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CHAPTER 7 PROPERTIES OF MULTIVARIATE POISSON FINITE MIXTURE MODELS AND APPLICATIONS 7.1 Introduction In this chapter, the properties of the multivariate Poisson finite mixture models are discussed. The importance of exploring the properties of the finite mixtures, the extension of these properties to the hidden Markov model and the application to other data sets are presented in the next sections. Even though there was more literature available on the analysis of count data, still only small portions of it deal with correlated counts. Holgate (1964) discussed the estimation problems of the bivariate Poisson distribution which does not support negative correlation between the two count variables. With the availability of powerful computing facilities Aitchison and Ho (1989) described how the multivariate lognormal mixture of the independent Poisson distributions could take into account the positive and negative correlation between the variables. A class of models proposed by Chib and Winkelmann (2001) can take into account the correlation among the counts. They developed an efficient Markov Chain Monte Carlo algorithm to estimate the model parameters. However, for these models, the computational burden was quite large. 138
Karlis and Meligkotsidou (2006) discussed the correlation structure of the multivariate Poisson mixture models. These mixture models allow for both negative correlations and overdispersion in addition to being computationally feasible. The multivariate Poisson distribution is discussed again in section 7.2, followed by the properties of the finite mixture models. In section 7.5, these properties were applied to both multivariate Poisson finite mixture models and multivariate Poisson hidden Markov models for several applications. 7.2 The multivariate Poisson distribution Consider a vector X = ( X1, X2,..., X k ) where X i ’s are independent and each follows a Poisson distribution with parameter λ j , j = 1,..., k . Suppose that matrix A has dimensions n × k with zeros and ones. Then the vector Y = Y , Y ,..., Y ) defined as the ( 1 2 n Y = AX follows a n-variate Poisson distribution. The most general form of a n-variate n Poisson distribution assumes that A is a matrix of size n× (2 − 1) of the form A=[A 1 , A 2 , A 3 ,…,A n ] where A , i ⎛n⎞ i = 1,..., n are matrices with n rows and ⎜ ⎟ columns. The matrix ⎝i ⎠ A i contains columns with exactly i ones and ( n− i) zeros, with no duplicate columns, for i = 1,...,n . Thus A n is the column vector of 1’s, while A 1 becomes the identity matrix of size n × n . For example, the fully structured multivariate Poisson model for three variables can be represented as follows: 139
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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
Page 101 and 102: 5.3.5 Estimation for the multivaria
Page 103 and 104: j j E[ X 12 i | Yi , Z ij = 1, Φ ]
Page 105 and 106: 5.4 Multivariate Poisson hidden Mar
Page 107 and 108: where P = Pr( S = k | S − 1 j), 1
Page 109 and 110: P jk n ∑ vˆ jk i= 2 = n m ∑∑
Page 111 and 112: 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: (a) Independent Contour 1 Contour 2
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 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
Page 181 and 182: 1400 1200 CPU time (1/100 second) 1
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
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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=
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# vˆ jk ( i) = P jk f ( y ; λ i
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