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Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation

Isaias Banales, Arturo Jaramillo

arXiv:2610.19588v1stat.MLcs.LG

TL;DR

Estimating a Hawkes background intensity is an infinite-dimensional likelihood problem when the triggering mechanism is fixed and the background is an unknown measure. The paper develops distribution-valued WFR dynamics and projects them onto statistical manifolds; its consistency result is conditional, and its Gaussian-mixture implementation is not covered by that theorem. In southern Mexico seismicity data, the reported optimal fit uses σ2 = 0.6 and 6 mixture components.

  • Problem

    Estimating the Hawkes background intensity is an infinite-dimensional likelihood problem when the triggering mechanism is fixed and the background is an unknown finite measure with a regular density.

    Source: §1 · p. 2

  • Method

    The paper formulates distribution-valued WFR dynamics for the likelihood and projects their action onto tangent spaces of statistical manifolds.

    Source: §3.2 · p. 10, §3.4 · p. 13

  • Results

    For southern Mexico seismicity data, the reported optimal fit uses σ2 = 0.6 and 6 mixture components.

    Source: §6.3 · p. 26

  • Takeaways & Limitations

    The Gaussian-mixture construction gives a concrete finite-dimensional scheme, while the consistency theorem applies only conditionally to schemes satisfying its assumptions.

    Source: §5.4 · p. 22, §7 · p. 27

    The consistency theorem does not establish consistency of the Gaussian-mixture implementation because tangent approximation and high-probability measurable empirical equilibria are not established for it.

    Source: §5.4 · p. 22

Abstract

from arXiv · show

We consider the problem of estimating the background intensity of a Hawkes process when the triggering mechanism is known, treating the background as an unknown finite positive measure with a regular density. We propose a likelihood-based variational approach based on the Wasserstein-Fisher-Rao geometry. In this setting, the relevant first variations are naturally distribution-valued and therefore fall outside the classical gradient-flow formulation. We develop a distributional framework for the resulting dynamics and give them a finite-dimensional meaning through projections onto statistical manifolds. We identify the true background as the unique zero of a limiting generalized gradient and, under suitable stability, compactness, integrability and approximation conditions, establish a conditional consistency theorem for measurable estimators satisfying the empirical projected equations on events whose probabilities tend to one. Finally, we specialize the construction to Gaussian-mixture manifolds and obtain an explicit finite-dimensional scheme.

1. Introduction

The paper develops likelihood-based, distribution-valued Wasserstein-Fisher-Rao dynamics for estimating a Hawkes background intensity, then projects them onto statistical manifolds. It motivates the construction against existing estimation and projection approaches, and outlines a conditional consistency result and Gaussian-mixture implementation.

1. Introduction

  • The background intensity is an unknown non-negative measure, or a density when absolutely continuous, making its non-parametric estimation an infinite-dimensional likelihood problem with the triggering mechanism fixed.The paper studies this problem for Hawkes-type models of seismic activity.Source: §1 · p. 2

1.1. Existing approaches to background intensity estimation.

  • Existing background estimators include parametric likelihood, declustering and EM, kernel or histogram methods, and Bayesian non-parametric models.The paper notes that flexible intensity representations still require choosing a representation or smoothing mechanism.Source: §1.1 · pp. 2–3

1.2. Variational inference approach.

  • The original Hawkes likelihood has a distribution-valued first variation, so the regular WFR gradient-flow formula cannot be applied directly; its singular event terms yield exact point evaluations when tested against smooth directions.The paper applies tangent-space projection to this weak action and calls the result a projected Wasserstein-Fisher-Rao likelihood flow.Source: §1.2 · p. 4

1.2.1. Projected Wasserstein-Fisher-Rao likelihood flows.

  • The projected flow matches the generalized gradient's action on tangent directions, producing finite-dimensional estimating equations rather than generally maximizing likelihood on the chosen manifold.Its statistical target is whether empirical projected equilibria identify the true background, not whether trajectories converge in algorithmic time.Source: §1.2.1 · p. 5

1.2.2. Equilibrium.

  • A convergent empirical projected trajectory yields an empirical equilibrium equation under the stated regularity and immersion assumptions, but this implication does not establish trajectory convergence.The equilibrium consequence is conditional on a solution existing for all algorithmic times and converging to a parameter.Source: §1.2.2 · p. 5

