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41. Felsenstein, J. and Churchill, G.A. (1996). A hidden Markov Model approach to variation among sites in rate of evolution, Molecular Biology and Evolution, Vol. 13, pp. 93-104. 42. Fishman, G.S., (1996). Monte Carlo: Concepts, algorithms, and applications, New York: Springer-Verlag. 43. Fjφrtoft, R., Boucher, J., Delignon, Y., Garello, R., Le Caillec, J., Maitre, H., Nicolas, J., Pieczynski, W., Sigelle, M., and Tupin, F. (2000). Unsupervised Classification of Radar Images Based on Hidden Markov Models and Generalized Mixture Estimation, Proc. EOS/SPIE Symposium on Remote Sensing, Conference on SAR Image Analysis, Modelling, and Techniques V, Vol. SPIE 4173, pp. 87-98. 44. Fjφrtoft, R., Delignon, Y., Pieczynski, W., Sigelle, M. and Tupin, F. (2003). Unsupervised Classification of Radar Images Using Hidden Markov Chains and Hidden Markov Fields, IEEE Transactions on Geoscience and Remote Sensing, Vol. 41(3), pp. 675-686. 45. Fjφrtoft, R., Wojciech Pieczynski and Yves Delignon (2001), Generalised Mixture Estimation and Unsupervised Classification Based on Hidden Markov Chains and Hidden Markov Random Fields, Proc. Scandinavian Conference on Image Analysis (SCIA'01), pp. 733-740 46. Forney, Jr. G.D. (1973). The Viterbi Algorithm, Proceedings of IEEE, Vol. 61(3), pp. 263-278. 47. Fraley, C. and Raftery, A.E. (1998). How many clusters? Which clustering method? Answers via Model based Cluster Analysis, Technical Report No. 329, 184
Department of Statistics, University of Washington, Box 354322, Seattle, WA, 98193-4322, USA. 48. Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 6(6), pp.721-741. 49. Hasselblad, V., (1969). Estimation of Finite Mixtures of Distributions from the Exponential Family, Journal of the American Statistical Association, Vol. 64(328), pp. 1459-1471. 50. Holgate P. (1964). Estimation for the bivariate Poisson distribution, Biometrika, Vol. 51 (1), pp.241-245. 51. Johnson N.L. and Kotz S. (1969). Distributions in Statistics: Discrete Distributions. Boston: Houghton Mifflin. 52. Johnson, N.L., Kemp, A.W. and Kotz, S. (2005). Univariate Discrete Distributions, Hoboken, New Jercy: Wiley-Interscience. 53. Johnson, N.L., Kotz, S. and Balakrisnan, N. (1997). Discrete Multivariate Distributions, New York: John Wiley and Sons, pp. 1-30, 124-152. 54. Juang, B.H. and Rabiner, L.R.(1991). Hidden Markov Models for Speech Recognition, Technometrics, American Statistical Association and the American Society for Quality Control, Vol. 33(3), pp. 251-272. 55. Kano, K. and Kawamura, K. (1991). On Recurrence Relations for the Probability Function of Multivariate Generalized Poisson distribution, Communications in Statistics: Theory and Methods, Vol. 20(1), pp. 165-178. 185
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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 -
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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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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
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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 and 194: REFERENCES 1. Aas, K., Eikvil, L. a
Page 195 and 196: 17. Bicego, M., Murino, V. & Figuei
Page 197: 33. Descombes, X., Morris, R.D., Ze
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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