The task of causal discovery from time series arises in many sciences, from climatology to econometrics. Most methods assume stationarity: links are considered constant over time. However, real systems often switch between regimes, and causal links can change direction, strength, or even sign. Neighboring works in our archive address evolution, horizontal gene transfer, and quantum correlations, but all of them, in one way or another, seek structure in data. This work stands out because it directly models discrete regimes with persistence and learns them without external labeling.
The method, named Regime-PCMCI, iteratively alternates two steps. In the first step, the PCMCI algorithm estimates the causal parents of each variable within the current regime partition, using conditional independence tests. In the second step, an optimization problem is solved: for each time point, the regime whose model best predicts the observed values is selected, subject to a constraint on the total number of switches. The process is repeated until convergence. To avoid local minima, several random initializations (simulated annealing) are run. In the experiments, linear models and partial correlation were used as the independence test.
On synthetic data with two regimes, the method correctly recovered the regimes on average in 94% of time steps for most scenarios, including changes in the sign of the link, its direction, and its lag. The hardest case was when only the strength of the link changed between regimes: there, accuracy dropped to 57%. For ten variables and two regimes, the method maintained high accuracy: the fraction of correctly found links was 0.94, and regimes were recovered with an error of 11.7%. On real data of El Niño and Indian rainfall, the method, without external information, identified two seasons: a summer regime with a strong negative link (coefficient −0.4) and a winter regime with almost no link, consistent with climatology.
This work shows that one can simultaneously learn hidden regimes and causal structures without requiring prior knowledge of the regimes. This distinguishes it from earlier approaches that either assumed known regimes or modeled continuous drift of links. Compared with neighboring works on evolution and quantum systems, the object here is different: discrete metastable states in macroscopic time series, and the method is for the first time brought to practical application on climate data. This extends the applicability of causal analysis to a broad class of nonstationary systems.
A natural extension is to move to nonlinear causal links, since PCMCI allows this, but regime optimization will become more challenging. Another direction is to account for contemporaneous causality, where links act at the same time instant. It may also be possible to improve the detection of changes in link strength, which currently performs worst. In applications, the method could be useful for finding regime-dependent links in climate, economics, or neuroscience.
The method could influence climatology, econometrics, and neurobiology, where regime switching is common. It also provides a tool for analyzing any time series with metastable states.
Immediate next steps include implementing a nonlinear version and validating it on other climate datasets. It is also worth investigating the theoretical convergence of the algorithm.
In the field of causal discovery, the question of how to handle hidden common causes (violation of causal sufficiency) remains open. Another unresolved problem is choosing the number of regimes without overfitting. This work partially solves the second problem via an information criterion, but the first remains.
In the climate example, the method, without any hints, split the year into two seasons corresponding to the summer monsoon and winter, even though it was given only the time series of El Niño and rainfall.
The idea of hidden regimes governing a system's behavior evokes Clifford Simak's 'Way Station', where different states of reality replace one another, and the observer sees only their consequences.