Fourier Analysis of Irregularly Spaced Data onRd

Abstract

The purpose of the paper is to propose a frequency domain approach for irregularly spaced data on R d . We extend the original definition of a periodogram for time series to that for irregularly spaced data and define non-parametric and parametric spectral density estimators in a way that is similar to the classical approach. Introduction of the mixed asymptotics, which are one of the asymptotics for irregularly spaced data, makes it possible to provide asymptotic theories to the spectral estimators.The asymptotic result for the parametric estimator is regarded as a natural extension of the classical result for regularly spaced data to that for irregularly spaced data. Empirical studies are also included to illustrate the frequency domain approach in comparisons with the existing spatial and frequency domain approaches.

Nonparametric Estimation of Probability Density Functions for Irregularly Observed Spatial Data

“…It may also be worth pointing out that our defined domainexpanding infill asymptotic by (2.3) and (2.4) is different from an alternative version of mixed asymptotic adopted by Hall and Patil (1994); see also Matsuda and Yajima (2009), who assumed in the notation of this article, that s i = (A 1 u i,1 , A 2 u i,2 ), where u i = (u i,1 , u i,2 ), i = 1, 2, . .…”

“…Our second real example is a larger dataset from Matsuda and Yajima (2009), who considered the parametric and nonparametric spectral estimation to land price data, collected by the Japanese Ministry of Land, Infrastructure and Transport in 2001. This dataset is the record of land prices (yen per square meter) irregularly spaced in the residential areas around Tokyo.…”

“…Hall and Patil (1994) looked at nonparametric estimation of a spatial covariance function using observations whose locations are generated by a probability distribution. Matsuda and Yajima (2009) had a similar model for generating observations irregularly in space and looked at the estimation of the spectral density nonparametrically and parametrically with the emphasis on the latter. The focus by Jenish and Prucha (2009) was also on parametric models.…”

Autoregressive spatial spectral estimates

“…Frequency domain techniques are commonly employed and irregular spatial lattices also considered (cf. Matsuda and Yajima (2009), Bandyopadhyay et al (2015)), but we focus on the regular case. Irregular spacing will disturb the Toeplitz property of the covariance matrix that we exploit for our results, but a more practical reason suggests itself.…”

“…High frequency spatial components may be interpreted as corresponding to phenomena (possibly noise phenomena) that change rapidly over the space, while low frequency components that change less frequently are more structural. For Tokyo land price data Matsuda and Yajima (2009) argue that accurately estimating the spectrum over low frequencies is more desirable than over high frequencies, interpreting the latter as noise and the former as the structural factors of interest. In this context they specify that high frequency noise can include environmental factors, air and noise pollution and sunshine.…”