Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

Abstract

We introduce a method to construct general multivariate positive definite kernels on a nonempty set $X$ that employs a prescribed bounded completely monotone function and special multivariate functions on $X$. The method is consistent with a generalized version of Aitken's integral formula for Gaussians. In the case in which $X$ is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation. In addition, it can be interpreted as an abstract multivariate version of the well-established Gneiting's model for constructing space-time covariances commonly highly cited in the literature. Many parametric models discussed in statistics can be interpreted as particular cases of the method.

Keywords

• positive definite kernels
• conditionally negative definite functions
• Aitken's integral
• Schur exponential
• Oppenheim's inequality
• Gneiting's model

References

1. R. Alexander, K. B. Stolarsky: Extremal problems of distance geometry related to energy integrals, Trans. Amer. Math. Soc., 193 (1974), 1-31.
2. A. Alfonsi, F. Klock and A. Schied: Multivariate transient price impact and matrix-valued positive definite functions, Math. Oper. Res., 41 (3) (2016), 914-934.
3. A. Belton, D. Guillot, A. Khare and M. Putinar: A Panorama of Positivity I: Dimension Free, In: Aleman A., Hedenmalm H., Khavinson D., Putinar M. (eds) Analysis of Operators on Function Spaces. Trends in Mathematics. Birkhäuser, Cham, (2019).
4. C. Berg, J. P. R. Christensen and P. Ressel: Harmonic analysis on semigroups. Theory of positive definite and related functions, Graduate Texts in Mathematics, 100. Springer-Verlag, New York, (1984).
5. M. Bourotte, D. Allard and E. Porcu: A flexible class of non-separable cross-covariance functions for multivariate spacetime data, Spat. Stat., 18 (2016), Part A, 125-146.
6. Y. K. Cho, D. Kim, K. Park and H. Yun: Schoenberg representations and Gramian matrices of Matérn functions, arXiv:1702.05894v1 [math.CA] (2017).
7. T. Gneiting: Nonseparable, stationary covariance functions for space-time data, J. Amer. Statist. Assoc., 97 (2002), 590-600.
8. I. S. Gradshteyn, I. Ryzhik: Table of integrals, series, and products, Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceitlin. Translated from Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey Academic Press, New York-London, (1965).

9. R. Horn, C. Johnson: Topics in matrix analysis, Corrected reprint of the 1991 original. Cambridge University Press, Cambridge, (1994).
10. Y. Kapil, R. Pal, A. Aggarwal and M. Singh: Conditionally negative definite functions, Mediterr. J. Math., 15 (5) (2018), No. 199, 12 p.
11. W. Kleiber, D. Nychka: Nonstationary modeling for multivariate spatial processes, J. Multivariate Anal., 112 (2012), 76-91.
12. V. A. Menegatto: Positive definite functions on products of metric spaces via generalized Stieltjes functions, Proc. Amer. Math. Soc., 148 (11) (2020), 4781-4795.
13. V. A. Menegatto: Positive definiteness on products via generalized Stieltjes and other functions, Math. Inequal. Appl., 24 (2) (2021), 477-490.
14. V. A. Menegatto: Strictly positive definite kernels on the Hilbert sphere, Appl. Anal., 55 (1994), 91-101.
15. V. A. Menegatto, C. P. Oliveira: An extension of Aitken’s integral for Gaussians and positive definiteness, Methods Appl. Anal., (2021), to appear.
16. V. Menegatto, C. Oliveira and E. Porcu: Gneiting class, semi-metric spaces and isometric embeddings, Constr. Math. Anal., 3 (2) (2020), 85-95.
17. C. A. Micchelli: Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), 11-22.
18. M. Micheli, M. A. Glaunés: Matrix-valued kernels for shape deformation analysis, Geometry, Imaging, and Computing, 1 (1) (2014), 57-139.
19. H. Q. Minh, L. Bazzani and V. Murino: A unifying framework in vector-valued reproducing kernel Hilbert spaces for manifold regularization and co-regularized multi-view learning, J. Mach. Learn. Res., 17 (2016), 1-72.
20. E. Porcu, R. Furrer and D. Nychka: 30 years of space-time covariance functions, Wiley Interdisciplinary Reviews: Computational Statistics, 13 (3) (2020), e1512.
21. R. Reams: Hadamard inverses, square roots and products of almost semidefinite matrices, Linear Algebra Appl., 288 (1-3) (1999), 35-43.
22. M. Schlather: Some covariance models based on normal scale mixtures, Bernoulli, 16 (2010), 780-797.
23. R. L. Schilling, R. Song and Z. Vondracek: Bernstein functions. Theory and applications, Second edition. De Gruyter Studies in Mathematics, 37. Walter de Gruyter & Co., Berlin, (2012).
24. S. R. Searle: Matrix algebra useful for statistics, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, (1982).
25. D. Wittwar, G. Santin and B. Haasdonk: Interpolation with uncoupled separable matrix-valued kernels, Dolomites research notes on approximation. Special issue of the “Seminari Padovani di Analisi Numerica 2018" (SPAN2018), Volume 11, 23-39, (2018).