Poisson limits and nonparametric estimation for pairwise interaction point processes

Abstract

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

Performance analysis of hypothesis testing for sparse pairwise interaction point processes

“…The preceding proof appeals to [3,Theorem 2.3]. The hypotheses of that theorem are identical to those of our limit theorem, except for our extra assumption that be positive almost everywhere.…”

“…The limit theorem of Appendix A shows that in the case of sparseness, the distribution of could be approximated by the distribution of the shot-noise random variable , and it is then shown that the distribution of could be computed using (6) or (12), depending on the form of the interaction function. While the analysis of (6) in Section IV-A is straightforward, the derivation of (12) in Section IV-B is more complicated, and the details are found in Appendix B. Extensions are treated in Appendix C. Finally, we note that our analysis of the sparseness conditions in Appendix D in the case of square regions leads to a simply computable value for the constant which appears in the characteristic function of in (5 (3) and (4)) and Now, the hypotheses of our theorem are sufficient for us to apply [3 …”

“…Then , and we have (3) Using this expression for , our goal is to approximate the probabilities in (1). Since the results below apply to both and , we simplify the notation by writing where has interaction , and it is understood that can be taken as either or as needed.…”

“…The sparseness conditions referred to in the paper are given for a sequence of regions whose areas grow as [3], [16]. The sparseness conditions are purely geometric constraints on how the region grows as the number of points increases.…”

Asymptotic distributions for the performance analysis of hypothesis testing of isolated-point-penalization point processes

“…Due to the complex nature of the joint pdf of interaction point processes, exact performance analysis of hypothesis testing is generally intractable even in the simple case of the Strauss process. Monte Carlo simulation and approximation theory have been extensively used in estimating the performance of hypothesis testing problems involving interaction point processes [3], [9].…”