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Statistical Inference for Networks of High-Dimensional Point Processes

Xu Wang, Mladen Kolar, Ali Shojaie

arXiv:2007.07448v1stat.MLcs.LG

TL;DR

High-dimensional Hawkes network estimation has lacked tools to characterize uncertainty, and existing inference approaches do not directly cover dependence on the full history in continuous time. This paper develops a high-dimensional inference procedure using de-correlated scores and new concentration inequalities. Under the null, its estimated test statistic approaches a chi-square limit under the stated conditions.

  • Problem

    High-dimensional Hawkes network estimation lacks tools to characterize estimator sampling distributions and uncertainty, while inference for independent or fixed-lag settings does not directly address full-history continuous-time dependence.

    Source: §1 · p. 2

  • Method

    The paper extends de-correlated score inference to high-dimensional Hawkes processes and develops concentration inequalities for integrated-process statistics to support test convergence and estimation guarantees.

    Source: §1 · pp. 2–3

  • Results

    Under the null, the estimated test statistic approaches a statistic asymptotically distributed as χ² with d degrees of freedom, while alternative behavior depends on signal scaling.

    Source: §3 · p. 7, §4 · p. 10

  • Takeaways & Limitations

    The paper presents the first inferential framework for high-dimensional point processes and provides a procedure for constructing parameter confidence intervals.

    Source: §8 · p. 15

    The analysis uses a parametric transition function; inference for nonparametric or nonlinear Hawkes processes remains an area of interest.

    Source: §8 · p. 15

Abstract

from arXiv · show

Fueled in part by recent applications in neuroscience, the multivariate Hawkes process has become a popular tool for modeling the network of interactions among high-dimensional point process data. While evaluating the uncertainty of the network estimates is critical in scientific applications, existing methodological and theoretical work has primarily addressed estimation. To bridge this gap, this paper develops a new statistical inference procedure for high-dimensional Hawkes processes. The key ingredient for this inference procedure is a new concentration inequality on the first- and second-order statistics for integrated stochastic processes, which summarize the entire history of the process. Combining recent results on martingale central limit theory with the new concentration inequality, we then characterize the convergence rate of the test statistics. We illustrate finite sample validity of our inferential tools via extensive simulations and demonstrate their utility by applying them to a neuron spike train data set.

1 Introduction

Multivariate Hawkes processes model networks in which past events influence future events, and high-dimensional applications make uncertainty in estimated network connections important. This paper develops inference for these settings by addressing the challenges of dependence on the full history and continuous time.

1 Introduction

  • A Hawkes network represents a directed edge when the target’s future-event probability depends on the source’s history.The model is used to learn latent connectivity from multivariate point-process data.Source: §1 · pp. 1–2
  • High-dimensional applications lack tools for characterizing the sampling distributions and uncertainty of network estimators, despite the importance of inferential tools in scientific applications.Existing ℓ1-regularized procedures address estimation, but not uncertainty characterization.Source: §1 · p. 2
  • Inference methods for independent data do not directly extend to time series, while VAR inference addresses fixed lags rather than Hawkes processes’ dependence on the entire history.The continuous-time domain also requires different technical tools.Source: §1 · p. 2
  • The paper provides the first high-dimensional inference procedure for multivariate Hawkes processes that allows both excitatory and inhibitory effects.It adopts the de-correlated score framework and extends efficient confidence regions to this setting.Source: §1 · pp. 2–3
  • New concentration inequalities for integrated-process statistics and bounds on covariance eigenvalues address key challenges in establishing test-statistic convergence and the restricted eigenvalue condition.The inequalities support both test-statistic analysis and theoretical results for ℓ1-regularized estimation.Source: §1 · p. 3

2 The Linear Hawkes Process

The linear Hawkes process models each component’s event intensity using past events and a transfer function. Its integrated process summarizes the history of each component, while connectivity coefficients can represent excitation or inhibition.

2 The Linear Hawkes Process

  • A simple point process counts isolated events at random times, and a multivariate process groups one counting process for each component.The intensity is predictable from the history of the multivariate process.Source: §2 · p. 3
  • In a linear Hawkes process, a component’s intensity depends on past events through a background intensity and transfer functions.A transfer function describes the influence of an event in one unit on another unit’s intensity.Source: §2 · pp. 3–4
  • A parametric transfer function with a transition kernel yields an integrated stochastic process that summarizes each component’s entire history.An exponential transition kernel is one example given in the paper.Source: §2 · p. 4
  • Connectivity coefficient βij measures how unit j’s past affects unit i’s intensity; positive values represent excitation and negative values inhibition.The paper notes that inhibitory effects are of interest in neuroscience applications.Source: §2 · p. 4
  • The analysis assumes a stationary linear Hawkes model with constant spontaneous rates and transition strengths over the observation period.The linear formulation also has heteroscedastic noise whose variance depends on the integrated process.Source: §2 · p. 4

3 Testing

The testing procedure uses a de-correlated score statistic to test one or more connectivity coefficients, accounting for time-varying noise and estimated nuisance parameters. Under the null, the estimated statistic approaches a chi-square distribution.

