Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2021, , 384 - 399, 13.12.2021
https://doi.org/10.33205/cma.964096

Öz

Kaynakça

  • R. Alexander, K. B. Stolarsky: Extremal problems of distance geometry related to energy integrals, Trans. Amer. Math. Soc., 193 (1974), 1-31.
  • 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.
  • 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).
  • 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).
  • 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.
  • Y. K. Cho, D. Kim, K. Park and H. Yun: Schoenberg representations and Gramian matrices of Matérn functions, arXiv:1702.05894v1 [math.CA] (2017).
  • T. Gneiting: Nonseparable, stationary covariance functions for space-time data, J. Amer. Statist. Assoc., 97 (2002), 590-600.
  • 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).
  • R. Horn, C. Johnson: Topics in matrix analysis, Corrected reprint of the 1991 original. Cambridge University Press, Cambridge, (1994).
  • Y. Kapil, R. Pal, A. Aggarwal and M. Singh: Conditionally negative definite functions, Mediterr. J. Math., 15 (5) (2018), No. 199, 12 p.
  • W. Kleiber, D. Nychka: Nonstationary modeling for multivariate spatial processes, J. Multivariate Anal., 112 (2012), 76-91.
  • V. A. Menegatto: Positive definite functions on products of metric spaces via generalized Stieltjes functions, Proc. Amer. Math. Soc., 148 (11) (2020), 4781-4795.
  • V. A. Menegatto: Positive definiteness on products via generalized Stieltjes and other functions, Math. Inequal. Appl., 24 (2) (2021), 477-490.
  • V. A. Menegatto: Strictly positive definite kernels on the Hilbert sphere, Appl. Anal., 55 (1994), 91-101.
  • V. A. Menegatto, C. P. Oliveira: An extension of Aitken’s integral for Gaussians and positive definiteness, Methods Appl. Anal., (2021), to appear.
  • V. Menegatto, C. Oliveira and E. Porcu: Gneiting class, semi-metric spaces and isometric embeddings, Constr. Math. Anal., 3 (2) (2020), 85-95.
  • C. A. Micchelli: Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), 11-22.
  • M. Micheli, M. A. Glaunés: Matrix-valued kernels for shape deformation analysis, Geometry, Imaging, and Computing, 1 (1) (2014), 57-139.
  • 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.
  • E. Porcu, R. Furrer and D. Nychka: 30 years of space-time covariance functions, Wiley Interdisciplinary Reviews: Computational Statistics, 13 (3) (2020), e1512.
  • R. Reams: Hadamard inverses, square roots and products of almost semidefinite matrices, Linear Algebra Appl., 288 (1-3) (1999), 35-43.
  • M. Schlather: Some covariance models based on normal scale mixtures, Bernoulli, 16 (2010), 780-797.
  • 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).
  • 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).
  • 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).

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

Yıl 2021, , 384 - 399, 13.12.2021
https://doi.org/10.33205/cma.964096

Öz

We introduce a method to construct general multivariate positive definite kernels on a nonempty set XX that employs a prescribed bounded completely monotone function
and special multivariate functions on XX. The method is consistent with a generalized version of Aitken's integral formula for Gaussians. In the case in which XX 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.

