图书馆订阅: Guest
Begell Digital Portal Begell 数字图书馆 电子图书 期刊 参考文献及会议录 研究收集
国际不确定性的量化期刊
影响因子: 4.911 5年影响因子: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN 打印: 2152-5080
ISSN 在线: 2152-5099

Open Access

国际不确定性的量化期刊

DOI: 10.1615/Int.J.UncertaintyQuantification.2020029614
pages 297-314

SENSITIVITY INDICES FOR OUTPUT ON A RIEMANNIAN MANIFOLD

R. Fraiman
Centro de Matemática, Facultad de Ciencias, Universidad de la República, Uruguay
F. Gamboa
Institut de Mathématiques de Toulouse, France
Leonardo Moreno
Departamento de Métodos Cuantitativos, FCEA, Universidad de la República, Uruguay

ABSTRACT

In the context of computer code experiments, sensitivity analysis of a complicated input-output system is often performed by ranking the so-called Sobol' indices. One reason for the popularity of the Sobol' approach relies on the simplicity of the statistical estimation of these indices using the so-called pick and freeze method. In this work we propose and study sensitivity indices for the case where the output lies on a Riemannian manifold. These indices are based on a Cramer-von Mises like criterion that takes into account the geometry of the output support. We propose a pick and freeze like estimator of these indices based on an U−statistic. The asymptotic properties of these estimators are studied. Further, we provide and discuss some interesting numerical examples.

REFERENCES

  1. Etienne, D.R., Devictor, N., and Tarantola, S., Uncertainty in Industrial Practice, New York: Wiley, 2008.

  2. Saltelli, A., Chan, K., and Scott, E., Sensitivity Analysis, Wiley Series in Probability and Statistics, Chichester, UK: Wiley, Ltd., 2000.

  3. Sobol, I.M., Sensitivity Estimates for Nonlinear Mathematical Models, Math. Model. Comput. Exp., 1(4):407-414,1993.

  4. Gamboa, F., Janon, A., Klein, T., Lagnoux, A., and Prieur, C., Statistical Inference for Sobol' Pick-Freeze Monte Carlo Method, Statistics, 50(4):881-902, 2016.

  5. Rugama, L.A.J. and Gilquin, L., Reliable Error Estimation for Sobol' Indices, Stat. Comp., 28(4):725-738,2018.

  6. Gamboa, F., Janon, A., Klein, T., and Lagnoux, A., Sensitivity Analysis for Multidimensional and Functional Outputs, Electron. J. Stat., 8(1):575-603, 2014.

  7. Marrel, A., Iooss, B., Jullien, M., Laurent, B., and Volkova, E., Global Sensitivity Analysis for Models with Spatially Dependent Outputs, Environmetrics, 22(3):383-397, 2011.

  8. Da Veiga, S., Global Sensitivity Analysis with Dependence Measures, J. Stat. Computat. Simul., 85(7):1283-1305, 2015.

  9. Borgonovo, E., Sensitivity Analysis: An Introduction for the Management Scientist, Vol. 251, Berlin: Springer, 2017.

  10. Borgonovo, E., Hazen, G.B., and Plischke, E., A Common Rationale for Global Sensitivity Measures and Their Estimation, Risk Anal, 36(10):1871-1895, 2016.

  11. Gamboa, F., Klein, T., andLagnoux, A., Sensitivity Analysis Based on Cramer-von Mises Distance, SIAM/ASAJ. Uncertainty Quantif, 6(2):522-548, 2018.

  12. Rao, C.R., Information and the Accuracy Attainable in the Estimation of Statistical Parameters, Bull. Calcutta Math., 37:81-91, 1945.

  13. Hendriks, H. and Landsman, Z., Asymptotic Data Analysis on Manifolds, Annals Stat., 35(1):109-131, 2007.

  14. Fefferman, C., Ivanov, S., Kurylev, Y., Lassas, M., and Narayanan, H., Fitting a Putative Manifold to Noisy Data, in Conf. on Learning Theory, pp. 688-720, 2018.

  15. Mardia,K.V., Statistics of Directional Data, New York: Academic Press, 1972.

  16. Bhattacharya, A. and Bhattacharya, R., Nonparametric Inference on Manifolds: with Applications to Shape Spaces, Vol. 2, Cambridge, UK: Cambridge University Press, 2012.

