Improvements on the distribution of maximal segmental scores in a Markovian sequence

TL;DR

This paper develops recursive formulas and new approximations for the distribution of maximal segmental scores in Markovian sequences, improving upon previous methods and validated through computational comparisons.

Contribution

It introduces novel recursive formulas and approximations for the distribution of maximal segmental scores in Markov chains, enhancing accuracy over existing methods.

Findings

New recursive formulas for the distribution of $S^+$.

Improved approximations for $Q_1$ and $M_n$ distributions.

Validation shows these methods outperform previous approaches.

Abstract

Let $(A_i)_{i \geq 0}$ be a finite state irreducible aperiodic Markov chain and $f$ a lattice score function such that the average score is negative and positive scores are possible. Define $S_0:=0$ and $S_k:=\sum_{i=1}^k f(A_i)$ the successive partial sums, $S^+$ the maximal non-negative partial sum, $Q_1$ the maximal segmental score of the first non-negative excursion and $M_n:=\max_{0\leq k\leq\ell\leq n} (S_{\ell}-S_k)$ the local score first defined by Karlin and Altschul (1990). We establish recursive formulae for the exact distribution of $S^+$ and derive new approximations for the distributions of $Q_1$ and $M_n$. Computational methods are presented in a simple application case and comparison is performed between these new approximations and the ones proposed by Karlin and Dembo (1992) in order to evaluate improvements.