Stochastic Environmental Research and Risk Assessment

, Volume 32, Issue 11, pp 3053–3066

# Admissible nested covariance models over spheres cross time

• Ana Peron
• Emilio Porcu
• Xavier Emery
Original Paper

## Abstract

Nested covariance models, defined as linear combinations of basic covariance functions, are very popular in many branches of applied statistics, and in particular in geostatistics. A notorious limit of nested models is that the constants in the linear combination are bound to be nonnegative in order to preserve positive definiteness (admissibility). This paper studies nested models on d-dimensional spheres and spheres cross time. We show the exact interval of admissibility for the constants involved in the linear combinations. In particular, we show that at least one constant can be negative. One of the implications is that one can obtain a nested model attaining negative correlations. We provide characterization theorems for arbitrary linear combinations as well as for nonconvex combinations involving two covariance functions. We illustrate our findings through several examples involving nonconvex combinations of well-known parametric families of covariance functions.

## Keywords

Covariance functions Nested models Negative covariance Spheres

## Notes

### Acknowledgements

Ana Peron was partially supported by São Paulo Research Foundation (FAPESP) under Grants 2016/03015-7 and 2016/09906-0. Emilio Porcu and Xavier Emery acknowledge the support of Grant CONICYT/FONDECYT/REGULAR/1170290 from the Chilean Commission for Scientific and Technological Research.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Ana Peron
• 1
• Emilio Porcu
• 2
• 3
• Xavier Emery
• 4
• 5
1. 1.Departamento de MatemáticaICMC-USPSão CarlosBrazil
2. 2.School of Mathematics and StatisticsUniversity of NewcastleNewcastle upon TyneUK
3. 3.Department of MathematicsUniversity of AtacamaCopiapóChile
4. 4.Department of Mining EngineeringUniversity of ChileSantiagoChile
5. 5.Advanced Mining Technology CenterUniversity of ChileSantiagoChile