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Causal Local States: Scalable Simultaneous Causal Network Inference and Forecasting for Dynamical Systems
Jonas Braun, Fabian Fischbach, Daniel Köglmayr, Sebastian Baur, Christoph Räth
TL;DR
Forecasting methods often hide interaction structure, while causal discovery does not optimize inferred networks for prediction. CLS jointly selects predictive local neighborhoods and forecasts each node from its selected inputs. Across three benchmarks, it reconstructs neighborhoods accurately and achieves forecasting performance on par with models supplied with the true network, although real-world validation remains open.
Problem
Existing forecasting methods are often black boxes, while causal discovery can infer interpretable networks without accounting for predictive utility or heterogeneous node-specific neighborhoods.
Method
CLS independently ranks candidate neighbors, selects the smallest near-optimal predictive neighborhood with a wrapper, removes redundancy, and combines local neighborhoods for whole-system forecasting.
Results
Across three benchmarks, CLS reconstructs neighborhoods accurately and produces forecasts on par with local-states models supplied with the true network.
Takeaways & Limitations
CLS provides a scalable framework that combines prediction with an explicit, inspectable approximation of causal structure in complex dynamical systems.
Takeaways & Limitations
CLS has not yet been applied to real-world datasets, and performance depends on selecting a wrapper model suited to the system.
Abstract
from arXiv · showhide
Machine learning methods predict many real-world systems with remarkable accuracy, but they are typically treated as black boxes that offer no insight into which interactions drive the dynamics. Causal discovery methods reconstruct the interaction network from observational data, but without regard to whether the inferred structure supports prediction. Existing approaches combining both tasks rely on a single global hyperparameter, such as a causal threshold or a fixed neighborhood size, which cannot recover the structure of heterogeneous systems. Here we introduce causal local states (CLS), a framework that simultaneously infers an approximate Granger-causal interaction network and forecasts the system dynamics. For each node independently, we select the smallest set of neighbors that allows a predictive model to forecast the node near-optimally, and the resulting neighborhoods are then combined for a forecast of the full system. On three benchmarks of increasing difficulty, we achieve reconstruction of the underlying networks with high fidelity and forecasts on par with a model that is supplied with the true network, providing a step toward explainable and scalable forecasting of complex systems.
Introduction
CLS addresses the gap between black-box forecasting and causal network reconstruction by making an interpretable interaction structure part of prediction. It replaces globally tuned neighborhood choices with per-variable selection based on predictive utility.
- Motivation: Forecasting models can predict complex systems accurately while remaining opaque about which variables interact.Structuring data into interaction networks can support both interpretability and variable selection.
- Motivation: Causal discovery reconstructs interpretable graphs from observational data, but typically does not account for whether the inferred structure improves prediction.Methods include Granger causality, transfer entropy, convergent cross mapping, and multivariate algorithms.
- Contribution: CLS jointly infers one local neighborhood per variable and uses those neighborhoods to forecast the whole system with an inspectable interaction network.The framework reports the variables each prediction relies on rather than hiding structure inside one monolithic model.
- Conceptual basis: CLS uses Granger’s predictive definition of causality to select variables whose past improves the target forecast after conditioning on relevant data.A wrapper-based procedure evaluates candidate variable sets according to prediction quality.
- Conceptual basis: Local-states methods make high-dimensional forecasting tractable by predicting each core node from a small neighborhood with its own model.CLS extends this decomposition by constructing neighborhoods through causal and predictive selection.
- Research gap: Existing hybrid approaches retain links above a causal threshold or choose the top q neighbors using one global hyperparameter for every node.This shared threshold or neighborhood size cannot accommodate heterogeneous variables with different predictive neighborhoods.
Results
CLS infers each node’s neighborhood by filtering causal candidates, selecting a predictive subset, and eliminating redundancy, then combines these local structures for system-wide forecasting. Across attractor, Lorenz96, and power-grid benchmarks, it recovers useful structure and achieves forecasting performance comparable to models given the true network, while wrapper choice remains important.
- Framework: CLS ranks candidate neighbors for each core node, evaluates nested candidates with a wrapper, removes redundant features, and assembles the resulting neighborhoods into a global matrix.Each node is then forecast from its own inferred neighborhood in parallel.
- Framework: The causal Markov condition motivates replacing one d-dimensional forecast with d local problems because a variable’s parents suffice for prediction.Additional non-parent neighbors do not invalidate the decomposition, but they increase computation and reduce interpretability.
