdescribedaboveisaniterativeprocedureinwhichtheiterates,theparametersofthemodel,convergetothevalueswhichmaximizeP(Oj)[ 25 ],[ 17 ],[ 9 ].Thusrepeatingthealgorithmgivesimprovedestimatesoftheparametersuntiloptimalvaluesfortheparametershasbeenobtained. 25 ];[ 17 ];[ 9 ](3{9)CurrentlyhiddenMarkovmodelsareappliedtoidenticationproblems.ThisisoftendonebyconstructingaHiddenMarkovModelforeachitemwhichistobeidentied. 35 PAGE 36 36 PAGE 37 25 ],[ 17 ],[ 9 ].ThatisonceXthasenteredstate2itwillneverreturntostate1.AhiddenMarkovmodelisthenconstructedfortheRepublicancandidate,andtheDemocraticcandidate.Bothmodelswillsharethesameunderlyingstatetransitionmatrix.Theobservationdistributionwillhoweverbedierentforeachcandidate.ThengivenasequenceofobservationvectorstheprobabilityofthesequencewillbecomputedconditionedontheHMMfortheRepublicanCandidate,andtheHMMfortheDemocraticcandidaterespectively.IftheprobabilityoftheobservationvectorisgreaterusingtheusingtheHMMfortheRepublicancandidateweconcludethevoterwillprobablyvotefortheRepublicancandidate.IftheprobabilityoftheobservationvectorisgreaterusingtheHMMfortheDemocraticcandidatethenwecanconcludethevoterwillprobablyvotefortheDemocraticcandidate.ThroughthissimpleexampletheconceptofidenticationusingHiddenMarkovModelshasbepresented.InChapter4someexamplesofcurrentapplicationsofHiddenMarkovModelstoidenticationsystemswillpresented. 37 PAGE 38 25 ],[ 17 ],[ 9 ].TheViterbialgorithmprovidesanecientmethodofndingsuchasequence.TheViterbiAlgorithmbeginsbydeningtwovariablestherstofwhichist(s)=maxsi0;si1;:::;sit12SP(O0=oo;O1=o1;:::;Ot=ot;X0=si0;X1=si1;:::;Xt=s)[ 25 ],[ 17 ],[ 9 ].Thisisthemaximumprobabilityofobservingthersttobservationsandbeinginstatesattimet.Thestateswhichmaximizet(s)areretainedusingthevariablet(s)[ 25 ],[ 17 ][ 9 ].Thisentrygivesthestateattimet1whichmaximizestheprobabilityofobservingthersttobservationsandbeinginstatesattimet.Thealgorithmbeginsbyinitializing0(s)ando(s)asfollows: 25 ];[ 17 ];[ 9 ](3{10) 25 ];[ 17 ];[ 9 ](3{11)Theremainingvaluesoft(s)andt(s)arecomputedasfollows: 25 ];[ 17 ];[ 9 ](3{12) 25 ];[ 17 ];[ 9 ](3{13)Oncet(s1)andt(s1)havebeencomputedforallvaluesofs1andt,thenalstateinthestatesequenceischosensuchthatst=argmaxs2S(t(s))[ 25 ],[ 17 ],[ 9 ].Thisisthestateattimewhichwhichmaximizestheprobabilityofobservingtheentireobservationsequenceandbeingthisstateatthelastobservation.Usingthisvalue,theremainingstatesinthe 38 PAGE 39 25 ];[ 17 ];[ 9 ](3{14)Statesattimet1arechosentomaximizetheprobabilityofobservingthesequencethroughtimetandbeinginthestatewhichmaximizedtheprobabilityofobservingthesequencethroughtimet+1andbeingintheappropriatestate.Inthebeginningofthissectionasituationwasdiscussedinwhichlocalocialsareworriedaboutthesuddenincreaseinthenumberofcasesofaparticularinfectiousdiseaseatalocalhospitalandwanttodeterminehowwidespreadthediseasemaybeamongthegeneralpublic.ThereisaclassicexampleinwhichthespreadofaninfectiousdiseaseinasmallpopulationismodeledusingaMarkovchain[ 15 ].Thenumberofpatientstreatedatthehospitalisanobservableresultofthenumberofpeopleinfectedwiththediseaseinthegeneralpopulation.ThusahiddenMarkovmodelmayproveasuitabletoolforanalyzingthesituation.Supposethehospitalservesacommunityof10,000andthatthebasedonpreviousexperiencelocalocialshavedevelopedavariousmethodsforstopping/controllingthespreadofthediseasebasedonhowwidespreadthediseaseisatanygiventime.Aspartofthisplanocialshavedevelopedascalerangingfrom1to10describingtheextenttowhichthediseasehasspread.Level1correspondstonormalcircumstances,whilelevel10correspondstoaseriousepidemic.LetXtbeaMarkovChainwith10statescorrespondingtothelevelofinfectionattimet.Thetimeparameterisassumedtobediscreteandrepresentsthetimeatwhichobservationsaretaken.SupposebasedonpreviousexperiencethetransitionprobabilitiesareknownanddenedbymatrixP.Notethattheseprobabilitiescorrespondtothechangeswhichoccurpriortoanyactiontakentostopthespreadofthedisease. 39 PAGE 41 1. Generateaninitialstateusingtheinitialdistribution. 2. ConditionedontheinitialstategeneratethersttransitionfromXt. 3. Generateanobservationfrombs(k)wheresisthestategeneratedinstep2. 4. Repeatsteps2and3togeneratethedesirednumberofobservations/states.[ 25 ],[ 17 ],[ 9 ]ThestatesequencecanbegeneratedusingthemethodpresentedinChapter2.Notethatthemethodusedtogeneratethestatesequencealsoworkstogeneratetheobservationsequenceprovidedtheobservationdistributionisdiscrete.Considerthefollowingexample.Supposeatownhasrecentlyobservedanincreaseintheamountoftraconthemainhighwaythroughtownandnowfrequentlyobservesheavyorstoppedtrac.Toalleviatethetracproblemsitissuggestedthatanadditionallanebeconstructedineachdirection.Thetownwantstoseeifthiswouldchangetheobservedtracintensity.Asimplied,ctional,HiddenMarkovModelispresentedto 41 PAGE 42 1. 2. 3. 4. 1. 2. 3. 4. PAGE 43 43 PAGE 44 44 PAGE 45 25 ],[ 11 ],[ 9 ].Aspokenwordorwordfragment,canbecharacterizedasaspeechsignal.Thissignalisasequenceofvectorswhichcharacterizethepropertiesofthespokenwordorwordfragmentovertime[ 11 ],[ 9 ].Foranysinglewordthereareavarietyofpossiblespeechsignals.Thisvariationcanbeattributedtodierencesinthesourceofsignal,i.e.thespeaker.Dialect,gender,aswellasenvironmentalfactorsallinuenceagivenspeechsignal.Automaticspeechrecognitionsystemsattempttomatchthesignalofaspokenwordtoastandardrepresentationofthatword.Aspeechsignalwillchangeovertimemakingitisnecessarytopartitionagivenspeechsignalintoregionsonwhichthepropertiesofthesignalareapproximatelystationary[ 11 ],[ 9 ].Thepartitionsofthesignalaresometimescalledframesandthelengthofpartitionstheframelength.Ifthesignalisframedintermsoftheindividualsoundsinawordthenthesignalproducedcanbeseenasaprobabilisticfunctionoftheintendedsound.AHiddenMarkovModelthenseemsparticularlywellsuitedtotheproblemofisolatedwordrecognition.InpracticeaHiddenMarkovModelisconstructedforeachwordinapredeterminedvocabulary.Givenaspeechsignal,theprobabilitythatthesignalrepresentsagivenwordisdeterminedbycomputingtheprobabilityoftheobservationsignalgiventhemodel 45 PAGE 46 25 ].Deningthestatesthiswayimpliesthatthemodelisaleftrightmodelinthatthesoundswhichcomposeawordfollowacertainorder.TheunderlyingMarkovChainforawordmodelneednotbestrictlyleftrightasdierentpronunciationsofawordmayutilizedierentsounds[ 25 ],[ 11 ].Thiscorrespondstoaleftrightmodelwithsomevariationintheprogressionofstates.Theobservationsassociatedwitheachstatecouldbeseenasaparticularframeofthesignal.ImplementingadiscreteHiddenMarkovModelrequiresthesetofpossibleobservationvectorsbereducedtoaniteset.Thisisaccomplishedusingaprocesscalledvectorquantizationwhichpartitionsthesetofallpossiblevectorsintoanitenumberofsubsets[ 9 ],[ 25 ],[ 11 ].Thecentroidofthesetisthenchosenastherepresentativeofthethatparticularsetofvectors[ 