High-recall causal discovery for autocorrelated time series with latent confounders

Andreas Gerhardus; Jakob Runge

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High-recall causal discovery for autocorrelated time series with latent confounders

Andreas Gerhardus, Jakob Runge

arXiv:2007.01884v3Jul 3, 2020stat.MEcs.LGstat.ML

TL;DR

Existing constraint-based methods have low recall for autocorrelated time series with latent confounders because conditional-independence tests can have low effect size. The paper proposes LPCMCI, which identifies ancestral relationships during edge removal and conditions on known parents; it is sound, complete, and order-independent in the oracle case, while simulations show higher recall with controlled false positives for autocorrelated continuous variables.

Problem
Existing causal discovery methods suffer from low recall when recovering lagged and contemporaneous links in autocorrelated time series with latent confounders.
Method
LPCMCI uses orientation rules during edge removal to identify ancestral relationships early, discards known non-ancestors from conditioning sets, and uses known parents as default conditions.
Results
LPCMCI achieves significantly improved recall over SVAR-FCI and SVAR-RFCI for autocorrelated continuous variables, with gains growing with autocorrelation and particularly strong for contemporaneous links.
Takeaways & Limitations
The method provides higher recall while maintaining false-positive control in the targeted autocorrelated continuous-variable setting, and its oracle output is sound, complete, and order-independent.
Takeaways & Limitations
The method relies on faithfulness and, like other constraint-based methods, cannot distinguish all members of Markov equivalence classes.

Abstract

from arXiv · show

We present a new method for linear and nonlinear, lagged and contemporaneous constraint-based causal discovery from observational time series in the presence of latent confounders. We show that existing causal discovery methods such as FCI and variants suffer from low recall in the autocorrelated time series case and identify low effect size of conditional independence tests as the main reason. Information-theoretical arguments show that effect size can often be increased if causal parents are included in the conditioning sets. To identify parents early on, we suggest an iterative procedure that utilizes novel orientation rules to determine ancestral relationships already during the edge removal phase. We prove that the method is order-independent, and sound and complete in the oracle case. Extensive simulation studies for different numbers of variables, time lags, sample sizes, and further cases demonstrate that our method indeed achieves much higher recall than existing methods for the case of autocorrelated continuous variables while keeping false positives at the desired level. This performance gain grows with stronger autocorrelation. At https://github.com/jakobrunge/tigramite we provide Python code for all methods involved in the simulation studies.

1 Introduction

Observational time-series causal discovery matters when interventions are infeasible, expensive, or unethical, but latent confounding, high dimensionality, and nonlinear dependencies make it challenging. The paper focuses on low recall when identifying lagged and contemporaneous links under autocorrelation.

Observational causal discovery supports understanding physical systems and predicting intervention effects when experiments are infeasible.
Causal graphs can guide interpretable variable selection for prediction and classification tasks.
Time-series causal discovery must address unobserved confounders, high dimensionality, nonlinear dependencies, and autocorrelation.
Existing latent causal discovery methods suffer from low recall when identifying lagged and contemporaneous causal links in autocorrelated time series.

2 Time series causal discovery in the presence of latent confounders

Time-series causal discovery with latent confounders combines stationarity, time-order, and faithfulness assumptions with graphical methods that represent directed, bidirected, and partially oriented links. A motivating example shows that existing methods can miss true links and produce false links or orientations because autocorrelation creates low-effect-size conditional-independence tests.

2.1 Preliminaries: The model assumes stationary structural vector-autoregressive processes, jointly independent noise variables, no causal cycles, no selection variables, and faithfulness.
2.1 Preliminaries: Latent variables are allowed, so the observed series form a subset of the full process and contemporaneous interactions remain acyclic.
2.1 Preliminaries: MAGs represent directed and bidirected links, while PAGs additionally allow partially oriented edges such as ◦→ and ◦−◦.
2.2 Existing methods: SVAR-FCI uses stationarity for edge removals and orientations, whereas SVAR-RFCI omits FCI’s potentially time-consuming second edge-removal phase.
2.3 Time-lag window: In the latent case, increasing the time-lag window can remove edges inside the original window, even with perfect statistical decisions.
2.4 Motivational example: In the motivating example, SVAR-FCI misses the true lagged link Yt−1→Zt, returns a false Yt−2→Zt link, and usually misses or misorients Xt↔Yt.
2.4 Motivational example: Low signal-to-noise ratios from unfavorable conditioning sets reduce effect size and CI-test power, causing missing links that can produce false positives and wrong orientations.

