multivariate poisson hidden markov models for analysis of spatial ...

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In this case, the results of the coin flips and which coins are being flipped are known. In general, which coins are being flipped is unknown. That is, the underlying model is known and the output sequence is observed, while the state sequence is “hidden.” In this case, the 2-biased-coins model is a hidden Markov model. In the above examples, the outcomes of the tossed coins are T or H and only two observations are possible. More general situation is explained below: considering a set of N urns and each urn consisting of several colored balls (M). The urn and ball model Consider the situation where there are N urns in a room, and within each urn there are M distinct colours of balls (Figure 2.5). The physical process for obtaining observations is as follows. A person is in a room, and using some random process, he (or she) chooses an initial urn. From this urn, a ball is chosen at random, and its color is recorded as the observation. The ball is then replaced in the urn from which it was selected. A new urn is then selected according to the random selection process associated with the current urn, and the ball selection process is repeated. This entire process generates a finite observation sequence of colours, which can be considered as the observational output of the HMM. Here, each state corresponds to a specific urn and the ball colour probability is defined for each state. The choice of urns is dictated by the state transition matrix of the HMM. 16
… Urn 1 Urn 2 Urn N P[Red]= b 1 (1) P[Red]= b (1) 2 P[Red]= b (1) P[Blue]= b 1 (2) P[Blue]= b (2) 2 P[Blue]= b (2) P[Green]= b (3) P[Green]= 1 b (3) 2 P[Green]= b (3) … P[Orange]= b( M ) P[Orange]= b ( M ) P[Orange]= ( ) 1 2 b M The observation sequence is Y= {Green, Green, Red, Yellow, Blue, …, Orange, Blue} N N N N Figure 2.5: The Urn and Ball Model 2.3 Definition of the hidden Markov model A hidden Markov Model is a doubly stochastic process, with an underlying stochastic process that is not observable (hidden), and can only be observed through another set of stochastic processes that produced the sequence of observations. Simply stated, a hidden Markov model is a finite set of states, each of them being associated with a probability distribution, and the transition between the states being covered by the transition probability. In particular, the observation can be generated according to the associated probability distribution so it is only the outcome that is 17
Page 1 and 2: MULTIVARIATE POISSON HIDDEN MARKOV
Page 3 and 4: ABSTRACT Multivariate count data ar
Page 5 and 6: ACKNOWLEDGEMENT First I would like
Page 7 and 8: TABLE OF CONTENTS PERMISSION TO USE
Page 9 and 10: 6.4 Data analysis..................
Page 11 and 12: Table 7.7: Loglikelihood and AIC to
Page 13 and 14: Figure 6.10: Loglikelihood, AIC and
Page 15 and 16: CHAPTER 1 GENERAL INTRODUCTION 1.1
Page 17 and 18: North Carolina, is modelled by Symo
Page 19 and 20: concept of Markov models to include
Page 21 and 22: egularity constraints on the underl
Page 23 and 24: CHAPTER 2 HIDDEN MARKOV MODELS ( HM
Page 25 and 26: Given the coin tossing experiment,
Page 27 and 28: P 11 P 22 P 12 1 2 P 21 P 32 P 13 P
Page 29: 0.8 0.6 0.2 1 0.4 2 (1) P [H]=2/3 (
Page 33 and 34: 5. The probability distribution of
Page 35 and 36: focus of this section. Random field
Page 37 and 38: Now consider a random field { X ( s
Page 39 and 40: for some real β . Again, the denom
Page 41 and 42: Since original HMMs were designed a
Page 43 and 44: CHAPTER 3 INFERENCE IN HIDDEN MARKO
Page 45 and 46: sequence. If we have several compet
Page 47 and 48: Using this equation we can calculat
Page 49 and 50: 3.2.2 Problem 2 and its solution Gi
Page 51 and 52: Letting Ut( S1 = i1, S2 = i2,..., S
Page 53 and 54: ξ α () iPβ ( jb ) ( y ) t ij t+
Page 55 and 56: ˆ ( n) = b j Expected Number of ti
Page 57 and 58: CHAPTER 4 HIDDEN MARKOV MODEL AND T
Page 59 and 60: 4.2.1 Wild Oats Figure 4.1: Wild Oa
Page 61 and 62: 4.2.2.1 Effects on crop quality Wil
Page 63 and 64: at each of the 150 grid locations (
Page 65 and 66: In the literature review (section 1
Page 67 and 68: estimated through the observations
Page 69 and 70: CHAPTER 5 MULTIVARIATE POISSON DIST
Page 71 and 72: Y Y Y 1 2 3 = X = X 1 = X 2 3 + X +
Page 73 and 74: educe the computational burden; how
Page 75 and 76: and y 3 as illustrated above. Again
Page 77 and 78: q In general, the number of paramet
Page 79 and 80: Similar to the fully structured mod
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ecause the former captures more of
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(Tsiamyrtzis and Karlis, 2004). The
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notation, the following recursive s
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5.2.2 The multivariate Poisson dist
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ypy ( ,0,0) = θ py ( − 1,0,0) y1
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ecurrence relationship ypy 1 ( 1 ,
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Raftery, 1998); identification of t
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maximization (EM) algorithm is appl
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the EM algorithms for use on very l
