Estimating orthant probabilities of high-dimensional Gaussian vectors with an application to set estimation. (English) Zbl 07498944
Summary: The computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low-dimensional term and a remainder. While the low-dimensional probability can be estimated by fast and accurate quadrature, the remainder requires Monte Carlo sampling. We further refine the estimation by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the remainder and we highlight cases where this approximation brings substantial efficiency gains. The proposed methods are compared against state-of-the-art techniques in a numerical study, which also calls attention to the advantages and drawbacks of the procedure. Finally, the proposed method is applied to derive conservative estimates of excursion sets of expensive to evaluate deterministic functions under a Gaussian random field prior, without requiring a Markov assumption. Supplementary material for this article is available online.
MSC:
| 62-XX | Statistics |
References:
| [1] | Abrahamson, I., Orthant Probabilities for the Quadrivariate Normal Distribution, The Annals of Mathematical Statistics, 35, 1685-1703 (1964) · Zbl 0125.09102 |
| [2] | Bayarri, M. J.; Berger, J. O.; Calder, E. S.; Dalbey, K.; Lunagomez, S.; Patra, A. K.; Pitman, E. B.; Spiller, E. T.; Wolpert, R. L., Using Statistical and Computer Models to Quantify Volcanic Hazards, Technometrics, 51, 402-413 (2009) |
| [3] | Bect, J.; Ginsbourger, D.; Li, L.; Picheny, V.; Vazquez, E., Sequential Design of Computer Experiments for the Estimation of a Probability of Failure, Statistics and Computing, 22, 773-793 (2012) · Zbl 1252.62081 |
| [4] | Bolin, D.; Lindgren, F., Excursion and Contour Uncertainty Regions for Latent Gaussian Models,, Journal of the Royal Statistical Society, 77, 85-106 (2015) · Zbl 1414.62332 |
| [5] | Botev, Z. I., TruncatedNormal: Truncated Multivariate Normal (2015) |
| [6] | ———, The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting,, Journal of the Royal Statistical Society, 79, 1-24 (2017) |
| [7] | Bratley, P.; Fox, B. L., Algorithm 659: Implementing Sobol’s Quasirandom Sequence Generator, ACM Transactions on Mathematical Software, 14, 88-100 (1988) · Zbl 0642.65003 |
| [8] | Chevalier, C., Fast Uncertainty Reduction Strategies Relying on Gaussian Process Models, (2013) |
| [9] | Chevalier, C.; Bect, J.; Ginsbourger, D.; Vazquez, E.; Picheny, V.; Richet, Y., Fast Kriging-Based Stepwise Uncertainty Reduction With Application to the Identification of an Excursion Set, Technometrics, 56, 455-465 (2014) |
| [10] | Chevalier, C.; Ginsbourger, D.; Bect, J.; Molchanov, I.; Uciński, D.; Atkinson, A.; Patan, C., mODa 10 Advances in Model-Oriented Design and Analysis, Estimating and Quantifying Uncertainties on Level Sets Using the Vorob’ev Expectation and Deviation With Gaussian Process Models,, 35-43 (2013), New York: Physica-Verlag HD, New York |
| [11] | Chevalier, C.; Picheny, V.; Ginsbourger, D., The KrigInv Package: An Efficient and User-Friendly R Implementation of Kriging-Based Inversion Algorithms, Computational Statistics and Data Analysis, 71, 1021-1034 (2014) · Zbl 1471.62043 |
| [12] | Cox, D. R.; Wermuth, N., A Simple Approximation for Bivariate and Trivariate Normal Integrals, International Statistical Review, 59, 263-269 (1991) · Zbl 0729.62637 |
| [13] | Craig, P., A New Reconstruction of Multivariate Normal Orthant Probabilities,, Journal of the Royal Statistical Society, 70, 227-243 (2008) · Zbl 05563352 |
| [14] | Dickmann, F.; Schweizer, N., Faster Comparison of Stopping Times by Nested Conditional Monte Carlo,, Journal of Computational Finance, 2, 101-123 (2016) |
| [15] | French, J. P.; Sain, S. R., Spatio-Temporal Exceedance Locations and Confidence Regions, Annals of Applied Statistics, 7, 1421-1449 (2013) · Zbl 1283.62226 |
| [16] | Genz, A., Numerical Computation of Multivariate Normal Probabilities, Journal of Computational and Graphical Statistics, 1, 141-149 (1992) |
| [17] | Genz, A.; Bretz, F., Comparison of Methods for the Computation of Multivariate t Probabilities, Journal of Computational and Graphical Statistics, 11, 950-971 (2002) |
| [18] | ———, Computation of Multivariate Normal and t Probabilities, 195 (2009), New York: Springer-Verlag, New York · Zbl 1204.62088 |