3 Latent PCMCI

LPCMCI increases CI-test effect sizes by restricting conditioning sets and adding known parents, identified early through interleaved edge removal and orientation. Its intermediate graph uses middle marks for sound causal interpretation, and the oracle algorithm is order-independent, sound, and complete.

3.1 Effect size in causal discovery: Effect size is the minimum CI-test statistic over all tested conditioning sets, so small minima can reduce detection power.LPCMCI addresses this by restricting tested sets and adding default conditions intended to increase the statistic without inducing spurious dependencies.
3.1 Effect size in causal discovery: LPCMCI discards conditioning sets containing known non-ancestors and adds known parents as default conditions.These choices are intended to increase effect size, reduce tested conditioning sets, and improve recall while avoiding inflated false positives through better-calibrated tests.
3.3 Orientations rules for LPCMCI-PAGs: LPCMCI-PAG middle marks give intermediate graphs an unambiguous causal interpretation while the algorithm is still removing and orienting edges.The intermediate representation preserves causal meaning at arbitrary interruption points; when the algorithm converges, the middle marks are empty and the graph is a PAG.
3.3 Orientations rules for LPCMCI-PAGs: The orientation rules may be applied at any point and in any order, unlike FCI and RFCI orientation phases that first require a completed PAG or collider orientation.This supports the algorithm’s order-independent output and early use of ancestral information.
3.1 Effect size in causal discovery: Novel orientation rules determine ancestral and non-ancestral relationships after only a few CI tests, before all incorrect links are removed.LPCMCI therefore interleaves edge removal and orientation to obtain the information needed for later conditioning sets.

4 Numerical experiments

The experiments evaluate LPCMCI across autocorrelation, model size, and maximum lag, comparing it with SVAR-FCI and SVAR-RFCI. LPCMCI generally achieves higher recall while controlling false positives, especially for autocorrelated continuous variables.

Experimental design: Datasets vary autocorrelation, cross-links, contemporaneous and lagged effects, observed-variable subsampling, and maximum time lag.Autocorrelations are sampled over ranges determined by a target a; models are stationary and use N = ⌈(1−λ)˜N⌉ observed variables with λ = 0.3.
Varying k: The largest recall and precision gain for LPCMCI occurs when k increases from 0 to 1, with further recall improvements converging by k = 4.Higher k increases conditioning-set cardinality and runtime, but cached CI tests and faster later l-steps keep the runtime increase marginal.
Varying autocorrelation: LPCMCI(k = 4) achieves more than 0.4 higher contemporaneous orientation recall and 0.1 higher lagged orientation recall than SVAR-FCI and SVAR-RFCI.LPCMCI(k = 0) already has higher adjacency and orientation recall as autocorrelation increases, while the baselines are on par at a = 0.
Varying model size: As the number of variables N increases, all methods lose adjacency and orientation recall, but LPCMCI starts from a much higher level and better controls false positives.For N = 3, both baselines fail to control false positives for lagged links, whereas false positives become controlled for larger N.
Varying maximum lag: Increasing maximum time lag τmax decreases lagged recall and precision for all methods, while different τmax values imply different true PAGs.Further lagged links appear for larger τmax, changing the evaluation target.
Additional experiments: Additional experiments report largely comparable relative performance across model parameters, sample sizes, nonlinear models with GPDC, and a residualization comparison.For non-time-series and discrete-variable models, LPCMCI is comparable to baselines at k = 0 but worsens with increasing k; further analysis is left for future research.

5 Application to real data

The application analyzes average daily river discharges in the upper Danube basin using measurements from the Iller, Danube, and Isar. The authors caution that heavy-rainfall extremes likely violate the stationarity assumption.

Application: LPCMCI is applied to average daily discharges from the Iller, Danube, and Isar rivers in the upper Danube basin.The measurements are provided by the Bavarian Environmental Agency.
Data caveat: Heavy-rainfall extreme events are expected to violate the stationarity assumption in the discharge data.The authors state that more detailed real-data applications and analyses of LPCMCI remain future work.

6 Discussion and future work

The discussion highlights LPCMCI’s recall gains for autocorrelated continuous variables and explains them through larger conditional-independence-test effect sizes. It also notes faithfulness and Markov-equivalence limitations, while motivating applications such as climate research.

