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Uncovering Causality from Multivariate Hawkes Integrated Cumulants

Massil Achab, Emmanuel Bacry, Stéphane Gaïffas, Iacopo Mastromatteo, Jean-Francois Muzy

arXiv:1607.06333v3stat.MLcs.LG

TL;DR

Estimating node-to-node causality from event timestamps has often required modeling Hawkes kernel shapes, despite potentially unrealistic shape assumptions and costly flexible fitting. The paper introduces NPHC, which estimates integrated kernels by matching cumulants; it reports robust results across kernel shapes and stronger MemeTracker performance than classical approaches.

  • Problem

    Recovering node influence and causality from timestamped events is important, but kernel-shape assumptions can be unrealistic and flexible kernel estimation can be computationally heavy.

    Source: §1 · pp. 1–3, §2.5 · p. 7

  • Method

    NPHC directly estimates the integrated-kernel matrix by matching empirical second- and third-order cumulants to those implied by candidate matrices.

    Source: §1 · p. 3

  • Results

    Across varied kernel shapes, NPHC provides fast, robust, reliable estimates, and on MemeTracker it outperforms classical EM and Wiener-Hopf approaches.

    Source: §5 · p. 20

  • Takeaways & Limitations

    The integrated-kernel matrix encodes self- and cross-influences and can be used to quantify system endogeneity and uncover network causality structure.

    Source: §5 · p. 20

    Scaling to d≫10^3 remains an open question; the cited state-of-the-art experiments use around 100 components or fewer.

    Source: §2.2 · p. 5

Abstract

from arXiv · show

We design a new nonparametric method that allows one to estimate the matrix of integrated kernels of a multivariate Hawkes process. This matrix not only encodes the mutual influences of each nodes of the process, but also disentangles the causality relationships between them. Our approach is the first that leads to an estimation of this matrix without any parametric modeling and estimation of the kernels themselves. A consequence is that it can give an estimation of causality relationships between nodes (or users), based on their activity timestamps (on a social network for instance), without knowing or estimating the shape of the activities lifetime. For that purpose, we introduce a moment matching method that fits the third-order integrated cumulants of the process. We show on numerical experiments that our approach is indeed very robust to the shape of the kernels, and gives appealing results on the MemeTracker database.

1 Introduction

Timestamp patterns in multivariate Hawkes processes can reveal mutual influence between nodes. The paper estimates integrated kernels directly with cumulant matching, avoiding the need to model the kernels’ shapes.

1 Introduction

  • Hawkes kernels encode the impact of events at node j on the activity of node i, including self- and cross-excitation.The paper frames timestamp patterns as a way to recover mutual influence among process components.Source: §1 · pp. 1–2
  • Parametric approaches often impose shared or exponential and power-law kernel shapes, which may miss pair-specific profiles or delayed peak impacts.Flexible nonparametric kernel estimation can also be computationally heavy because it fits kernels in an over-parameterized space.Source: §1 · p. 2
  • The integrated kernel matrix G represents mean direct triggering counts and encodes causality relationships between nodes.The paper relates these integrals to Granger causality.Source: §1 · p. 3
  • NPHC estimates G by matching second- and third-order cumulants, without modeling kernel shapes, and the paper proves estimator consistency.It estimates cumulants, then minimizes the L2 difference between their estimates and the cumulants implied by a candidate G; the authors describe it as the first nonparametric cumulant-matching approach for Hawkes processes.Source: §1 · p. 3

2 NPHC: The Non Parametric Hawkes Cumulant method

NPHC estimates integrated Hawkes kernels directly, using integrated cumulants to recover a matrix that represents direct influences and causal relationships. Its method uses third-order cumulants to resolve an ambiguity left by covariance information alone.

