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

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1988). However, due to sampling errors and alignment efficiencies, the observed locations are
better modeled as a non-homogeneous Poisson process (NHPP). Not all maps are successfully
aligned, and we observe locations, up to errors in the alignment, only for those that do align.
Alignment algorithms are not uniformly sensitive, and the chance that the origin of a map
is correctly identified depends on the origin. For example, maps from regions where markers
are sparse are less likely to align, while maps with rare patterns score higher and are more
likely to align (see Figure 3.10). Copy number alteration in a region of the sampled genome is
manifested as further change in the non-homogeneous alignment rate. To detect fluctuations
that are due to copy number changes, we must first adjust for the normal fluctuations due
to the inherent but unknown non-homogeneity.
Notation: The availability of optical map data from normal cells can be used to establish
normal variation in the absence of CNP, but this requires some preliminary work. Formally,
let {Y 1 < · · · < Y n } be aligned locations of the optical maps of interest, and {X 1 < · · · < X m }
be similar aligned locations for ‘normal’ optical maps. We model {X k } m 1
as realizations of
a non-homogeneous Poisson process with rate λ 0 (·) and {Y k } n 1
as realizations of a nonhomogeneous
Poisson process with rate λ(·). If the two genomes are identical, we have
λ(c) = κλ 0 (c), where c denotes position along the reference genome, and κ represents a
constant of proportionality reflecting the possibility of different overall coverage in the two
genomes. κ can be estimated in this case by the ratio n/m of the number of aligned maps. If
copy number alterations exist in the genome, κ is defined to be the constant of proportionality
in regions of normal copy number.
Interval counts: It is natural to work with interval counts when dealing with Poisson
process data, so we discretize the sample space into intervals G i = (a i , b i ], i = 1, . . .,L tiling
the reference genome. The choice of G i ’s is important and is discussed below. Assume for
the moment that λ 0 and κ are known. For the i th interval, define the counts
N i = ∑ k
I {ai