Robust Causal Discovery in Real-World Time Series with Power-Laws

TL;DR

This work tackles robust causal discovery in real-world time series by exploiting power-law spectral structure. The authors propose PLaCy, a spectral-feature mapping that fits a power-law model to each process’ spectrum and applies Granger causality to the resulting intercepts and slopes, preserving causal structure under the transform. A theoretical invariance result supports using transformed spectral features for causal graph recovery, and extensive experiments on synthetic OU-based data and real rivers and air-quality datasets show improved F1 and competitive or higher TNR versus state-of-the-art baselines. The approach offers practical robustness to noise and non-stationarity, highlighting the value of frequency-domain analysis for causal inference in scale-free systems, and points to avenues for extending to non-VAR settings and latent confounders.

Abstract

Exploring causal relationships in stochastic time series is a challenging yet crucial task with a vast range of applications, including finance, economics, neuroscience, and climate science. Many algorithms for Causal Discovery (CD) have been proposed, but they often exhibit a high sensitivity to noise, resulting in misleading causal inferences when applied to real data. In this paper, we observe that the frequency spectra of typical real-world time series follow a power-law distribution, notably due to an inherent self-organizing behavior. Leveraging this insight, we build a robust CD method based on the extraction of power -law spectral features that amplify genuine causal signals. Our method consistently outperforms state-of-the-art alternatives on both synthetic benchmarks and real-world datasets with known causal structures, demonstrating its robustness and practical relevance.

Figures (5)

• Figure 1: Schematic illustration of the proposed methodology. The original time series, here $\mathbf{x}_1$ and $\mathbf{x}_2$, are segmented into overlapping windows (step *[height=1.8ex,charshrink=0.65]1). Then, for each window $k$, the amplitudes ($a^k_1$, $a^k_2$) and the exponents ($\lambda^k_1$, $\lambda^k_2$) of the power-law distributed spectra are computed (step *[height=1.8ex,charshrink=0.65]2). These give rise to new, multi-dimensional, time series: ($\mathbf{a}_1$,$\boldsymbol{\lambda}_1$) for $\mathbf{x}_1$ and ($\mathbf{a}_2$,$\boldsymbol{\lambda}_2$) for $\mathbf{x}_2$ respectively (step *[height=1.8ex,charshrink=0.65]3). Finally, multivariate Granger causality tests are performed on these new series, and the causal graph is constructed (step *[height=1.8ex,charshrink=0.65]4).
• Figure 2: Results on synthetic datasets, with $N=5, \ C = 0.5, \ \sigma_g^a = 1.0$.
• Figure 3: Stride and window length analysis.
• Figure 4: Generated synthetic processes.
• Figure 5: Non linear process. $N = 5$, $C = 1.0$, $s_g = 1.0$, $s_b = 1.0$.

Theorems & Definitions (12)

• Theorem 1: Invariance of the Causal Graph under Spectral Transformations
• proof : Proof sketch
• Theorem 2: Markovianity of Spectral Features Derived from a Markov Process
• proof
• Theorem 3: Asymptotic Gaussianity and Stationarity under Colored Noise
• proof
• Theorem 4: Stationarity of Intercept Sequence for Certain Non-Stationary Processes
• proof
• Theorem 5: Preservation of Linear Dependence under Spectral Transformation
• proof