A Probability Model for Analyzing Regime Shifts Over Time

Graphical and numerical techniques exist to represent the duration, magnitude, and intensity of climatic events. We have now developed statistical decision and limit methodologies for quantifying time series fluctuations and regime shifts. Here we analyze a reconstructed annual index of the Pacific Decadal Oscillation (PDO) by fitting parametric models to represent the duration and magnitude of climatic episodes. Duration is defined as the number of consecutive years PDO is above or below its overall median, and magnitude is the sum of PDO index values for any given duration. Our parametric model admits a natural interpretation and can be easily applied to many time series data sets. Assuming that a regime shift can occur every year, independently of prior years, with some small probability, we naturally obtain a class of standard waiting time distributions (waiting times for the regime shift). Because magnitude can be expressed as a random sum of N random variables (where N is the duration of the episode), its probability distribution can be approximated by a limiting distribution for random sums. We explicitly describe these distributions, and estimate their parameters from the PDO data obtaining a reasonably good fit. Given these statistical models for duration and magnitude of climatic episodes, we can now compute climatological probabilities, such as the likelihood of magnitude exceeding any given value. These models also enable us to quantify the statistical significance of any climatic episode, and help us decide whether two episodes are significantly different from one another.