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

More documents

Recommendations

Info

over the range of the data points. For each set of starting values, the algorithm was run for 200 iterations without caring about any convergence criterion. Then, the set of initial starting values with the largest loglikelihood was selected. The EM iteration were continued with these selected initial values until the convergence criterion is satisfied, i.e. until the relative change of the loglikelihood between two successive iterations was smaller than 12 10 − . This procedure is repeated 7 times for each value of k . The selection of number of clusters were based on the most well-known information criterions (section 5.3.4), i.e. the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). The AIC is given as AIC= L k − d k and the BIC is given as BIC= Lk − ln( n) dk / 2 where L k is the value of the maximized loglikelihood for a model with k components and d k is the number of free parameters of the model. For the restricted covariance, the independent and the common covariance models d k is d k 2 2 2 = 6 k + k − k , = 3 k + k − k and = 4 k + k − k respectively. d k d k Figure 6.9 illustrates the evolution of the loglikelihood for the different components (k=1,…,7) of the local independence multivariate Poisson hidden Markov model. This figure illustrates that the AIC and the BIC selects five states as the optimal number of states. Similarly, Figure 6.10 and Figure 6.11 illustrate the evolution of the loglikelihood for the different components of the common covariance and the restricted covariance models respectively. Based on the AIC and the BIC criterion, the five states for the 124
common covariance and the four states for the restricted covariance model were selected as optimal number of states. -575 -625 -675 Loglikelihood -725 -775 -825 -875 -925 -975 -1025 1 2 3 4 5 6 7 k (the Nnumber of States) Loglikelihood AIC BIC Figure 6.9: Loglikelihood, AIC and BIC against the number of states for the local independent multivariate Poisson hidden Markov model 125
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 and 12:
Table 7.7: Loglikelihood and AIC to
Page 13 and 14:
Figure 6.10: Loglikelihood, AIC and
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
Page 63 and 64:
at each of the 150 grid locations (
Page 65 and 66:
In the literature review (section 1
Page 67 and 68:
estimated through the observations
Page 69 and 70:
CHAPTER 5 MULTIVARIATE POISSON DIST
Page 71 and 72:
Y Y Y 1 2 3 = X = X 1 = X 2 3 + X +
Page 73 and 74:
educe the computational burden; how
Page 75 and 76:
and y 3 as illustrated above. Again
Page 77 and 78:
q In general, the number of paramet
Page 79 and 80:
Similar to the fully structured mod
Page 81 and 82:
ecause the former captures more of
Page 83 and 84:
(Tsiamyrtzis and Karlis, 2004). The
Page 85 and 86:
notation, the following recursive s
Page 87 and 88: 5.2.2 The multivariate Poisson dist
Page 89 and 90: ypy ( ,0,0) = θ py ( − 1,0,0) y1
Page 91 and 92: ecurrence relationship ypy 1 ( 1 ,
Page 93 and 94: Raftery, 1998); identification of t
Page 95 and 96: maximization (EM) algorithm is appl
Page 97 and 98: the EM algorithms for use on very l
Page 99 and 100: Another alternative is to use the p
Page 101 and 102: 5.3.5 Estimation for the multivaria
Page 103 and 104: j j E[ X 12 i | Yi , Z ij = 1, Φ ]
Page 105 and 106: 5.4 Multivariate Poisson hidden Mar
Page 107 and 108: where P = Pr( S = k | S − 1 j), 1
Page 109 and 110: P jk n ∑ vˆ jk i= 2 = n m ∑∑
Page 111 and 112: E[ X | Y, u ( i) = 1, Φ ] = d = y
Page 113 and 114: The bootstrap method is a powerful
Page 115 and 116: eplications are generally sufficien
Page 117 and 118: Y and assumes a probability distrib
Page 119 and 120: For the case of two categorical var
Page 121 and 122: ejected. In this situation, the (sm
Page 123 and 124: 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: Table 6.8: Parameter estimates (boo
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 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 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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?