Electronic Journal of Statistics

Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation

Xavier Bay, Laurence Grammont, and Hassan Maatouk

Full-text: Open access

Abstract

In this paper, we extend the correspondence between Bayes’ estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of convex constraints such as boundedness, monotonicity or convexity. In the unconstrained interpolation case, the mean of the posterior distribution of a Gaussian Process (GP) given data interpolation is known to be the optimal interpolation function minimizing the norm in the RKHS associated to the GP. In the constrained case, we prove that the Maximum A Posteriori (MAP) or Mode of the posterior distribution is the optimal constrained interpolation function in the RKHS. So, the general correspondence is achieved with the MAP estimator and not the mean of the posterior distribution. A numerical example is given to illustrate this last result.

Article information

Source
Electron. J. Statist. Volume 10, Number 1 (2016), 1580-1595.

Dates
Received: February 2016
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1464710242

Digital Object Identifier
doi:10.1214/16-EJS1149

Mathematical Reviews number (MathSciNet)
MR3507374

Zentralblatt MATH identifier
1348.60055

Subjects
Primary: 60G15: Gaussian processes 60G25: Prediction theory [See also 62M20] 62K20: Response surface designs

Keywords
Correspondence interpolation inequality constraints reproducing Kernel Hilbert space Gaussian process Bayesian estimation maximum a posteriori

Citation

Bay, Xavier; Grammont, Laurence; Maatouk, Hassan. Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation. Electron. J. Statist. 10 (2016), no. 1, 1580--1595. doi:10.1214/16-EJS1149. https://projecteuclid.org/euclid.ejs/1464710242.


Export citation

References

  • [1] N. Aronszajn. Theory of reproducing kernels., Transactions of the American Mathematical Society, 68, 1950.
  • [2] X. Bay, L. Grammont, and H. Maatouk. A New Method For Interpolating In A Convex Subset Of A Hilbert Space. hal -01136466, 2015.
  • [3] A. Cousin, H. Maatouk, and D. Rullière. Kriging of financial term-structures., European Journal of Operational Research, April 2016.
  • [4] G. Kimeldorf and G. Wahba. A correspondence between Bayesian estimation on stochastic processes and smoothing by splines., The Annals of Mathematical Statistics, pages 495–502, 1970.
  • [5] H. Maatouk., Correspondence between Gaussian process regression and interpolation splines under linear inequality constraints. Theory and applications. PhD thesis, École des Mines de St-Étienne, 2015.
  • [6] H. Maatouk and X. Bay. Gaussian Process Emulators for Computer Experiments with Inequality Constraints. in revision, https://hal.archives-ouvertes.fr/hal-01096751/file/Hassan December 2014.
  • [7] H. Maatouk and Y. Richet. constrKriging, 2015. R package available online at, https://github.com/maatouk/constrKriging.
  • [8] C. Micchelli and F. Utreras. Smoothing and interpolation in a convex subset of a Hilbert space., SIAM Journal on Scientific and Statistical Computing, 9(4):728–746, 1988.
  • [9] E. Parzen., Statistical Inference on Time Series by Hilbert Space Methods. Stanford University, 1959.
  • [10] J. Quinonero-Candela, C. E. Rasmussen, and C. K. I. Williams. Approximation methods for Gaussian process regression., Large-Scale Kernel Machines, pages 203–223, 2007.
  • [11] C. E. Rasmussen and C. K. I. Williams., Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning). The MIT Press, 2005.
  • [12] G. F. Trecate, C. K. I. Williams, and M. Opper. Finite-dimensional approximation of Gaussian processes. In, Proceedings of the 1998 conference on Advances in neural information processing systems II, pages 218–224. MIT Press, 1999.
  • [13] F. Utreras. Smoothing noisy data under monotonicity constraints existence, characterization and convergence rates., Numerische Mathematik, 47(4):611–625, 1985.