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The Ising model Assuming χ = { −1, + 1} and the neighbourhood structure to the left in Figure 2.6, an often used local specification is exp( β x( s) x( t)) t∈Ν s PX [ ( s) = x( s)| X( Νs) = x( Νs)] = exp( − β x( t)) + exp( β x( t)) for some real β ; note that the denominator does not depend on x() s and is only a normalizing factor to make the right hand side a proper distribution, summing to unity. This model is called the Ising model (McCoy et al., 1973; Binder, 1979; Binder et al., 1992), after German physicist Ising who invented it with the original purpose of using it as an idealized model of a ferromagnetic material. The sum in the exponent is positive if x() s has the same sign as the most of its neighbours. Hence, if β >0 the sites interact such that configurations x with many neighbours of the same sign will have large probabilities. On the contrary, if β
for some real β . Again, the denominator does not depend on x( s ) and is only a normalizing factor. This model is called the Potts model (Binder, 1979; Wu, 1982; Binder et al., 1992). Note that, #{ t∈ N : x( t) ≠ x( s )} is the number of neighbours of s s that have values different from x( s ) . Hence, if β >0 this model gives large probabilities to configurations x in which there are many neighbours with different values. If β
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 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: Now consider a random field { X ( s
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
Page 81 and 82: ecause the former captures more of
Page 83 and 84: (Tsiamyrtzis and Karlis, 2004). The
Page 85 and 86: notation, the following recursive s
Page 87 and 88: 5.2.2 The multivariate Poisson dist
Page 89 and 90:
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
Page 129 and 130:
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
Page 141 and 142:
-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
Page 149 and 150:
illustrates the contour plot of the
Page 151 and 152:
(a) Independent Contour 1 Contour 2
Page 153 and 154:
Karlis and Meligkotsidou (2006) dis
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The mean vector and the covariance
Page 157 and 158:
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
Page 165 and 166:
The estimated covariance matrix (AI
Page 167 and 168:
Table 7.6 Bacterial counts by 3 sam
Page 169 and 170:
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
Page 181 and 182:
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
Page 193 and 194:
REFERENCES 1. Aas, K., Eikvil, L. a
Page 195 and 196:
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
Page 201 and 202:
66. Li C.S., Lu J.C., Park J., Kim
Page 203 and 204:
85. Petrie T. (1969). Probabilistic
Page 205 and 206:
104. University of Manitoba, Depart
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z0
Page 209 and 210:
threep2[i]
Page 211 and 212:
loglike[nit]
Page 213 and 214:
theta33
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} prob[g1+1, g2 + 1]
Page 217 and 218:
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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