Wavelet Analysis Of Irregularly Spaced Data And Its Spatio-Temporal Extension

This paper proposes an orthogonal basis expansion by Haar wavelets for irregularly spaced spatial data and conducts its spatio-temporal extension when spatial points are not necessarily the same at each temporal point. One crucial contribution of this paper is discovery of algorithm to construct orthogonal basis for any irregularly spaced data via 2 dimensional Haar wavelets. We construct a spatial model by imposing a prior distribution on coefficients of the orthogonal expansion. In conducting spatio-temporal extension, we fit a simple autoregressive model not to data itself but to the coefficients of the orthogonal expansion, which makes it possible to handle spatial sampling points that are not necessarily the same at each temporal point. The spatio-temporal model can express nonstationary spatial behaviors by allowing parameters dependent spatially. Advantage of the models is superb parameter estimation efficiency that is attained by independency of wavelet transform providing efficient Gaussian likelihood evaluation. Finally we demonstrate the wavelet modeling for land price data in Tokyo from 2003 till 2014, yearly series of public land prices per m2 over about 8000 sampling points scattered irregularly all over Tokyo areas.