Mining Statistically Significant Substrings using the Chi-Square Statistic

TL;DR

This paper introduces an efficient algorithm to identify the most statistically significant substring in a sequence using the chi-square statistic, improving computational complexity from quadratic to sub-quadratic, with applications in cryptology, finance, and sports.

Contribution

The paper presents a novel O(n^{3/2}) algorithm for finding the most significant substring based on chi-square, outperforming previous methods with O(n^2) complexity.

Findings

Algorithm runs in O(n^{3/2}) time with high probability

Effective in detecting significant patterns in various domains

Outperforms existing heuristics in experiments

Abstract

The problem of identification of statistically significant patterns in a sequence of data has been applied to many domains such as intrusion detection systems, financial models, web-click records, automated monitoring systems, computational biology, cryptology, and text analysis. An observed pattern of events is deemed to be statistically significant if it is unlikely to have occurred due to randomness or chance alone. We use the chi-square statistic as a quantitative measure of statistical significance. Given a string of characters generated from a memoryless Bernoulli model, the problem is to identify the substring for which the empirical distribution of single letters deviates the most from the distribution expected from the generative Bernoulli model. This deviation is captured using the chi-square measure. The most significant substring (MSS) of a string is thus defined as the substring having the highest chi-square value. Till date, to the best of our knowledge, there does not exist any algorithm to find the MSS in better than O(n^2) time, where n denotes the length of the string. In this paper, we propose an algorithm to find the most significant substring, whose running time is O(n^{3/2}) with high probability. We also study some variants of this problem such as finding the top-t set, finding all substrings having chi-square greater than a fixed threshold and finding the MSS among substrings greater than a given length. We experimentally demonstrate the asymptotic behavior of the MSS on varying the string size and alphabet size. We also describe some applications of our algorithm on cryptology and real world data from finance and sports. Finally, we compare our technique with the existing heuristics for finding the MSS.