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An Adaptive Importance Sampling for Locally Stable Point Processes

Hee-Geon Kang, Sunggon Kim

TL;DR

This work addresses the challenge of estimating E_f[K(X)] for locally stable point processes in bounded regions. It introduces an adaptive importance sampling scheme that confines importance samples to homogeneous Poisson point processes and uses cross-entropy minimization to select an optimal intensity, achieving almost-sure consistency and asymptotic normality of the estimator. The authors establish rigorous convergence and CLT results, develop stopping criteria, and demonstrate the method on intensity estimation for stationary and inhomogeneous Strauss point processes, comparing performance against MH and perfect sampling. The numerical results show AIS often dramatically reduces time-variance and computation time, making it a practical alternative for estimating functionals of locally stable point processes, with some caveats when the nominal distribution is very distant from Poisson. The approach broadens the toolkit for spatial point process inference and opens avenues for sequential importance resampling to further enhance efficiency.

Abstract

The problem of finding the expected value of a statistic of a locally stable point process in a bounded region is addressed. We propose an adaptive importance sampling for solving the problem. In our proposal, we restrict the importance point process to the family of homogeneous Poisson point processes, which enables us to generate quickly independent samples of the importance point process. The optimal intensity of the importance point process is found by applying the cross-entropy minimization method. In the proposed scheme, the expected value of the function and the optimal intensity are iteratively estimated in an adaptive manner. We show that the proposed estimator converges to the target value almost surely, and prove the asymptotic normality of it. We explain how to apply the proposed scheme to the estimation of the intensity of a stationary pairwise interaction point process. The performance of the proposed scheme is compared numerically with the Markov chain Monte Carlo simulation and the perfect sampling.

An Adaptive Importance Sampling for Locally Stable Point Processes

TL;DR

This work addresses the challenge of estimating E_f[K(X)] for locally stable point processes in bounded regions. It introduces an adaptive importance sampling scheme that confines importance samples to homogeneous Poisson point processes and uses cross-entropy minimization to select an optimal intensity, achieving almost-sure consistency and asymptotic normality of the estimator. The authors establish rigorous convergence and CLT results, develop stopping criteria, and demonstrate the method on intensity estimation for stationary and inhomogeneous Strauss point processes, comparing performance against MH and perfect sampling. The numerical results show AIS often dramatically reduces time-variance and computation time, making it a practical alternative for estimating functionals of locally stable point processes, with some caveats when the nominal distribution is very distant from Poisson. The approach broadens the toolkit for spatial point process inference and opens avenues for sequential importance resampling to further enhance efficiency.

Abstract

The problem of finding the expected value of a statistic of a locally stable point process in a bounded region is addressed. We propose an adaptive importance sampling for solving the problem. In our proposal, we restrict the importance point process to the family of homogeneous Poisson point processes, which enables us to generate quickly independent samples of the importance point process. The optimal intensity of the importance point process is found by applying the cross-entropy minimization method. In the proposed scheme, the expected value of the function and the optimal intensity are iteratively estimated in an adaptive manner. We show that the proposed estimator converges to the target value almost surely, and prove the asymptotic normality of it. We explain how to apply the proposed scheme to the estimation of the intensity of a stationary pairwise interaction point process. The performance of the proposed scheme is compared numerically with the Markov chain Monte Carlo simulation and the perfect sampling.
Paper Structure (16 sections, 7 theorems, 141 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 16 sections, 7 theorems, 141 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Monte Carlo methods for locally stable point processes
  3. Metropolis-Hastings algorithm for the estimation of $E_f[K(X)]$
  4. Perfect sampling for the estimation of $E_f[K(X)]$
  5. The proposed estimator
  6. Importance sampling for the estimation of $E_f[K(X)]$
  7. An adaptive importance sampling estimator
  8. Asymptotic properties of the proposed estimator
  9. Convergence of the proposed estimator
  10. Asymptotic normality of the proposed estimator
  11. Stopping criteria
  12. Estimation of the intensity of a stationary pairwise interaction point process
  13. Numerical results
  14. Estimation of the intensity of a stationary Strauss point process
  15. Estimation of the expected number of points for an inhomogeneous Strauss point process
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that $\int_{S} \phi^2(\xi)d\xi < \infty$ and $E_{q_1}[K(X)^2] < \infty$. Then, for a point process $X$ following Poi$(S,1)$,

Figures (2)

  • Figure 1: Realizations of stationary Strauss point processes on the unit square with varying values of the parameters $\beta$, $\gamma$, $R$.
  • Figure 2: Plots of realizations for point processes on the unit square: (a) the stationary Strauss point process with $\beta = 100$ and $\gamma = 0.2$, (b) the stationary Strauss point process with $\beta = 100$ and $\gamma = 0.8$, (c) the Poisson point process with $\rho = 34$, and (d) the Poisson point process with $\rho = 62$.

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 4 more