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

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affect inference. If necessary, the degree of change can be assessed by dividing the data by
time of collection and compare estimates across time periods.
To illustrate estimation techniques, we use the GM07535 data set and the in silico reference
map derived from the human genome sequence (Build 35).
2.2.1 Desorption
Random truncation: We begin with the estimation of desorption rates, since this can
be achieved without alignments, under certain assumptions. For a fragment in the true
restriction map spanned by an observed optical map, let Z be a random variable indicating
whether that particular fragment was observed, and let Y be its measured length had it been
observed. The desorption rate is quantified by the probability that a fragment is observed
(not truncated), given by
π(y) = P (Z = 1|Y = y)
Suppose that the marginal density of the unobserved (pre-truncation) random variable Y is
g. Let X represent the length of an observed (truncated version of Y ) fragment, i.e. X = Y
if Z = 1. Then, the marginal density of X is given by
h(x) = 1 K
π(x) g(x)
where K is the normalizing constant
K =
∫ ∞
0
π(t) g(t) dt
As formulated, π(·) only identifiable up to scale since π ′ (y) := α π(y), 0 < α ≤ 1 induces the
same h from g. Desorption is known to affect small fragments only, so we may additionally
assume that lim y→∞ π(y) = 1, making π(·) identifiable. Empirically, π(y) = 1 for y > 15 Kb.
Length distribution: Let us consider the distribution of the unobserved random variable
Y . Suppose the true recognition sites are realizations of a homogeneous Poisson process with
rate θ, true recognition sites are observed independently with probability p, and false cuts