Efficient evaluation of the pdf of a random variable through the kernel density maximum entropy approach

International Journal for Numerical Methods in Engineering

Volumn 75, Issue 13, 2008, Pages 1511-1548

  Alibrandi, U ^a   Ricciardi, G ^a  

^a UNIVERSITY OF MESSINA   (Italy)

Author keywords

Maximum entropy distribution; Moment problem; Probabilistic methods; Uncertainty modelling; Variational methods

Indexed keywords

CLUSTER ANALYSIS; DISTRIBUTION FUNCTIONS; NUMERICAL METHODS; PROBABILITY DISTRIBUTIONS; PROBLEM SOLVING; RANDOM PROCESSES; RANDOM VARIABLES; RISK ASSESSMENT; WIND EFFECTS;

MAXIMUM ENTROPY DISTRIBUTION; MOMENT PROBLEM; PROBABILISTIC METHODS; UNCERTAINTY MODELLING; VARIATIONAL METHODS;

PROBABILITY DENSITY FUNCTION;

EID: 53749105519     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2300     Document Type: Article

References (41)

1. Hausdorff F. Summationsmethoden und Momentfolgen. Mathematische Zeitschrift 1921; 16:74-109, 280-299.
2. Hausdorff F. Moment probleme fur ein endliches Intervall. Mathematische Zeitschrift 1923; 9:220-248.
3. Pearson ES, Johnson NL, Burr IW. Comparison of the percentage points of distributions with the same first four moments, chosen from eight different systems of frequency curves. Communications in Statistics B 1979; 8:191.
4. Charlier CVL. Über das Fehlergesetz. Arkiv foer Matematik, Astronomy och Fysik 1905; 2(8):1-9.
5. van der Marel MP, Franx M. A new method for the identification of non-Gaussian line profiles in elliptical galaxies. Astrophysical Journal 1993; 407(2):525-539.
6. Edgeworth FY. The law of error. Transactions of Cambridge Philosophical Society 1905; 20:36-66, 113-141.
7. Blinnikov S, Moessner R. Expansions for nearly Gaussian distributions. Astronomy and Astrophysics Supplement Series 1998; 130:193-205.
8. Powell MJD. Approximation Theory and Methods. Cambridge University Press: Cambridge, MA, 1981.
9. Charlier CVL. A new form of the frequency function. Meddelande, Lunds Astronomiska Observatorium Series II 1928; 51:1-28.
10. Muscolino G, Ricciardi G, Vasta M. Stationary and non-stationary probability density function for non-linear oscillators. International Journal of Non-Linear Mechanics 1997; 32(6):1051-1064.
11. Muscolino G, Ricciardi G. Probability density function of MDOF structural systems under non-normal delta-correlated inputs. Computer Methods in Applied Mechanics and Engineering 1999; 168:121-133.
12. Guo-Kang Er. A method for multi-parameter pdf estimation of random variables. Structural Safety 1998; 20:25-36.
13. Jaynes ET. Information theory and statistical mechanics. Physical Review 1957; 106:620-630.
14. Jaynes ET. Where do we stand on maximum entropy? In The Maximum Entropy Formalism, Levine RD, Tribus MD (eds). MIT Press: Cambridge, MA, 1978; 15-118.
15. Kapur JN, Kesavan HK. Entropy Optimization Principles with Applications. Academic Press: San Diego, NY, 1992.
16. Athanassoulis GA, Gavriliadis PN. The truncated Hausdorff moment problem solved by using kernel density functions. Probabilistic Engineering Mechanics 2002; 17:273-291.
17. Athanassoulis GA, Gavriliadis PN. Moment data can be analytically completed. Probabilistic Engineering Mechanics 2003; 18:329-338.
18. Karlin S, Shapley LS. Geometry of Moments Spaces, vol. 12. AMS Memoirs: Providence, RI, 1953.
19. Stieltjes T. Recherches sur les fractions continues. Annals de la Faculté de Science de l'Université de Toulouse pour les Sciences Mathématiques et les Sciences Physiques 1894; 8:1-22.
20. Stieltjes T. Recherches sur les fractions continues. Annals de la Faculté de Science de l'Université de Toulouse pour les Sciences Mathématiques et les Sciences Physiques 1895; 9:1-47.
21. Hamburger H. Beitrãge zur Konvergenztheorie der Stieltjesschen Kettenbrüche. Mathematische Zeitschrift 1919; 4:186-222.
22. Hamburger H. Über eine Erweiterung des Stieltjesschen Momentproblems. Mathematische Annalen 1920; 81: 235-319.
23. Hamburger H. Über eine Erweiterung des Stieltjesschen Momentproblems. Mathematische Annalen 1921; 82: 120-164, 168-187.
24. Karlin S, Studden WJ. Tchebyshev Systems. Interscience: NY, 1966.
25. Krein MG, Nudelman AA. The Markov Moment Problem and External Problems. Translations of Mathematical Monographs, vol. 50. American Mathematical Society (AMS), 1977.
26. Prékopa A. Stochastic Programming. Kluwer: Dordrecht, Boston, 1995.
27. Shoat JA. Tamarkin JD. The Problem of Moments. American Mathematical Society: Providence, RI, 1963.
28. Prékopa A, Szedmak S. On the numerical solution of the univariate discrete moment problems. Rutcor Research Report RRR, 32, 2003.
29. GNU Open Source Community. GNU MP, the GNU Multiple Precision Arithmetic Library. http://www.swox.com/gmp/, 2007.
30. Bailey D, Chatterjee S. MPFUN++, a multiple precision floating point computation package in C++. http://www.cs.unc.edu/Research/HARPOON/ mpfun++, 2007.
31. Fasino D. Spectral properties of Hankel matrices and numerical solutions of finite moment problems. Journal of Computational and Applied Mathematics 1995; 65:145-155.
32. Krein MG. The ideas of P.L. Chebyshev and A.A. Markov in the theory of limiting values of integrals and their further development. American Mathematical Society Translations Series 2 1959; 12:1-122.
33. Pandey MD, Ariaratnam ST. Crossing rate analysis of non-Gaussian response of linear systems. Journal of Engineering Mechanics 1996; 122(6):507-511.
34. Tagliani A. Existence and stability of a discrete probability distribution by maximum entropy approach. Applied Mathematics and Computation 2000; 110:105-114.
35. Tagliani A. Discrete probability distributions in the generalized moment problem. Applied Mathematics and Computation 2000; 112:333-343.
36. Kennedy CA, Lennox WC. Solution to the practical problem of moments using non-classical orthogonal polynomials, with application for probabilistic analysis. Probabilistic Engineering Mechanics 2000; 15:371-379.
37. Mathur AK. Statistical analysis of concrete compression results: a comparative study of different codal provisions. Proceedings of National Seminar on Modern Trends in Building Materials, Design and Construction, Institute of Engineers, Allahabad, India, 1987.
38. Kottegoda NT, Rosso R. Statistics, Probability and Reliability for Civil and Environmental Engineers. McGraw-Hill: New York, 1998.
39. Press WH, Flannery BP, Teukolsy SA, Wetterling WT. Numerical Recipes in Fortran. Cambridge University Press: Cambridge, MA, 1992.
40. Vanmarcke EH, Lai SP. Strong-motion duration and RMS amplitude of earthquake records. Bulletin of the Seismological Society of America 1980; 70(4):1293-1307.
41. Lai SP. Statistical characteristics of strong ground motions using power spectral density function. Bulletin of the Seismological Society of America 1982; 1:259-274.