| [19] | Genz, A.; Bretz, F.; Miwa, T.; Mi, X.; Leisch, F.; Scheipl, F.; Hothorn, T., mvtnorm: Multivariate Normal and t Distributions (2017) |
| [20] | Geweke, J., Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, Efficient Simulation From the Multivariate Normal and Student-t Distributions Subject to Linear Constraints and the Evaluation of Constraint Probabilities,, 571-578 (1991) |
| [21] | Hajivassiliou, V.; McFadden, D.; Ruud, P., Simulation of Multivariate Normal Rectangle Probabilities and Their Derivatives Theoretical and Computational Results, Journal of Econometrics, 72, 85-134 (1996) · Zbl 0849.62064 |
| [22] | Hajivassiliou, V. A.; McFadden, D. L., The Method of Simulated Scores for the Estimation of LDV Models, Econometrica, 66, 863-896 (1998) |
| [23] | Hammersley, J., Conditional Monte Carlo, Journal of the ACM (JACM), 3, 73-76 (1956) |
| [24] | Hammersley, J.; Morton, K., Mathematical Proceedings of the Cambridge Philosophical Society, 52, A New Monte Carlo Technique: Antithetic Variates,, 449-475 (1956), Cambridge University Press · Zbl 0071.35404 |
| [25] | Horrace, W. C., Some Results on the Multivariate Truncated Normal Distribution, Journal of Multivariate Analysis, 94, 209-221 (2005) · Zbl 1065.62098 |
| [26] | Kahn, H., Random Sampling (Monte Carlo) Techniques in Neutron Attenuation Problems-I, Nucleonics, 6, 27-33 (1950) |
| [27] | Kahn, H.; Marshall, A. W., Methods of Reducing Sample Size in Monte Carlo Computations, Journal of the Operations Research Society of America, 1, 263-278 (1953) · Zbl 1414.90373 |
| [28] | Kushner, H. J., A New Method of Locating the Maximum Point of an Arbitrary Multipeak Curve in the Presence of Noise, Journal of Basic Engineering, 86, 97-106 (1964) |
| [29] | Lemieux, C., Monte Carlo and Quasi-Monte Carlo Sampling (2009), New York: Springer, New York · Zbl 1269.65001 |
| [30] | Miwa, T.; Hayter, A.; Kuriki, S., The Evaluation of General Non-Centred Orthant Probabilities,, Journal of the Royal Statistical Society, 65, 223-234 (2003) · Zbl 1063.62082 |
| [31] | Molchanov, I., Theory of Random Sets (2005), London: Springer, London · Zbl 1109.60001 |
| [32] | Moran, P., The Monte Carlo Evaluation of Orthant Probabilities for Multivariate Normal Distributions, Australian Journal of Statistics, 26, 39-44 (1984) · Zbl 0549.62039 |
| [33] | Owen, D. B., Tables for Computing Bivariate Normal Probabilities, The Annals of Mathematical Statistics, 27, 1075-1090 (1956) · Zbl 0073.13405 |
| [34] | Pakman, A.; Paninski, L., Exact Hamiltonian Monte Carlo for Truncated Multivariate Gaussians, Journal of Computational and Graphical Statistics, 23, 518-542 (2014) |
| [35] | Picheny, V.; Ginsbourger, D.; Richet, Y.; Caplin, G., Quantile-Based Optimization of Noisy Computer Experiments With Tunable Precision, Technometrics, 55, 2-13 (2013) |
| [36] | Rasmussen, C. E.; Williams, C. K., Gaussian Processes for Machine Learning (2006), Cambridge, MA: MIT Press, Cambridge, MA · Zbl 1177.68165 |
| [37] | R Core Team, R: A Language and Environment for Statistical Computing (2017), Vienna, Austria: R Foundation for Statistical Computing, Vienna, Austria |
| [38] | Ridgway, J., Computation of Gaussian Orthant Probabilities in High Dimension, Statistics and Computing, 26, 899-916 (2016) · Zbl 1505.62339 |
| [39] | Robert, C.; Casella, G., Monte Carlo Statistical Methods (2013), New York: Springer, New York |
| [40] | Robert, C. P., Simulation of Truncated Normal Variables, Statistics and Computing, 5, 121-125 (1995) |
| [41] | Rossi, P., Bayesm: Bayesian Inference for Marketing/Micro-Econometrics (2015) |
| [42] | Sacks, J.; Welch, W. J.; Mitchell, T. J.; Wynn, H. P., Design and Analysis of Computer Experiments, Statistical Science, 4, 409-435 (1989) · Zbl 0955.62619 |
| [43] | Schervish, M. J., Algorithm AS 195: Multivariate Normal Probabilities With Error Bound,, Journal of the Royal Statistical Society, 33, 81-94 (1984) · Zbl 0547.65097 |
| [44] | Tong, Y. L., The Multivariate Normal Distribution (2012), New York: Springer, New York · Zbl 0689.62036 |
| [45] | Wickham, H., ggplot2: Elegant Graphics for Data Analysis (2009), New York: Springer-Verlag, New York · Zbl 1170.62004 |
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