k-order additive discrete fuzzy measures and their representation
Fuzzy Sets and Systems - Special issue on fuzzy measures and integrals
Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms
Fuzzy Sets and Systems - Special issue on triangular norms
Fuzzy Sets and Systems
Limit properties of quasi-arithmetic means
Fuzzy Sets and Systems
Flexible spatio-temporal stationary variogram models
Statistics and Computing
Spatio-temporal stationary covariance models
Journal of Multivariate Analysis
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Weighted composite likelihood-based tests for space-time separability of covariance functions
Statistics and Computing
Journal of Multivariate Analysis
Nonstationary modeling for multivariate spatial processes
Journal of Multivariate Analysis
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The 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 quasi-arithmetic 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.