Quasi-arithmetic means of covariance functions with potential applications to space–time data

Abstract

a b s t r a c tThe theory of quasi-arithmetic means represents a powerful tool in the study of covariance functions across space-time. In the present study we use quasi-arithmetic functionals to make inferences about the permissibility of averages of functions that are not, in general, permissible covariance functions. This is the case, e.g., of the geometric and harmonic averages, for which we obtain permissibility criteria. Also, some important inequalities involving covariance functions and preference relations as well as algebraic properties can be derived by means of the proposed approach. In particular, quasi-arithmetic covariances allow for ordering and preference relations, for a Jensen-type inequality and for a minimal and maximal element of their class. The general results shown in this paper are then applied to the study of spatial and spatio-temporal random fields. In particular, we discuss the representation and smoothness properties of a weakly stationary random field with a quasi-arithmetic covariance function. Also, we show that the generator of the quasiarithmetic means can be used as a link function in order to build a space-time nonseparable structure starting from the spatial and temporal margins, a procedure that is technically sound for those working with copulas. Several examples of new families of stationary covariances obtainable with this procedure are shown. Finally, we use quasi-arithmetic functionals to generalise existing results concerning the construction of nonstationary spatial covariances, and discuss the applicability and limits of this generalisation.

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“…We first recall the basic definition of generalized means 17 that generalizes the usual arithmetic and geometric means. For a strictly continuous and monotonous function f , the generalized mean [48], [12], [8] of a sequence V of n real positive numbers V = {v 1 , ..., v n } is defined as…”

“…Note that since f is injective, its reciprocal function f −1 is properly defined. Further, since f is monotonous, it is noticed that the generalized mean is necessarily bounded between the extremal set elements min i v i and max i v i : min i∈{1,...,n} 17 Studied independently in 1930 by Kolmogorov and Nagumo, see [48]. A more detailed account is given in [49], Chapter 3.…”

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