1.2.3. Conditional consistency.

  • The limiting generalized gradient has the true background as its unique zero, and selected empirical equilibrium densities converge in probability under the theorem's assumptions.The convergence is in Hβ(Z) and L1(Z), with total masses converging as well; assumptions include tangent approximation and high-probability existence of measurable empirical equilibria.Source: §1.2.3 · pp. 5–6

1.3. Gaussian-mixture realization and numerical study.

  • The implementation uses fixed-bandwidth Gaussian mixtures, whose explicit tangent calculations yield a finite-dimensional ODE without discretizing the spatial domain.The numerical study varies component count, initialization, bandwidth, and the Wasserstein and Fisher-Rao weights; component count is an implementation parameter, not an asymptotic refinement scheme.Source: §1.3 · pp. 6–7

2. Model and background likelihood

The section fixes the triggering mechanism and estimates the time-independent background in a Hawkes/ETAS model. It isolates the background-dependent likelihood while noting that unrestricted maximization can be ill posed.

2.1. Hawkes processes and the ETAS model.

  • The ground process has a predictable intensity combining a time-independent background with excitation from earlier events, and the triggering mechanism stays fixed across candidate backgrounds.The excitation uses only events strictly before the current time.Source: §2.1 · p. 7, §2 · p. 7
  • In ETAS, productivity, the temporal distribution of direct descendants, and a mark-dependent spatial density specify the triggering kernel.The background is nonnegative with finite mass; the statistical analysis imposes additional conditions for a stationary version.Source: §2.1 · pp. 7–8
  • The finite-sample construction does not require stationarity; the statistical analysis uses conditions yielding a stationary, mixing version with finite mean rate and assumes no spatial stationarity.The stated statistical assumptions include α < β_GR.Source: §2.1 · p. 8

2.2. Likelihood for the background intensity.

  • The background likelihood uses observed event locations and their past excitation, while the mark-density factor is independent of the candidate background and can be omitted.Omitting that factor is likelihood factorization, not conditioning on the entire observed mark list.Source: §2.1 · p. 8, §2.2 · p. 8
  • The normalized objective has the same equilibria as the unnormalized likelihood, but unrestricted concentration at observed points can make likelihood arbitrarily large with bounded mass.The paper therefore uses projected estimating equations on smooth statistical manifolds rather than assuming an unrestricted maximizer exists.Source: §2.2 · p. 8

3. Projected Wasserstein-Fisher-Rao construction

The construction interprets the empirical likelihood’s distribution-valued first variation as a weak WFR direction, then realizes that direction on smooth finite-dimensional statistical manifolds. Sobolev regularity and tangent-space projection provide the conditions and mechanism for these dynamics.

3.1. Wasserstein Fisher Rao dynamics.

  • WFR geometry selects transport and reaction fields through a cost, and its formal gradient-ascent direction increases a functional with a regular first variation.The transport and reaction fields are determined by the spatial gradient and value of that first variation.Source: §3.1 · pp. 9–10

3.2. Distributional Wasserstein Fisher Rao gradient.

  • The empirical first variation contains Dirac event terms, so integration by parts transfers spatial derivatives to test functions and yields a meaningful weak WFR action.The weak action can then use the distribution-valued first variation without differentiating its Dirac masses.Source: §3.2 · pp. 10–11

3.3. Sobolev regularity.

  • For β > d/2 + 2, the weak empirical gradient extends continuously to H^β and is Lipschitz on bounded sets with event intensities bounded below.The Lipschitz bound is in H^−β and requires the candidate densities to satisfy the stated norm and event-intensity bounds.Source: §3.3 · pp. 11–12

3.4. Projected dynamics.

  • On a chosen smooth statistical manifold, the projected velocity is the tangent vector whose L2 pairings reproduce the ambient generalized-gradient action.This is not the gradient for the WFR metric restricted to the manifold, so likelihood ascent or maximization is not implied.Source: §3.4 · pp. 13–14
  • An empirical projected equilibrium satisfies g_T(θ) = 0, but the analysis does not assert that a global trajectory has a limit.The statistical question instead concerns empirical equilibria as the observation horizon and available tangent directions increase.Source: §3.4 · p. 14

4. Gaussian-mixture realization

The paper specializes the projected dynamics to fixed-bandwidth Gaussian mixtures, computing their tangent directions, Gram matrix, and generalized-gradient coefficients for numerical implementation.