  17. Patrangenaru, V. and Ellingson, L., Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis, Boca Raton, FL: CRC Press, 2015.

  18. Chastaing, G., Gamboa, F., and Prieur, C., Generalized Sobol' Sensitivity Indices for Dependent Variables: Numerical Methods, J.Stat. Comput. Simul., 85(7):1306-1333,2015.

  19. Bezanson, J., Edelman, A., Karpinski, S., and Shah, V.B., Julia: A Fresh Approach to Numerical Computing, SIAM Rev, 59(1):65-98, 2017.

  20. R Core Team, A Language and Environment for Statistical Computing [Internet], Vienna, Austria, 2017.

  21. do Carmo, M., Riemannian Geometry: Mathematics, Boston Basel Berlin: Birkhauser, 1992.

  22. Petersen, P., Riemannian Geometry, Vol. 171, Berlin: Springer, 2006.

  23. Folland, G.B., Real Analysis: Modern Techniques and Their Applications, New York: Wiley,2013.

  24. Pennec, X., Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements, J. Math. Imag. Vis, 25(1):127,2006.

  25. Christensen, J.P.R., On Some Measures Analogous to Haar Measure, Math. Scand., 26(1):103-106, 1970.

  26. Fraiman, R., Gamboa, F., and Moreno, L., Connecting Pairwise Geodesic Spheres by Depth: DCOPS, J. Multivariate Anal, 169:81-94,2019.

  27. Nash, J., C1-Isometric Imbedding, Annal. Math, 60:383-396, 1954.

  28. Sobol, I.M., Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates, Math. Comput. Simul., 55(1):271-280, 2001.

  29. Janon, A., Klein, T., Lagnoux, A., Nodet, M., and Prieur, C., Asymptotic Normality and Efficiency of Two Sobol' Index Estimators, ESAIM: Probab. Stat, 18:342-364, 2014.

  30. Ibragimov, R. and Sharakhmetov, S., Analogues of Khintchine, Marcinkiewicz-Zygmund and Rosenthal Inequalities for Symmetric Statistics, Scand. J. Stat., 26(4):621-633, 1999.

  31. Arcones, M.A. and Gine, E., On the Bootstrap of U and V Statistics, Annal. Stat., 20:655-674, 1992.

  32. Kendall, D.G., Pole-Seeking Brownian Motion and Bird Navigation, J. R. Stat. Soc., B36:365-417, 1974.

  33. Watson, G.S., Statistics on Spheres, New York: Wiley, 1983.

  34. Landau,L.D. andLifshitz, E.M., Theory of Elasticity, Moscow: Nauka, 1965.

  35. Moakher, M., A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices, SIAM J. Matrix Anal. Appl., 26(3):735-747,2005.

  36. Billingsley, P., Convergence of Probability Measures, New York: Wiley, 2013.

  37. Hoeffding, W., Probability Inequalities for Sums of Bounded Random Variables, J. Am. Stat. Assoc., 58(301):13-30, 1963.

  38. Serfling, R.J., Approximation Theorems of Mathematical Statistics, Vol. 162, New York: Wiley, 2009.


Articles with similar content:

Kinematic and Dynamical Models of Mechatronic Systems
Journal of Automation and Information Sciences, Vol.29, 1997, issue 4-5
Fedor A. Sopronyuk, M. F. Kirichenko
Estimation of State and Parameters of Dynamic System with the Use of Ellipsoids at the Lack of a Priori Information on Estimated Quantities
Journal of Automation and Information Sciences, Vol.46, 2014, issue 4
Nikolay N. Salnikov
Analysis of Parallel Algorithm of Empirical Models Synthesis on Principles of Genetic Algorithms
Journal of Automation and Information Sciences, Vol.48, 2016, issue 2
Alla N. Lazoriv , Vera M. Medvedchuk , Mikhail I. Gorbiychuk
GLOBAL SENSITIVITY ANALYSIS: AN EFFICIENT NUMERICAL METHOD FOR APPROXIMATING THE TOTAL SENSITIVITY INDEX
International Journal for Uncertainty Quantification, Vol.6, 2016, issue 1
Matieyendou Lamboni
On a Problem of Numerical Simulating the Derivative of Discrete Time Series with Approximate Values
Journal of Automation and Information Sciences, Vol.47, 2015, issue 12
Elena M. Kiseleva, Pavel A. Dovgay, Lyudmila L. Hart