Parametric families for complex valued covariance functions: Some results, an overview and critical aspects

Complex valued random fields, a natural generalization of real valued random fields, represent a powerful tool for modeling phenomena which evolve in time, spatial vectorial data in two dimensions and spatio-temporal vectorial data (i.e., a wind field). However, only a few efforts have been proposed in the literature to address some relevant aspects of spatial and spatio-temporal geostatistical analysis concerning complex valued random functions. For example, it is well known that covariance is, in general, a complex valued function; for this purpose, the main aim of this paper concerns the construction of parametric models for complex valued covariance functions, which are able to model wide families of complex valued random fields. Indeed, a previous class of complex covariance functions, which were built by simply translating an even spectral density function, was able to model just correlation structures characterized by damped oscillations because of the necessary presence of the cosine and sine functions in the real and imaginary part, respectively, of the complex covariance function. The wide class of covariance models, proposed in this paper, can be considered the basic building blocks which could be utilized in a spatial context, in several dimensional spaces, as well as in a spatio-temporal domain to model the correlation structure of complex valued random fields. Moreover, in order to provide a complete overview of the subject, a brief outline of the construction of the subset of real covariance functions has also been given; in particular, it has been shown how some parametric families of real valued covariance functions, whose spectral representation cannot provide an analytic solution, can be constructed using the formalism of differential equations.