A0728
Title: Pseudo-spectra of multivariate inhomogeneous spatial point processes
Authors: Junho Yang - Academia Sinica (Taiwan) [presenting]
Abstract: Spectral analysis is a technique that characterizes the second-order structure of stationary time series, random fields, and point processes. However, in spatial point processes, a stationary assumption, especially homogeneity of the first-order intensity assumption, is often considered too stringent. A new spectral analysis is proposed for a multivariate inhomogeneous point process observed on $\mathbb{R}^d$. A key idea is the asymptotic behavior of the discrete Fourier transform (DFT) of the observed spatial point pattern and its tapered version. It is shown that, even in the case of inhomogeneous processes, the expectation of the periodogram converges to some deterministic matrix-valued function, which is Hermitian and positive definite. This limit is referred to as the pseudo-spectrum. It is shown that the defined pseudo-spectrum has an interpretation in terms of integrating the local spectra. The consistent estimator of the pseudo-spectrum is derived via periodogram smoothing, and methods are proposed to select the bandwidth. Finally, the proposed estimator is demonstrated through simulations and real data analysis.
