On detecting anomalous behaviour in runs

Suppose that values in a finite sequence of data are labelled as either +1 or -1 depending, respectively, on whether they are either above or below the median. Using this sequence of +1 and -1 s a unit-lag autocorrelation coefficient may be determined. The present study establishes the probability distribution for the number of runs of a fixed length of -1 s (or +1 s) occurring in the labelled data. This distribution is calculated both with and without the constraint of the sample autocorrelation. The distribution is compared with observed stream flow data to illustrate its use in detecting both deficits and excesses of low or high flows of a given duration. The use of this distribution, which is not restricted to stream flow data, provides an extremely convenient alternative to the more traditional methods of detecting anomalous behaviour and avoids requiring knowledge of the form of the parent distribution.