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A=[A 1 , A 3 ] ⎡1 0 0 1⎤ A = ⎢ 0 1 0 1 ⎥ ⎢ ⎥ ⎢⎣ 0 0 1 1⎥⎦ where A A 1 3 ⎡1 0 0⎤ = ⎢ 0 1 0 ⎥ ⎢ ⎥ ⎢⎣0 0 1⎥⎦ ⎡⎤ 1 = ⎢⎥ ⎢⎥ 1. ⎢⎥ ⎣⎦ 1 Although the covariance structure is limited compared to the general definition of the multivariate Poisson, there are a convenient number of parameters to deal with. Moreover, trivariate Poisson distribution PY [ 1 = y1, Y2 = y2, Y3 = y3] can now be obtained as the marginal distribution from PY [ 1 = y1 , Y2 = y2, Y3 = y2, X123 = x123] as follows: min( y1, y2, y3) ∑ PY [ = y, Y = y , Y = y] = PY [ = y, Y = y , Y = y, X = x ]. (5.3) 1 1 2 2 3 3 1 1 2 2 3 3 123 123 (3) x = 0 Substituting the X’s for the Y’s in (5.3) results in: (4) 1 = 1 2 = 2 3 = 3 = ∑ L 1 = 1 2 = 2 3 = 3 123 = 123 (3) x = 0 PY [ y, Y y , Y y] P[ X x, X x, X x, X x ] with all X ’s (4) independent univariate Poisson distributions and = [ ] thus: L where L5 = min( y1, y2, y3) , L 5 1 1 2 2 3 3 (4) L ∑∏ PY [ = y, Y = y , Y = y] = P[ X = ( y − x)] P[ X = x] = x L (3) (4) ∑ = 0 e ∑ ∏ j j l i i (3) x = 0 j∈R1 l∈R3 i∈R3 − ( θ + θ + θ + θ ) 1 2 3 123 ∏ j∈R ∏ j∈R θ ( y − x ) j j ∑ l l∈R3 ∏ i∈R θ 1 3 . ( y − x )! x ! ∑ ∏ 1 3 3 xi i j l i l∈R i∈R 64
Similar to the fully structured model presented in the previous section, the general k- components q-variate Poisson mixture model requires the estimation of ( k −1) + k× ( q+ 1) parameters. The number of parameters increases linearly both with the number of the variables and components being considered. 5.1.3 The multivariate Poisson model with local independence In the previous sections, it was revealed that in the case of three weed species, the joint probability of observing multiple outcomes PY [ 1i = y1 i, Y2i = y2i, Y3i = y3i] for an ‘i ’ th location is distributed according to the multivariate Poisson distribution. On the other hand, under the assumption of local independence of the weed count rates within each mixture component, this joint probability reduces to the product of the weed speciesspecific densities, that is, PY [ = y , Y = y , Y = y ] = PY [ = y ] × PY [ = y ] × PY [ = y ]. 1i 1i 2i 2i 3i 3i 1i 1i 2i 2i 3i 3i This means that the following representation is obtained for the Y i ’s: Y Y Y = X 1 1 = X 2 2 3 = X . 3 In matrix notations, this model can be presented as: Y = AX A=[A 1 ] 65
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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,
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: q In general, the number of paramet
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
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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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