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

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mixing proportions and the transition probabilities were selected according to uniform random numbers and rescaled them to sum up to 1. First for each set of initial values, the algorithm was run for 100 iterations without any convergence criterion. Then, the model parameters with the largest likelihood were selected for the further analysis. For all five sets, once the suitable initial values have been selected the computer time (CPU time-central processing unit time) was recorded after running the algorithm 200 iterations. The time spent for simulating the samples was not included in this calculation. All the calculations were carried with a PC with a Pentium microprocessor, which has 2 GB of random access memory (RAM). CPU time severely depends on the RAM of the computer. The results of the two models, (a) the multivariate finite mixture model (MFM) and (b) the multivariate hidden Markov model (HMM) were reported for the different components and for the different covariance structures. The CPU times were recorded to the order of 1/100 second. 8.3 Results of computational efficiency Table 8.1- Table 8.3 and Figure 8.1-Figure 8.3 illustrate the independent, the common, and the restricted covariance structure results, respectively, for the models (a) and (b). It is clear that when the sample size increases, the CPU time (in 1/100 second) also increases exponentially for all models. For small sample sizes ( n = 50, 100 and 200) models (a) and (b) have similar CPU times (some cases hidden Markov model take more time) regardless of the number of parameters to be estimated. However, when sample sizes increased ( n = 500, 1000) it revealed that the hidden Markov model takes 162
less time compared to the multivariate finite mixture model even though hidden Markov models have more parameters to be estimated. This is due to the computational procedures involved in these two models. Therefore, in terms of computational efficiency it can be concluded that for small sample sizes, two models, (a) and (b) have same computational efficiency and for large sample sizes, the multivariate Poisson hidden Markov model is more efficient compared to the multivariate Poisson finite mixture model. Table 8.1: Independent covariance structure –CPU time (of the order of 1/100 second) k =1(components/states) k =2 (components/states) k =3 (components/states) n MFM HMM MFM HMM MFM HMM 50 22.64 23.72 43.63 45.37 67.61 67.34 100 39.63 43.98 76.58 80.58 115.55 118.20 200 85.17 87.98 164.14 168.22 247.35 260.56 500 216.70 212.46 419.98 416.37 599.51 601.13 1000 470.78 427.33 855.31 843.81 1268.11 1212.84 163
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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
Page 125 and 126: 0 1 2 3 4 Wild Buckwheat species109
Page 127 and 128: Table 6.4: The frequency of occurre
Page 129 and 130: 6.4.1 Results for the different mul
Page 131 and 132: Proportion 1.0 0.9 0.8 0.7 0.6 0.5
Page 133 and 134: -600 -650 -700 Loglikelihood -750 -
Page 135 and 136: Figure 6.7 illustrates the evolutio
Page 137 and 138: Table 6.8: Parameter estimates (boo
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: CHAPTER 8 COMPUTATIONAL EFFICIENCY
Page 179 and 180: 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
Page 187 and 188: indication of the relative goodness
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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