Scanning for Clusters of Large Values in Time Series: Application of the Stein-Chen Method

Change Detection by Monitoring Residuals from Time Series Models

“…The SC bound does not seem to be commonly used; however, related references are available [5][6][7][8][9][10]. For example, Ref.…”

“…The Poisson pdf with mean parameter λ ¼ N À 2 ðÞ p p provides an approximation Y to the true pdf W for the number of times {1 Â 1} occurs in a series of length n [5,6]. The value n À 2 ðÞ is used instead of n because the length 3 pattern could only be found starting at index 1, 2, … , n À 2:…”

“…The Stein-Chen (SC) method approximates the probability density function (pdf) that assigns probabilities to the number of times that a pattern such as I t , I tþ1 , I tþ2 ¼ 101 fg occurs, starting at position t in a binary time series of length n. In example 1, the original time series that is converted to binary is assumed to consist of a sequence of independent iid residuals that result from fitting any type of time series model. Recently the SC method was shown to provide an accurate Poisson-based approximation and corresponding total variation distance bounds in a time series context [5]. The binary values.…”

“…The probability p is the probability that the original time series X exceeds a threshold, and the I notation denotes an indicator or binary variable. As an aside, the SC method can also be applied if p is not constant over time, but the independence assumption is difficult to avoid [5][6][7][8][9][10]. Any type of time series model [1,2] can be fit, and then the resulting residuals become the original series that is thresholded to convert to binary; therefore, the application is quite general.…”

“…Nevertheless, Ref. [5] showed 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. Consider scanning for {1 x 1} with x = 0 or 1, with p = P(I i = 1) being quite small, such as 0.10 or less in a residual series of length n = 10.…”