Evaluating nearly singular multinormal expectations with application to wave distributions

Methodology and Computing in Applied Probability

Volumn 8, Issue 1, 2006, Pages 65-91

  Brodtkorb, Per A ^a  

^a NORWEGIAN DEFENCE RESEARCH ESTABLISHMENT FFI   (Norway)

Author keywords

Multivariate normal probabilities; Numerical integration; Singular distribution; Wavecrest amplitude and crest velocity

Indexed keywords

EID: 33747596627     PISSN: 13875841     EISSN: 15737713     Source Type: Journal    
DOI: 10.1007/s11009-006-7289-y     Document Type: Article

References (53)

1. A. Ambartzumian, A. Der Kiureghian, V. Ohanian, and H. Sukiasian, "Multinormal probability by sequential conditioned importance sampling: Theory and application," Probalistic Engineering Mechanics vol. 13(4) pp. 299-308, 1998.
2. A. Baxevani, K. Podgórski, and I. Rychlik, "Velocities for moving random surfaces," Probalistic Engineering Mechanics vol. 18(3) pp. 251-271, 2003.
3. M. Beckers and A. Haegemans, "A comparision of numerical integration techniques for multivariate normal integrals." Technical Report preprint, Comp. Sci. Dept., Cath. Univ. Leuven, Belgium, 1992.
4. J. Berntsen, T. Espelid, and A. Genz, "Algortihm 698: DCUHRE - an adaptive multidimensional integration routine for vector of integrals," ACM Transactions on Mathematical Software vol. 17 pp. 452-456, 1991.
5. P. Brodtkorb, "Evaluating multinormal probabilities with product correlation structure." Technical Report No. 28, ISSN 1403 9338, Math. Stat., Center for Math. Sci., Lund Univ., Sweden, 2004.
6. P. Brodtkorb, P. Johannesson, G. Lindgren, I. Rychlik, J. Rydén, and E. Sjö, "WAFO - a Matlab toolbox for the analysis of random waves and loads." In: Proc. 10'th Int. Offshore and Polar Eng. Conf., ISOPE, Seattle, USA, vol. 3. pp. 343-350, 2000.
7. T. C. Chan and H. A. van der Vorst, "Approximate and incomplete factorizations." Technical Report 871, Dept. of Math., Univ. of Utrecht, The Netherlands, http://www.citeseer.ist.psu.edu/ chan94approximate.html, 1994.
8. W. J. Cody, "Rational Chebyshev approximations for the error function," Mathematics Competitions pp. 631-638, 1969.
9. R. Cranley and T. Patterson, "Randomization of number theoretic methods for multiple integration," SIAM Journal on Numerical Analysis vol. 13 pp. 904-914, 1976.
10. P. I. Davies and N. J. Higham, "Numerically stable generation of correlation matrices and their factors." Technical Report 354, Manchester Centre for Comp. Math., Manchester, England, 1999.
11. P. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, pp. 482-483, 1984.
12. I. Deák, "Three digit accurate multiple normal probabilities," Numerical Mathematics vol. 35 pp. 369-380, 1980.
13. I. Deák, "Computing probabilities of rectangles in case of multinormal distribution." Journal of Statistical Computation and Simulation vol. 26, pp. 101-114, 1986.
14. I. Deák, Random Number Generation and Simulation, Akadémiai Kiadó: Budapest, Chapter 7, 1990.
15. I. Deák, "Probabilities of simple n-dimensional sets for the normal distribution," IIE Transactions vol. 35 pp. 285-293, 2003.
16. O. Ditlevsen, R. Melchers, and H. Gluver, "General multi-dimensional probability integration by directional simulation," Computers and Structures vol. 36(2) pp. 355-368, 1990.
17. C. W. Dunnett, "Multivariate normal probability integrals with product correlation structure," Applied Statistics vol. 38(3) pp. 564-571, 1989.
18. H. Gassmann, "Multivariate normal probabilities: Implementing an old idea of Plackett's," Journal of Computational and Graphical Statistics vol. 12(3) pp. 731-752, 2003.
19. H. Gassmann, I. Deák, and T. Szantai, "Computing multivariate normal probabilities: A new look," Journal of Computational and Graphical Statistics vol. 11(4) pp. 920-949, 2002.
20. A. Genz, "Numerical computation of multivariate normal probabilities," Journal of Computational and Graphical Statistics vol. 1 pp. 141-149, 1992.
21. A. Genz, "Comparison of methods for the computation of multivariate normal probabilities," Computing Science and Statistics vol. 25 pp. 400-405, 1993.
22. A. Genz, "Numerical computation of rectangular bivariate and trivariate normal and t-probabilities," Statistics and Computing vol. 14 pp. 151-160, 2004.
23. A. Genz and K.-S. Kwong, "Numerical evaluation of singular multivariate normal distributions," Journal of Statistical Computation and Simulation vol. 68 pp. 1-21, 2000.
24. A. Genz and F. Bretz, "Methods for the computation of multivariate t-probabilities," Journal of Computational and Graphical Statistics vol. 11 pp. 950-971, 2002.
