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The parameters θ ( j ∈ Rm , m = 2, 3) correspond to a m-way covariance in a similar j way to the m-way interactive terms, and thus, they impose structure on the data. Mardia (1970) introduced the multivariate reduction technique to create the multivariate Poisson distribution. This reduction technique has been used extensively for the construction of multivariate models. The idea of the method is to start with some independent random variables and to create new variables by considering some functions of the original variables. Then, since the new variables contain jointly some of the original ones, a kind of structure is imposed creating multivariate models (Tsiamyrtzis et al., 2004). We can represent this model using following matrix notations. Assume that X i , i = 1,...,k are independent Poisson random variables and A is a n× k matrix with zeros and ones. Then the vector Y = Y , Y ,..., Y ) defined as Y = AX follows a n -variate ( 1 2 n Poisson distribution. The most general form assumes that A is a matrix of size n n× (2 − 1) of the form A=[A 1 , A 2 , A 3 ,…,A n ] where A i , ⎛n⎞ i = 1,..., n are matrices with n rows and ⎜ ⎟ ⎝i ⎠ columns. The matrix A i contains columns with exactly i ones and n− i zeros, with no duplicate columns, for i = 1,...,n . Thus A n is the column vector of 1’s while A 1 becomes the identity matrix of size n× n. For example, the fully structured multivariate Poisson model for three variables can be represented as follows: 56
Y Y Y 1 2 3 = X = X 1 = X 2 3 + X + X + X 12 12 13 + X + X + X 13 23 23 + X + X + X 123 123 123 Y = AX A=[A 1 , A 2 , A 3 ] ⎡1 A = ⎢ ⎢ 0 ⎢⎣ 0 0 1 0 0 0 1 1 1 0 1 0 1 0 1 1 1⎤ 1 ⎥ ⎥ 1⎥⎦ n×k where A A A 1 2 3 ⎡1 0 0⎤ = ⎢ 0 1 0 ⎥ ⎢ ⎥ ⎢⎣0 0 1⎥⎦ ⎡1 1 0⎤ = ⎢ 1 0 1 ⎥ ⎢ ⎥ ⎢⎣0 1 1⎥⎦ ⎡⎤ 1 = ⎢⎥ ⎢⎥ 1. ⎢⎥ ⎣⎦ 1 Also we can write Y= A X + A X + A X where (1) (2) (3) 1 2 3 Y = AX in more detailed as below. ⎡Y ⎤ Y , 1 = ⎢ Y ⎥ ⎢ 2 ⎥ ⎢⎣Y ⎥ 3 ⎦ ⎡X ⎤ 1 (1) X ⎢ X ⎥ 2 , = ⎢ ⎥ ⎢⎣X ⎥ 3 ⎦ ⎡ X ⎤ 12 (2) ⎢ X ⎥ (3) X 13 , and = [ X 123 ] = ⎢ ⎥ ⎢⎣ X ⎥ 23 ⎦ X . Moreover, this model enables us to construct some interesting submodels by appropriately defining the set R . For example: 57
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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
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: CHAPTER 5 MULTIVARIATE POISSON DIST
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
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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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loglikelihood providing at least a
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illustrates the contour plot of the
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(a) Independent Contour 1 Contour 2
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Karlis and Meligkotsidou (2006) dis
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The mean vector and the covariance
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The simple moments of B are polynom
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and E ( Y ) = AM where ⎡λ1 ⎤
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7.5 Applications In addition to wee
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The estimated covariance matrix and
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The estimated covariance matrix (AI
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Table 7.6 Bacterial counts by 3 sam
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Table 7.8: Loglikelihood and AIC to
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(c) Finite mixture with the five co
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(i) Hidden Markov model with the fi
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CHAPTER 8 COMPUTATIONAL EFFICIENCY
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less time compared to the multivari
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1400 1200 CPU time (1/100 second) 1
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1400 1200 CPU time (1/100 second) 1
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In this thesis, three species count
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In the applications of the HMMs, li
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indication of the relative goodness
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underlying data. The advantage of t
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9.6 Further research We can present
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REFERENCES 1. Aas, K., Eikvil, L. a
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17. Bicego, M., Murino, V. & Figuei
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33. Descombes, X., Morris, R.D., Ze
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Department of Statistics, Universit
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66. Li C.S., Lu J.C., Park J., Kim
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85. Petrie T. (1969). Probabilistic
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104. University of Manitoba, Depart
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z0
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threep2[i]
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loglike[nit]
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theta33
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} prob[g1+1, g2 + 1]
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theta131
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threep22[i]
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dens=matrix(0,nrow=T,ncol=N) alpha=
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# vˆ jk ( i) = P jk f ( y ; λ i
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