A note on smoothness measures for space-time surfaces

摘要

The differentiability of a random field has a direct relationship with the differentiability of its covariance function. We review the concept of differentiability of space-time covariance models and random fields, and its implications on predictions. We analyze the change of behavior of the covariance function at the origin and at different space-time lags away from the origin, by using the concept of smoothness which can be considered the geometrical view of the differentiability. We propose a way to measure the smoothness of any covariance function, and apply it to purely spatial and space-time covariance functions.