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

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are realizations of a homogeneous Poisson process with rate ζ. If we assume independence of
these errors and no error in sizing, then the observed cut sites are realizations of a homogeneous
Poisson process with rate p θ+ζ. Consequently, the fragment sizes Y are exponentially
distributed. The assumption of no sizing error is of course unrealistic; Valouev et al. (2006)
show that the exponential distribution holds approximately even with reasonable sizing error
models.
Exponential rate: The rate of the relevant exponential distribution depends on unknown
parameters. Fortunately, this rate can be estimated directly from the data, thanks to the
memoryless property of the exponential distribution, namely that
P(Y > t + s | Y > t) = P(Y > s)
when Y is exponentially distributed, or equivalently, Y |Y > t has the same distribution as
Y + t. In other words, left truncation of exponential variates is equivalent to an additive
shift. Since it is known empirically that π(y) = 1 for y > 15 Kb, the truncated observations
X|X > 15 has the same distribution as Y |Y > 15, i.e., an exponential truncated at 15 Kb.
A robust estimate of the rate can be obtained from the interquartile range of the truncated
observations. Empirical evidence is provided by a Q-Q plot of the observed values of X in
Figure 2.2.
Non-parametric estimation: A naive non-parametric estimate of π is given by
̂π(t) ∝ ĥ(t)
g(t)
where ĥ is the estimated density of observed fragment lengths X and g is the known density of
Y . X is a positive random variable, so usual kernel density estimates are inappropriate, but
alternatives such as zero-truncated kernel density estimates and log-spline density estimates
exist. More interestingly, the non-parametric MLE of π can be obtained under the additional
assumption that π is increasing. This is reasonable since longer fragments are less likely to
desorb. The MLE follows from the existence of the MLE of a monotone density, given by