4.1. Tangent directions.

  • The mixture parameters are log-weights and component locations, with tangent directions restricted to configurations where they are linearly independent.Coinciding or redundant components can violate this condition.Source: §4.1 · pp. 14–15, §4 · p. 14

4.2. Gram matrix.

  • The L2 Gram matrix of the Gaussian-mixture tangent directions is computed explicitly from Gaussian overlap identities.The derivation uses completed-square Gaussian moments for the weight, mixed, and location blocks.Source: §4.2 · p. 15

4.3. Projected generalized gradient.

  • The projected generalized-gradient coefficients are obtained by differentiating the Gaussian tangent directions and evaluating their pairings using Gaussian overlaps.These coefficients supply the right-hand side of the projected system.Source: §4.3 · pp. 15–16

4.4. Finite-dimensional implementation.

  • The implementation solves G(θ)v = g_T(θ), while optional regularization changes the trajectory but not the exact equilibrium equations.The coordinate-tied regularization does not remove geometric degeneracy of the original parametrization.Source: §4.4 · p. 16

5. Conditional consistency

The consistency argument links a deterministic limiting generalized gradient with the restricted equations satisfied by projected equilibria. Under compactness, integrability, tangent-approximation, and measurable-equilibrium conditions, those equilibria converge to the true background; Gaussian-mixture compatibility alone does not establish all these conditions.

5.1. Heuristic considerations.

  • The proof combines a deterministic limit, uniform convergence over candidates, and tangent-space approximation to relate projected equilibria to the full test-space condition.The tangent approximation lets restricted equilibrium equations control the generalized gradient on admissible test functions.Source: §5.1 · p. 17

Thus (5.3) holds whenever

  • The event-integrability bound uses a logarithmic-density derivative and does not require a global reciprocal bound or a uniform lower density profile.For Gaussian mixtures, the logarithmic-derivative bound is independent of component count and requires no lower bound on individual component weights.Source: §5.2 · p. 18, Thus (5.3) holds whenever · p. 22

5.2. Deterministic limit and identification.

  • For fixed candidates, stationary ergodic averages yield the empirical operator's deterministic limit, whose only zero is the true background when κFR > 0.The proof compares the stationary process with the actual empty-history observations.Source: §5.2 · pp. 18–20

5.3. Uniform convergence.

  • Uniform convergence over candidates and a test-function unit ball makes the empirical approximation applicable to data-dependent equilibria.The accompanying distributional norm bound supports control of tangent-approximation error.Source: §5.3 · p. 20

5.4. Tangent approximation and consistency.

  • With tangent approximation and measurable empirical equilibria on events whose probabilities tend to one, selected densities converge in probability in Hβ and L1, and their total masses converge.The result is conditional; the tangent-approximation property and high-probability equilibrium existence are not established for the Gaussian-mixture implementation.Source: §5.4 · pp. 21–22

5.5. Gaussian compatibility.

  • Fixed-bandwidth Gaussian mixtures with centers in a fixed ball and total mass in a positive bounded interval form a compact ambient class, but this verifies only analytical class conditions.Membership of the true density, tangent approximation, and high-probability equilibrium existence still require verification; fixed-bandwidth mixtures are not asserted to approximate arbitrary true densities.Source: §5.5 · p. 22, Thus (5.3) holds whenever · p. 22

6. Numerical experiments

The experiments apply the Gaussian-mixture scheme to a simulated Poisson process and Southern Mexico seismicity, with component merging yielding parsimonious intensity representations. The seismicity fit agrees with regional estimates and is described as physically preferable to the benchmark optimum, though it takes longer to fit.

6.1. Numerical considerations.

  • Nearby Gaussian components are merged at a threshold because center collapse makes the Gram matrix noninvertible; the resulting collapse also reduces model complexity without separate model selection.The merged component is placed at the average of the two centers and receives their combined weight.Source: §6.1 · p. 23
  • Across experiments, κW = κFR = 1; mixture variance is calibrated by grid search, with AIC and BIC selecting the same models in the reported experiments.The implementation uses ADAM to accelerate convergence and avoid numerical issues from differing gradient scales for weights and means.Source: §6.1 · p. 23

6.2. Simulated Poisson Process.

  • For the simulated Poisson process, σ^2 = 1.1 is selected by AIC and BIC, and the estimated intensity is described as a highly satisfactory approximation.The estimate uses 8 Gaussian components, versus 695 evaluations for standard KDE; the grid search took 166.24 seconds.Source: §6.2 · p. 24

