  The application of geostatistical and machine learning methods based on
Gaussian processes to big space-time data is beset by the requirement for
storing and numerically inverting large and dense covariance matrices.
Computationally efficient representations of space-time correlations can be
constructed using local models of conditional dependence which can reduce the
computational load. We formulate a stochastic local interaction model for
regular and scattered space-time data that incorporates interactions within
controlled space-time neighborhoods. The strength of the interaction and the
size of the neighborhood are defined by means of kernel functions and adaptive
local bandwidths. Compactly supported kernels lead to finite-size local
neighborhoods and consequently to sparse precision matrices that admit explicit
expression. Hence, the stochastic local interaction model's requirements for
storage are modest and the costly covariance matrix inversion is not needed. We
also derive a semi-explicit prediction equation and express the conditional
variance of the prediction in terms of the diagonal of the precision matrix.
For data on regular space-time lattices, the stochastic local interaction model
is equivalent to a Gaussian Markov Random Field.
