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Table 7.4: Loglikelihood, AIC and BIC together with the number of components for the common covariance multivariate Poisson hidden Markov Model Number of Number of free Loglikelihood AIC BIC components ( k ) parameters 1 3 -450.6038 -453.6038 -457.5115 2 8 -398.6838 -406.6838 -417.1045 3 15 -377.4649 -392.4649 -412.0036 4 24 -361.7565 -385.7565 -417.0185 5 35 -348.8014 -383.8014 -429.3918 6 48 -340.3936 -388.3936 -450.9177 Similarly, the loglikelihood, the AIC and the BIC values for the common covariance and the independent model for the Markov-dependent bivariate Poisson finite mixture models are given in Table 7.4 and Table 7.5, respectively. The corresponding estimated covariance matrices and the correlation coefficients between x 1 and x 2 are also presented. The estimated covariance matrix (AIC selection) and the correlation coefficient are ⎡ 6.7930 ⎢ ⎣− 0.3909 − 0.3909⎤ 6.4713 ⎥ ⎦ and r = -0.0590 respectively. The estimated covariance matrix (BIC selection) and the correlation coefficient are ⎡ 6.6645 ⎢ ⎣− 0.6649 − 0.6649⎤ 5.0389 ⎥ ⎦ and r = -0.1147 respectively. Table 7.5: Loglikelihood, AIC and BIC together with the number of components for the local independence multivariate Poisson hidden Markov Model Number of Number of free Loglikelihood AIC BIC components ( k ) parameters 1 2 -450.6038 -452.6038 -455.2089 2 6 -398.7847 -404.7847 -412.6002 3 12 -377.4630 -389.4630 -405.0940 4 20 -368.0097 -388.0097 -414.0614 5 30 -359.2484 -389.2484 -428.3259 6 42 -350.8904 -392.8904 -447.5989 150
The estimated covariance matrix (AIC selection) and the correlation coefficient are ⎡ 7.3839 ⎢ ⎣−1.1433 −1.1433⎤ 5.6561 ⎥ ⎦ and r = -0.1769 respectively. The estimated covariance matrix (BIC selection) and the correlation coefficient are ⎡ 6.6645 ⎢ ⎣− 0.6649 − 0.6649⎤ 5.0389 ⎥ ⎦ and r = -0.1147 respectively. Comparing all these models for the bivariate Poisson finite mixture and the hidden Markov models, this goodness of fit criterion suggest that the local independence bivariate Poisson finite mixture model is the best fitted model with respect to the estimated covariance and the correlation parameters. The observed correlation for lenses count data given in Table 7.1 was -0.1796. Using the multivariate Poisson-log normal distribution Aitchison and Ho (1989) found that the best model with loglikelihood –426.40 and the estimated count correlation between x 1 and x 2 was –0.21. Multivariate Poisson finite mixtures provide a better fit compared to Aitchison and Ho models, having larger loglikelihoods (-420.6121, -420.2611, and - 423.6535). It also demonstrates that estimated correlation coefficient is very much close to the observed correlations (AIC selections) except the hidden Markov common covariance model, compared to Aitchison and Ho models. Note that the maximum likelihood estimation of the parameters of the multivariate Poisson-log normal model was obtained by the combination of the Newton-Raphson and the steepest ascent method (Aitchison and Ho, 1989). 151
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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
Page 113 and 114: The bootstrap method is a powerful
Page 115 and 116: eplications are generally sufficien
Page 117 and 118: Y and assumes a probability distrib
Page 119 and 120: For the case of two categorical var
Page 121 and 122: ejected. In this situation, the (sm
Page 123 and 124: the Poisson distribution is well su
Page 125 and 126: 0 1 2 3 4 Wild Buckwheat species109
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
Page 133 and 134: -600 -650 -700 Loglikelihood -750 -
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: The estimated covariance matrix and
Page 167 and 168: Table 7.6 Bacterial counts by 3 sam
Page 169 and 170: Table 7.8: Loglikelihood and AIC to
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]
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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=
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# vˆ jk ( i) = P jk f ( y ; λ i
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