Main findings: LPCMCI has significantly improved recall over SVAR-FCI and SVAR-RFCI for autocorrelated continuous variables, with gains increasing with autocorrelation.The improvement is particularly strong for contemporaneous links, while LPCMCI with k > 0 also has better-calibrated CI tests and false-positive control.
Interpretation: The method increases CI-test effect size by including causal parents in conditioning sets and entangles edge removal with orientation.The paper attributes the recall improvement in simulations to this design.
Limitations: The faithfulness assumption limits the method’s scope, and constraint-based methods cannot distinguish all members of Markov equivalence classes.The authors contrast this with SCM-based methods that distinguish more equivalence-class members but impose stronger dependency assumptions.
Motivation: Observational causal discovery is motivated by settings such as climate research and neuroscience where experimental manipulation may be impossible or unethical.The paper links causal climate analysis from observational satellite datasets to understanding and modeling climate change.

Supplementary material

The supplementary material defines the time-series and latent-confounding graph representations, reviews FCI, and details LPCMCI’s iterative edge-removal and orientation procedure. It also states guarantees for the evolving graph and records a computational-design limitation.

Graphical representations: The time series graph is an infinite, repeating DAG-like representation with nodes for variables and directed links encoding causal relationships across time.Parents, ancestors, descendants, adjacencies, lagged links, contemporaneous links, autodependencies, and cross links are defined from this graph.
Graphical representations: With latent variables and no selection variables, maximal ancestral graphs use directed and bidirected edges, while partial ancestral graphs represent their Markov equivalence classes.Observed-variable d-separation corresponds to m-separation in the latent-variable graph representation.
FCI background: FCI starts from a complete graph, removes edges using conditional-independence tests and separating sets, then orients unshielded triples and applies further orientation rules.The algorithm targets a maximally informative PAG under faithfulness, acyclicity, and an underlying structural causal model.
LPCMCI procedure: LPCMCI extends FCI-style reasoning with middle marks, orientation rules, and parent-based conditioning sets that can identify ancestral relationships during edge removal.Its ordering tests autodependencies first, then cross links by increasing lag, and does not depend on the ordering of the time-series variables.
Limitations: The authors use the standard R4 rule instead of the modified discriminating-path rule because order-independent conflict-resolved application of the latter is computationally intensive.They report that these choices work well in practice but do not claim they are optimal.
Guarantees: The evolving LPCMCI graph retains an unambiguous, sound causal interpretation even if the algorithm is interrupted at an arbitrary point.The stated lemmas establish that the graph remains an LPCMCI-PAG throughout the procedure and that certain oriented edges are preserved in the corresponding PAG.

S7 Definition and relevance of apdst and napdst sets

LPCMCI uses apdst and napdst sets to construct tighter conditioning sets for CI testing, exploiting inferred ancestry, time order, and graph structure. The resulting procedure supports sound, complete, and order-independent causal discovery while improving recall in several autocorrelated settings.

Set definitions: apdst and napdst sets provide conditioning-set candidates for LPCMCI's CI tests.Algorithms S2 and S3 test conditional independence using subsets of these sets.
Set construction: The sets are built from potentially directed paths, collider configurations, time order, and restrictions on edge marks and intermediate nodes.These structural constraints define which variables can enter the approximate D-Sep sets.
Set relevance: apdst sets contain D-Sep sets when A is an ancestor of B, while napdst sets contain them after non-ancestorships are identified and rule R0′a is exhausted.This establishes why the sets can replace broader Possible-D-Sep or pdst sets in relevant cases.
Effect-size criterion: LPCMCI's effect-size criterion compares conditional-dependence values after default conditions and candidate conditioning subsets are considered.The criterion selects sets minimizing the conditional-independence statistic and checks whether removing default conditions changes the dependence measure.
Theoretical guarantees: Under oracle assumptions, LPCMCI is sound and complete, and its output does not depend on the ordering of the time-series variables.The guarantees assume faithfulness, no causal cycles, no selection variables, and perfect CI decisions.
Numerical behavior: In simulations, LPCMCI generally achieved higher recall than SVAR-FCI and SVAR-RFCI, especially for contemporaneous links and stronger autocorrelation.The reported gains sometimes came with precision or false-positive trade-offs, particularly for higher preliminary-iteration settings.