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Hierarchical Representations for Evolving Acyclic Vector Autoregressions (HEAVe)

Cameron Cornell, Lewis Mitchell, Matthew Roughan

TL;DR

HEAVe introduces a hierarchical evolutionary framework to fit acyclic VAR models, enabling direct modeling of hierarchical causal structure in time series. By evolving node hierarchies and constraining VAR coefficients, HEAVe produces DAG-consistent networks that preserve most predictive power while enhancing interpretability and structural insight. Across synthetic simulations and a 100-cryptocurrency empirical study, HEAVe achieves strong predictive performance with substantially pruned networks and accurate recovery of hierarchical relationships, suggesting a largely hierarchical underlying dynamic. The approach offers a flexible baseline for DAG identification in econometrics and broader time-series applications, balancing accuracy, interpretability, and scalability.

Abstract

Causal networks offer an intuitive framework to understand influence structures within time series systems. However, the presence of cycles can obscure dynamic relationships and hinder hierarchical analysis. These networks are typically identified through multivariate predictive modelling, but enforcing acyclic constraints significantly increases computational and analytical complexity. Despite recent advances, there remains a lack of simple, flexible approaches that are easily tailorable to specific problem instances. We propose an evolutionary approach to fitting acyclic vector autoregressive processes and introduces a novel hierarchical representation that directly models structural elements within a time series system. On simulated datasets, our model retains most of the predictive accuracy of unconstrained models and outperforms permutation-based alternatives. When applied to a dataset of 100 cryptocurrency return series, our method generates acyclic causal networks capturing key structural properties of the unconstrained model. The acyclic networks are approximately sub-graphs of the unconstrained networks, and most of the removed links originate from low-influence nodes. Given the high levels of feature preservation, we conclude that this cryptocurrency price system functions largely hierarchically. Our findings demonstrate a flexible, intuitive approach for identifying hierarchical causal networks in time series systems, with broad applications to fields like econometrics and social network analysis.

Hierarchical Representations for Evolving Acyclic Vector Autoregressions (HEAVe)

TL;DR

HEAVe introduces a hierarchical evolutionary framework to fit acyclic VAR models, enabling direct modeling of hierarchical causal structure in time series. By evolving node hierarchies and constraining VAR coefficients, HEAVe produces DAG-consistent networks that preserve most predictive power while enhancing interpretability and structural insight. Across synthetic simulations and a 100-cryptocurrency empirical study, HEAVe achieves strong predictive performance with substantially pruned networks and accurate recovery of hierarchical relationships, suggesting a largely hierarchical underlying dynamic. The approach offers a flexible baseline for DAG identification in econometrics and broader time-series applications, balancing accuracy, interpretability, and scalability.

Abstract

Causal networks offer an intuitive framework to understand influence structures within time series systems. However, the presence of cycles can obscure dynamic relationships and hinder hierarchical analysis. These networks are typically identified through multivariate predictive modelling, but enforcing acyclic constraints significantly increases computational and analytical complexity. Despite recent advances, there remains a lack of simple, flexible approaches that are easily tailorable to specific problem instances. We propose an evolutionary approach to fitting acyclic vector autoregressive processes and introduces a novel hierarchical representation that directly models structural elements within a time series system. On simulated datasets, our model retains most of the predictive accuracy of unconstrained models and outperforms permutation-based alternatives. When applied to a dataset of 100 cryptocurrency return series, our method generates acyclic causal networks capturing key structural properties of the unconstrained model. The acyclic networks are approximately sub-graphs of the unconstrained networks, and most of the removed links originate from low-influence nodes. Given the high levels of feature preservation, we conclude that this cryptocurrency price system functions largely hierarchically. Our findings demonstrate a flexible, intuitive approach for identifying hierarchical causal networks in time series systems, with broad applications to fields like econometrics and social network analysis.
Paper Structure (24 sections, 16 equations, 4 figures, 2 tables)

This paper contains 24 sections, 16 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. HEAVe Methodology
  4. Graphical Structures
  5. Evolutionary Algorithm
  6. Representation
  7. Parent selection
  8. Recombination
  9. Mutation
  10. Vector Autoregression
  11. Acyclic VAR and evolutionary objective function
  12. Metrics
  13. Simulation Study
  14. Simulating VAR processes
  15. Generating $\Sigma_\epsilon$:
  16. ...and 9 more sections

Figures (4)

  • Figure 1: Canonicalisation of a hierarchy drops nodes to their lowest possible position.
  • Figure 2: Illustration of HEAVe’s indirect traversal of A-space through H-space EA and local VAR optimizations.
  • Figure 3: Performance metrics per generation, showing both the sample mean and bootstrapped 95% confidence intervals for this mean across the several hierarchical representations as well as our order-based comparison. The three subplots represent: (a) Mean objective function value, (b) Mean hierarchical score, and (c) Mean F1 link classification score.
  • Figure 4: Empirical Network node attributes.