Probabilistic Structure of Extreme Run Lengths in a Stationary Hydrologic Time Series

Abstract

Current probabilistic methods for maximum and minimum run lengths in a hydrologic time series need further development. Using the first-order Markov chain model and an extreme-value theory for randomly occurring events, this study investigates probabilistic properties of both extreme run lengths in a stationary hydrologic time series. The interaction between negative run length, positive run length, total run length, life span, and number of total runs was analyzed. As a result, a joint probabilistic space was defined for negative (or positive) and total run lengths, and a corrector was established for an existing cumulative distribution function (CDF) of negative run length. Lengths of incomplete runs were considered appropriately, especially when no total run occurred during the life span. A generalized CDF was formulated for each extreme value, and simplified versions of the generalized CDF were proposed as practical approximate methods, including expectation approximations and limiting approximations. Results of computational examples indicated that expectation approximations generally provided better results than limiting approximations for maximum negative run lengths, and the non-negligible probability of no total run occurring during the life span may have a significant influence on CDFs.