- Lorenz63–Rössler: Candidate neighborhoods containing all same-attractor true neighbors produce one-step MSEs several orders of magnitude below candidates containing neither.This monotone separation supports using predictive loss as the wrapper criterion, although exhaustive search becomes combinatorially infeasible.
- Lorenz63–Rössler: In 90-dimensional Lorenz63–Rössler systems, post-elimination TPR remains above 0.8 across tested sizes while FPR is suppressed to near zero.The per-node inference procedure scales linearly in the number of core nodes when parallelized.
- Lorenz96: On Lorenz96, CLS retains every true parent across ten seeds with TPR = 1.000±0.000 and reduces FPR from 0.187±0.024 to 0.146±0.007.The inferred neighborhoods contain additional nearby lattice nodes beyond the three governing-equation parents.
- UK power grid: In the UK power-grid benchmark, Kuramoto-NGRC reduces FPR from 10.62 ± 0.49% to 1.87 ± 0.15% while TPR changes from 98.1 ± 0.5% to 96.1 ± 1.1%.Backward elimination removes many spurious neighbors selected by the CCM filter while largely preserving true neighbors.
- UK power grid: CLS with Kuramoto-NGRC achieves prediction quality comparable to a local-states model using the true underlying network.The benchmark also shows that polynomial NGRC can diverge on oscillator systems, making wrapper-model choice important.
Discussion
CLS jointly infers causal neighborhoods and forecasts high-dimensional dynamical systems, with benchmark performance supporting accurate, scalable, and more interpretable prediction. Its practical scope remains bounded by model-selection and hyperparameter challenges, along with unresolved extensions to temporal feature selection and real-world datasets.
- Contribution: CLS simultaneously infers each node’s causal neighborhood and forecasts the full system, making the inferred interaction structure an inspectable output.The framework combines filter- and wrapper-based inference and uses the resulting neighborhoods for prediction.
- Results: Across benchmarks, CLS accurately reconstructed neighborhoods and achieved forecasting performance on par with local-states models given the true network.Results included an almost exact UK power-grid adjacency matrix with Kuramoto-NGRC and comparable short-term accuracy.
- Scalability: Node-wise neighborhood inference allows both inference and prediction to parallelize, supporting scaling to high-dimensional systems.Each neighborhood is inferred separately from the full system.
- Open challenges: The authors identify application to real-world datasets as an unresolved challenge despite potential relevance to finance, power grids, and weather forecasting.These domains require both predictive performance and interpretability.
- Open challenges: CLS currently identifies causal neighbors but does not select relevant temporal features, limiting potential insight into system dynamics and predictive improvement.The authors suggest tree-based NGRC models with built-in feature selection as a possible direction.
- Open challenges: CLS uses shared hyperparameter configurations across local models, although this is unlikely to be optimal and efficient optimization remains open.Each local model may require its own hyperparameter set, but exhaustive search is impractical.
Methods
CLS infers a local neighborhood for each core node by combining causal filtering, predictive wrapper selection, and backward elimination, then uses the neighborhoods for full-system forecasting. The procedure is parallelizable, uses NGRC models, and is limited by combinatorial feature-space growth in high dimensions.
- Causal Local States: CLS decomposes a d-dimensional system into d core nodes, infers one local neighborhood per node, and combines them into a global neighborhood matrix for prediction.Inference is performed independently for each core node, making the procedure trivially parallelizable.
- Causal filtering: The filter step ranks candidate neighbors with a directed causal measure and restricts exhaustive neighborhood search to dmax + 1 nested candidates.This reduces the search space from O(2^d) to dmax + 1 candidates, with causal-matrix computation having O(d^2) complexity.
- Wrapper selection: Wrapper selection trains a predictive model for each candidate neighborhood and chooses the smallest candidate achieving near-optimal core-node performance under threshold α1.Wrapper loss is evaluated on a held-out wrapper test set and reflects prediction of the core node alone.
- Backward elimination: Backward elimination removes the neighbor whose deletion least degrades prediction when the relative deterioration remains within threshold α2.The procedure is motivated by the causal Markov condition: removing spurious neighbors should harm prediction less than removing true causal neighbors.
- Forecasting model: NGRC serves as both wrapper and per-node forecasting model, constructing features explicitly from time-delay coordinates and polynomial nonlinearities rather than a random recurrent reservoir.The nonlinear expansion improves accuracy but causes combinatorial feature-space growth as neighborhood dimension and polynomial order increase.
