# CR 23:23-30 (2002) - doi:10.3354/cr023023

## Stochastic modeling of regime shifts

### Franco Biondi^{1,}*, Tomasz J. Kozubowski^{2}, Anna K. Panorska^{2}

ABSTRACT: 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 geometric 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.

KEY WORDS: Climatological probabilities · Environmental change · Proxy records · Pacific Decadal Oscillation · Random sums

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