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Another alternative is to use the p
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5.3.5 Estimation for the multivaria
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j j E[ X 12 i | Yi , Z ij = 1, Φ ]
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5.4 Multivariate Poisson hidden Mar
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where P = Pr( S = k | S − 1 j), 1
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P jk n ∑ vˆ jk i= 2 = n m ∑∑
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E[ X | Y, u ( i) = 1, Φ ] = d = y
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The bootstrap method is a powerful
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eplications are generally sufficien
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Y and assumes a probability distrib
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For the case of two categorical var
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ejected. In this situation, the (sm
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the Poisson distribution is well su
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0 1 2 3 4 Wild Buckwheat species109
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Table 6.4: The frequency of occurre
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6.4.1 Results for the different mul
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Proportion 1.0 0.9 0.8 0.7 0.6 0.5
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-600 -650 -700 Loglikelihood -750 -
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Figure 6.7 illustrates the evolutio
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Table 6.8: Parameter estimates (boo
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common covariance and the four stat
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-600 -700 -800 Loglikelihood -900 -
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Table 6.11: Parameter estimates (bo
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6.5 Comparison of the different mod
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loglikelihood providing at least a
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illustrates the contour plot of the
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(a) Independent Contour 1 Contour 2
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Karlis and Meligkotsidou (2006) dis
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The mean vector and the covariance
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The simple moments of B are polynom
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and E ( Y ) = AM where ⎡λ1 ⎤
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7.5 Applications In addition to wee
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The estimated covariance matrix and
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The estimated covariance matrix (AI
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Table 7.6 Bacterial counts by 3 sam
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Table 7.8: Loglikelihood and AIC to
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(c) Finite mixture with the five co
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(i) Hidden Markov model with the fi
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CHAPTER 8 COMPUTATIONAL EFFICIENCY
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less time compared to the multivari
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1400 1200 CPU time (1/100 second) 1
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1400 1200 CPU time (1/100 second) 1
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In this thesis, three species count
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In the applications of the HMMs, li
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indication of the relative goodness
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underlying data. The advantage of t
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9.6 Further research We can present
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REFERENCES 1. Aas, K., Eikvil, L. a
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17. Bicego, M., Murino, V. & Figuei
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33. Descombes, X., Morris, R.D., Ze
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Department of Statistics, Universit
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66. Li C.S., Lu J.C., Park J., Kim
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85. Petrie T. (1969). Probabilistic
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104. University of Manitoba, Depart
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z0
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threep2[i]
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loglike[nit]
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theta33
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} prob[g1+1, g2 + 1]
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theta131
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threep22[i]
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dens=matrix(0,nrow=T,ncol=N) alpha=
Page 223 and 224:
# vˆ jk ( i) = P jk f ( y ; λ i
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