Probability distributions of time series with temporal correlation: From frequently occurring to extreme values

In many cases, an analytical distribution accurately represents the probabilities of either the frequently occurring or extreme values of a time series dataset, but not both. This situation leads practitioners to regularly use separate distributions to represent the same dataset, which is cumbersome, and generally inconsistent to have two representations of different portions of the same dataset. A new practical method is proposed that couples the distribution of the extreme values of a time series dataset based on Extreme Value Theory with a distribution that models the frequently occurring values. An original feature of the method is that it estimates probabilities of the extreme values of time series without requiring that these values be modeled as independent, a key assumption of Extreme Value Theory. This is useful because many time series, such as offshore wind speeds sampled at an hourly interval, include significant temporal correlation. The core of this method is a normalized exceedance ratio curve defined as the ratio of the exceedance probabilities between the time series and its subset of extreme values. This paper provides a detailed procedure to implement this method and includes two examples: a numerical experiment and a metocean hindcast of wind and wave.