A Non-Parametric Approach to Detect Patterns in Binary Sequences

Abstract: In many circumstances, given an ordered sequence of one or more types of elements or symbols, the objective is to determine the existence of any randomness in the occurrence of one specific element, say type 1. This method can help detect non-random patterns, such as wins or losses in a series of games. Existing methods of tests based on total number of runs or tests based on length of longest run (Mosteller (1941)) can be used for testing the null hypothesis of randomness in the entire sequence, and not a specific type of element. Moreover, the Runs Test often yields results that contradict the patterns visualized in graphs showing, for instance, win proportions over time. This paper develops a test approach to address this problem by computing the gaps between two consecutive type 1 elements, by identifying patterns in occurrence and directional trends (increasing, decreasing, or constant), applies the exact Binomial test, Kendall's Tau, and the Siegel-Tukey test for scale problems. Further modifications suggested by Jan Vegelius(1982) have been applied in the Siegel Tukey test to adjust for tied ranks and achieve more accurate results. This approach is distribution-free and suitable for small sample sizes. Also comparisons with the conventional runs test demonstrates the superiority of the proposed method under the null hypothesis of randomness in the occurrence of type 1 elements.