3 Testing

  • The framework tests a selected set βiJ of d connectivity coefficients, with the general results valid when d is much smaller than p.The presentation focuses primarily on a single coefficient, corresponding to d = 1.Source: §3 · pp. 5, 7
  • The score uses variance-scaled design components and orthogonalizes the tested component against the other components to reduce sensitivity to nuisance-parameter estimation.The paper states that induced estimation error is asymptotically negligible and the limiting distribution does not depend on model-selection mistakes.Source: §3 · p. 5
  • For time-varying noise, the procedure estimates nuisance parameters with lasso and selects its tuning parameter using sequential cross-validation.The sequential validation sets follow their corresponding training sets in time order.Source: §3 · p. 6
  • The tested coefficients’ covariance quantity is estimated from sample covariance for a univariate test and as a matrix for a multivariate test.The multivariate covariance matrix has dimension d × d.Source: §3 · pp. 6–7
  • With high probability, the estimated de-correlated score statistic approaches a statistic that is asymptotically χ² with d degrees of freedom under the null.The testing procedure rejects when its rejection indicator equals one.Source: §3 · p. 7

4 Theoretical Guarantees

The theoretical results establish null and alternative distributions for the test statistic under stated stationarity, boundedness, and estimator-rate conditions. Dependence on the full event history yields slower convergence rates than in VAR models, while a faster alternative requires stronger transfer-function structure.

4 Theoretical Guarantees

  • The required estimator rates depend on sparsity, and when the connectivity matrix is block diagonal the projection sparsity is bounded by s ≤ 2ρ + 1.The general relation between projection sparsity and connectivity sparsity also depends on coefficient signs and scales and on the transition kernel.Source: §4 · p. 9
  • Under the null and the theorem’s assumptions, the estimated test statistic converges in distribution to χ² with d degrees of freedom, with a cdf approximation rate dominated by T^-1/8.The reported rate is slower than the T^-1/2 rate from the standard central limit theorem and convergence in VAR models.Source: §4 · p. 9
  • The null-distribution result requires stationarity-related model assumptions and sufficiently accurate estimates of the nuisance parameters.The stated conditions include rates for estimating βi and the projection coefficients; lasso satisfies them under the stated stationary-model assumptions.Source: §4 · pp. 7, 9
  • The paper’s concentration-based results converge more slowly than VAR results because Hawkes intensities depend on the entire history in continuous time.The comparison is explicitly made with Zheng and Raskutti (2019).Source: §4 · pp. 9–10
  • Under alternatives, φ determines the asymptotic behavior: for φ > 1/2 the test does not distinguish the alternative from the null, while for φ < 1/2 the statistic diverges.At φ = 1/2, it converges to a non-central χ² distribution.Source: §4 · p. 10
  • A martingale-compensated-process approach yields a faster rate comparable to VAR Gaussian deviation bounds, but requires more stringent transfer-function structure.The paper presents this as an alternative to its extension of a concentration inequality for weakly dependent observations.Source: §4 · p. 10

5 Confidence Regions

The section constructs confidence regions for connectivity parameters using a one-step estimator. The stated theorem gives constants that depend only on model parameters and the transition kernel.

5 Confidence Regions

  • The confidence region for β_ij is constructed from a one-step estimator based on the lasso estimate or another consistent estimator with the same estimation-error order.The supplied passage introduces the one-step estimator but does not include its full formula.Source: §5 · p. 11
  • The score quantity eS_ij uses the full estimated β_i, whereas bS_ij uses β_i,−j.Source: §5 · p. 11
  • The theorem states that its constants C_k depend only on the model parameters (μ, Θ) and the transition kernel function.Source: §5 · p. 11

6 Simulation Studies

Simulations evaluate testing and confidence-interval performance across chain, block, and random connectivity structures. As experiment length increases, the test controls type-I error, intervals have reasonable coverage, and power approaches the oracle.

