Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Causal Structure Learning in Hawkes Processes with Complex Latent Confounder Networks

Songyao Jin, Biwei Huang

TL;DR

This paper shows that continuous-time event sequences can be represented by a discrete-time causal model as the time interval shrinks, and proposes a two-phase iterative algorithm that alternates between inferring causal relationships among discovered subprocesses and uncovering new latent subprocesses, guided by path-based conditions that guarantee identifiability.

Abstract

Multivariate Hawkes process provides a powerful framework for modeling temporal dependencies and event-driven interactions in complex systems. While existing methods primarily focus on uncovering causal structures among observed subprocesses, real-world systems are often only partially observed, with latent subprocesses posing significant challenges. In this paper, we show that continuous-time event sequences can be represented by a discrete-time causal model as the time interval shrinks, and we leverage this insight to establish necessary and sufficient conditions for identifying latent subprocesses and the causal influences. Accordingly, we propose a two-phase iterative algorithm that alternates between inferring causal relationships among discovered subprocesses and uncovering new latent subprocesses, guided by path-based conditions that guarantee identifiability. Experiments on both synthetic and real-world datasets show that our method effectively recovers causal structures despite the presence of latent subprocesses.

Causal Structure Learning in Hawkes Processes with Complex Latent Confounder Networks

TL;DR

This paper shows that continuous-time event sequences can be represented by a discrete-time causal model as the time interval shrinks, and proposes a two-phase iterative algorithm that alternates between inferring causal relationships among discovered subprocesses and uncovering new latent subprocesses, guided by path-based conditions that guarantee identifiability.

Abstract

Multivariate Hawkes process provides a powerful framework for modeling temporal dependencies and event-driven interactions in complex systems. While existing methods primarily focus on uncovering causal structures among observed subprocesses, real-world systems are often only partially observed, with latent subprocesses posing significant challenges. In this paper, we show that continuous-time event sequences can be represented by a discrete-time causal model as the time interval shrinks, and we leverage this insight to establish necessary and sufficient conditions for identifying latent subprocesses and the causal influences. Accordingly, we propose a two-phase iterative algorithm that alternates between inferring causal relationships among discovered subprocesses and uncovering new latent subprocesses, guided by path-based conditions that guarantee identifiability. Experiments on both synthetic and real-world datasets show that our method effectively recovers causal structures despite the presence of latent subprocesses.

Paper Structure

This paper contains 73 sections, 15 theorems, 29 equations, 10 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Partially Observed Multivariate Hawkes Process-based Causal Model
  4. Multivariate Hawkes Process
  5. Model Definition
  6. Structure Identification in Partially Observed Hawkes Processes
  7. From Continuous-Time to Discrete-Time Representation
  8. Structure Discovery Through Rank Constraints
  9. From Observed Parents to Latent Confounder Subprocesses
  10. Characterizing Latent Confounders via Rank Constraints
  11. Rank-based Discovery Algorithm
  12. Phase i: Identifying Causal Relations
  13. Phase ii: Discovering New Latent Subprocesses
  14. Experiments
  15. Synthetic Data
  16. ...and 58 more sections

Key Result

Proposition 3.4

Subprocess $N_i$ is locally independent (defined in Hawkes Process) of $\mathcal{N}_\mathcal{G} \backslash \mathcal{P}_\mathcal{G}$ given $\mathcal{P}_\mathcal{G}$ if and only if $\mathcal{P}_\mathcal{G}$ is the parent-cause set of $N_i$ in $\mathcal{N}_\mathcal{G}$.

Figures (10)

  • Figure 1: Figure 1: Illustration of multivariate Hawkes processes. (a) Point process representation with three subprocesses $N_1, N_2, N_3$, where the continuous timeline is partitioned into intervals of length $\Delta$. (b) The corresponding summary causal graph, the central object of this paper, with causal relations $N_1 \leftarrow N_2 \leftrightarrow N_3$ and self-loops on all nodes. (c) The window causal graph, showing the underlying time-lagged causal mechanism: each node denotes the count in one interval of length $\Delta$, modeled as a weighted sum of lagged parent nodes plus noise (Eq. \ref{['equ1']}). (d) A minimal example with a latent subprocess $L_1$ confounding $O_1$ and $O_2$, highlighting the primary focus of this paper.
  • Figure 2: Examples of causal graphs with latent confounder subprocesses. (a) Summary graph where $O_1,O_2,O_3,O_4$ are observed and $L_1$ is latent. (Unlike \ref{['fig1:subfig4']}, $O_1,O_2$ are shown without self-loops to simplify the derivation.) (b) Corresponding window causal graph among $O_1, O_2$, and $L_1$ with two effective lags. (c) More complex case where $L_1$ connects $O_1,O_2$ via intermediate latent subprocesses $L_2,L_3$. All subprocesses have self-loops except $L_2$ and $L_3$. (d) An even more intricate case, extending (c) with more complex intermediate latent subprocess paths and an additional edge $O_2 \to L_1$. All subprocesses except the intermediate latent ones have self-loops.
  • Figure 3: Illustrative examples of interactions among inferred latent confounder and the remaining observed subprocesses. In (a)--(c), assume $L_1$ has been inferred from its observed effects $\{O_1, O_2\}$. (a) $O_3$ causes $L_1$. (b) Both $L_1$ and $O_3$ cause $O_4$. (c) $L_1$ causes $L_4$, where $L_4$ can be inferred from $\{O_3, O_4\}$. (d) $L_1$ serves as the latent confounder of both inferred latent confounder $L_2$ and $L_3$.
  • Figure 4: F1-score comparisons for first four synthetic causal graphs (Cases 1--4), corresponding to the structures in Figs. \ref{['fig:subfig2']}, \ref{['fig3:subfig1']}, \ref{['fig5:subfig1']} and \ref{['fig5:subfig2']}. See \ref{['appendix_additional_results']} for additional cases.
  • Figure 5: Inferred causal subgraph from the cellular network dataset, where Alarm_id=7 is successfully identified as a latent subprocess.
  • ...and 5 more figures

Theorems & Definitions (37)

  • Definition 3.1: Partially Observed Multivariate Hawkes Process-based Causal Model (PO-MHP)
  • Definition 3.2: Cause and Effect
  • Definition 3.3: Parent-Cause Set
  • Proposition 3.4: Parent-Cause Set and Local Independence
  • Theorem 4.1: Hawkes Process as a Linear Autoregressive Model
  • Lemma 4.2: D-separation and Rank Constraints in the Window Graph
  • Proposition 4.3: Identifying Observed Parent-Cause Set
  • Definition 4.4: Symmetric Acyclic Path Situation
  • Proposition 4.5: Identifying Latent Confounder from Observed Effects
  • Definition 4.6: Observed Effects as Surrogates
  • ...and 27 more