Fuzzy Logic and Space-Time Interaction Parameter in Covariance Model

In the literature, several efforts have been made in order to define a comprehensive procedure for the identification of the most appropriate class of space-time covariance functions for the observations at hand. Several statistical tests have been proposed in order to check the relevant aspects presented by the variable under study, which are particularly significant to select an appropriate class of space-time covariance functions. Among these aspects, it is crucial the identification of the space-time interaction parameter, which discriminates between separable and non-separable covariance models. Space-time interaction produces effects in the behavior of the covariance/variogram surface. In particular, it can be observed that the spatio-temporal correlation is stronger (weaker) than the theoretical separable correlation (or correlation without space-time interaction). This can happen for all lags or just some of them. However, the knowledge of the interaction parameter is imprecise, thus it might be useful to recall the fuzzy logic. In the literature, the fuzzy logic was applied in order to address imprecise information on variogram parameters (sill, nugget, range) in spatial kriging, while in the context of spatio-temporal geostatistics fuzzy theory was used to treat fuzzy data in very few cases. In this paper, a novel approach based on the fuzzy logic is proposed to assess the space-time interaction parameter of the covariance function. Hence, the vector of possible parameters of the spatio-temporal covariance is considered to be a fuzzy set and each parameter vector is assigned a membership value in this set. A case study on environmental variable is thoroughly discussed.