Pseudo-spectra of multivariate inhomogeneous spatial point processes

In this article, we propose a spectral method for a class of multivariate inhomogeneous spatial point processes, namely the second-order intensity reweighted stationary processes. A key ingredient of our approach is utilizing the asymptotic behavior of the periodogram. For second-order stationary point processes, the periodogram is an asymptotically unbiased estimator of the spectrum. By calculating the moment, we show that under inhomogeneity, the expectation of the periodogram 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 admits interpretation in terms of local parameters. We derive a consistent nonparametric estimator of the pseudo-spectrum via kernel smoothing and propose two bandwidth selection methods. The performance and utility of the proposed methods are demonstrated through simulation studies and an application to rainforest point pattern data.