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

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32 µ (52.1,448] (37.3,52.1] (29.9,37.3] (24.9,29.9] (21.2,24.9] (18.2,21.2] (15.6,18.2] (13.6,15.6] (11.7,13.6] (10,11.7] (X − µ) µ (X − µ) µ (X − µ) τ 2 µ 2 + σ 2 µ(τ 2 + 1) −4 −2 0 2 −0.6 −0.4 −0.2 0.0 0.2 0.4 −6 −4 −2 0 2 4 6 Figure 2.4 Variance models for observed fragments sizes. The dependence of the variance on the true fragment length µ is difficult to study because the true fragment lengths are unknown. However, given a reasonable alignment scheme, the true lengths are known if highly significant alignments are assumed to be true. Here, we use significant alignments of the GM07535 data, leaving out all fragments less than 10 Kb since smaller fragments may have a different variance model. To remove possible effects of within-map dependence, only one fragment is randomly selected from each map. The plots are box and whisker plots of lengths standardized according to different variance models, compared across different ranges of µ. Clearly, the first two variance models do not completely capture the systematic dependence on µ, but the last one does. The large proportion of ‘outliers’ suggests non-normality, which is explored further in Figure 2.5. −4 −2 0 2 4 Normal t with 5 d.f. X − µ τ 2 µ 2 + σ 2 µ(τ 2 + 1) 2 0 −2 −4 −2 0 2 4 Theoretical Quantiles Figure 2.5 Distribution of standardized fragment length errors. The data used in Figure 2.4 are used again in Q-Q plots to illustrate that the observed lengths are non-normal. Empirically, the t distribution seems to be a better fit.
33 by rejecting maps that do not align, as well as assuming that the significant alignments are completely correct, is uncertain. Often it is instructive instead to assess a model by some diagnostic plots, as described next. 2.3 Diagnostics Due to the complexity of the model and the interplay between its various aspects, it is next to impossible to estimate all the parameters separately. However, given a particular set of parameter values, maps simulated from that model can be used to indirectly test goodness of fit. Specifically, simulated maps should have characteristics that are similar to observed maps, be they numerical summaries or graphical diagnostics. In Figures 2.6, 2.7 and 2.8, we present three diagnostic plots based on the marginal distributions of observed restriction fragment lengths, and the number of fragments in a map. The data being modeled is the set of GM07535 optical maps; maps are simulated from the in silico reference map with a combination of values for p (0.70, 0.75 and 0.80) and ζ (0.001, 0.003 and 0.005), keeping all other components fixed. The rate of desorption is determined by the function π α (t) = 1 −e −αt . The first plot suggests that the effect of desorption has been well modeled. Considered together, the three plots suggest that p = 0.7 and ζ = 0.005 come closest to modeling the observed data. It is important to note that these are only a few examples, and other similar diagnostic plots could be useful for similar purposes. None of these plots require alignments, but plots analogous to Figure 2.4 that do depend on alignment may also be useful.
Page 1 and 2: ON THE ANALYSIS OF OPTICAL MAPPING
Page 3 and 4: To my parents. i
Page 5 and 6: DISCARD THIS PAGE
Page 7 and 8: iv Page 3.3 Results . . . . . . . .
Page 9 and 10: v LIST OF TABLES Table Page 1.1 Sum
Page 11 and 12: vi LIST OF FIGURES Figure Page 1.1
Page 13 and 14: ON THE ANALYSIS OF OPTICAL MAPPING
Page 15 and 16: 1 Chapter 1 Overview of Optical Map
Page 17 and 18: 3 hard, do not always have a unique
Page 19 and 20: 5 for microbial and other small gen
Page 21 and 22: 7 Figure 1.2 Close-up of a typical
Page 23 and 24: 9 0.96 0.98 1.00 1.02 1.04 Offset a
Page 25 and 26: 11 direct glimpse at the underlying
Page 27 and 28: 13 which is not surprising since we
Page 29 and 30: 15 Gapped alignments: The above des
Page 31 and 32: Figure 1.5 A visualization of align
Page 33 and 34: 19 Assembly: For these examples, th
Page 35 and 36: 21 Chapter 2 Modeling Optical Map D
Page 37 and 38: 23 Alternatively, it can be thought
Page 39 and 40: 25 and V (X i ) = E(V (Y i R i |R i
Page 41 and 42: 27 affect inference. If necessary,
Page 43 and 44: 29 Quantiles of fragment lengths (K
Page 45: 31 as a function of the parameters.
Page 49 and 50: 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 and 102:
87 Appendix B: Hidden Markov Model
Page 103 and 104:
89 which can be shown to have highe
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