Kaynakça

  • R. Alexander, K. B. Stolarsky: Extremal problems of distance geometry related to energy integrals, Trans. Amer. Math. Soc., 193 (1974), 1-31.
  • 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.
  • 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).
  • 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).
  • 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.
  • Y. K. Cho, D. Kim, K. Park and H. Yun: Schoenberg representations and Gramian matrices of Matérn functions, arXiv:1702.05894v1 [math.CA] (2017).
  • T. Gneiting: Nonseparable, stationary covariance functions for space-time data, J. Amer. Statist. Assoc., 97 (2002), 590-600.
  • 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).
  • R. Horn, C. Johnson: Topics in matrix analysis, Corrected reprint of the 1991 original. Cambridge University Press, Cambridge, (1994).
  • Y. Kapil, R. Pal, A. Aggarwal and M. Singh: Conditionally negative definite functions, Mediterr. J. Math., 15 (5) (2018), No. 199, 12 p.
  • W. Kleiber, D. Nychka: Nonstationary modeling for multivariate spatial processes, J. Multivariate Anal., 112 (2012), 76-91.
  • V. A. Menegatto: Positive definite functions on products of metric spaces via generalized Stieltjes functions, Proc. Amer. Math. Soc., 148 (11) (2020), 4781-4795.
  • V. A. Menegatto: Positive definiteness on products via generalized Stieltjes and other functions, Math. Inequal. Appl., 24 (2) (2021), 477-490.
  • V. A. Menegatto: Strictly positive definite kernels on the Hilbert sphere, Appl. Anal., 55 (1994), 91-101.
  • V. A. Menegatto, C. P. Oliveira: An extension of Aitken’s integral for Gaussians and positive definiteness, Methods Appl. Anal., (2021), to appear.
  • V. Menegatto, C. Oliveira and E. Porcu: Gneiting class, semi-metric spaces and isometric embeddings, Constr. Math. Anal., 3 (2) (2020), 85-95.
  • C. A. Micchelli: Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx., 2 (1986), 11-22.
  • M. Micheli, M. A. Glaunés: Matrix-valued kernels for shape deformation analysis, Geometry, Imaging, and Computing, 1 (1) (2014), 57-139.
  • 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.
  • E. Porcu, R. Furrer and D. Nychka: 30 years of space-time covariance functions, Wiley Interdisciplinary Reviews: Computational Statistics, 13 (3) (2020), e1512.
  • R. Reams: Hadamard inverses, square roots and products of almost semidefinite matrices, Linear Algebra Appl., 288 (1-3) (1999), 35-43.
  • M. Schlather: Some covariance models based on normal scale mixtures, Bernoulli, 16 (2010), 780-797.
  • 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).
  • 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).
  • 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).
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik
Bölüm Makaleler
Yazarlar

Valdir Menegatto 0000-0002-4213-8759

Claudemir Oliveira 0000-0002-6079-2236

Yayımlanma Tarihi 13 Aralık 2021
Yayımlandığı Sayı Yıl 2021

Kaynak Göster

APA Menegatto, V., & Oliveira, C. (2021). Matrix valued positive definite kernels related to the generalized Aitken’s integral for Gaussians. Constructive Mathematical Analysis, 4(4), 384-399. https://doi.org/10.33205/cma.964096
AMA Menegatto V, Oliveira C. Matrix valued positive definite kernels related to the generalized Aitken’s integral for Gaussians. CMA. Aralık 2021;4(4):384-399. doi:10.33205/cma.964096
Chicago Menegatto, Valdir, ve Claudemir Oliveira. “Matrix Valued Positive Definite Kernels Related to the Generalized Aitken’s Integral for Gaussians”. Constructive Mathematical Analysis 4, sy. 4 (Aralık 2021): 384-99. https://doi.org/10.33205/cma.964096.
EndNote Menegatto V, Oliveira C (01 Aralık 2021) Matrix valued positive definite kernels related to the generalized Aitken’s integral for Gaussians. Constructive Mathematical Analysis 4 4 384–399.
IEEE V. Menegatto ve C. Oliveira, “Matrix valued positive definite kernels related to the generalized Aitken’s integral for Gaussians”, CMA, c. 4, sy. 4, ss. 384–399, 2021, doi: 10.33205/cma.964096.
ISNAD Menegatto, Valdir - Oliveira, Claudemir. “Matrix Valued Positive Definite Kernels Related to the Generalized Aitken’s Integral for Gaussians”. Constructive Mathematical Analysis 4/4 (Aralık 2021), 384-399. https://doi.org/10.33205/cma.964096.
JAMA Menegatto V, Oliveira C. Matrix valued positive definite kernels related to the generalized Aitken’s integral for Gaussians. CMA. 2021;4:384–399.
MLA Menegatto, Valdir ve Claudemir Oliveira. “Matrix Valued Positive Definite Kernels Related to the Generalized Aitken’s Integral for Gaussians”. Constructive Mathematical Analysis, c. 4, sy. 4, 2021, ss. 384-99, doi:10.33205/cma.964096.
Vancouver Menegatto V, Oliveira C. Matrix valued positive definite kernels related to the generalized Aitken’s integral for Gaussians. CMA. 2021;4(4):384-99.