6.3. Seismicity data in southern Mexico.

  • The seismicity analysis estimates spatial background intensity alongside ETAS parameters from the USGS catalog covering 2000–2016, with cutoff magnitude 4.3.The reported grid search across σ^2 values took 734.79 seconds.Source: §6.3 · pp. 25–26
  • The benchmark optimum is supercritical and may be explosive, whereas the authors’ subcritical optimum keeps the aftershock cascade finite and stable.The authors report that their algorithm is not faster, but achieves a parsimonious, physically meaningful optimum without latent variables.Source: §6.3 · pp. 26–27
  • The Southern Mexico ETAS fit uses σ^2 = 0.6 and 6 mixture components; its estimated background agrees with regional estimates and the map from.The reported fit took 71 seconds, compared with 16.79 seconds for.Source: §6.3 · pp. 26–27

7. Discussion

The discussion presents distribution-valued Wasserstein-Fisher-Rao dynamics and their statistical-manifold projections as a framework for likelihood-based estimation. It identifies conditional consistency results and Gaussian-mixture applications, while leaving several extensions and convergence rates for future work.

7. Discussion

  • The framework feeds the distribution-valued first variation into a formal Wasserstein-Fisher-Rao gradient flow, with closed-form tangent projections enabling tractable numerical schemes.Gaussian mixtures illustrate the construction, and the paper reports a conditional consistency theorem for projected equilibria.Source: §7 · p. 27
  • Gaussian-mixture manifolds illustrate intensity inference for Poisson and Hawkes processes, while nearby components are merged to produce parsimonious estimates.The paper also reports convergence of the empirical generalized gradient under suitable assumptions.Source: §7 · p. 27
  • Joint estimation of a non-parametric triggering kernel and background intensity, and extension to time-dependent backgrounds, are left for future investigation.The paper specifically notes the identifiability of the triggering and background components as a concern for joint estimation.Source: §7 · p. 27
  • Determining convergence rates for the proposed estimators remains an open question.Source: §7 · p. 28

Appendix A. Stationarity and prehistory estimates

The appendix constructs a stationary Hawkes process and couples it to the observed empty-history process to justify the limiting operator used in the consistency argument. Under subcritical ETAS conditions, the stationary process is mixing under time translations, and pre-zero immigrant families have vanishing average contribution.

Appendix A. Stationarity and prehistory estimates

  • Under subcritical ETAS conditions and finite total background mass, the model admits a stationary version that is mixing under time translations, so the unit-time shift is ergodic.Stationarity is only in time; no spatial stationarity is imposed.Source: Appendix A · p. 28
  • The Hawkes–Oakes construction yields finite expected cluster size 1/(1−n), so clusters are finite almost surely and bounded time intervals contain finitely many events.The argument uses Poisson immigrants and independent branching clusters.Source: Appendix A · p. 28
  • Finite mean event rate ensures finite stationary excitation almost everywhere and positive candidate denominators at observed events almost surely.Source: Appendix A · pp. 28–29

A.1. Contribution of the initial history.

The appendix couples the stationary and empty-history processes by isolating post-zero descendants of pre-zero immigrants. Bounds on those families show that their contribution vanishes in time averages, supporting comparison of the empirical operators.

A.1. Contribution of the initial history.

  • The stationary and empty-history processes differ on positive times only through post-zero events descended from immigrants before time zero.The coupling classifies families by immigrant ancestor, and the prehistory excitation includes events from those families before zero.Source: §A.1 · p. 29
  • The pre-zero-family bounds make both the average excitation discrepancy and the average contribution of additional events vanish, without requiring a temporal first moment.The estimates use the decaying, bounded function Ψ derived from the offspring-generation contributions.Source: §A.1 · p. 29

Appendix B. Identification of the background intensity

Appendix B establishes that the limiting generalized gradient is well-defined and continuous, and that its zero uniquely identifies the true background under κFR > 0. Appendix C sets up uniform empirical approximation and a bound for the tangent-approximation error.

