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LIST OF FIGURES Figure 2.1: 1- coin model ..............................................................................11 Figure 2.2: 2- coins model.............................................................................12 Figure 2.3: 3- coins model.............................................................................13 Figure 2.4: 2-biased coins model...................................................................15 Figure 2.5: The urn and ball model ...............................................................17 Figure 2.6: Two different neighbourhood structures and their corresponding cliques.................................................................22 Figure4.1: Wild Oats .....................................................................................45 Figure 4.2: Wild Buckwheat..........................................................................46 Figure 4.3: Dandelion....................................................................................48 Figure 4.4: Distribution of weed counts in field #1.......................................49 Figure 4.5: Data collection locations from field #1.......................................50 Figure 4.6: Distribution of Weed Counts and Different States (clusters) in Field #1...................................................................................50 Figure 4.7: Scanning method: Line Scan ......................................................51 Figure 5.1: Flat algorithm (stage 1)...............................................................76 Figure 5.2: Calculating p (2,2,2) using the Flat algorithm ..........................76 Figure 5.3: Flat algorithm (stage 2)...............................................................77 Figure 5.4: Calculating p (2,2) using the Flat algorithm..............................77 Figure 6.1: Histograms of species counts (a) Wild Buckwheat, (b) Dandelion and (c) Wild Oats .................................................109 Figure 6.2: Scatter plot matrix for three species............................................111 Figure 6.3: Loglikelihood, AIC and BIC against the number of components for the local independence multivariate Poisson finite mixture model.................................116 Figure 6.4: The mixing proportions for model solutions with k = 2 to 7 components for the local independence multivariate Poisson finite mixture model .....................................................117 Figure 6.5: Loglikelihood, AIC and BIC against the number of components for the common covariance multivariate Poisson finite mixture model .....................................................119 Figure 6.6: The mixing proportions for model solutions with k = 2 to 7 components for the common covariance multivariate Poisson finite mixture model .....................................................120 Figure 6.7: Loglikelihood, AIC and BIC against the number of components for the restricted covariance multivariate Poisson finite mixture model ......................................................121 Figure 6.8: The mixing proportions for model solutions with k = 2 to 7 components for the restricted covariance multivariate Poisson finite mixture model......................................................122 Figure 6.9: Loglikelihood, AIC and BIC against the number of states for the local independent multivariate Poisson hidden Markov model .............................................................................125 xi
Figure 6.10: Loglikelihood, AIC and BIC against the number of states for the common covariance multivariate Poisson hidden Markov model ............................................................................126 Figure 6.11: Loglikelihood, AIC and BIC against the number of states for the restricted covariance multivariate Poisson hidden Markov model................................................................127 Figure 6.12: Loglikelihood against the number of components ( k ) for the multivariate Poisson finite mixture models ..................131 Figure 6.13: Loglikelihood against the number of components ( k ) for the multivariate Poisson hidden Markov models................132 Figure 6.14: Contour plot of clusters for the (a) independent, (b) common and (c) restricted covariance multivariate Poisson finite mixture models .........................................................................136 Figure 6.15: Contour plot of clusters for the (a) independent, (b) common and (c) restricted covariance multivariate Poisson hidden Markov models........................................................................137 Figure 8.1: Sample Size vs CPU time for different models of the Independent covariance structure ...............................................165 Figure 8.2: Sample Size vs CPU time for different models of the common covariance structure.....................................................................166 Figure 8.3: Sample Size vs CPU time for different models of the restricted covariance structure.....................................................................167 xii
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: Table 7.7: Loglikelihood and AIC to
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
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
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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
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REFERENCES 1. Aas, K., Eikvil, L. a
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17. Bicego, M., Murino, V. & Figuei
Page 197 and 198:
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
Page 205 and 206:
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=
Page 223 and 224:
# vˆ jk ( i) = P jk f ( y ; λ i
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