11 ],[ 9 ].Classifyinganewvectoristhenmerelytheprocessofndingthesetwhoserepresentativeistheclosestindistancetothenewvector[ 11 ],[ 9 ].Thesetofrepresentativesofeachclassformacodebook.OnceacodebookhasbeendesignedaHiddenMarkovModelforeachwordinthevocabularyisconstructedfromatrainingsetconsistingofspokenversionsoftheword[ 11 ],[ 9 ].VectorquantizationallowsfordiscreteprobabilitydistributionstobeusedandthemethodsofChapter3aredirectlyapplicable.Themajordisadvantagetovectorquantizationisdistortioncausedbypartitioningthesetofallpossiblevectorstoanitesetwhereeachsethasachosenrepresentative[ 11 ],[ 9 ].AnalternativetovectorquantizationistoapplyHiddenMarkovModelsusingcontinuousobservationdensities.Oneofthemostcommondistributions,forallcontinuousHMMs,isamixtureofmultivariatenormaldistributionswherebi(o)=PMk=1cikN(o;ik;ik)[ 25 ],[ 9 ].Hereoisanobservationvector,cikarethemixing 46 PAGE 47 9 ],[ 11 ].Followingparameterestimationtheidenticationprocessissimilartothatinthediscretecase.TheprecedingparagraphshopefullygaveanideaastohowHiddenMarkovModelscanbeappliedtoisolatedspeechrecognition.AninterestedreadercanconsultbooksbyHuang,ArrikiandJack[ 9 ]orbyRabinerandJuang[ 11 ]forfurtherinformationonthetopic. 16 ].PresentedinapaperbyLiandPorikli,thesecondexampleisaHiddenMarkovModelusedinanautomatedmethodofidentifyinghighwaytracpatternsusingvideoimagesofthehighway[ 19 ].Bothofthesemodelswillbediscussedinthefollowingparagraphs.Considerasmallblockofpixelsofanimage.Thepurposeoftherstexample,[ 16 ],istodetermineiftheblockofpixelsisshowingthebackgroundofanimage,theforegroundofanimage,orashadow.Thiswasaccomplishedwithmoderatesuccessbyimplementinga3stateHiddenMarkovModel.Thestatesofthemodelare"Background,shadow,andforeground"[ 16 ].Theobservationsare2dimensionalvectorscharacterizingtheimageintheblockofpixels.Thestatetransitionprobabilitiesarecharacterizedbya3by3transitionmatrixandastartingstatedistribution[ 16 ].Theobservationprobabilitiesforthebackgroundstateandtheshadowstateareapproximatedbymultivariatenormalmixturedensitiesofthetypedescribedintheprevioussectionwhiletheobservationdensityfortheforegroundstateisapproximatedbyauniformdistribution[ 16 ]. 47 PAGE 48 16 ].Theresultspresentedinthepaperwerefairlyimpressive.Itwasseenthatthemodelidentiedlightcoloredcarseasilybuthadmoredicultywithdarkandgraycoloredcarsthoughincorporatingfeaturesofthesurroundingregiondidimprovetheresultsobtainedfromdarkandgraycars[ 16 ].ThesecondapplicationofHiddenMarkovModelstotracmodelinginvolvedidentifyinghighwaytracpatternsbyanalyzingvideoimages.DierentthanmostidenticationsystemsimplementedusingHiddenMarkovModelsthedetectionsystemdescribedin[ 19 ]isimplementedusingasingle6statehiddenMarkovModel.Thestatesofthemodelcorrespondtonumberofvehiclesonthehighwayandthespeedatwhichtheyaretraveling[ 19 ].Statesareclassiedas"heavycongestion,highdensitywithlowspeed,highdensitywithhighspeed,lowdensitywithlowspeed,lowdensitywithhighspeed,andvacancy"[ 19 ].Theobservationsaredenedbya4dimensionalvectorwhichdescribesapieceoftheimageinaframeofvideoatagiveninstant[ 19 ].TransitionsbetweenstatesaredenedbyatransitionmatrixandtheobservationprobabilitiesareeitherdenedbyasinglenormaldistributionoraGaussianmixture,withtrainingaccomplishedusingtheBaumWelchalgorithm[ 19 ].Followingtraining,theViterbialgorithmwasusedtoidentifythemostlikelystatesequencegivenanobservationsequenceandidentifyagiventracpattern[ 19 ]. 