Peaks over threshold

If X(i) is known with higher accuracy, a threshold criterion may be applied to detect extremes.

{Tout(j),xout(j)}7=1 = {T(i),X(i)|X(i) > u}n=1 (6.2)

is a rule for detecting maxima with a constant threshold, u. The extension to detecting minima is straightforward.

The peaks-over-threshold (POT) data can be analysed in two ways. Occurrence rate estimation (Section 6.3.2) uses the sample {t0ut(j)}j'L1 to infer trends in the occurrence of extremes. Fitting a generalized Pareto distribution (Section 6.2.2) to {x0ut(j)}j'l1 is helpful for studying the risk of an event of pre-defined size, prob(X(i) > u + v) with v > 0.

In climatology it is also useful to consider a time-dependent threshold to take into account effects of trends in mean, Xtrend(T), and variability, S(T). To fulfill the assumption of mutual independence of the POT data, imposing further criteria than passing the threshold may be necessary. Example: volcanic peaks in the NGRIP sulfate record (continued)

Outlier/extremes detection in the NGRIP sulfate record (Fig. 4.16) employed a time-dependent threshold, Xtrend(i) + z ■ S(i), and robust estimates of trend ("background") and variability, to take into account variable oceanic input. A second criterion was the absence of contemporaneous Ca and Na peaks to extract the extremes caused by volcanic eruptions (Fig. 1.4). To satisfy the independence assumption, further threshold exceedances closely neighboured in time were discarded (third criterion). In general, the size of such a neighbourhood can be estimated using persistence models (Chapter 2). Instead of taking {X0ut(j) }mm=1 from {X(i)}n=1, one may also collect scaled extremes {X0ut(j)}j'l1 from {[X(i) — Xtrend(i)]/S(i)}n=1. Scaling is one form of taking nonstation-arity into account (Section 6.3).

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