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• If * R = R 1 then the model reduces to an independence model (referred to as the local independence model). • If = ∪ then the model reduces to a model with one common covariance * R R1 R3 term (referred to as the common covariance model). • If the model assumes that = ∪ then it allows only two-way covariances * R R1 R2 (referred to as the restricted covariance model). Note that omitting the set of parameters θ ( j ∈ ), is equivalent to setting θ = 0 . j The submodels can be formed by assuming that the corresponding θ ’s equal zero. Now, denote the cardinality of R as J which, for a trivariate model, equals J =7. Then, using the above notation, and considering the most general model with all the covariance terms (though it imposes unnecessarily large structure), the joint probability density of the corresponding multivariate Poisson distribution is given as R m j p( y | θ ) = PY [ = y, Y = y , Y = y | θ , j∈R] 1 1 2 2 3 3 J = ∑.... ∑∏ Po( x | θ ) , where the summation is extended over all the combinations of j∈R j j j x j such that ∑ y ≥ , i x k k k ∈ R and k contains the subscript i . The fully-structured covariance model which is illustrated in section 5.1.1 needs four summations for the trivariate case, which obviously implies a large computational burden. The major problem of the use of the probability distribution in its general form is the calculation difficulty of the probability mass function. Kano and Kawamura (1991) described recursive schemes (section 5.2) to 58
educe the computational burden; however the calculation remains computationally time-consuming for large dimensions. This computational burden problem brings out the idea to create multivariate distributions with selected covariances, that is, not to include all the possible covariance terms, but only to select the covariance terms that are useful. In reality, using all the m- fold covariance terms imposes too much structure, while complicating the whole procedure without adding any further insight into the data. For this reason, after a preliminary assessment, one may identify interesting covariance terms that may be included into the model. This selection corresponds to fixing the value of the Poisson parameters, that is, the corresponding θ ’s. Based on this general description of the multivariate Poisson distribution and the relationship with more suitable submodels, a detailed description of each model is provided in the next few sections. 5.1.1 The fully- structured multivariate Poisson model The theoretical development of the fully structured multivariate Poisson model will be illustrated by weed species: Wild Buckwheat, Dandelion and Wild Oats. Suppose the objective is to cluster weed count data based on the mean counts in a set of three weed species, that is, Wild Buckwheat ( Y 1 ), Dandelion ( Y 2 ), Wild Oats ( Y 3 ). Following the 59
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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
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: Y Y Y 1 2 3 = X = X 1 = X 2 3 + X +
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
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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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