  Inferring causal relations from observational time series data is a key
problem across science and engineering whenever experimental interventions are
infeasible or unethical. Increasing data availability over the past decades has
spurred the development of a plethora of causal discovery methods, each
addressing particular challenges of this difficult task. In this paper we focus
on an important challenge that is at the core of time series causal discovery:
regime-dependent causal relations. Often dynamical systems feature transitions
depending on some, often persistent, unobserved background regime, and
different regimes may exhibit different causal relations. Here, we assume a
persistent and discrete regime variable leading to a finite number of regimes
within which we may assume stationary causal relations. To detect
regime-dependent causal relations, we combine the conditional
independence-based PCMCI method with a regime learning optimisation approach.
PCMCI allows for linear and nonlinear, high-dimensional time series causal
discovery. Our method, Regime-PCMCI, is evaluated on a number of numerical
experiments demonstrating that it can distinguish regimes with different causal
directions, time lags, effects and sign of causal links, as well as changes in
the variables' autocorrelation. Further, Regime-PCMCI is employed to
observations of El Ni\~no Southern Oscillation and Indian rainfall,
demonstrating skill also in real-world datasets.
