The world's fastest genomic segmentation algorithm.

Peakolator solves the following problem:

Given a vector (x_1, x_2, ..., x_n), find the highest density interval [i, j], where density is defined by some function f(x, k), where x = x_i + ... + x_j and k = j - i + 1. Typically n is quite large, say 100 million or so.

The density function must comply with some simple properties and in exchange, Peakolator can typically arrive the solution to this problem quite quickly.

The algorithm centers around a density function (my terminology). A density function is any function f(k, x) that obeys the following two properties.

1. f(x', k) >= f(x, k) for all x' >= x.
2. f(x, k') <= f(x, k) for all k' >= k.

The first property says that density is non-decreasing with mass (x), and the second that it is non-increasing with volume (k). This should hopefully fit with one's intuitive notion of "density".

There is an obvious density function (namely, f(x, k) = x/k) that obeys these two properties, but many other functions do as well.

One example is the traditional frequentist scan statistics methodologies. Suppose we have a vector (x_1, ..., x_n) and a null hypothesis that x_i ~ Poisson(a), and thus x_i + ... + i_j ~ Poisson((j - i + 1) * a). We wish to find intervals in the vector in which the null hypothesis has very low-likelihood.

This sort of approach is used very frequently with "sliding window" algorithms, but sliding window algorithms involve choosing one or more fixed window lengths. Peakolator can search for significant windows of all length. We can set f(x, k) = 1 - P(x | k * a) where P is the Poisson CDF function. This is a proper density function and can be used with Peakolator!

A wide variety of useful functions are density functions, but it is also a limiting notion. Often we have some idea of the length of the regions we are search for. Peakolator thus supports specifying an arbitrary prior distribution over interval lengths.

This is a useful thing to be able to do in genomics. First of all, if we can find the highest density region efficiently, we can perform clustering or segmentation like so.

1. Find the highest density region.
2. Subtract it from the search space.
3. Repeat.

With the right vector and density function this can be used to solve many bread-and-butter problems in genomics. E.g. discover CpG islands or highly conserved regions. It can also be applied to sequencing data to discover novel transcribed regions from RNA-Seq data, or peaks in ChIP-Seq data.

Peakolator is a heuristic search algorithm: it has a quadratic worst case performance, but is almost always extremely fast. It combines two mainstays of heuristic search: branch-and-bound and best-first.

TODO: Write this.

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