Stochastic modeling of regime shifts

Probabilistic methods for modeling the distribution of regimes and their shifts over time are developed by drawing on statistical decision and limit theory of random sums. Multi-annual episodes of opposite sign are graphically and numerically represented by their duration, magnitude, and intensity. Duration is defined as the number of consecutive years above or below a reference line, magnitude is the sum of time series values for any given duration, and intensity is the ratio between magnitude and duration. Assuming that a regime shift can occur every year, independently of prior years, the waiting times for the regime shift (or regime duration) are naturally modeled by a geomet- ric distribution. Because magnitude can be expressed as a random sum of N random variables (where N is duration), its probability distribution is mathematically derived and can be statistically tested. Here we analyze a reconstructed time series of the Pacific Decadal Oscillation (PDO), explicitly describe the geometric, exponential, and Laplace probability distributions for regime duration and magnitude, and estimate parameters from the data obtaining a reasonably good fit. This stochastic approach to modeling duration and magnitude of multi-annual events enables the computation of probabilities of climatic episodes, and it provides a rigorous solution to deciding whether 2 regimes are significantly different from one another.