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Results suggest that the multivariate Poisson hidden Markov model, with five states and an independent covariance structure, gives a reasonable fit to this dataset. Since this model deals with overdispersion and spatial information, it will help to get an insight about weed distribution for herbicide applications. This model may lead researchers to find other factors such as soil moisture, fertilizer level, etc., to determine the states, which govern the distribution of the weed counts. Keywords: Multivariate Poisson distribution, multivariate Poisson hidden Markov model, Weed species counts, EM algorithm. iii
ACKNOWLEDGEMENT First I would like to acknowledge and express my sincere thanks and gratitude to my supervisor Dr. William H. Laverty for his availability, continual guidance, valuable suggestions and encouragement throughout the course of study. Next, I would like to thank the members of my advisory committee, Prof. R. Srinivasan, Prof. C. E. Soteros, Prof. M.J. Miket and Prof. I.W. Kelly for their valuable suggestions and advice in many aspects of my thesis completion. I am also grateful for comments and suggestions from my external examiner, Prof. Peter MacDonald. My special thanks to Dr. Dimitris Karlis, Athens University of Economics, Athens, Greece for his valuable advice and help me solve the problems I had with multivariate Poisson distributions. I am very grateful for the funding provided by College of Graduate Studies and Research and Depaerment of Mathematics and Statistics. Without their support and resources, it is impossible to complete this thesis. I would also like to thank Ms. Jessica Antonio of the Department of English, University of Saskatchewan for proof reading this thesis. Finally, my heartfelt thanks go to my dear parents, and especially my husband, Sumith Priyashantha, who always wished and encouraged me to successfully complete my study program in Canada. iv
Page 1 and 2: MULTIVARIATE POISSON HIDDEN MARKOV
Page 3: ABSTRACT Multivariate count data ar
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 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 (
Page 65 and 66:
In the literature review (section 1
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estimated through the observations
Page 69 and 70:
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
Page 87 and 88:
5.2.2 The multivariate Poisson dist
Page 89 and 90:
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
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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
Page 145 and 146:
6.5 Comparison of the different mod
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loglikelihood providing at least a
Page 149 and 150:
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
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
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The estimated covariance matrix and
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The estimated covariance matrix (AI
Page 167 and 168:
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
Page 175 and 176:
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
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
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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
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
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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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