Appendix B. Identification of the background intensity

  • The stationary expectation defines A(ρ) in H−β, and A is continuous in the candidate density.The expectation is finite by an integrable bound; continuity follows by dominated convergence.Source: Appendix B · pp. 30–31
  • The score representation follows from the stationary compensator identity, with the divergence term integrating to zero.The score vanishes only when ρ = ρ⋆, because the reciprocal expectation in the score formula is strictly positive almost everywhere.Source: Appendix B · pp. 30, 32
  • At a zero of A, local regularity permits testing the distributional equation, and the cutoff argument yields ∫ρ|v| dx = 0.The needed regularity is derived at a zero of A without differentiating the excitation field.Source: Appendix B · pp. 31–32
  • The equation A(ρ) = 0 identifies the true background: the converse rules out nonzero scores, giving v = 0 almost everywhere and hence ρ = ρ⋆ almost everywhere, with continuous representatives agreeing everywhere.The converse uses local regularity to test the resulting equation and a cutoff argument on the whole spatial domain.Source: Appendix B · pp. 31–32

Appendix C. Uniform convergence of the empirical generalized gradient

  • Appendix C targets data-dependent candidates by combining compactness, stationary uniform convergence, coupling to empty-history observations, and a uniform tangent-approximation error bound.These steps provide the approximation and error control used in Proposition 5.2.Source: Appendix C · p. 32

C.1. Compactness of the test class.

This subsection establishes compactness of the test-function parameter space and uniform convergence of the stationary empirical generalized gradient, extending the result to arbitrary horizons.

C.1. Compactness of the test class.

  • The inclusion Xβ+r into Hβ is compact, while Xβ+r is dense in Hβ.Rellich compactness, a tail bound, and interpolation yield compactness; smooth compactly supported functions give density.Source: §C.1 · p. 32
  • The product P = K × Br is compact, enabling the uniform ergodic theorem to establish the stationary uniform limit at integer horizons.The unit-block contribution is a measurable continuous random function on P with the required first-moment continuity.Source: §C.1 · pp. 32–33
  • The uniform limit extends from integer horizons to arbitrary T, and the scalar ergodic theorem also gives convergence for a fixed test function.The deterministic mass term cancels in the horizon comparison, and the remaining partial-block terms vanish.Source: §C.1 · pp. 33–34

C.2. Comparison with the observed process.

This subsection transfers the stationary uniform limit to the empirical operator computed from the observed process, yielding convergence in probability under the original data law.

C.2. Comparison with the observed process.

  • The observation-to-stationary comparison uses a fixed spatial ball to control denominators and the integrability assumption to control events outside it.The constants bounding candidate densities and their gradients on the ball need not be uniform as its radius grows.Source: §C.2 · p. 34
  • The comparison also holds in X−(β+r) for every r ≥ 0, without a temporal first moment or a relation between the localization radius and T.The limit is taken first in T for fixed radius, then as the radius tends to infinity.Source: §C.2 · p. 35
  • The two conclusions of Proposition 5.2 supply the bounds used in (5.14), with the uniform bound obtained by Markov’s inequality.The relevant suprema are measurable by separability and continuity of the pairings.Source: §C.2 · pp. 35–36
  • The empirical operator converges uniformly to A in probability under the original data law.The stationary limit holds almost surely on the coupling, while the observation-to-stationary comparison vanishes in mean and hence in probability.Source: §C.2 · pp. 35–36

Appendix D. Gaussian classes and spatial integrability

Appendix D verifies analytical conditions for the Gaussian-mixture class and derives a sufficient stationary spatial-moment condition from the background and offspring displacements. Appendix E gives pseudocode for alternating background and kernel fitting.

Appendix D. Gaussian classes and spatial integrability

  • The Gaussian calculations verify analytical assumptions but do not establish tangent approximation or the existence of empirical equilibria.This is the stated scope boundary of the compatibility result.Source: Appendix D · p. 36
  • Allowed Gaussian mixing measures generate a compact class of strictly positive densities in the specified density metric.Weak compactness of the mixing measures transfers to convergence in Hβ ∩ W1,1 and hence in dK.Source: Appendix D · p. 36
  • A bound on the stationary event integral follows from the Gaussian center bound and the spatial moment condition, verifying the event-integrability requirement.The logarithmic-derivative ratio is a weighted mean of centers of norm at most D.Source: Appendix D · pp. 36–37
  • A finite first spatial moment of the background and offspring displacements yields the required stationary spatial moment when h = n < 1.Only normalization of the temporal kernel is needed, not a temporal first moment.Source: Appendix D · p. 37

Appendix E. Pseudocode

  • The pseudocode alternates WFR background-density optimization with Hawkes self-excitation parameter fitting, and includes component merging and pruning.It initializes Gaussian centers and log-weights, updates them using the background velocity, then merges nearby components and removes low-weight components.Source: Appendix E · p. 38

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