On Fractional Moment Estimation from Polynomial Chaos Expansion

• URL: http://arxiv.org/abs/2403.01948v1
• Date: Mon, 4 Mar 2024 11:34:12 GMT
• Title: On Fractional Moment Estimation from Polynomial Chaos Expansion
• Authors: Luk\'a\v{s} Nov\'ak and Marcos Valdebenito and Matthias Faes
• Abstract summary: This paper presents a novel approach for the estimation of fractional moments directly from chaos expansions. The proposed approach achieves a superior performance in estimating the distribution of the response, in comparison to a standard Latin hypercube sampling.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Fractional statistical moments are utilized for various tasks of uncertainty quantification, including the estimation of probability distributions. However, an estimation of fractional statistical moments of costly mathematical models by statistical sampling is challenging since it is typically not possible to create a large experimental design due to limitations in computing capacity. This paper presents a novel approach for the analytical estimation of fractional moments, directly from polynomial chaos expansions. Specifically, the first four statistical moments obtained from the deterministic PCE coefficients are used for an estimation of arbitrary fractional moments via H\"{o}lder's inequality. The proposed approach is utilized for an estimation of statistical moments and probability distributions in three numerical examples of increasing complexity. Obtained results show that the proposed approach achieves a superior performance in estimating the distribution of the response, in comparison to a standard Latin hypercube sampling in the presented examples.

Related papers

• Robust Estimation for Kernel Exponential Families with Smoothed Total Variation Distances [2.317910166616341]
  In statistical inference, we commonly assume that samples are independent and identically distributed from a probability distribution. In this paper, we explore the application of GAN-like estimators to a general class of statistical models.
  arXiv  Detail & Related papers  (2024-10-28T05:50:47Z)
• Unveiling the Statistical Foundations of Chain-of-Thought Prompting Methods [59.779795063072655]
  Chain-of-Thought (CoT) prompting and its variants have gained popularity as effective methods for solving multi-step reasoning problems. We analyze CoT prompting from a statistical estimation perspective, providing a comprehensive characterization of its sample complexity.
  arXiv  Detail & Related papers  (2024-08-25T04:07:18Z)
• Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution [2.1146241717926664]
  We show that the Wasserstein GAN, constrained to left-invertible push-forward maps, generates distributions that avoid replication and significantly deviate from the empirical distribution. Our most important contribution provides a finite-sample lower bound on the Wasserstein-1 distance between the generative distribution and the empirical one. We also establish a finite-sample upper bound on the distance between the generative distribution and the true data-generating one.
  arXiv  Detail & Related papers  (2023-07-31T06:11:57Z)
• Efficient CDF Approximations for Normalizing Flows [64.60846767084877]
  We build upon the diffeomorphic properties of normalizing flows to estimate the cumulative distribution function (CDF) over a closed region. Our experiments on popular flow architectures and UCI datasets show a marked improvement in sample efficiency as compared to traditional estimators.
  arXiv  Detail & Related papers  (2022-02-23T06:11:49Z)
• Probabilistic learning inference of boundary value problem with uncertainties based on Kullback-Leibler divergence under implicit constraints [0.0]
  We present a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a boundary value problem from a prior probability model. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional homogenization of heterogeneous linear elastic media.
  arXiv  Detail & Related papers  (2022-02-10T16:00:10Z)
• Sampling from Arbitrary Functions via PSD Models [55.41644538483948]
  We take a two-step approach by first modeling the probability distribution and then sampling from that model. We show that these models can approximate a large class of densities concisely using few evaluations, and present a simple algorithm to effectively sample from these models.
  arXiv  Detail & Related papers  (2021-10-20T12:25:22Z)
• Divergence Frontiers for Generative Models: Sample Complexity, Quantization Level, and Frontier Integral [58.434753643798224]
  Divergence frontiers have been proposed as an evaluation framework for generative models. We establish non-asymptotic bounds on the sample complexity of the plug-in estimator of divergence frontiers. We also augment the divergence frontier framework by investigating the statistical performance of smoothed distribution estimators.
  arXiv  Detail & Related papers  (2021-06-15T06:26:25Z)
• Statistically significant tests of multiparticle quantum correlations based on randomized measurements [0.0]
  We introduce hierarchies of multi-qubit criteria, satisfied by states which are separable with respect to partitions of different size. We analyze in detail the statistical error of the estimation in experiments and present several approaches for estimating the statistical significance.
  arXiv  Detail & Related papers  (2020-12-22T17:14:28Z)
• Distributionally Robust Parametric Maximum Likelihood Estimation [13.09499764232737]
  We propose a distributionally robust maximum likelihood estimator that minimizes the worst-case expected log-loss uniformly over a parametric nominal distribution. Our novel robust estimator also enjoys statistical consistency and delivers promising empirical results in both regression and classification tasks.
  arXiv  Detail & Related papers  (2020-10-11T19:05:49Z)
• Instability, Computational Efficiency and Statistical Accuracy [101.32305022521024]
  We develop a framework that yields statistical accuracy based on interplay between the deterministic convergence rate of the algorithm at the population level, and its degree of (instability) when applied to an empirical object based on $n$ samples. We provide applications of our general results to several concrete classes of models, including Gaussian mixture estimation, non-linear regression models, and informative non-response models.
  arXiv  Detail & Related papers  (2020-05-22T22:30:52Z)
• Minimax Optimal Estimation of KL Divergence for Continuous Distributions [56.29748742084386]
  Esting Kullback-Leibler divergence from identical and independently distributed samples is an important problem in various domains. One simple and effective estimator is based on the k nearest neighbor between these samples.
  arXiv  Detail & Related papers  (2020-02-26T16:37:37Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.