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conditional on rates, which are generated from an independent mixing distribution for finite mixture models. The counts for multivariate Poisson hidden Markov models are assumed to follow independent Poisson distributions, conditional on rates with Markov dependence. Finite mixture models can be particularly attractive because they provide plausible explanations for variation in the data (Leroux and Puterman, 1992). 1.2 Literature review 1.2.1 Introduction to finite mixture models The main question here is determining the structure of clustered data when no information other than the observed values is available. Finite mixture models have been proposed for quite sometime as a basis for studying the clustered data (Symons, 1981; McLachan, 1982; McLachlan et al., 1988). In this approach, the data are viewed as coming from a mixture of probability distributions, each representing a different cluster. Recently, finite mixture model analysis have been used in several practical applications: character recognition (Murtagh and Raftery, 1984); tissue segmentation (Banfield and Raftery, 1993); minefield and seismic fault detection (Dasgupta and Raftery, 1998); identification of textile flaws from images (Campbell et al., 1997); and classification of astronomical data (Celeux et al., 1995). Most of these examples are based on Gaussian finite mixture models. There are some examples of Poisson finite mixtures. Leroux and Puterman (1992) describe a univariate Poisson finite mixture model for fetal movement data. The clustering of cases of a rare disease (sudden infant death syndrome), on the basis of the number of cases observed for various counties in 2
North Carolina, is modelled by Symons et al. (1983) using a mixture of two Poisson distributions, which describe the two groups of high and low risk counties. Very recently, a multivariate Poisson finite mixture model was used for a marketing application (Brijs et al., 2004). Brijs describes a multivariate Poisson finite mixture model for clustering supermarket shoppers based on their purchase frequency in a set of product categories. In this thesis, the multivariate Poisson finite mixture model is applied for the first time to weed species counts in an agricultural field. Also, we developed a multivariate Poisson hidden Markov model and applied it to analyze the weed species data. The goodness of fit measure of the model is also evaluated (Chapter 7). Details about multivariate Poisson finite mixture models and multivariate Poisson hidden Markov models are given in Chapter 5. The history of hidden Markov models is presented in the next section. 1.2.2 History of hidden Markov models Hidden Markov Models (HMMs) are statistical models that are widely used in many areas of probabilistic modeling. These models have received increasing attention (Rabiner and Juang, 1986, 1991 and Rabiner, 1989), partially because of their mathematical properties (they are rich in mathematical structure), but mostly because of their applications to many important areas in scientific research. 3
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: CHAPTER 1 GENERAL INTRODUCTION 1.1
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
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
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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
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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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