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A New Covariance Function and Spatio‐Temporal Prediction (Kriging) for A Stationary Spatio‐Temporal Random Process
Author(s) -
Subba Rao T.,
Terdik Gyorgy
Publication year - 2017
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/jtsa.12245
Subject(s) - mathematics , covariance function , covariance , rational quadratic covariance function , discrete fourier series , fourier series , stationary process , law of total covariance , fourier transform , spectral density estimation , discrete fourier transform (general) , short time fourier transform , mathematical analysis , fourier analysis , covariance intersection , statistics
Consider a stationary spatio‐temporal random processY ts ; s ∈ R d , t ∈ Zand letY ts i; i = 1 , 2 , … , m ; t = 1 , … , nbe a sample from the process. Our object here is to predict, given the sample,Y ts ofor all t at the location s o . To obtain the predictors, we define a sequence of discrete Fourier transformsJs iω j; i = 1 , 2 , … , musing the observed time series. We consider these discrete Fourier transforms as a sample from the complex valued random variableJ sω. Assuming that the discrete Fourier transforms satisfy a complex stochastic partial differential equation of the Laplacian type with a scaling function that is a polynomial in the temporal spectral frequency ω , we obtain, in a closed form, expressions for the second‐order spatio‐temporal spectrum and the covariance function. The spectral density function obtained corresponds to a non‐separable random process. The optimal predictor of the discrete Fourier transformJs oωis in terms of the covariance functions. The estimation of the parameters of the spatio‐temporal covariance function is considered and is based on the recently introduced frequency variogram method. The methods given here can be extended to situations where the observations are corrupted by independent white noise. The methods are illustrated with a real data set.