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Table 7.7: Loglikelihood and AIC together with the number of components for the local independence multivariate Poisson finite mixture Model Number of Number of free Loglikelihood AIC components ( k ) parameters 1 3 -472.8759 -475.8759 2 7 -422.3424 -429.3424 3 11 -409.9689 -420.9689 4 15 -402.9517 -417.9517 5 19 -395.5478 -414.5478 6 23 -390.3447 -413.3447 7 27 -382.2335 -409.2335 8 31 -381.3817 -412.3817 9 35 -381.3717 -416.3817 The estimated covariance matrix and the estimated correlation matrix are given below: ⎡15.9583 ⎢ − 0.2698 ⎢ ⎢⎣ − 3.3550 − 0.2698 13.3753 − 7.9948 − 3.3550⎤ − 7.9948 ⎥ ⎥ 33.4182 ⎥⎦ ⎡1 ⎢ ⎢ ⎢⎣ − 0.0185 1 − 0.1453⎤ − 0.3781 ⎥ . ⎥ 1 ⎥⎦ The covariance between X 1 and X 2 samplers does not seem to be in the right direction; however, other parameters are close to the observed covariance matrix. Similar analyses were carried out with our proposed model (local independence multivariate Poisson hidden Markov model) and the results are given in Table 7.8. According to the AIC criterion, the model with five components (loglikelihood – 382.9375) gives a better fit compared to other component models. 154
Table 7.8: Loglikelihood and AIC together with the number of components for the local independence multivariate Poisson hidden Markov Model Number of Number of free Loglikelihood AIC components ( k ) parameters 1 3 -472.8759 -475.8759 2 8 -421.5801 -429.5801 3 15 -406.6949 -421.6949 4 24 -395.4245 -419.4245 5 35 -382.9375 -417.9375 6 48 -375.2770 -423.2770 7 63 -372.5788 -435.5788 The estimated covariance matrix and the estimated correlation matrix are given below: ⎡14.0247 ⎢ 1.4131 ⎢ ⎢⎣ − 3.1730 1.4131 11.5828 − 8.2055 − 3.1730⎤ − 8.2055 ⎥ ⎥ 32.9665 ⎥⎦ ⎡1 ⎢ ⎢ ⎢⎣ 0.1109 1 − 0.1476⎤ − 0.4199 ⎥ . ⎥ 1 ⎥⎦ The coefficients of covariance matrix demonstrate that all the pairs of covariance are in the correct direction and some estimates seem to be underestimated and some are overestimated. Since the correlations between ( X 1 , X 2 ) and ( X 1 , X 3 ) were not significantly different from zero, these parameter estimates were reasonable for both models and give a better fit as compared to the Poisson-log normal model (loglikelihood –397.8; Aitchison and Ho, 1989). 155
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MULTIVARIATE POISSON HIDDEN MARKOV
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ABSTRACT Multivariate count data ar
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ACKNOWLEDGEMENT First I would like
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TABLE OF CONTENTS PERMISSION TO USE
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6.4 Data analysis..................
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Table 7.7: Loglikelihood and AIC to
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Figure 6.10: Loglikelihood, AIC and
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CHAPTER 1 GENERAL INTRODUCTION 1.1
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North Carolina, is modelled by Symo
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concept of Markov models to include
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egularity constraints on the underl
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CHAPTER 2 HIDDEN MARKOV MODELS ( HM
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Given the coin tossing experiment,
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P 11 P 22 P 12 1 2 P 21 P 32 P 13 P
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0.8 0.6 0.2 1 0.4 2 (1) P [H]=2/3 (
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… Urn 1 Urn 2 Urn N P[Red]= b 1 (
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5. The probability distribution of
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focus of this section. Random field
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Now consider a random field { X ( s
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for some real β . Again, the denom
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Since original HMMs were designed a
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CHAPTER 3 INFERENCE IN HIDDEN MARKO
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sequence. If we have several compet
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Using this equation we can calculat
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3.2.2 Problem 2 and its solution Gi
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Letting Ut( S1 = i1, S2 = i2,..., S
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ξ α () iPβ ( jb ) ( y ) t ij t+
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ˆ ( n) = b j Expected Number of ti
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CHAPTER 4 HIDDEN MARKOV MODEL AND T
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4.2.1 Wild Oats Figure 4.1: Wild Oa
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4.2.2.1 Effects on crop quality Wil
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at each of the 150 grid locations (
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In the literature review (section 1
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estimated through the observations
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CHAPTER 5 MULTIVARIATE POISSON DIST
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Y Y Y 1 2 3 = X = X 1 = X 2 3 + X +
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educe the computational burden; how
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and y 3 as illustrated above. Again
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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
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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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Page 119 and 120: For the case of two categorical var
Page 121 and 122: ejected. In this situation, the (sm
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Page 127 and 128: Table 6.4: The frequency of occurre
Page 129 and 130: 6.4.1 Results for the different mul
Page 131 and 132: Proportion 1.0 0.9 0.8 0.7 0.6 0.5
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Page 135 and 136: Figure 6.7 illustrates the evolutio
Page 137 and 138: Table 6.8: Parameter estimates (boo
Page 139 and 140: common covariance and the four stat
Page 141 and 142: -600 -700 -800 Loglikelihood -900 -
Page 143 and 144: Table 6.11: Parameter estimates (bo
Page 145 and 146: 6.5 Comparison of the different mod
Page 147 and 148: 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
Page 155 and 156: The mean vector and the covariance
Page 157 and 158: The simple moments of B are polynom
Page 159 and 160: and E ( Y ) = AM where ⎡λ1 ⎤
Page 161 and 162: 7.5 Applications In addition to wee
Page 163 and 164: The estimated covariance matrix and
Page 165 and 166: The estimated covariance matrix (AI
Page 167: Table 7.6 Bacterial counts by 3 sam
Page 171 and 172: (c) Finite mixture with the five co
Page 173 and 174: (i) Hidden Markov model with the fi
Page 175 and 176: CHAPTER 8 COMPUTATIONAL EFFICIENCY
Page 177 and 178: less time compared to the multivari
Page 179 and 180: 1400 1200 CPU time (1/100 second) 1
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Page 183 and 184: In this thesis, three species count
Page 185 and 186: In the applications of the HMMs, li
Page 187 and 188: indication of the relative goodness
Page 189 and 190: underlying data. The advantage of t
Page 191 and 192: 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
Page 207 and 208: z0
Page 209 and 210: threep2[i]
Page 211 and 212: loglike[nit]
Page 213 and 214: theta33
Page 215 and 216: } prob[g1+1, g2 + 1]
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threep22[i]
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dens=matrix(0,nrow=T,ncol=N) alpha=
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# vˆ jk ( i) = P jk f ( y ; λ i
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