An extreme value theory for sequence matching
Suppose there are two finite sequences \(X_ 1,X_ 2,...,X_ m\) and \(Y_ 1,Y_ 2,...,Y_ n\) where the letters \(\{X_ i\}\) and \(\{Y_ j\}\) are chosen i.i.d. on a countable alphabet with \(p=P\{X_{\ell}=Y_{\ell}\}\in (0,1)\). This paper studies the distribution of the longest contiguous run of matches between the X's and Y's, allowing at most k mismatches. The distribution is closely approximated by that of the maximum of (1-p)mn i.i.d. negative binomial random variables. The latter is shown to behave like the integer part of an extreme value distribution. Approximations to the expectation and the variance are found. The paper concludes with an example in which segments taken from the DNA sequence of the bacteriophage lambda are compared. The authors attribute the lack of fit from asymptotic prediction to the small sample properties of the distribution of the maximum k-interrupted match run length. However, this reviewer's experience is that the i.i.d. assumption is usually not valid for such data, and so this may be the cause of the lack of fit. Corresponding results for the more difficult case of insertions and deletions are still outstanding.
distribution of the longest contiguous run of matches
scientific article; zbMATH DE number 3971966