# Journal of Time Series Analysis

## A New Covariance Function and Spatio‐Temporal Prediction (Kriging) for A Stationary Spatio‐Temporal Random Process

### Journal Article

Consider a stationary spatio‐temporal random process $\left\{{Y}_{t}\left(\mathbf{s}\right);\mathbf{s}\in {\mathbb{R}}^{d},\phantom{\rule{1em}{0ex}}t\in \mathbb{Z}\right\}$ and let $\left\{{Y}_{t}\left({\mathbf{s}}_{i}\right);i=1,2,\dots ,m;t=1,\dots ,n\right\}$ be a sample from the process. Our object here is to predict, given the sample, $\left\{{Y}_{t}\left({\mathbf{s}}_{o}\right)\right\}$ for all t at the location so. To obtain the predictors, we define a sequence of discrete Fourier transforms $\left\{{J}_{{\mathbf{s}}_{i}}\left({\omega }_{j}\right);i=1,2,\dots ,m\right\}$ using the observed time series. We consider these discrete Fourier transforms as a sample from the complex valued random variable $\left\{{J}_{\mathbf{s}}\left(\omega \right)\right\}$. 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 transform ${J}_{{\mathbf{s}}_{o}}\left(\omega \right)$ is 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.

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