[Paper Review] A space-time covariance function for spatio-temporal random processes and spatio-temporal prediction (kriging)

T. Subba Rao, György Terdik|arXiv (Cornell University)|Nov 8, 2013

Soil Geostatistics and Mapping24 references3 citations

TL;DR

This paper proposes a novel space-time covariance function for stationary spatio-temporal processes using Discrete Fourier Transforms (DFTs) of spatial data, enabling efficient kriging prediction via a frequency-domain approach. By modeling the DFTs as complex-valued processes governed by a Complex Stochastic Partial Differential Equation (CSPDE), the method avoids time discretization and enables fast, robust estimation using the Frequency Variogram, outperforming classical likelihood and variogram methods in computational speed and robustness to non-Gaussianity.

ABSTRACT

We consider a stationary spatio-temporal random process and assume that we have a sample. By defining a sequence of discrete Fourier transforms at canonical frequencies at each location, and using these complex valued random varables as observed sample, we obtain expressions for the spatio-temporal covariance functions and the spectral density functions of the spatio-temporal random processes. These spectra correspond to non separable class of random processes. The spatio-temporal covariance functions, obtained here are functions of the spatial distance and the temporal frequency and are similar to Matern class. These are in terms of modified Bessel functions of the second kind. and the parameters are in terms of the second order spectral density functions of the random proces and the spatial distances. We consider the estimation of the parameters of the covariance function and also briefly mention their asymptotic properties. The estimation of the entire data at a known location, and also the estimation of a value given the above sample is also considered. The predictors are obtained using the vectors of Discrete Fourier Transforms. We also describe a statistical test for testing the independence of the m spatial time series (testing for spatial independence) using the Finite Fourier Transforms and it is based on the likelihood ratio test of complex valued random variables The methods are illustrated with real data. Keywords: Discrete Fourier Transforms, Covariance functions, Spectral density functions, Space-Time Processses, Prediction(kriging) Laplacian operators, Frequency Variogram, Tests for independence, Whittle likelihood.

Motivation & Objective

• To develop a computationally efficient and robust method for spatio-temporal prediction (kriging) of stationary random processes with both spatial and temporal dependence.
• To address the challenge of estimating missing data at unobserved locations by modeling the spatio-temporal covariance structure using frequency-domain techniques.
• To avoid the computational burden of likelihood-based methods and the sensitivity of variogram methods by introducing a Frequency Variogram approach based on DFTs.
• To derive a non-separable, positive-definite space-time covariance function that generalizes existing models like those in [11], [12], and [20].
• To enable efficient parameter estimation without inverting large covariance matrices, leveraging fast Fourier transform (FFT) algorithms.

Proposed method

• The method transforms spatio-temporal time series data at multiple spatial locations into Discrete Fourier Transforms (DFTs) at Fourier frequencies, treating the DFTs as complex-valued observations.
• It models the DFTs as a complex-valued stationary process governed by a Complex Stochastic Partial Differential Equation (CSPDE) of the Laplacian type in spatial coordinates only.
• The spectral representation of the process is derived using Fourier transforms, leading to a covariance function that depends on spatial distance and temporal frequency, yielding a non-separable class of processes.
• The method uses the Frequency Variogram approach for parameter estimation, which avoids matrix inversion and is robust to non-Gaussianity.
• The DFTs are asymptotically approximated using the Fejér kernel, enabling the spectral representation to converge to a form involving the wave number space integral.
• Prediction at unobserved locations is performed using the estimated spectral density and covariance function, with prediction intervals derived from the kriging framework.

Experimental results

Research questions

• RQ1 Can a frequency-domain approach based on DFTs provide a computationally efficient alternative to classical likelihood and variogram methods for spatio-temporal kriging?
• RQ2 How can a complex stochastic partial differential equation (CSPDE) in space be used to model the DFTs of spatio-temporal data and yield a valid, non-separable space-time covariance function?
• RQ3 Does the proposed Frequency Variogram estimation method provide robust and accurate parameter estimates without requiring inversion of large covariance matrices?
• RQ4 To what extent does the model generalize existing spatio-temporal covariance functions, such as those in [11], [12], and [20]?
• RQ5 Can the method effectively handle missing data in real-world spatio-temporal datasets, such as air pollution measurements with large gaps?

Key findings

• The derived spatio-temporal covariance function is non-separable and depends on both spatial lag and temporal frequency, providing a more flexible model than separable alternatives.
• The method achieves computational efficiency by leveraging Fast Fourier Transform (FFT) algorithms, significantly reducing computation time compared to classical likelihood and variogram-based approaches.
• The Frequency Variogram estimation method is robust to departures from Gaussianity and avoids the need to invert large-dimensional covariance matrices.
• The model includes the spatio-temporal processes of [11], [12], and [20] as special cases, demonstrating its generality and unifying potential.
• In the real-data application to PM2.5 levels in New York City, the method successfully estimated missing values at location 11 with accurate prediction intervals, showing strong empirical performance.
• The asymptotic approximation of the DFTs using the Fejér kernel ensures that the spectral representation converges to a form involving the wave number space integral, enabling theoretical consistency.