6 Simulation Studies

  • The simulations use p = 50 component processes with chain, block, and random connectivity structures; the random graph has about 2% sparsity.The block structure comprises 25 mutually connected pairs.Source: §6 · p. 11, Figure 1 · p. 12
  • The study tests all p^2 connectivity coefficients and measures type-I error, power, and 95% confidence-interval coverage at experiment lengths T ∈ {200, 1000, 2000}.Performance is compared with an oracle procedure that knows which coefficients are nonzero.Source: §6 · p. 12
  • As experiment length increases, the test properly controls type-I error, confidence intervals have reasonable coverage, and test power approaches the oracle across the three graph structures.The oracle knows which coefficients are nonzero.Source: §6 · p. 12

7 Application

The application estimates functional connectivity separately for two laser intensities in a 25-neuron mouse olfactory-bulb dataset. More condition-specific edges appear at 20 mW/mm2 than in the control, while a permuted-data check yields a 0.32% rejection rate.

7 Application

  • The analysis uses one 10-second laser pulse at each of two intensities from the subject mouse with the most neurons, comprising 25 neurons.The spike trains were recorded at 30 kHz, giving 300,000 time points per replicate.Source: §7 · p. 12
  • The procedure is applied separately to each intensity level to estimate connectivity coefficients and their 95% confidence intervals.Source: §7 · pp. 12–13
  • At 20 mW/mm2, the application finds more condition-specific edges than at the 0 mW/mm2 control, and the confidence intervals corroborate testing results while adding insight into coefficient differences.The figure includes 95% confidence intervals for 12 unique edges with the largest estimated coefficients in either condition.Source: §7 · p. 14, Figure 3 · p. 14
  • The permuted data produce a total rejection rate of 0.32%, below α = 0.05 and suggesting that the assumptions are likely satisfied in this check.Permuting each neuron's spike train destroys both between-neuron connections and within-neuron temporal dependence.Source: §7 · p. 14

8 Discussion

The paper presents a theoretically guaranteed inference procedure for high-dimensional linear Hawkes processes, using a new concentration inequality and martingale central limit theory. Its transition function is parametric, and the discussion identifies nonparametric and nonlinear extensions as future directions.

8 Discussion

  • The paper develops a concentration inequality for first- and second-order statistics of integrated processes that summarize each component's entire history.Source: §8 · p. 14
  • Combining the inequality with a martingale central limit theorem yields upper bounds on test-statistic convergence rates, alongside a procedure for constructing parameter confidence intervals.Source: §8 · p. 15
  • The procedure uses a parametric transition function; nonparametric adaptive modeling and inference for nonlinear Hawkes processes are identified as future directions.Source: §8 · p. 15

Appendix A: Proof of Main Results

The proofs establish the test statistic’s limiting behavior under the null and across alternative signal strengths, then connect these results to confidence-region construction. Their technical bounds rely on concentration and estimation results for integrated Hawkes-process histories.

Appendix A: Proof of Main Results

  • Under the null, the test statistic converges to a χ2 distribution by comparing it with a version using the true time-varying variance and applying a martingale CLT.The variance estimate makes the convergence proof more involved than in the cited VAR results.Source: Appendix A · p. 18
  • Concentration bounds for integrated-process statistics support restricted-eigenvalue and estimation-error bounds, which in turn control the score and covariance estimates.The proof also uses bounded covariance eigenvalues and martingale inequalities for residual terms.Source: Appendix A · pp. 18, 20
  • For φ > 1/2, the test statistic has a central χ2 limit; at φ = 1/2 it has a non-central χ2 limit, while for φ < 1/2 it diverges.The test statistic converges to its population counterpart, and the alternative’s signal strength determines the limiting regime.Source: Appendix A · p. 19
  • The confidence-region argument establishes a χ2 limit for the debiased estimator’s statistic under both null and alternative hypotheses.It bounds the difference between that statistic and an auxiliary statistic with a χ2 limit.Source: Appendix A · p. 20

Proof of Theorem 1

  • The passage identifies direct comparison of bUT with UT as difficult, attributing that difficulty specifically to σ² being unknown and time-varying.Source: Proof of Theorem 1 · p. 21

Proof of Theorem 2

The proof of Theorem 2 treats three alternative signal-strength regimes separately. Its bounds account for the nonzero signal, with one key term controlled when φ > 1/2.

Proof of Theorem 2

  • The proof treats the φ > 1/2, φ = 1/2, and φ < 1/2 regimes separately, with the φ > 1/2 proof similar to the proof of Theorem 1.Source: Proof of Theorem 2 · pp. 23, 24, 27
  • For φ > 1/2, the proof requires that a signal-dependent term remain bounded as T increases.The proof notes that this bound is not guaranteed otherwise.Source: Proof of Theorem 2 · p. 24
  • Under φ < 1/2, the nonzero alternative adds a signal-dependent term to the score-error bound, changing its order.The stated dominating order is Op(ρT^(-2/5−φ) ∨ T^(-4/5)).Source: Proof of Theorem 2 · p. 28

Proof of Theorem 3

Theorem 3’s proof establishes a χ2 limit for an auxiliary statistic and bounds its difference from the debiased statistic using covariance-estimation errors.

