Detecting dynamical changes in nonlinear time series using locally linear state‐space models

Abstract

Interest is growing in methods for predicting and detecting regime shifts—changes in the structure of dynamical processes that cause shifts among alternative stable states. Here, we use locally linear, autoregressive state‐space models to statistically identify nonlinear processes that govern the dynamics of time series. We develop both time‐varying and threshold models. In time‐varying autoregressive models with p time lags, AR(p), and vector autoregressive models for n‐dimensional systems of order p = 1, VAR(1), we assume that coefficients vary with time. We can infer an approaching regime shift if the coefficients indicate critical slowing down of the local dynamics of the system. In self‐excited threshold models, we assume that the time series is governed by two autoregressive processes; the state variable switches between them when the time series crosses a threshold value. We use the existence of a statistically significant threshold as an indicator of alternative stable states. All models are fit to data using a state‐space form that incorporates measurement error, and maximum likelihood estimation allows for statistically testing alternative hypotheses about the processes governing dynamics. Our model‐based approach for forecasting regime shifts and identifying alternative stable states overcomes limitations of other common metric‐based approaches and is a useful addition to the toolbox of methods for analyzing nonlinear time series.