Kernel density maximum entropy method with generalized moments for evaluating probability distributions, including tails, from a small sample of data

Summary In this paper, a novel method to determine the distribution of a random variable from a sample of data is presented. The approach is called Generalized Kernel Density Maximum Entropy Method (GKDMEM), because it adopts a Kernel Density (KD) representation of the target distribution, while its free parameters are determined through the principle of Maximum Entropy (ME). Here, the ME solution is determined by assuming that the available information is represented from generalized moments, which include as their subsets the power and the fractional ones. The proposed method has several important features: (i) applicable to distributions with any kind of support, (ii) computational efficiency because the ME solution is simply obtained as a set of systems of linear equations, (iii) good trade-off between bias and variance, and (iv) good estimates of the tails of the distribution, in presence of samples of small size. Moreover, the joint application of GKDME with a bootstrap resampling allows to define credible bounds of the target distribution. The method is first benchmarked through an example of stochastic dynamic analysis. Subsequently, it is used to evaluate the seismic fragility functions of a reinforced concrete frame, from the knowledge of a small set of available ground motions.

[1]  Bruce R. Ellingwood,et al.  Earthquake risk assessment of building structures , 2001, Reliab. Eng. Syst. Saf..

[2]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[3]  Henryk Gzyl,et al.  Determination of the probability of ultimate ruin by maximum entropy applied to fractional moments , 2013 .

[4]  F. Laurent,et al.  An extended quadrature method of moments for population balance equations , 2003 .

[5]  Umberto Alibrandi,et al.  Efficient evaluation of the pdf of a random variable through the kernel density maximum entropy approach , 2008 .

[6]  Pierfrancesco Cacciola,et al.  New insights in the analysis of the structural response to response-spectrum-compatible accelerograms , 2014 .

[7]  G. Talenti.  Recovering a function from a finite number of moments , 1987 .

[9]  R. Baker,et al.  Probability estimation and information principles , 1990 .

[10]  P. L. Novi Inverardi,et al.  Maximum Entropy Density Estimation from Fractional Moments , 2003 .

[11]  Gerassimos A. Athanassoulis,et al.  The truncated Hausdorff moment problem solved by using kernel density functions , 2002 .

[12]  Umberto Alibrandi,et al.  TAIL EQUIVALENT LINEARIZATION METHODS FOR SEISMIC RESPONSE SPECTRUM ANALYSIS , 2015 .

[13]  Khalid M. Mosalam,et al.  PEER Performance-Based Earthquake Engineering Methodology, Revisited , 2013 .

[14]  Pierfrancesco Cacciola,et al.  Combination of modal responses consistent with seismic input representation , 2004 .

[15]  Khalid M. Mosalam,et al.  Response Spectrum Code-Conforming PEER PBEE using Stochastic Dynamic Analysis and Information Theory , 2018 .

[16]  C. Yuan,et al.  Conditional quadrature method of moments for kinetic equations , 2011, J. Comput. Phys..

[17]  S. Winterstein.  Nonlinear Vibration Models for Extremes and Fatigue , 1988 .

[18]  E. Jaynes.  On the rationale of maximum-entropy methods , 1982, Proceedings of the IEEE.

[19]  M. Pandey,et al.  Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method , 2013 .

[20]  J. N. Kapur,et al.  Entropy Optimization Principles and Their Applications , 1992 .

[21]  H Y Kim,et al.  STATISTICAL ANALYSIS OF FRAGILITY CURVES , 2000 .

[22]  Ying Min Low,et al.  A new distribution for fitting four moments and its applications to reliability analysis , 2013 .

[23]  M. Shinozuka,et al.  Digital simulation of random processes and its applications , 1972 .

[24]  J. Wheeler,et al.  Modified moments and Gaussian quadratures , 1974 .

[25]  N. L. Johnson,et al.  Comparisons of the percentage points of distributions with the same first four moments, chosen from eight different systems of frequency curves , 1979 .

[26]  P. C. Jennings.  Simulated earthquake motions for design purposes , 1968 .

[27]  Aldo Tagliani,et al.  Discrete probability distributions in the generalized moment problem , 2000, Appl. Math. Comput..

[28]  Aldo Tagliani.  Hausdorff moment problem and fractional moments: A simplified procedure , 2011, Appl. Math. Comput..

[29]  D. W. Scott,et al.  Multidimensional Density Estimation , 2005 .

[30]  M. Shinozuka,et al.  Simulation of Stochastic Processes by Spectral Representation , 1991 .

[31]  Robert McGraw,et al.  Description of Aerosol Dynamics by the Quadrature Method of Moments , 1997 .

[32]  Aldo Tagliani,et al.  INVERSE z TRANSFORM AND MOMENT PROBLEM , 2000, Probability in the Engineering and Informational Sciences.