SpaceTime: Causal Discovery from Non-Stationary Time Series

TL;DR

SpaceTime addresses causal discovery in non-stationary, multi-context time series by unifying regime detection, context partitioning, and causal graph learning under a Minimum Description Length (MDL) framework. It employs Gaussian processes to model nonparametric causal mechanisms and uses an edge-greedy search to build a window causal graph, while kernel-based tests identify invariant contexts and regimes and a PELT-based method detects changepoints. Theoretical consistency is established under assumptions of regime persistence and independent mechanism changes, and the algorithm is validated on synthetic data and two real-world hydrology/meteorology case studies, revealing spatially varying causal strengths and regime-dependent structures. This approach enables joint inference of dynamic causal networks across space and time, providing insights into complex systems such as river discharge dynamics and biosphere-atmosphere interactions, with practical implications for understanding environmental and climate processes.

Abstract

Understanding causality is challenging and often complicated by changing causal relationships over time and across environments. Climate patterns, for example, shift over time with recurring seasonal trends, while also depending on geographical characteristics such as ecosystem variability. Existing methods for discovering causal graphs from time series either assume stationarity, do not permit both temporal and spatial distribution changes, or are unaware of locations with the same causal relationships. In this work, we therefore unify the three tasks of causal graph discovery in the non-stationary multi-context setting, of reconstructing temporal regimes, and of partitioning datasets and time intervals into those where invariant causal relationships hold. To construct a consistent score that forms the basis of our method, we employ the Minimum Description Length principle. Our resulting algorithm SPACETIME simultaneously accounts for heterogeneity across space and non-stationarity over time. Given multiple time series, it discovers regime changepoints and a temporal causal graph using non-parametric functional modeling and kernelized discrepancy testing. We also show that our method provides insights into real-world phenomena such as river-runoff measured at different catchments and biosphere-atmosphere interactions across ecosystems.

Figures (6)

• Figure 1: Left: A temporal causal graph representing the data-generating causal mechanism of three variables ($X_{(1)}$, $X_{(2)}$, and $X_{(3)}$). The bold edges indicate a local mechanism change across contexts ($g_1$ or $g_2$) or over time ($f_1$ or $f_2$). Right: Variable measurements from different contexts ($C_1$ and $C_2$) and under different temporal regimes ($R_1$ and $R_2$).
• Figure 2: Causal Discovery, Changepoint Detection and Regime Partitioning. We evaluate the methods on multiple time series with causal mechanism shifts across time and datasets, with non-linear functional form, Gaussian noise, where $|\mathcal{G}_{\boldsymbol \tau }|=5, |\mathcal{T}| = 200, |\mathcal{C}| =2, |\mathcal{R}| = 3, |\mathcal{L}| = 2,$ and a fraction $s=\frac{1}{2}$ of intervened edges.
• Figure 3: Drivers of River Discharge. Given temperature (T), precipitation (P) and river discharge (Q) in 307 catchments across Europe, we discover a common WCG $\mathcal{G}_{\boldsymbol \tau }$ for $\boldsymbol \tau =2$ (a) with a lagged edge $P\to \emph{Q}$, and illustrate its variability in causal strength between catchments (b).
• Figure 4: Changepoints of the Interaction between Precipitation and Runoff. We show the regime changepoints $\mathcal{L}$ and partition $\mathcal{R}$ that we discover for $P \to \emph{Q}$ with SpaceTime for selected catchments $d_1$ (CH), $d_2$ (PL), $d_3$ (GB). The colors denote different regimes.
• Figure 5: Biosphere-Atmosphere Interactions. For the FLUXNET data from 64 locations $\mathcal{D}$ in multiple years $\mathcal{T}$, we show the DAG $\mathcal{G}$ (a) that SpaceTime discovers. We visualize the causal strengths of the seven edges in $\mathcal{G}$ in two dimensions using $t$-SNE (b), where each sample $(x_1, x_2)$ corresponds to a fixed location $d_i$, month $m_j$, and year $y_k$. We recover regions that correspond to distinct underlying meteorological conditions (precipitation $P$, global radiation $R_g$).

Theorems & Definitions (6)

• Definition 1: Contexts
• Definition 2: Regimes
• Definition 3: Non-stationary Time Series
• Definition 4: Temporal Causal Graph assad:22:ts_survey
• Definition 5: Window Causal Graph assad:22:ts_survey
• Theorem 1: Consistency