25. G. Gibson, C. A. Glasbey, and D. Elston, "Monte-Carlo evaluation of multivariate normal integrals." Technical Report preprint, Scottish Agricultural Stat. Service, 1992.
26. G. Golub and C. Van Loan, Matrix Computations (2nd edn.), Johns Hopkins University Press, 1989.
27. K. Hasselmann, T. Barnett, E. Buows, H. Carlson, D. Carthwright, K. Enke, J. Ewing, H. Gienapp, D. Hasselmann, P. Kruseman, A. Meerburg, A. Müller, D. Olbers, K. Richter, W. Sell, and H. Walden, "Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project," Deutschen Hydrografischen Zeitschrift vol. 12 pp. 9-95, 1973.
28. M. Healy, "Algorithm AS 6: Triangular decomposition of a symmetric matrix," Applied Statistics vol. 17 pp. 195-196, 1968.
29. N. J. Higham, "Analysis of the Cholesky decomposition of a semi-definite matrix." In M. Cox and S. Hammarling (eds.), Reliable Numerical Computation, Oxford Universtiy Press, pp. 161-185, 1990.
30. M. Hohenbichler and R. Rackwitz, "First-order concepts in system reliability," Structural Safety vol. 1(3) pp. 177-188, 1983.
31. H. Hong and F. Hickernell, "Implementing scrambled digital sequences," ACM Transactions on Mathematical Software vol. 29 pp. 95-109, 2003.
32. P. Keast, "Optimal parameters for multidimensional integration," SIAM Journal on Numerical Analysis vol. 10 pp. 831-838, 1973.
33. G. Lepage, "A new algorithm for adaptive multidimensional integration," Journal of Computational Physics vol. 27 pp. 192-203, 1978.
34. G. Lindgren and I. Rychlik, "How reliable are contour curves? Confidence sets for level countours," Bernoulli vol. 1 pp. 301-319, 1995.
35. J. Monahan and A. Genz, "Spherical-radial integration rules for Bayesian computation," Journal of the American Statistical Association vol. 92 pp. 664-674, 1997.
36. M. Ochi and K. Ahn, "Non-Gaussian probability distribution of coastal waves." In: Proc. 24'th Int. Conf. on Coastal Eng., ICCE, Kobe, Japan, vol. 1. pp. 482-496, 1994.
37. K. Podgórski, I. Rychlik, and U. Machado, "Exact distributions for apparent waves in irregular seas," Ocean Engineering vol. 27(1) pp. 979-1016, 2000a.
38. Podgórski, K. Rychlik, I. Rydén, J. Sjó, E. "How big are the big waves?" International Journal of Offshore and Polar Engineering vol. 10 pp. 161-169, 2000b.
39. K. Podgórski, I. Rychlik, and E. Sjö, "Statistics for velocities of Gaussian waves," International Journal of Offshore and Polar Engineering vol. 10(2) pp. 91-98, 2000c.
40. M. Rosenblatt, "Remarks on a multivariate transformation," Annals of Mathematical Statistics vol. 23 pp. 470-472, 1952.
41. I. Rychlik, "A note on Durbin's formula for the first-passage density," Statistics and Probability Letters vol. 5 pp. 425-428, 1987.
42. I. Rychlik, "Confidence bands for linear regressions," Communications in Statistics-Simulation and Computation vol. 21(2) pp. 333-352, 1992.
43. I. Rychlik, P. Johannesson, and M. R. Leadbetter, "Modelling and statistical analysis of ocean-wave data using transformed Gaussian process," Marine Structures, Design, Construction and Safety vol. 10 pp. 13-47, 1997.
44. J. Rydén, S. van Iseghem, M. Olagnon, and I. Rychlik, "Evaluating height-length joint distributions for the crests of ocean waves," Applied Ocean Research vol. 24(2) pp. 189-201, 2002.
45. M. Schervish, "Multivariate normal probabilities with error bound," Applied Statistics vol. 33 pp. 81-87, 1984.
46. W. Schroeder, K. Martin, and B. Lorensen, "The visualization toolkit." An Object-Orientated Approach to 3D Graphics., ISBN 0-13-954694-4. Prentice-Hall. http://public.kitware.com/VTK/, 1998.
47. W. Schroeder, L. Avila, K. Martin, W. Hoffman, and C. Law, The visualization toolkit user's guide, ISBN 1-930934-05-X. Kitware, inc. http://public.kitware.com/VTK/, 2001.
48. E. Sjö, "Simultaneous distributions of space-time wave characteristics in a Gaussian sea," Extremes vol. 3 pp. 263-288, 2001.
49. P. Somerville, "Numerical computation of multivariate normal and multivariate-t probabilities over Convex regions," Journal of Computational and Graphical Statistics vol. 7(4) pp. 529-544, 1998.
50. G. Stewart, "The efficient generation of random orthogonal matrices with an application to condition estimators," SIAM Journal on Numerical Analysis vol. 17 pp. 403-409, 1980.
51. Y. Tong, The Multivariate Normal Distribution, Springer-Verlag, 1990.
52. S. Winterstein, "Nonlinear vibration models for extremes and fatigue," Journal of Engineering Mechanics, ASCE vol. 114(10) pp. 1772-1790, 1988.
53. n) transformation," Mathematical Tables and other Aids to Computation vol. 10 pp. 91-96, 1956.