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

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.