Multiple window discrete scan statistic for higher-order Markovian sequences

• Published in: Journal of applied statistics Vol. 42; no. 8; pp. 1690 - 1705
• Main Authors: Coleman, Deidra A., Martin, Donald E.K., Reich, Brian J.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 03.08.2015; Taylor & Francis Ltd
• Subjects: Approximation; Athletes; Business; Cluster analysis; Clusters; Markov analysis; Markov chains; Markov processes; one-dimensional scan statistics; p-values; Professional basketball; Shot; Statistical methods; Statistics; Studies; Tracking; variable windows; United States > US
• ISSN: 0266-4763; 1360-0532
• DOI: 10.1080/02664763.2015.1005061

Accurate and efficient methods to detect unusual clusters of abnormal activity are needed in many fields such as medicine and business. Often the size of clusters is unknown; hence, multiple (variable) window scan statistics are used to identify clusters using a set of different potential cluster sizes. We give an efficient method to compute the exact distribution of multiple window discrete scan statistics for higher-order, multi-state Markovian sequences. We define a Markov chain to efficiently keep track of probabilities needed to compute p-values for the statistic. The state space of the Markov chain is set up by a criterion developed to identify strings that are associated with observing the specified values of the statistic. Using our algorithm, we identify cases where the available approximations do not perform well. We demonstrate our methods by detecting unusual clusters of made free throw shots by National Basketball Association players during the 2009-2010 regular season.