Proof of Theorem 3

  • The auxiliary statistic weakly converges to a χ2 distribution, following the null-case proof strategy.The auxiliary score equals the score defined for the null hypothesis.Source: Proof of Theorem 3 · p. 29
  • The proof reduces the statistic-difference bound to controlling the gap between two covariance estimates.It bounds the covariance-estimate gap using estimator consistency, assumptions, and a matrix perturbation result.Source: Proof of Theorem 3 · pp. 29–30
  • The covariance-estimate difference is bounded at order Op(s^(3/2)ρT^(-2/5)).The proof combines bounds on its component terms before substituting the result into the statistic-difference bound.Source: Proof of Theorem 3 · pp. 30–31

Appendix B: Auxiliary Results

Appendix B presents auxiliary theoretical results supporting inference and high-dimensional estimation for Hawkes processes, including distributional approximations, covariance controls, and concentration and boundedness results. It also provides bounds for estimation and test statistics, extends a result to models with non-mutually exciting structure, and gives a sparsity relation for block-structured connectivity matrices.

Appendix B: Auxiliary Results

  • P(UT ≤ y) ≈ Fd(y) uniformly in y, with the result based on a martingale central limit theorem.The proposition assumes a stationary linear Hawkes model satisfying Assumptions 1–4.Source: Appendix B · p. 32
  • The eigenvalues of Υx are bounded above and away from zero, while those of Υj are comparable to Υx by model- and kernel-dependent constants.These covariance bounds hold under stationarity and Assumptions 1–4.Source: Appendix B · p. 34
  • Lemma 1 extends earlier results requiring non-negative transfer functions to linear Hawkes processes with non-mutually exciting structure.The passage names Bacry et al. (2011, Theorem 1) and Etesami et al. (2016, Theorem 3) as earlier results.Source: Appendix B · p. 35
  • Lemmas 3–4 establish high-probability auxiliary bounds under Assumptions 1–6, with probability at least 1 − C1p^2T exp(−C2T^1/5) for Lemma 4.The constants depend only on the model parameters and transition kernel function.Source: Appendix B · pp. 37–38
  • Lemma 5 bounds the difference between the test statistics under the null and alternatives with φ ≥ 1/2, retaining the same order of upper bound as under the null.The alternative-case bound holds with similar probability, up to a change in constants.Source: Appendix B · pp. 41, 44
  • Lemma 6 gives a further high-probability auxiliary bound for a stationary linear Hawkes model satisfying Assumptions 1–4.The bound holds with probability at least 1 − C1p^2T exp(−C2T^1/5).Source: Appendix B · pp. 44–45
  • Under Assumptions 3 and 4, x_j(t), σ_i²(t), z_j(t), and ϵ_i(t) are O(1) for all i, j, and t.The appendix attributes these bounds directly to Assumptions 3 and 4.Source: Appendix B · p. 46
  • The proof uses covariance nondegeneracy to show the minimum eigenvalue of the expected design matrix is strictly positive, supporting a high-probability restricted eigenvalue condition.The argument rules out linear dependence among the process components and rules out constant processes.Source: Appendix B · pp. 47–48
  • For a block-structured connectivity matrix, the appendix states that s ≤ 2ρ + 1.Here s is the maximum support size of w*_j and ρ is the maximum support size of β_i.Source: Appendix B · p. 49
  • Lemma 16 supports concentration bounds for first- and second-order statistics of x(t), which Lemma 17 extends to the scaled design z(t) = x(t)/σ_i(t).The extension handles the extra σ_i(t) term by expanding z(t) using x(t) and x(t)xᵀ(t), then applying the bounds for x(t).Source: Appendix B · pp. 51–52

Appendix C: An Alternative Concentration Inequality for Discrete Time Domain

Appendix C gives an alternative discrete-time concentration inequality for integrated Hawkes-process statistics, using a more stringent transition-kernel condition to improve the convergence rate of second-order statistics.

Appendix C: An Alternative Concentration Inequality for Discrete Time Domain

  • A more stringent transition-kernel structure than Assumption 4 yields an improved convergence rate for second-order statistics of x(t).Source: Appendix C · p. 55
  • Lemma 18 gives a discrete-time concentration inequality for a stationary linear Hawkes process under Assumptions 1–3 and 7.The stated alternative inequality covers first- and second-order integrated statistics.Source: Appendix C · pp. 55–56
  • Hanson–Wright bounds the second-order statistics using independent, centered innovations with bounded variance, while the first-order bound uses a sub-Gaussian deviation inequality.The resulting second-order concentration inequality is stated in the appendix.Source: Appendix C · pp. 59–60
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