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

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as a function of the parameters. An analytical solution exists for some simple families,
including one that is commonly used, given by π α (t) = 1 − e −αt , α > 0.
Caveats: Although the analysis described above is appealing, it depends critically on the
assumption that Y is exponentially distributed, which in turn depends on the sizing error
model used. Unfortunately, sizing error is less stable for small fragments, which is precisely
the region of interest. It should also be noted that even if the exponential distribution
holds marginally, fragment lengths can be considered independent only conditional on the
underlying restriction map. The marginal dependence is weak for large genomes, but cannot
be ignored for smaller ones.
2.2.2 Length errors
Parameters: Recall that the marginal distribution of the measured length of an optical
map fragment of true length µ is characterized by mean µ and variance σ 2 (τ 2 + 1)µ+τ 2 µ 2 . σ
and τ can be estimated indirectly given a reasonable alignment scheme, provided we assume
highly significant alignments to be true. The values of σ and τ used in Figure 2.4 were
derived using an informal method of moments estimator, with the observed variances for
subsets of the data defined by various ranges of µ used as the response in a linear regression
with terms µ and µ 2 . It should be noted that these estimates are based on maps with
significant alignments and are thus likely to be biased to some extent.
Normality: Figure 2.4 suggests that the distribution of the standardized lengths have
heavier tails than normal. This is clearer in the normal Q-Q plot in Figure 2.5. Empirically,
a t distribution appears to be a much better fit, which is not surprising since the calculation
of X i involves an estimated scale (see Section 2.1.2).
2.2.3 Cut errors
The probability p of a true cut being observed and the rate ζ of spurious cuts can be
similarly estimated using significant alignments. However, the amount of bias introduced