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is reached. The result of this procedure gives us the maximum likelihood estimator of the hidden Markov model (Baum et al., 1970). It should be noted that the Baum-Welch method leads to a local maximum of λ only. In practice, to get a good solution, the initial guess λ 0 is very important. Usually several sets of starting guesses of λ 0 are used and one with the greatest likelihood value is chosen. Laird (1978) suggested a grid search method, which divides the searching domain into equally spaced small grids and starts from each of the intersections. Leroux and Puterman (1992) argue that the grid method would generate too many initial points when high dimensional spaces are involved and as such, they suggested a clustering algorithm. 42
CHAPTER 4 HIDDEN MARKOV MODEL AND THEIR APPLICATIONS TO WEED COUNTS 4.1 Introduction Weed management makes a significant contribution to the harvesting of crops. Controlling weeds can improve the crop yield. It is also interesting to determine the relationship among more common weeds in the field. Another main factor of weed management is to find out whether there are different patterns or distributions within the field due to physical factors such as soil types, soil moisture and other reasons. The importance of these findings leads to better weed control practices. In an agricultural survey conducted by Agriculture Canada, there were several fields considered without any treatments in Prairie Provinces. There were different kinds of weeds present in these fields, and some of the most common weeds found were Stinkweed, Wild Oats, Canada Thistle, Wild Buckwheat, Perennial Sow Thistle, Wild Mustard, Green Foxtail and Dandelion. In this thesis, one field has been selected (namely field #1; note that exact site location is not available), and the two most common weed types and one less frequent weed type are selected for analysis. The most 43
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MULTIVARIATE POISSON HIDDEN MARKOV
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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: ˆ ( n) = b j Expected Number of ti
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
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where P = Pr( S = k | S − 1 j), 1
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P jk n ∑ vˆ jk i= 2 = n m ∑∑
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E[ X | Y, u ( i) = 1, Φ ] = d = y
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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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