48 PAGE 49 26 ]formodelingrainfallatdierentsitesthroughoutaregionusingregionalrainfallstatisticsfortheregionwillbediscussed.Rainfalloccurrenceinagivenareacanbeviewedasaresultofanunderlyingweatherpatterninthatarea.BuildingonthisideaHughesandGuttorpsuggestaHiddenMarkovModelwhichviewstheoccurrenceofrainfallatanetworkofdetectionstationsasanobservableresultofthecurrentstateofweatherinthatarea.HughesandGuttorp,[ 14 ],[ 13 ].viewchangesinweatherstatesasaresultofthepreviousweatherstateandcurrentatmosphericconditions[ 14 ].Thus,statetransitionscannotbemodeledasasimpleMarkovChainastheMarkovpropertyisnotsatised.InsteadHughesandGuttorpsuggestanonhomogeneousMarkovChaintomodelthetransitionsbetweenweatherstates[ 14 ].ThusthemodelisreferredtoasanonhomogeneousHiddenMarkovModel.Anobservationconsistsofanndimensionalvector,Rtwhichcorrespondstoanetworkofsiteswhererainfallattimetatsiteiisindicatedbya1inpositionioftheobservationvectorandalackofrainfallattimetatsiteiisindicatedbya0inpositionioftheobservationvector[ 14 ].TheweatherstateattimetisnotatedSt[ 14 ].AtmosphericconditionsattimetarerepresentedbyXt[ 14 ].Twoassumptionsaremaderegardingthemodel:rstitisassumedthatobservationsaredependentonlyonthecurrentweatherstate,seconditisassumedthattheprobabilityofbeinginagivenweatherstateattimet+1isdependentonlyontheweatherstateattimetandatmosphericconditionsattimet+1[ 14 ].Theirmodelisthendenedbyttingtwoprobabilitydistributions[ 14 ].Therstdistributionwhichmustbedeterminedistheprobabilityofacertainrainfallobservationgivenacertainweatherstate[ 14 ].Theseconddistributionwhichmustbedeterminedistheprobabilityofacertainweatherstategiventhepreviousweatherstateandcurrentatmosphericconditions[ 14 ].Theauthorsprovidesuggestionsfordistributions, 49 PAGE 50 13 ]and[ 14 ]forfurtherinformationonthetechnicalaspectsofimplementingthismodel.ApplyingavariationofthemodelsuggestedbyHughesandGuttorp,Robertson,InesandHansenhaveshownthatanonhomgeneousHiddenMarkovModelcanbeusedtomodelrainfalloccurrencesatasetofsitesinaregionbasedonaveragerainfallstatisticsfortheregion[ 26 ].Inadditiontomodelingrainfalloccurrenceatanumberofsitesthroughoutaregionthemodelalsomodelstheamountofrainfallataparticularsite[ 26 ].Fortheirmodeltheauthorsassumethatrainfallonaparticulardayisindependentofallotherstatesandrainfallonanypreviousdays[ 26 ].Inadditiontheauthorsassumethatrainfallindierentlocationsisindependentgiventhecurrentweatherstate[ 26 ].Transitionbetweenstatesisafunctionoftheaverageobserveddailyrainfall[ 26 ].Followingconstructionofthemodelitwasappliedtocropsimulationanddecodingstatesequencescorrespondingtohistoricalrainfalldata[ 26 ]. 50 PAGE 51 8 ].Havingnodirectmethodfortestingthishypothesis,Herschelsuggestedanindirectmethod.Reasoningthatthegrowthofvegetationisaectedbyheatandlightfromthesun,Herschelarguedthatifitcouldbeshownthatadecreaseinthehealthofacropcorrespondedtotimeswhentherearenovisiblesunspots,itmightsuggestthatthesunisoperatingdierentlyduringthesetimeperiodsthanduringtimeperiodswhensunspotsarepresent[ 8 ].Usingthepriceofwheatasanindicatorofthehealthofthewheatcrop,Herschelcomparedtheaveragepriceofwheatduringtimeperiodswhichrecordsindicatethatnosunspotswerevisiblewiththeaveragepriceofwheatoverthetimeperiodofequallengthdirectlyfollowingthistimeperiodandin2casesprecedingit[ 