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Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions
Michael Eichler, Rainer Dahlhaus, Johannes Dueck
TL;DR
The paper addresses how to represent and estimate Granger-causal structure in multivariate Hawkes processes without imposing parametric link functions. It proves the structure is encoded in the kernels and introduces a consistent discretization-based estimator using infinite-order autoregression.
Problem
Causal discovery in partially observed Hawkes systems requires tractable estimation despite the extremely large number of models needed for many variables.
Method
The paper discretizes the point process and estimates nonparametric Hawkes link functions with an infinite-order autoregression.
Results
The Granger causality structure is fully encoded in the Hawkes kernels, and the proposed estimator is shown to be consistent.
Takeaways & Limitations
The estimator is easy and fast to compute in higher dimensions, supporting causal-search procedures that fit many full and submodels.
Takeaways & Limitations
The paper establishes consistency but does not prove asymptotic normality, and the neural application cannot model inhibitory connections or refractory periods.
Abstract
from arXiv · showhide
Hawkes (1971) introduced a powerful multivariate point process model of mutually exciting processes to explain causal structure in data. In this paper it is shown that the Granger causality structure of such processes is fully encoded in the corresponding link functions of the model. A new nonparametric estimator of the link functions based on a time-discretized version of the point process is introduced by using an infinite order autoregression. Consistency of the new estimator is derived. The estimator is applied to simulated data and to neural spike train data from the spinal dorsal horn of a rat.
multivariate Hawkes processes.
For multivariate Hawkes processes, Granger-causality structure is encoded by the link functions and supports global Markov properties for causal analysis. The paper introduces a consistent, computationally efficient nonparametric estimator and applies it to neural spike trains despite model misspecification.
- Graphical structure: The graphical framework supports causal discovery by relating Granger noncausality to pathwise separation and addressing spurious causation from unobserved variables.Brute-force causal discovery would require fitting 2^d−d−1 models for d variables, making iterative estimation methods impractical.
- Graphical structure: Stationary multivariate Hawkes processes satisfy the global Granger causal Markov property, and every subprocess satisfies a corresponding global Markov property.The process is also Markov with respect to the moral graph derived from its Granger-causality graph.
- Nonparametric estimation: The paper introduces a nonparametric Hawkes-kernel estimator based on time discretization and infinite-order autoregression, without assuming an exponential link-function form.Consistency is established, and the estimator is described as easy and fast to compute in higher dimensions.
- Neural spike-train application: For ten spinal dorsal horn neurons observed for 100s, the analysis used h = 0.5 and k = 100 and found self-inhibition plus excitatory effects in five neurons.Excitatory effects appeared after delays ranging from 125 ms for neuron 5 to almost 500 ms for neuron 10.
- Neural spike-train application: Only 10% of the 90 possible directed links showed clear positive peaks, typically near 17 ms with intensity approximately 0.38 spikes per millisecond.Nine clearly non-zero link functions were represented as directed Granger-causality edges.
- Neural spike-train application: The neural-data application appeared reasonably fitted except for neurons 1 and 9, while refractory periods and inhibitory connections remained limitations of the Hawkes model.The estimation method was reported as robust against misspecification caused by refractory periods.
Appendix A. Pr oofs
The appendix proves that Hawkes-process likelihood factorization yields the global Markov property for the moral graph and establishes the technical bounds needed for estimator consistency. It concludes that the estimated link functions converge in probability to their discretized targets.
- Proof of Theorem 3.5: The likelihood factorizes over cliques of the moral graph, implying the process satisfies its global Markov property.The factorization is obtained from conditional measurability and remains valid as t0 tends to −∞.
- Proof of Theorem 3.5: The graph-separation argument shows that NA does not Granger cause NB relative to the subprocess NA∪B∪C.Paths from A to B are blocked by B ∪C under the stated moral-graph condition.
- Technical lemmas: Lemma A.1 establishes technical relationships between empirical and mean-substituted covariance quantities used in Theorem 4.1.The proof compares ˆΓh,k with ˜Γh,k through deviations of ¯Y h,k from its mean-based counterpart.
- Technical lemmas: Lemma A.2 supplies covariance, spectral, cumulant, and convergence bounds for the discretized process Y h.The proof uses stationarity, positive definiteness, bounded spectral quantities, and cumulant estimates of order O(h).
- Proof of Theorem 4.1: Theorem 4.1 follows because the three error terms are oP(h), yielding ∥ˆφh,k −φh,k∥2 = oP(1).The remaining terms are controlled using Lemmas A.1 and A.2 and converge to zero in probability.