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4.3 Problem of interest and proposed solution ..................................48 4.4 Goals of the thesis .......................................................................53 5 MULTIVARIATE POISSON DISTRIBUTION, MULTIVARIATE POISSON FINITE MIXTURE MODEL AND MULTIVARIATE POISSON HIDDEN MARKOV MODEL 5.1 The multivariate Poisson distribution: general description .........55 5.1.1 The fully- structured multivariate Poisson model ............................................................................59 5.1.2 The multivariate Poisson model with common covariance structure.......................................63 5.1.3 The multivariate Poisson model with local independence .......................................................65 5.1.4 The multivariate Poisson model with restricted covariance.....................................................................66 5.2 Computation of multivariate Poisson probabilities .....................68 5.2.1 The multivariate Poisson distribution with common covariance......................................................70 5.2.2 The multivariate Poisson distribution with restricted covariance ....................................................73 5.2.3 The Flat algorithm ........................................................75 5.3 Multivariate Poisson Finite mixture models................................78 5.3.1 Description of model-based clustering.........................79 5.3.2 Model-based cluster estimation....................................82 5.3.3 ML estimation with the EM algorithm.........................82 5.3.3.1 Properties of the EM algorithm .....................84 5.3.4 Determining the number of components or states........85 5.3.5 Estimation for the multivariate Poisson finite mixture models ..........................................................................87 5.3.5.1 The EM algorithm .........................................87 5.4 Multivariate Poisson hidden Markov models..............................91 5.4.1 Notations and description of multivariate setting.........91 5.4.2 Estimation for the multivatiate Poisson hidden Markov models.............................................................92 5.4.2.1 The EM algorithm .........................................93 5.4.2.2 The forward-backward algorithm..................95 5.5 Bootstrap approach to standard error approximation .................98 5.6 Splus/R code for multivariate Poisson hidden Markov model ...........................................................................................102 5.7 Loglinear analysis........................................................................102 6 RESULTS OF MULTIVARIATE POISSON FINITE MIXTURE MODELS AND MULTIVARIATE POISSON HIDDEN MARKOV MODELS 6.1 Introduction .................................................................................108 6.2 Exploratory data analysis ............................................................108 6.3 Loglinear analysis........................................................................112 vii
6.4 Data analysis................................................................................114 6.4.1 Results for the different multivariate Poisson finite mixture models....................................................115 6.4.2 Results for the different multivariate Poisson hidden Markov models .................................................123 6.5 Comparison of different models..................................................131 7 PROPERTIES OF THE MULTIVARIATE POISSON FINITE MIXTURE MODELS 7.1 Introduction .................................................................................138 7.2 The multivariate Poisson distribution..........................................139 7.3 The properties of multivariate Poisson finite mixture models ....141 7.4 Multivariate Poisson-log Normal distribution.............................145 7.4.1 Definition and the properties ........................................145 7.5 Applications.................................................................................147 7.5.1 The lens faults data.......................................................147 7.5.2 The bacterial count data................................................152 7.5.3 Weed species data.........................................................156 8 COMPUTATIONAL EFFICIENCY OF THE MULTIVARIATE POISSON FINITE MIXTURE MODELS AND MULTIVARIATE POISSON HIDDEN MARKOV MODELS 8.1 Introduction .................................................................................161 8.2 Calculation of computer time ......................................................161 8.3 Results of computational efficiency ............................................162 9 DISCUSSION AND CONCLUSION 9.1 General summary.........................................................................168 9.2 Parameter estimation ...................................................................170 9.3 Comparison of different models..................................................171 9.4 Model application to the different data sets.................................174 9.5 Real world applications ...............................................................174 9.6 Further research ...........................................................................177 REFERENCES ..............................................................................................179 APPENDIX ...................................................................................................192 A. Splus/R code for Multivaraite Poisson Hidden Markov Model- Common Covariance Structure ........................................................192 B. Splus/R code for Multivaraite Poisson Hidden Markov Model- Restricted and Independent Covariance Structure................200 viii
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: TABLE OF CONTENTS PERMISSION TO USE
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 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
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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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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 ⎤
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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
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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
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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
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17. Bicego, M., Murino, V. & Figuei
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33. Descombes, X., Morris, R.D., Ze
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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]
Page 213 and 214:
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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