8 ].Herschel'sobservationsaswellastheyearsofrelevantsolarminimums,thetimeperiodduringwhichthefewestsunspotsarevisiblearegiveninthetables51,52,53[ 23 ],[ 8 ].Ineachcasethepriceofwheatishigherduringtimeperiodswhensunspotswerenotpresentthanduringtimeperiodsofequallengthdirectlyfollowingthesetimeperiods.Additionallyin2casesthepriceofwheatislowerinthetimeperioddirectlyprecedingtheperiodofnosunspotactivitythanduringtheperiodofnosunspotactivity.Thissuggeststheremightbearelationshipbetweentheappearanceofsunspotsandthehealthofwheatcropssinceadecreaseinsupplywithoutacorrespondingdecreaseindemandshouldcauseanincreaseinprice. 51 PAGE 52 8 ].Inthe1970'swasdiscoveredthattheremaybearelationshipbetweenspecictimeperiodsinthesunspotcycleanddroughtperiodsincertainregions[ 3 ].Itwasalsofoundthatdroughtperiodstendtobeassociatedwithlowercornyields,afactwhichmayeasilycarryovertowheatyields[ 2 ].Ifitisknownthatdroughtperiodstendoccurduringspecictimeperiodsinthesolarcycleandthatduringthesedroughtperiodsyieldstendtobeloweritisquitepossiblethatsunspotscouldprovideanindicationofprocessestakingplaceinthesunwhichwillimpactvegetation.ConsideringtheevidencepresentedbyWilliamHerschelaswellasmoremoderninvestigationsitseemspossiblethattheremaybearelationshipbetweensunspotactivityandtheprice/healthofwheatcrops. 52 PAGE 53 18 ].Sunspotswhichwereobserveddidnotappeartofollowthecurrentapproximately11yearcycle[ 18 ].TheMaunderMinimumspansHerschel'sentireobservationsetandthereforeitispossiblethattheobservationsonwhichHerschelbasedhisclaimarevalid.Furtherinvestigationisneededbeforeanyconclusionscanbereached. 22 ].TheaverageannualwheatpriceandyieldfortheUnitedStateswereobtainedfromtheUnitedStatesDepartmentofAgricultureWheatDataYearbook[ 30 ].Initially,itwasnecessarytoidentifythetimeperiodswhichhadtheleastsolaractivityasindicatedbysunspotactivity.Itwasfoundthattheaverageannualsunspotnumbertendedtobethelowestintheyearinwhichasolarminimumoccurred,andtheyearsprecedingandfollowingthatyear.So,the3yearperiodsurroundingtheyearinwhichasolarminimumoccurswillbeusedastheperiodoflowsunspotactivity.Havingfoundtheperiodwhich,onaverage,hastheleastsunspotactivity,theaveragepriceofwheatduringtheperiodoflowsunspotactivitywillbecomparedtotheaveragepriceofwheatinthe3yearsprecedingand3yearsfollowingthistimeperiod.Similarly, 53 PAGE 54 7 ].Asummaryoftheresultsobtained,usingpricesfrom1922to2000,isgivenintable54.Fornotationalconveniencethesamplemeanforthe3yearperiodcontainingasolarminimumwillbecalled1=$5.22withcorrespondingsamplevarianceS1=$5.51.Thesamplemeanforthe3yearperiodbeforetheperiodcontainingasolarminimumwillbecalled2=$5.81withcorrespondingsamplevarianceS2=$5.51.Thesamplemeanforthe3yearperiodaftertheperiodcontainingasolarminimumwillbecalled3=$5.43withcorrespondingsamplevarianceS3=$6.67.Fromtheseresultsitisseenthat2>1and3>1.Thusitseemsthattheaveragepriceofwheatislowerduringtheperiodoflowsunspotactivitythanduringtimeperiodsdirectlybeforeandafterthistimeperiod.Itremainshowevertobedeterminedifthedierencesinthesamplearesignicantenoughtoinferthatthesamplesrepresentdierentpopulations.Independenceofthesamplescannotbeassumedasthepricesinconsecutive3yearperiodshavesomedegreeofdependence.Becauseofthisdependencethedierencesbetweenpairsofobservationswillbeexaminedtodeterminewhetherthepricesaresignicantlydierent.Lackingconclusiveevidencethatthedierencesinthepairsof 