DecoR: Deconfounding Time Series with Robust Regression

TL;DR

This work introduces Deconfounding by Robust regression (DecoR), a novel approach that estimates the causal effect using robust linear regression in the frequency domain using distributional assumptions on the covariates and proves upper bounds for the estimation error of DecoR that imply consistency.

Abstract

Causal inference on time series data is a challenging problem, especially in the presence of unobserved confounders. This work focuses on estimating the causal effect between two time series that are confounded by a third, unobserved time series. Assuming spectral sparsity of the confounder, we show how in the frequency domain this problem can be framed as an adversarial outlier problem. We introduce Deconfounding by Robust regression (DecoR), a novel approach that estimates the causal effect using robust linear regression in the frequency domain. Considering two different robust regression techniques, we first improve existing bounds on the estimation error for such techniques. Crucially, our results do not require distributional assumptions on the covariates. We can therefore use them in time series settings. Applying these results to DecoR, we prove, under suitable assumptions, upper bounds for the estimation error of DecoR that imply consistency. We demonstrate DecoR's effectiveness through experiments on both synthetic and real-world data from Earth system science. The simulation experiments furthermore suggest that DecoR is robust with respect to model misspecification.

Figures (9)

• Figure 1: Due to the hidden confounder, regressing $Y$ on $X$ in the time-domain (left) does not yield a consistent estimator of the true causal effect (dashed, yellow). The idea of DecoR (green, see Section \ref{['sec:sparse_deconfounding_robust']}) is to consider the data in the frequency domain. Even though it is unknown, which of the data points correspond to confounded (orange) and unconfounded (gray) frequencies, robust regression techniques can be used to estimate the causal effect. We prove that DecoR is consistent under weak assumptions if the confounding is sparse, see Section \ref{['sec:theoretical_g']}.
• Figure 2: Directed acyclic graph covered by Setting \ref{['set:robust_improved']}. The dashed arrow from $U_t$ to $Y_t$ indicates that we assume $U_t$ to be sparse and the effect on $Y_t$ to be additive,
• Figure 3: Synthetic experiment where $\epsilon_x$ and $U$ are generated by two independent band-limited processes (left) or two independent Ornstein-Uhlenbeck processes (right) and where we choose $\phi$ to be the cosine basis. For this experiment DecoR-Tor is used.
• Figure 4: Histogram of the frequencies excluded by DecoR in the final iteration (left) and centred average 5-year precipitation in the alps over time (right); here, the methods' outputs estimate the precipitation without external forcing. The "ground truth" data is the precipitation provided by the ClimEx project leduc2019climex.
• Figure 5: Visualization of DecoR (Section \ref{['sec:sparse_deconfounding_robust']}): in the time domain (left), the processes are confounded, so estimators based on least squares will generally be biased. Assuming that the confounding is sparse in the frequency domain (middle), we can apply robust regression methods to estimate the causal effect (right). We prove that DecoR is consistent under weak assumptions if the confounding is sparse; see Section \ref{['sec:theoretical_g_an']}.

Theorems & Definitions (25)

• Definition 1: $(\phi, G)$-sparse process
• Theorem 1
• Lemma 1
• Theorem 2
• Corollary 1: sub-Gaussian noise
• Theorem 3: Convergence properties of DecoR-BFS
• Theorem 4: Convergence properties of DecoR-Tor
• Proposition 1
• Definition 2: Cosine basis
• Definition 3: Haar basis haar1909theorie