A finite element method for density estimation with Gaussian process priors

SIAM Journal on Numerical Analysis

Volumn 47, Issue 6, 2009, Pages 4759-4792

  Griebel, Michael ^a   Hegland, Markus ^b  

^a UNIVERSITY OF BONN   (Germany)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Density estimation; Finite elements; Galerkin; Newton

Indexed keywords

ERROR ANALYSIS; FINITE ELEMENT METHOD; GALERKIN METHODS; GAUSSIAN DISTRIBUTION; GAUSSIAN NOISE (ELECTRONIC); NEWTON-RAPHSON METHOD;

DENSITY ESTIMATION; DENSITY ESTIMATOR; GAUSSIAN PROCESS PRIORS; MAXIMUM A POSTERIORI METHODS; MULTILINEAR FUNCTIONS; NEWTON'S METHODS; NUMERICAL EXPERIMENTS; VARIATIONAL PROBLEMS;

NUMERICAL METHODS;

EID: 79961038273     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/080736478     Document Type: Article

References (35)

1. E. PARZEN, An approach to time series analysis, Ann. Math. Statist., 32 (1961), pp. 951-989.
2. R. O. DUDA AND P. E. HART, Pattern Classification and Scene Analysis, John Wiley and Sons, New York, 1973.
3. I. GOOD AND R. GASKINS, Nonparametric roughness penalties for probability densities, Biometrika, 58 (1971), pp. 255-277.
4. T. LEONARD, Density estimation, stochastic processes and prior information, J. Roy. Statist. Soc. Ser. B, 40 (1978), pp. 113-146.
5. R. H. CAMERON AND W. T. MARTIN, Transformations of Wiener integrals under translations, Ann. of Math. (2), 45 (1944), pp. 386-396.
6. V. I. BOGACHEV, Gaussian measures, Mathematical Surveys and Monographs 62, American Mathematical Society, Providence, RI (1998).
7. M. HEGLAND, Approximate maximum a posteriori with Gaussian process priors, Constr. Approx., 26 (2007), pp. 205-224.
8. M. G. BULMER, Principles of Statistics, 2nd ed., Dover, New York, 1979.
9. D. W. SCOTT, Multivariate density estimation, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley and Sons, New York, 1992.
10. R. FISHER, Two new properties of mathematical likelihood, Proc. Roy. Soc. A, 144 (1934), pp. 285-307.
11. Y. YAMASAKI, Measures on infinite-dimensional spaces, Series in Pure Mathematics 5, World Scientific, Singapore, 1985.
12. G. WAHBA, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.
13. M. SEEGER, Gaussian processes for machine learning, Internat. J. Neural Syst., 14 (2004), pp. 69-106.
14. R. J. ADLER, The geometry of random fields, Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons, New York, 1981.
15. I. EKELAND AND R. TÉMAM, Convex analysis and variational problems, English ed., Classics in Applied Mathematics 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.
16. J. GARCKE, Maschinelles Lernen durch Funktionsrekonstruktion mit verallgemeinerten dünnen Gittern, Doktorarbeit, Institut für Numerische Simulation, Universität Bonn, 2004.
17. C. T. KELLEY, Solving nonlinear equations with Newton's method, Fundamentals of Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.
18. P. DEUFLHARD AND M. WEISER, Global inexact Newton multilevel FEM for nonlinear elliptic problems, Multigrid Methods V (Stuttgart, 1996), Lect. Notes in Comput. Sci. Engrg. 3, Springer, Berlin, 1998, pp. 71-89.
19. P. DEUFLHARD, Newton methods for nonlinear problems, Affine invariance and adaptive algorithms Springer Series in Computational Mathematics 35, Springer-Verlag, Berlin, 2004.
20. P. DEUFLHARD AND M. WEISER, Local inexact Newton multilevel FEM for nonlinear elliptic problems, Computational science for the 21st century, M. O. Bristeau, G. Etgen, W. Fitzgibbon, J. L. Lions, J. Periaux, and M. Wheeler, eds., John Wiley and Sons, New York 1997, pp. 129-138.
21. M. GRIEBEL AND S. KNAPEK, Optimized tensor-product approximation spaces, Constr. Approx., 16 (2000), pp. 525-540.
22. S. KNAPEK, Approximation und Kompression mit Tensorprodukt- Multiskalenräumen, Doktorarbeit, Universität Bonn, 2000.
23. M. GRIEBEL AND S. KNAPEK, Optimized general sparse grid approximation spaces for operator equations, Math. Comput. (2008), submitted. Also available as SFB611 preprint 402.
24. D. BRAESS, Finite Elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Translated from the German by Larry L. Schumaker.
25. R. A. DEVORE AND B. J. LUCIER, Wavelets, Acta Numer., Cambridge University Press, Cambridge, 1992, pp. 1-56.
26. P. OSWALD, Multilevel Finite Element Approximation, Teubner Skripten zur Numerik, Teubner, Stuttgart, 1994.
27. M. GRIEBEL AND P. OSWALD, Tensor product type subspace splitting and multilevel iterative methods for anisotropic problems, Adv. Comput. Math., 4 (1995), pp. 171-206.
28. B. W. SILVERMAN, Density estimation for statistics and data analysis, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1986.
29. R. A. TAPIA AND J. R. THOMPSON, Nonparametric probability density estimation, Johns Hopkins Series in the Mathematical Sciences, Vol. 1, Johns Hopkins University Press, Baltimore, MD, 1978.
30. P. HAHNEN, Nichtlineare numerische Verfahren zur multivariaten Dichteschätzung. Master's thesis, Institut für numerische Simulation, Universität Bonn, 2006.
31. A. AZZALINI AND A. W. BOWMAN, A look at some data on the Old Faithful Geyser, Appl. Statist., 39 (1990), pp. 357-365.
32. D. SCOTT AND J. THOMPSON, Probability density estimation in higher dimensions, Computer Science and Statistics: Proceedings of the Fifteenth Symposium on the Interface, Amsterdam, North-Holland, 1983, pp. 173-179.
33. J. GARCKE, Regression with the optimised combination technique, in Proceedings of the 23rd ICML '06, W. Cohen and A. Moore, eds., ACM Press, New York, 2006, pp. 321-328.
34. M. HEGLAND, J. GARCKE, AND V. CHALLIS, The combination technique and some generalisations, Linear Algebra Appl., 420 (2007), pp. 249-275.
35. J. GARCKE, M. GRIEBEL, AND M. THESS, Data mining with sparse grids, Comput. 67 (2001), pp. 225-253.