2.1 Branching structure and Granger causality

  • Each gij represents the average number of type-i events directly descended from a type-j event, so the matrix encodes direct causal relationships.The paper relates learning kernel integrals to uncovering causality and Granger causality.Source: §2.1 · pp. 3–4

2 NPHC: The Non Parametric Hawkes Cumulant method

  • NPHC estimates the integrated-kernel matrix by matching observed integrated cumulants to cumulants implied by a candidate matrix, without modeling kernel shapes.The loss combines covariance and contracted third-order cumulant estimates, scaled rather than using the computationally impractical optimal GMM weighting matrix.Source: §2.2 · p. 5

2.2 Integrated cumulants of the Hawkes process

  • Covariance alone supplies d(d + 1)/2 coefficients for d2 unknowns, while selected third-order cumulants provide enough information to identify the matrix.The method selects d2 components Kiij; their skewness information can break the covariance symmetry.Source: §2.2 · p. 5

2.3 Estimation of the integrated cumulants

  • The covariance and third-order cumulant estimators are biased, but the paper reports that this bias is small in practice and establishes consistency when H does not grow too fast relative to T.The integration domain is truncated to [−H, H]; the approximation is considered good when kernel support is smaller than H and the process is stable.Source: §2.3 · p. 6

2.4 The NPHC algorithm

  • NPHC minimizes a non-convex degree-six loss with AdaGrad, initializing from the covariance-matching family using the identity orthogonal matrix.The starting point is important for convergence; the paper reports that this choice leads to good convergence results.Source: §2.4 · pp. 6–7

2.5 Complexity of the algorithm

  • NPHC has complexity O(nd2 + Niter d3) and outperforms kernel-estimating baselines when n≫d; scaling to d≫10^3 remains open.Here n is the maximum event count per component.Source: §2.2 · p. 5, §2.5 · p. 7

2.6 Theoretical guarantee: consistency

  • As T grows, the estimated moment function converges in probability to its population counterpart, supporting the consistency result under the stated assumptions.The paper identifies stability as mandatory and assumptions 3 and 4 as sufficient for consistency of the integrated-cumulant estimators.Source: §2.6 · pp. 8–9

3 Numerical Experiments

The experiments compare NPHC with established baselines on simulated Hawkes processes, MemeTracker, and financial order-book data. NPHC performs better across the reported datasets and metrics, while its estimated financial influence matrix yields interpretable patterns.

3 Numerical Experiments

  • Tests covered rectangular, power-law, and exponential kernels, alongside MemeTracker’s top 100 sites and an eight-dimensional DAX order book.The simulated processes were 10-dimensional for rectangular and power-law kernels and 100-dimensional for exponential kernels; MemeTracker contains about 7 million events.Source: §3 · pp. 9–10
  • NPHC consistently outperformed all baselines on synthetic, MemeTracker, and financial data in Kendall rank correlation and relative error, and ran roughly 50 times faster.The reported comparison spans Rect10, PLaw10, Exp100, MemeTracker, and financial data.Source: §3 · p. 11, Table 2 · p. 11, Table 3 · p. 12
  • On Rect10, PLaw10, and Exp100, NPHC approximately recovered each block’s true value; its rank correlation was comparable or slightly better, with much lower relative error.On Exp100, both estimates recover the three blocks, but ADM4 overestimates two blocks while NPHC gives the same value on each.Source: Figure 2 · p. 10, §3 · p. 11
  • On MemeTracker, NPHC achieved 7% relative error versus 9%–19% for the baselines and attained a much better Kendall rank correlation.The paper notes that MemeTracker kernels are neither exponential nor power law, partly explaining NPHC’s performance.Source: §3 · p. 11
  • The DAX order-book estimate shows symmetric ask and bid roles, cross-excited price moves, and strong self-excitation in market, limit, and cancel orders.The authors relate self-excitation to persistent order flows and meta-order splitting, and report that bid-side cancel increases cause downward price moves.Source: Figure 3 · p. 11, §3 · p. 12

4 Technical details

This section supplies derivations for the stated equations and integrated-cumulant estimators, derives the objective’s scaling coefficient κ, and proves the paper’s theorem.

4 Technical details

  • The section derives the equations stated earlier and the estimators of the integrated cumulants.Source: §4 · p. 12
  • It derives the scaling coefficient κ that appears in the objective function.Source: §4 · p. 12
  • The section also proves the paper’s theorem.Source: §4 · p. 12

4.1 Proof of Equation (8)

The proof uses a characterization of second-order Hawkes-process statistics and evaluates a relation involving Laplace transforms at z = 0.

