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A nonparametric Bayesian analysis of independent and identically distributed observations of covariate-driven Poisson processes

Patric Dolmeta, Matteo Giordano

TL;DR

The paper develops a nonparametric Bayesian framework to estimate covariate-driven Poisson process intensities from i.i.d. realizations, modeling the intensity as $\lambda_\rho(x)=\rho(Z(x))$ with the reparameterization $\rho(z)=\rho^*\,\sigma\big(w(z)\big)$. A hierarchical multi-bandwidth Gaussian process prior with ARD kernels enables automatic adaptation to anisotropic smoothness, and posterior contraction rates are proven to be optimal in the $d_Z$-distance at rate $n^{-\alpha_0/(2\alpha_0+1)}$ up to log factors, where $\alpha_0=1/\sum_j\alpha_j^{-1}$. Inference is implemented via a Metropolis-within-Gibbs sampler employing a discretized high-dimensional $w$ and pCN updates, with likelihoods computed by numerical quadrature. Simulations demonstrate accurate recovery of anisotropic intensities and favorable comparison with kernel methods, while an application to Canadian wildfire data shows the approach effectively leveraging multiple covariates and years to reveal spatiotemporal risk patterns. The work advances adaptive, covariate-aware Bayesian nonparametrics for fixed-domain covariate-driven Cox processes, with practical relevance for environmental and ecological event data.

Abstract

An important task in the statistical analysis of inhomogeneous point processes is to investigate the influence of a set of covariates on the point-generating mechanism. In this article, we consider the nonparametric Bayesian approach to this problem, assuming that $n$ independent and identically distributed realizations of the point pattern and the covariate random field are available. In many applications, different covariates are often vastly diverse in physical nature, resulting in anisotropic intensity functions whose variations along distinct directions occur at different smoothness levels. To model this scenario, we employ hierarchical prior distributions based on multi-bandwidth Gaussian processes. We prove that the resulting posterior distributions concentrate around the ground truth at optimal rate as $n\to\infty$, and achieve automatic adaptation to the anisotropic smoothness. Posterior inference is concretely implemented via a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm that incorporates a dimension-robust sampling scheme to handle the functional component of the proposed nonparametric Bayesian model. Our theoretical results are supported by extensive numerical simulation studies. Further, we present an application to the analysis of a Canadian wildfire dataset.

A nonparametric Bayesian analysis of independent and identically distributed observations of covariate-driven Poisson processes

TL;DR

The paper develops a nonparametric Bayesian framework to estimate covariate-driven Poisson process intensities from i.i.d. realizations, modeling the intensity as with the reparameterization . A hierarchical multi-bandwidth Gaussian process prior with ARD kernels enables automatic adaptation to anisotropic smoothness, and posterior contraction rates are proven to be optimal in the -distance at rate up to log factors, where . Inference is implemented via a Metropolis-within-Gibbs sampler employing a discretized high-dimensional and pCN updates, with likelihoods computed by numerical quadrature. Simulations demonstrate accurate recovery of anisotropic intensities and favorable comparison with kernel methods, while an application to Canadian wildfire data shows the approach effectively leveraging multiple covariates and years to reveal spatiotemporal risk patterns. The work advances adaptive, covariate-aware Bayesian nonparametrics for fixed-domain covariate-driven Cox processes, with practical relevance for environmental and ecological event data.

Abstract

An important task in the statistical analysis of inhomogeneous point processes is to investigate the influence of a set of covariates on the point-generating mechanism. In this article, we consider the nonparametric Bayesian approach to this problem, assuming that independent and identically distributed realizations of the point pattern and the covariate random field are available. In many applications, different covariates are often vastly diverse in physical nature, resulting in anisotropic intensity functions whose variations along distinct directions occur at different smoothness levels. To model this scenario, we employ hierarchical prior distributions based on multi-bandwidth Gaussian processes. We prove that the resulting posterior distributions concentrate around the ground truth at optimal rate as , and achieve automatic adaptation to the anisotropic smoothness. Posterior inference is concretely implemented via a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm that incorporates a dimension-robust sampling scheme to handle the functional component of the proposed nonparametric Bayesian model. Our theoretical results are supported by extensive numerical simulation studies. Further, we present an application to the analysis of a Canadian wildfire dataset.

Paper Structure

This paper contains 33 sections, 6 theorems, 92 equations, 30 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Main notation
  4. Multi-bandwidth Gaussian process methods for covariate-based intensities
  5. The prior model
  6. Adaptive anisotropic posterior contraction rates
  7. Posterior contraction rates in the case of stationary covariates
  8. Posterior sampling via a Metropolis-within-Gibbs algorithm
  9. Simulation studies
  10. Results for univariate covariates
  11. Results for bivariate covariates
  12. Applications to a Canadian wildfire dataset
  13. Exploratory univariate analysis
  14. Full analysis
  15. Summary and discussion
  16. ...and 18 more sections

Key Result

Theorem 2.3

For $\alpha = (\alpha_1,\dots,\alpha_d)\in (0,\infty)^d$, let $\rho_0\in C^\alpha(\mathcal{Z})$ satisfy $\inf_{z\in\mathcal{Z}}\rho_0(z)>0$, and consider data $D^{(n)}\sim P_{\rho_0}^{(n)}$ arising as described at the beginning of Section Sec:GPMethods. Let $\Pi$ be a prior for $\rho$ constructed as in $P^{(\infty)}_{\rho_0}$-probability as $n\to\infty$, for all sufficiently large $L>0$ and some l

Figures (30)

  • Figure 1: Independent realizations of the covariates and the point pattern with intensity \ref{['Eq:1DTruth']}.
  • Figure 2: Left to right: Posterior means (solid blue), pointwise $95\%$-credible intervals (shaded blue) and averaged kernel estimates (solid red) for $n = 250, 500, 1000$. The ground truth $\rho_0$ from \ref{['Eq:1DTruth']} is shown in solid black in each plot for comparison.
  • Figure 3: Left to right: Posterior means for $n = 50, 250, 1000$ and the anisotropic ground truth \ref{['Eq:2DTruth']}.
  • Figure 4: Top row, left to right: Average temperatures (in Celsius), precipitations (in $\text{mm/m}^2$) and wind speeds (in km/h) in Ontario during June 2013. Bottom row: Observations for 2021. The wildfires are represented by black dots (respectively, $34$ and $130$ in total).
  • Figure 5: Left to right: Posterior means (solid line) and point-wise $95\%$-credible intervals (shaded region) for the wildfire intensity as a function of the average temperature, precipitation level and wind speed, respectively. The dashed lines show the kernel-based estimates.
  • ...and 25 more figures

Theorems & Definitions (17)

  • Remark 2.1: Repeated observations of covariates and points
  • Remark 2.2: Bounded covariate spaces
  • Theorem 2.3
  • Remark 2.4: Bounded away from zero intensities
  • Remark 2.5: Deterministic covariates
  • Remark 2.6: Discrete covariates
  • proof : (of Theorem \ref{['Theo:GPRates']})
  • Lemma A.2
  • proof
  • Lemma A.3
  • ...and 7 more