Scanning for Clusters of Large Values in Time Series: Application of the Stein-Chen Method
Abstract: The purpose of this application paper is to apply the Stein-Chen (SC) method to provide a Poisson-based approximation and corresponding total variation distance bounds in a time series context. The SC method that is used approximates the probability density function (PDF) defined on how many times a pattern such as It,It+1,It+2 = {1 0 1} occurs starting at position t in a time series of length N that has been converted to binary values using a threshold. The original time series that is converted to binary is assumed to consist of a sequence of independent random variables, and could, for example, be a series of residuals that result from fitting any type of time series model. Note that if {1 0 1} is known to not occur, for example, starting at position t = 1, then this information impacts the probability that {1 0 1} occurs starting at position t = 2 or t = 3, because the trials to obtain {1 0 1} are overlapping and thus not independent, so the Poisson distribution assumptions are not met. Nevertheless, the results shown in four examples demonstrate that Poisson-based approximation (that is strictly correct only for independent trials) can be remarkably accurate, and the SC method provides a bound on the total variation distance between the true and approximate PDF.
Cite this paper: Burr, T. and Henderson, B. (2021) Scanning for Clusters of Large Values in Time Series: Application of the Stein-Chen Method. Applied Mathematics, 12, 1031-1037. doi: 10.4236/am.2021.1211067.
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