4.1 Proof of Equation (8)

  • The proof uses the characterization of second-order statistics, first formulated by Hawkes (1971) and generalized by Bacry and Muzy (2016).Source: §4.1 · p. 12
  • Using Id + Lz(Ψ) = (Id − Lz(Φ))^-1, the derivation sets z = 0 to obtain the next relation.Source: §4.1 · p. 13
  • The resulting relation yields Cij from the corresponding entry of the preceding equation.Source: §4.1 · p. 13

4.2 Proof of Equation (9)

The proof of Equation (9) begins by grouping terms from Equations (48)–(51) of Jovanović et al. (2015).

4.2 Proof of Equation (9)

  • The proof starts from Equations (48)–(51) of Jovanović et al. (2015) and groups some terms.Source: §4.2 · p. 13

4.3 Integrated cumulants estimators

The estimator derivations restrict integration to a finite interval and replace expectations with empirical counterparts; the section also addresses the remaining integral estimate.

4.3 Integrated cumulants estimators

  • The preceding derivation concludes by proving Equation (9).Source: §4.3 · p. 13
  • Restricting the integration domain in Equations (4) and (5) to (−H, H) allows the integrals and expectations to be permuted.Source: §4.3 · p. 13
  • Estimators (11) and (12) are obtained by replacing expectations with their empirical counterparts.Source: §4.3 · p. 13
  • The section also uses the preceding remark to estimate the last integral.Source: §4.3 · p. 14

4.4 Choice of the scaling coefficient κ

The section replaces the usual inverse weighting matrix with a projection onto scalar multiples of the identity, then uses the resulting eigenvalue sums to form the loss.

4.4 Choice of the scaling coefficient κ

  • Rather than compute the inverse weighting matrix, the method projects it onto matrices of the form αId.The projection is based on the weighting matrix evaluated at θ1.Source: §4.4 · p. 14
  • The projection sets α to the mean eigenvalue of the weighting matrix evaluated at θ1.Source: §4.4 · p. 14
  • The two residual blocks contribute squared-norm eigenvalue sums; setting R1 = 0 yields the loss in Equation (10).Source: §4.4 · p. 14

4.5 Proof of the Theorem

The proof adapts GMM consistency arguments to estimated integrated cumulants, and establishes consistency under the theorem’s stated assumptions. It also sets out the filtering and event notation used in the proof.

4.5 Proof of the Theorem

  • The proof relaxes standard GMM moment conditions and uses filtered cumulant estimates that become asymptotically unbiased as T grows.The global covariance and skewness integrals are not easily estimated without bias; the filtered covariance estimate is asymptotically unbiased.Source: §4.5 · pp. 14–15

4.5.1 Notations

  • The notation section uses “covariance” and “skewness” as shorthand for their integrated forms and defines a compactly supported filtering window.It also specifies the event sets used in the estimators.Source: §4.5.1 · p. 15

4.5.2 Consistency

  • Under the theorem’s assumptions, the estimator converges in probability to θ0.The stated conditions include a positive semi-definite weighting matrix with an identifying moment condition, kernel norm spectral radius below 1, and filtering constraints.Source: §4.5.2 · p. 16
  • For compact Θ, the consistency argument reduces to convergence of the estimates of L, C, and Kc.The proof then establishes convergence in probability for those estimates.Source: §4.5.2 · p. 16

5 Conclusion

NPHC estimates the integrated-kernel matrix by matching second- and third-order integrated cumulants. Experiments report robustness across kernel shapes and better MemeTracker results than classical approaches.

5 Conclusion

  • The NPHC algorithm estimates the integrated-kernel matrix by matching empirical cumulants of orders two and three, without estimating kernel shapes.Source: §5 · p. 20
  • The matrix can quantify system endogeneity and uncover network causality through the self- and cross-influences it represents.Source: §5 · p. 20
  • NPHC provides fast, robust, and reliable results when kernel shapes vary, while parametric and nonparametric baselines are sensitive to misspecification and numerically expensive.Source: §5 · p. 20
  • On MemeTracker, NPHC outperforms classical approaches based on EM algorithms or Wiener–Hopf equations; its financial-data results are consistent with known stylized facts.Source: §5 · p. 20
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