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

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Figure 4.5: Data collection locations from field #1 Figure 4.6: Distribution of weed counts and different states (clusters) in field #1 50
In the literature review (section 1.2.3), it is given that there is a very close relation with Markov random fields and Markov chains, and we explained that a 2-D (twodimensional) Markov random field can be easily transformed into a 1-D (onedimensional) Markov chain (Fjφrtoft et al., 2003 and Aas et al., 1999). There are different types of scanning methods (Memon et al., 2000) available to transfer 2-D data into 1-D data (e.g. Line Scan, Hilbert-Peano scan etc.). The most common way is a scan line order, where the grid locations are traversed horizontally line by line (Figure 4.7). This method is called the line scan method (Memon et al., 2000; Bar-Joseph and Cohenor, 2003). a x b grid Figure 4.7: Scanning method: Line Scan To illustrate how the line scan method can approximate spatial data, consider monthly (January-December) data across several years. This data can be arranged in a rectangular array (2-D) with rows representing years and columns representing months. Time series (1-D) periodic models exist where data from both neighbouring months in the same year and successive years in the same months are highly correlated. These 1-D 51
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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
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: at each of the 150 grid locations (
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
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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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-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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