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

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hidden because we do not know exactly which coin (state) led to the head or tail of each observation. P 11 P 22 1-P 11 1 1-P 22 2 P 11 - The probability of staying in state 1 P 22 - The probability of staying in state 2 1-P 11 - The probability of leaving state 1 1-P 22 - The probability of leaving state 2 Y = H H T T H T H H T T H…… S = 2 1 1 2 2 2 1 2 2 1 2….. (1) P[H]=P 1 (2) P(H)=P 2 P[T]=1-P 1 P[T]=1-P 2 , where (1) is the probability distribution of heads and tails in state 1 and (2) is the probability distribution of heads and tails in state 2. Figure 2.2: 2- coins model 3-coins model: The third form of the HMM for explaining the observed sequence of coin tossing outcomes is given in Figure 2.3. This model corresponds to the example using three biased coins, and choosing from among the three based on some probabilistic event. 12
P 11 P 22 P 12 1 2 P 21 P 32 P 13 P 31 P 23 3 P 33 P 11 - The probability of staying in state 1 P 22 - The probability of staying in state 2 P 33 - The probability of staying in state 3 P 12 - The probability of leaving state 1 and reaching state 2 P 21 - The probability of leaving state 2 and reaching state 1 P 13 - The probability of leaving state1 and reaching state 3 P 31 - The probability of leaving state 3 and reaching state 1 P 32 - The probability of leaving state 3 and reaching state 2 P 23 - The probability of leaving state 2 and reaching state 3 Y = H H T T H T H H T T H…… S = 3 1 2 3 3 1 1 2 3 1 3….. (1) P[H]=P 1 (2) P[H]=P 2 (3) P[H]=P 3 P[T]=1-P 1 P[T]=1-P 2 P[T]=1-P 3 , where (1) is the probability distribution of heads and tails in state 1, (2) is the probability distribution of heads and tails in state 2 and (3) is the probability distribution of heads and tails in state 3. Figure 2.3: 3- coins model 13
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: Given the coin tossing experiment,
Page 29 and 30: 0.8 0.6 0.2 1 0.4 2 (1) P [H]=2/3 (
Page 31 and 32: … Urn 1 Urn 2 Urn N P[Red]= b 1 (
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
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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
Page 87 and 88:
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 ⎤
Page 161 and 162:
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
Page 167 and 168:
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
Page 183 and 184:
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
Page 197 and 198:
33. Descombes, X., Morris, R.D., Ze
Page 199 and 200:
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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