- Causal filtering: CCM can replace transfer entropy as the causal filter score, with fixed embedding settings and library size used to rank candidate influences.The paper uses CCM in an adapted fixed-library ranking role rather than its usual convergence analysis.
Author contributions statement
The authors describe individual roles in algorithm development, experimentation, conceptualization, coding, supervision, interpretation, and manuscript writing.
- J. B. developed the algorithm and performed the experimental studies.
- F. F. and S. B. contributed to conceptualization of the initial algorithm and coding.
- C. R. initiated and supervised the work, while all authors interpreted the findings and wrote the manuscript.
Declaration of Interests
The paper reports a related patent application and financial interests involving authors connected to commercialization of reservoir computing.
- The German Aerospace Center has filed a patent application related to the work.
- C. R., S. B., and D. K. have financial interests as co-founders of Entrox Systems, which is commercializing RC.
- The remaining authors declare no competing interests.
Supplementary Note 1: A global causal threshold cannot reconstruct even a simple network
A single global neighborhood size or causal threshold cannot recover heterogeneous interaction structure because causal scores differ across subsystems and can rank true and spurious neighbors inconsistently.
- Global neighborhood size: q = 1 misses the true Rössler parent Rz, while q = 5 adds three independent Lorenz variables ranked above it.TE scores are Ry = 1.03, Rz = 0.37, Lx = 0.46, Lz = 0.43, and Ly = 0.42.
- Global neighborhood size: No global neighborhood size q recovers the neighborhood of Rx correctly in the composite Lorenz63–Rössler system.The example shows that a fixed neighborhood size cannot separate the two attractors despite the system’s small dimension.
- Global causal threshold: A global threshold also fails because the two attractors evolve on different scales, making TE values incomparable across subsystems.At threshold 0.28, the Lx neighborhood includes spurious Ry and Rx links with TE = 0.39.
Supplementary Note 2: Why a network reconstruction with causal-discovery algorithms is not optimal for prediction
Causal-discovery methods optimize structural correctness, whereas the structure that best supports prediction can differ in finite or real data. Edge importance varies, and predictive but noncausal links can sometimes improve forecasts.
- Causal-discovery algorithms target structural correctness rather than forecast accuracy, so their returned graph need not be prediction-optimal.
- Global significance levels, thresholds, or neighborhood sizes cannot recover heterogeneous neighborhoods without false negatives or spurious edges.
- Importance of edges for prediction: Finite-data errors make equal-priority edge recovery harmful when edges differ substantially in their effects on prediction.
- Importance of edges for prediction: Removing one edge from the UK power-grid model can affect different numbers of nodes and change their valid prediction steps by different amounts.
- Spurious links can improve prediction: A hidden-confounder link can be spurious yet predictive because observed variables may echo the same unmeasured state.
Supplementary Note 3: Choice of the wrapper model and self-influence
The wrapper model must distinguish useful neighborhoods from spurious ones without letting the core node’s own past mask neighbor contributions. NGRC inference mode provides that separation and is used for neighborhood selection, while the framework remains model-agnostic.
- The framework compares reservoir computing and next-generation reservoir computing as wrapper models, with NGRC offering comparable accuracy at lower training cost and fewer hyperparameters.
- Sample strategies: The one-step strategy resets the neighborhood every step, whereas the multi-step strategy permits forecast errors to accumulate between periodic resets.
- Sample strategies: NGRC inference mode predicts the next core state from neighbors alone, excluding the core node’s own past from the feature vector.
- Wrapper choice: Only NGRC inference mode cleanly separates true-neighbor neighborhoods from spurious ones because one-step self-prediction can mask neighbor contributions.
- Self-influence: The full-system forecast restores core-node time delays as inputs, while NGRC remains replaceable because the framework is model-agnostic.
Supplementary Note 4: Simulation and model hyperparameters
The supplementary material documents the simulation, causal-measure, wrapper, prediction, and neighborhood-selection settings for the three benchmarks. It also specifies data splitting and notes that hyperparameters were handpicked rather than deeply optimized.
- The benchmark configurations record dynamical-system parameters, RK4 integration, causal measures, NGRC variants, and CLS thresholds α1 and α2.
- Wrapper evaluation uses a separate wrapper train/test split so the final prediction test set remains unseen during neighborhood selection.
- Hyperparameters were handpicked for interpretable, nondivergent behavior rather than selected through deep hyperparameter search.
- The three supplementary tables provide hyperparameters for the composite Lorenz63–Rössler, Lorenz96, and UK power-grid benchmarks.