Abstract: |
The differentiability of a stochastic process has a direct relationship with the differentiability of covariance function. Space-time geostatistics have had recently a great development, and new families of covariance functions have been proposed following different methodologies. However, although they are positive definite, not all of them represent the reality of a particular phenomenon of study. Therefore, the analysis of other properties is necessary before choosing the suitable model. Here we presented a review of the concept of differentiability of both, space--time covariance models and stochastic process, and its implications on correlations of linear combinations underlying observations, specifically, in the increments. We analyze the change of the function of covariance from the origin and as lag grows. The predictions depend on the values that the covariance function takes. So, by using the concept of smoothness of a covariance function, which can be considered as the geometrical view of the differentiability, we determine some characteristics of the predictions obtained with these covariance functions. We propose two ways of measuring the smoothness of any covariance function. For iIlustrative purposes, we apply them to purely spatial covariance functions and to several space-time covariance models, and we show a characterization of these models according to their smoothness. |
