A Space -Time stochastic model, space- time prediction( Kriging) of Spatio-Temporal data: A Frequency Domain Approach Seminar

In view of inclusion of temporal dimension, the prediction, estimation, etc. become complicated and this is due to high dimensional matrices of dimensions $mn\times mn$. In order to circumvent these problems, we suggest an alternative methodology based on Discrete Fourier Transforms. It is well known that if the process is temporally stationary, under certain conditions, DFT's are aymptotically uncorrelated over the frequencies and also asymptotically distributed as complex Gaussian. This property will be very useful in modelling and estimation. There is a one to one correspondence between the data and its DFT. We construct predictors of the data at location $s_0$ by obtaining its predictors of DFT's

and then inverting the DFT. Since this requires a space-time covariance function, we model the DFTs using Laplacian model of the form relating to a stochastic partial differential equation. One can obtain an explicit expression for the second order space-time spectrum and also for space -time covariance (of the DFt's) which is required for prediction. We discuss the significance of the model in

describing the structure of the process and also discuss the properties of the Frequency variogram (FV) recently introduced, which is a measure of dissimilarity between the time series defined at two distinct locations of distance say $h$. This measure can be defined under weaker conditions than stationarity. The FV increases as $h$ increases and tends to zero as $h$ tends to 0. It is conditionally negative definite. We discuss some of the properties and show how it can be used for estimation of the parameters.

Reference:

1. T Subba Rao and Gy Terdik(2012) statistical analysis of spatio-temporal models and their applications. In Handbook of Statistics -Vol 30, Elsevier, 521-540

2. T Subba Rao and S Das and G.Boshnakov ( 2014) A frequency domain approach for the estimation of parameters of spatio-temporal random processes .Jour Of Time Series Analysis, vol 35, 357-377

3. T Subba Rao and Gy.Terdik (2016) A space time stochastic model and a covariance function for the stationary temporal random process and spatio-temporal prediction (Kriging) ( Submitted for publication)

4. T Subba Rao and Gy.Terdik (2016) On the Frequency variogram and on Frequency Domain methods for the analysis of spatio-temporal data (Submitted to Priestley Memorial volume)