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9. Akobundu, I.O., Ekeleme, F. and Chikoye, D. (1999). Influence of fallow management systems and frequency of cropping on weed growth and crop yield. Weed Research,Vol. 39, pp.241–256. 10. Archer G.E.B. and Titterington D.M. (2002). Parameter estimation for hidden Markov chains, Journal of Statistical Planning and Inference, Vol. 108(1), pp.365-390. 11. Banfield, J.D. and Raftery, A.E. (1993). Model-Based Gaussian and Non- Gaussian Clustering, Biometrics, Vol. 49, pp. 803-821. 12. Bar-Joseph, Z. and Cohen-Or, D. (2003). Hierarchical Context-based Pixel Ordering, Computer Graphics Forum, Proceedings of Eurographics, Vol. 22(3), pp. 1-10. 13. Basford, K.E., Greenway, D.R., McLachlan, G.J. and Peel, D. (1997a). Standard errors of fitted means under normal mixture models, Computational Statistics, Vol. 12, pp. 1-17. 14. Baum, L.E. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains, Annals of Mathematical Statistics, Vol. 37, pp.1554-1563. 15. Baum, L.E., Peterie, T., Souled, G. and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains, Annals of Mathematical Statistics, Vol. 41(1), pp. 164-171. 16. Baum, L.E. and Egon, J.A. (1967). An inequality with applications to statistical estimation for probabilistic functions of a Markov process and to a model for ecology, Bulletin of the American Meteorological Society, Vol.73, pp. 360-363. 180
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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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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 -
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
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: REFERENCES 1. Aas, K., Eikvil, L. a
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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