multivariate poisson hidden markov models for analysis of spatial ...

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CHAPTER 9 DISCUSSION AND CONCLUSION 9 .1 General summary Multivariate count data occur in different areas of science. Examples of count data can be found in agriculture (weed species counts in a field), in epidemiology (death count from a disease), in marketing (purchases of different products), in production (different types of faults in a production system), in criminology (different type of crimes in different areas), in accident analysis (different types or different time periods of accidents), and many others. There are a variety of methods available to model the multivariate normal data and the multivariate categorical data. Multivariate count data has small counts with many zeros. Therefore, a normal approximation may not be adequate. The different approaches can be used to handle the multivariate count data. In this study, several more attractive types of models, multivariate Poisson models, were used to overcome the above mentioned problem. 168
In this thesis, three species counts from an agriculture field were selected for analysis. The main objective was to determine the model for the distribution of these multivariate counts. The estimation involves finding out the mean and covariance structures of the distribution. The data are recorded in a grid, and this data can be considered as a twodimensional Markov random field. At the same time, an agricultural field has a large neighbourhood system compared to an image. That is, the distance between the neighbouring points or coordinates in an agricultural field is large compared to the distance between the neighbouring points or coordinates in an image. A drawback of the models based on a Markov random field is that they can only be used for small neighbourhoods in an image, due to the computational complexity and the modeling problems posed by large neighbourhoods (Aas et al., 1999). These data can be transformed into a one-dimensional chain. Therefore, as a first step, the grid data were converted into a sequence or a one-dimensional chain using line scan (Chapter 4). The analysis of these data involves two methods, (a) the multivariate Poisson finite mixture model and (b) the multivariate Poisson hidden Markov model. The multivariate Poisson finite mixture model has been used in many other applications (e.g. marketing Brijs et al., 2004). However, the multivariate Poisson hidden Markov model is a new application to this kind of data (agricultural field data) with Poisson counts. For both models, the computation of the multivariate Poisson probabilities was studied according to Mahamunulu’s recurrence relations (see section 5.2.2). The preliminary loglinear analysis suggests that there were no significant two-way interactions. It can be 169
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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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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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Page 135 and 136: Figure 6.7 illustrates the evolutio
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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 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
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Page 185 and 186: In the applications of the HMMs, li
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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]
Page 217 and 218: theta131
Page 219 and 220: threep22[i]
Page 221 and 222: 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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