On the Analysis of Optical Mapping Data - University of Wisconsin ...

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Q(θ|θ t ) = ∑ Π 88 Baum-Welch updates: The parameter estimation step of the HMM needs some calculations specific to the negative binomial model. To state the results, we first need some notation, which roughly follows Durbin et al. (1998). The data is assumed to be a sequence (x i ) L i=1 of observed hits in L successive intervals along the genome, with the corresponding sequence of random variables being denoted by (X i ) L i=1 . In practice, the data will be in the form of several sequences (e.g. one for every chromosome). The derivations done below generalize trivially to this situation, provided we assume that all the sequences are generated by the same model. Each observation X i has an associated hidden state Π i . We will often abbreviate (x i ) L i=1 by x, (X i) L i=1 by X and (Π i) L i=1 by Π. The unobserved sequence (Π i) L i=1 is a time-homogeneous stationary Markov process with a finite state space S = {1, 2, . . ., K}. The distribution of X i is entirely determined by Π i . This distribution is defined by the emission probabilities e k (b) = P (X i = b|Π i = k) (B.1) The evolution of the process (Π i ) L i=1 is governed by the transition probability matrix P, which has entries a k,l = P (Π i+1 = l|Π i = k) (B.2) and stationary distribution π 0 . The parameters in the model consist of the transition probabilities a = ((a k,l )) along with any parameters involved in the emission distribution, which we denote by η. Collectively, the parameters are denoted by θ = (a, η). Estimation of the parameters, often referred to as ‘training’, can be accomplished by using the Baum-Welch algorithm, which can also be stated in terms of the more familiar EM algorithm. It is an iterative procedure generally described in terms of observed data, missing data and parameters. In our case, the observed data are x and the missing data are the hidden states Π. Given a current estimate for θ, say θ t , the E-step involves computing the function P ( Π|X = x, θ t) log P (X = x,Π|θ) (B.3) and the M-step involves obtaining the next iterate θ t+1 = arg max Q(θ|θ t ) θ (B.4)
89 which can be shown to have higher likelihood than θ t . The details of these calculations depend on the emission model. We consider two specific models for the emission distribution (B.1). In the first model, η = (λ 1 , . . .,λ K , σ), which defines the emission distributions X i |Π i = k ∼ N B (λ k , σ) (B.5) In words, the emission distribution in each state is negative binomial with a different mean and a common size parameter. The second model attempts to ensure that copy number changes are limited to simple fractions (e.g. 2:3, 1:2, 1:1, 2:1, 3:2) by fixing them relative to each other. Specifically, η = (λ, σ) and X i |Π i = k ∼ N B (λα k , σ) (B.6) where α 1 , . . .,α K are fixed known constants. Theorem 1 gives the Baum-Welch updates for these two models in terms of the quantities A k,l = B k = N k,b = ∑L−1 i=1 ∑L−1 i=1 ∑L−1 i=1 I {Πi =k,Π i+1 =l} I {Πi =k} I {Πi =k,x i =b} and Ā k,l (θ) = E (A k,l |x, θ) ¯B k (θ) = E (B k |x, θ) ¯N k,b (θ) = E (N k,b |x, θ) The proof of the theorem is long but straightforward, and will not be given here.
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ON THE ANALYSIS OF OPTICAL MAPPING
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To my parents. i
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DISCARD THIS PAGE
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iv Page 3.3 Results . . . . . . . .
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v LIST OF TABLES Table Page 1.1 Sum
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vi LIST OF FIGURES Figure Page 1.1
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ON THE ANALYSIS OF OPTICAL MAPPING
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1 Chapter 1 Overview of Optical Map
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3 hard, do not always have a unique
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5 for microbial and other small gen
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7 Figure 1.2 Close-up of a typical
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9 0.96 0.98 1.00 1.02 1.04 Offset a
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11 direct glimpse at the underlying
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13 which is not surprising since we
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15 Gapped alignments: The above des
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Figure 1.5 A visualization of align
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19 Assembly: For these examples, th
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21 Chapter 2 Modeling Optical Map D
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23 Alternatively, it can be thought
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25 and V (X i ) = E(V (Y i R i |R i
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27 affect inference. If necessary,
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29 Quantiles of fragment lengths (K
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31 as a function of the parameters.
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33 by rejecting maps that do not al
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35 30 0.700 − 0.005 0 50 100 150
Page 51 and 52: 37 Chapter 3 Significance of Optica
Page 53 and 54: 39 using optical mapping data from
Page 55 and 56: 41 1.0 0.8 0.6 0.4 0.2 0.0 1.0 0.8
Page 57 and 58: 43 3.3.2 Simplifications Direct app
Page 59 and 60: 45 Mean spurious score 0 −10 −2
Page 61 and 62: 47 3.3.3 Simulation Given a generat
Page 63 and 64: 49 3.4 Discussion 3.4.1 Uses Alignm
Page 65 and 66: 51 The ability to simulate from the
Page 67 and 68: 53 maps, where the separation betwe
Page 69 and 70: Figure 3.10 Schematic representatio
Page 71 and 72: 57 especially a short noisy one, to
Page 73 and 74: 59 Test statistics: Variability due
Page 75 and 76: 61 in sequence assembly and validat
Page 77 and 78: 63 1988). However, due to sampling
Page 79 and 80: 65 and rate parameters Λ i = E(N i
Page 81 and 82: 67 with mean µ k for the k th stat
Page 83 and 84: 69 Estimated Copy Number in simulat
Page 85 and 86: 71 Posterior probabilities 1.0 0.8
Page 87 and 88: 73 (a) Observed counts and decoded
Page 89 and 90: 75 Conclusion: Copy number alterati
Page 91 and 92: 77 well in its current form, but th
Page 93 and 94: 79 Change in score 15 10 5 0 0.9 1.
Page 95 and 96: 81 will rarely be homozygous. It ma
Page 97 and 98: 83 E.T. Dimalanta, A. Lim, R. Runnh
Page 99 and 100: 85 Appendix A: Score functions for
Page 101: 87 Appendix B: Hidden Markov Model
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