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Also the calculations of the multivariate Poisson probabilities of higher dimensions (four or more) can be carefully studied and programmed for further research of the hidden Markov model. Even though our model provides overall positive and negative correlation of the variables (in this case, species) it is not able to provide the negative interdependence within the components. Therefore, one can study further about this area for Poisson count variables. The models presented in this thesis do not include any other covariates. The covariates may help us to explain the differences between the distributions of the component. The above issue is high on the list of topics for further research. 178
REFERENCES 1. Aas, K., Eikvil, L. and Huseby, R. (1999). Applications of hidden Markov chains in image analysis, Patten Recognition, Vol. 32(4), pp. 703-713. 2. Agresti, A., (2002). Categorical Data Analysis, New Jersey: John Wiley & Sons, Inc. 3. Aitchison J. and Ho C.H. (1989). The multivariate Poisson-log normal distribution, Biometrika, Vol. 76 (4), pp. 643-653. 4. Aitkin, M. and Aitkin, I. (1996). An hybrid EM/Gauss-Newton algorithm for maximum likelihood in mixture distributions, Statistics and Computing, Vol. 6, pp. 127-130. 5. Aittokallio T. and Uusipaikka E. (2000). Computation of Standard Errors for maximum likelihood estimating in Hidden Markov models, Turku Centre for computer Science, Finland, TUCS Technical Report No. 379, ISBN 952-12- 0760-4. 6. Aiyer A., Pyun K., Huang Y. and Gray D.B.O.R. (2001). Lloyd clustering of gauss mixture models for image compression and classification, IEEE 2001 International conference on Image Processing. [on line] 7. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B.N. Petrov and F. Csake (eds.), Second International Symposium on Information Theory. Budapest: Akademiai Kiado, pp. 267-281. 8. Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, Vol. AC-19, pp.716-723. 179
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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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Page 145 and 146: 6.5 Comparison of the different mod
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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
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Page 161 and 162: 7.5 Applications In addition to wee
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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
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Page 175 and 176: CHAPTER 8 COMPUTATIONAL EFFICIENCY
Page 177 and 178: less time compared to the multivari
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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: 9.6 Further research We can present
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
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Page 213 and 214: theta33
Page 215 and 216: } prob[g1+1, g2 + 1]
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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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