54 PAGE 55 27 ].Thenullhypothesisforthetestisthatthesamplescomefromthesamepopulationwhichimpliesthattheprobabilityofobservingapositivedierenceshouldbe.5[ 27 ].Thealternativehypothesisisthatthesamplesdonotcomefromthesamepopulationwhich,inthiscase,impliestheprobabilityofobservingapositivedierenceisgreaterthan.5[ 27 ].Todeterminewhetherthetheteststatisticliesintherejectionregion,i.e.theprobabilityofobservingagivennumberofpositivedierences,callitx,inasampleofnsigneddierencesissignicantlysmallsoastoimplythenullhypothesisislikelyfalse,onecomputestheprobabilityofobservingatleastxpositivedierencesinasampleofnsigneddierences,assumingtheprobabilityofpositivedierenceis.5,usingthebinomialdistribution[ 27 ].Iftheprobabilityobservingatleastxpositivedierencesinasampleofnsigneddierencesissignicantlysmall,lessthan.05,thenullhypothesiscanberejectedinfavorofthealternativehypothesis[ 27 ].WhentheBinomialSignTestisappliedtothedierencesinthepriceofwheatinthe3yearperiodprecedingtheperiodoflowsunspotactivityandtheperiodoflowsunspotactivitythenullhypothesisisthatthesamplesrepresentdierentpopulations,i.e.theaveragepriceintheperiodoflowsunspotactivityisnotsignicantlydierentthantheaveragepriceintheprecedingtimeperiod.Thealternativehypothesisisthatthesamplesrepresentdierentpopulations,i.e.theaveragepriceinthepreceding3yeartimeperiodishigherthanthepriceinthe3yeartimeperiodaroundaminimum.ApplyingtheBinomialSignTestitwasfoundthattherewere5positivedierencesoutof8signeddierences.Theprobabilityofobserving5,6,7,or8positivedierencesis.3632andthusthenull 55 PAGE 56 56 PAGE 57 57 PAGE 58 58 PAGE 59 YearsofrelevantsolarminimaandaveragewheatpricesreportedbyWilliamHerscheloverperiodsofnoreportedsunspotactivity. Year(s)ofsolarminimaYear(s)ofnoobservedsunspotsAverageprice 1655,166616501670$2.10s.51 29 1d.167916771684$2.7.7d.168916861688$1.15s.2 3d.169816951670$3.3s.31 5d.171217101713$2.17s.4d. W.Herschel,\ObservationsTendingtoInvestigatetheNatureoftheSun,inOrdertoFindtheCausesorSymptomsofitsVariableofLightandHeat;WithRemarksontheUseThatMayPossiblybeDrawnfromSolarObservations,"PhilosophicalTransactionsoftheRoyalSocietyofLondon,vol.91,pp.265318,1801.NationalAtmosphericandOceanographicAdministration/NationalGeophysicalDataCenter,\MinimaandMaximaofSunspotNumberCycles,"NGDC.NOAA.gov,2008,7Jan.2008,ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SUNSPOT NUMBERS/maxmin.new. Table52. YearsofrelevantsolarminimaandaveragewheatpricesreportedbyWilliamHerscheloverperiodsdirectlybeforeperiodsofnoreportedsunspotactivity. Year(s)ofsolarminimaTimepriortoyear(s)ofnoobservedsunspotsAverageprice 168916901694$2.9s.44 5d.169817061709$2.3s.71 2d. W.Herschel,\ObservationsTendingtoInvestigatetheNatureoftheSun,inOrdertoFindtheCausesorSymptomsofitsVariableofLightandHeat;WithRemarksontheUseThatMayPossiblybeDrawnfromSolarObservations,"PhilosophicalTransactionsoftheRoyalSocietyofLondon,vol.91,pp.265318,1801.NationalAtmosphericandOceanographicAdministration/NationalGeophysicalDataCenter,\MinimaandMaximaofSunspotNumberCycles,"NGDC.NOAA.gov,2008,7Jan.2008,ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SUNSPOT NUMBERS/maxmin.new. 59 PAGE 60 YearsofrelaventsolarminimaandaveragewheatpricesreportedbyWilliamHerscheloverperiodsdirectlyafterperiodsofnoreportedsunspotactivity. Year(s)ofsolarminimaTimeafteryear(s)ofNoobservedsunspotsAverageprice 