The meteorological phenomenon of hot spells or heat waves is the focus of much research. I examine whether the concept has any statistical basis as well. Ideally, statistical modeling of extreme high temperatures should be based on extreme value theory. The effect of clustering at high levels can be quantified in terms of the "extremal index." One interpretation of this index is as the reciprocal of the mean cluster length. An extremal index of one indicates no clustering at high levels, whereas the farther the index falls below one the greater the degree of clustering.

A temporally dependent time series may still have no clustering at high levels (i.e., an extremal index of one). In particular, a first-order Gaussian process has no clustering at high levels, no matter how high the first-order autocorrelation coefficient. Such a process might be used as a statistical model for time series of daily maximum temperature. Despite this asymptotic property, it may well be that substantial enough clustering remains, at least in a pre-asymptotic sense, at levels of practical significance.

In practice, it is difficult to estimate the extremal index, as apparent clustering that gradually vanishes as the threshold increases can confuse the issue. The "intervals estimator," based on the times between consecutive exceedances of a high threshold, is one of the few techniques that does not require the a priori identification of clusters. We apply this estimator to synthetic time series generated from a first-order Gaussian process, as well as to observed time series of daily maximum temperature during the summer season. It is demonstrated that maximum temperatures do exhibit a substantial degree of clustering at high levels, more than could be produced in a pre-asymptotic sense by a Gaussian process. Hence the phenomenon of hot spells or heat waves has not only a meteorological basis, but a statistical one as well.

Reference

Furrer, E.M., R.W. Katz, M.D. Walter, and R. Furrer, 2010: "Statistical modeling of hot spells and heat waves." Climate Research, V. 43, pp. 191-205.