🤖 AI Summary
This paper addresses the challenge of spectral modeling for multivariate inhomogeneous spatial point processes—particularly second-order intensity-reweighted stationary processes. Methodologically, it introduces a novel spectral analysis framework centered on the definition and estimation of a “pseudo-spectrum”: a matrix-valued, locally interpretable spectral density estimator that inherits the asymptotic properties of the classical stationary spectrum. The work extends periodogram asymptotics—previously established only for stationary settings—to multivariate nonstationary spatial point processes, yielding a consistent pseudo-spectrum estimator. It further proposes two data-driven, adaptive bandwidth selection strategies. The approach integrates kernel smoothing, spectral-domain modeling, and rigorous asymptotic analysis. Extensive simulation studies and application to real-world tropical forest tree species distribution data demonstrate the method’s effectiveness and robustness, substantially advancing the frequency-domain analysis capability for nonstationary spatial point processes.
📝 Abstract
In this article, we propose a spectral method for multivariate inhomogeneous spatial point processes. A key ingredient is utilizing the asymptotic behavior of the periodogram. The periodogram is an asymptotically unbiased estimator of the spectrum of a second-order stationary point process. By extending this property, we show that under inhomogeneity, the expectation of the periodogram also converges to a matrix-valued function, which we refer to as the pseudo-spectrum. The pseudo-spectrum shares similar properties with the spectrum of stationary processes and can be interpreted using local parameters. We derive a consistent estimator of the pseudo-spectrum through kernel smoothing and propose two bandwidth selection methods. The performance and utility of our frequency domain methods are illustrated through simulation studies and a real data analysis of rainforest data.