1679,168916711691$2.4s.42 3d.168916851691$1.17s.13 4d.168916891692$1.12s.102 3d.169817001704$1.17s.111 5d.171217141717$2.6s.9d. W.Herschel,\ObservationsTendingtoInvestigatetheNatureoftheSun,inOrdertoFindtheCausesorSymptomsofitsVariableofLightandHeat;WithRemarksontheUseThatMayPossiblybeDrawnfromSolarObservations,"PhilosophicalTransactionsoftheRoyalSocietyofLondon,vol.91,pp.265318,1801.NationalAtmosphericandOceanographicAdministration/NationalGeophysicalDataCenter,\MinimaandMaximaofSunspotNumberCycles,"NGDC.NOAA.gov,2008,7Jan.2008,ftp://ftp.ngdc.noaa.gov/STP/SOLAR DATA/SUNSPOT NUMBERS/maxmin.new. Table54. MeanandvarianceofthepriceofwheatintheUnitedStatesduringperiodsoflowsunspotactivityandtimeperiodsdirectlyprecedingandfollowingtimeperiodsoflowsunspotactivity. Preceding3yearperiod3YearperiodsurroundingaminimumFollowing3yearperiod. Mean$5.81$5.22$5.43 Variance$5.51$4.18$6.67 Table55. Meanandvarianceofwheatyieldsduringperiodsoflowsunspotactivityandperiodsdirectlyprecedinganddirectlyfollowingtimeperiodsoflowsunspotactivity. Preceding3yearperiod3YearperiodsurroundingaminimumFollowing3yearperiod. Mean20.5220.6121.87 Variance97.6888.32109.73 60 PAGE 61 22 ].ThesunspotnumberisgivenbyR=k(10*g+s)wheregisthenumberofsunspotgroups,sisthenumberofindividualsunspotsandkistheobservationconstantwhichtakesintoaccounttheconditionsunderwhichtheobservationsweremade[ 22 ].AcombinationofthesunspotnumberandthepositioninthesolarcyclewillbeusedinconstructingtheMarkovchainrepresentingthesystem.AfewassumptionsweremadeinconstructingtheMarkovchain.Therstassumptionisauniform,discretetimestepof1day.Thesecondassumptionthatwasmadeisonestepdependence.Namelyitisassumedthattheprobabilityoftransitioningtoanewstateonthenextdayisdependentonlyonthecurrentstate.Sincesunspotgroupstendto 61 PAGE 62 22 ].ItshouldbenotedthatthedataisreportedbytheNOAAwebsitebutwascompiledbytheSolarInuencesDataCenter,WorldDataCenterforTheSunspotIndex,attheRoyalObservatoryofBelgium[ 22 ].Eachsunspotcyclewasdividedinto100approximatelyequalpartitions.Thecodeusedtopartitionthecyclescanbefoundintheappendix.Thepositioninthesunspotcycleisthendescribedashundredthsofthewaythroughasunspotcycle.ThesecondcomponentofthestatesoftheMarkovChainisthesunspotnumber.DuetolimiteddataitisnecessarytogroupsunspotnumberstodecreasethenumberofstatesintheMarkovChain.Forthisreasonthesunspotnumbersaregroupedinsetsof20.The 62 PAGE 63 63 PAGE 64 NumberofSunspotCycles(6{1)Itshouldbenotedthattheinthetrainingdataonlyonesunspotcycledidnotstartinstate1;itstartedinstate2.Thusthereareonlytwopossiblestartingstates. 64 PAGE 65 65 PAGE 66 66 PAGE 67 67 PAGE 68 Averagesunspotnumberoverthecourseofthesunspotcyclefromsimulatedsunspotcyclesandtheaveragesunspotnumberoverthecourseofthesunspotcyclefromactualdata(18561996). 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PAGE 84 ChristopherC.KrutwasborninBeaver,Pennsylvaniain1983.RaisedintheMonongahelaValley,WhiteOakPennsylvaniatobeexact,hegraduatedfromMcKeesportAreaHighSchoolin2002.AfterreceivinghisB.S.inappliedmathematicsfromtheUniversityofPittsburgh,Greensburg,in2006,heacceptedateachingassistantshipintheMathematicsDepartmentattheUniversityofFloridawhichallowedhimtopursueanM.S.inappliedmathematics.AftercompletinghisM.S.inAppliedMathematicsChristopherwillbeginworkingonanM.S.andPhD.instatisticsatNorthCarolinaStateUniversity. 84 