scholarly journals Improving Kernel Methods for Density Estimation in Random Differential Equations Problems

2020 ◽  
Vol 25 (2) ◽  
pp. 33
Author(s):  
Juan Carlos Cortés López ◽  
Marc Jornet Sanz
Keyword(s):  
Monte Carlo Simulation ◽  
Differential Equations ◽  
Density Estimation ◽  
Kernel Methods ◽  
Random Quantity ◽  
Parametric Method ◽  
Kernel Density ◽  
Random Variable ◽  
Random Parameters ◽  
Random Differential Equations

Kernel density estimation is a non-parametric method to estimate the probability density function of a random quantity from a finite data sample. The estimator consists of a kernel function and a smoothing parameter called the bandwidth. Despite its undeniable usefulness, the convergence rate may be slow with the number of realizations and the discontinuity and peaked points of the target density may not be correctly captured. In this work, we analyze the applicability of a parametric method based on Monte Carlo simulation for the density estimation of certain random variable transformations. This approach has important applications in the setting of differential equations with input random parameters.

Statistical Gas Distribution Modeling Using Kernel Methods

2011 ◽  
pp. 153-179 ◽  
Author(s):  
Sahar Asadi ◽  
Matteo Reggente ◽  
Cyrill Stachniss ◽  
Christian Plagemann ◽  
Achim J. Lilienthal
Keyword(s):  
Density Estimation ◽  
Kernel Density Estimation ◽  
Kernel Methods ◽  
Kernel Density ◽  
Estimation Algorithm ◽  
Three Dimensions ◽  
Sensor Data ◽  
Gas Distribution ◽  
Distribution Models ◽  
Gas Measurements

Gas distribution models can provide comprehensive information about a large number of gas concentration measurements, highlighting, for example, areas of unusual gas accumulation. They can also help to locate gas sources and to plan where future measurements should be carried out. Current physical modeling methods, however, are computationally expensive and not applicable for real world scenarios with real-time and high resolution demands. This chapter reviews kernel methods that statistically model gas distribution. Gas measurements are treated as random variables, and the gas distribution is predicted at unseen locations either using a kernel density estimation or a kernel regression approach. The resulting statistical models do not make strong assumptions about the functional form of the gas distribution, such as the number or locations of gas sources, for example. The major focus of this chapter is on two-dimensional models that provide estimates for the means and predictive variances of the distribution. Furthermore, three extensions to the presented kernel density estimation algorithm are described, which allow to include wind information, to extend the model to three dimensions, and to reflect time-dependent changes of the random process that generates the gas distribution measurements. All methods are discussed based on experimental validation using real sensor data.

scholarly journals Modeling the Distribution of Precipitation Forecasts from the Canadian Ensemble Prediction System Using Kernel Density Estimation

2008 ◽  
Vol 23 (4) ◽  
pp. 575-595 ◽  
Author(s):  
Syd Peel ◽  
Laurence J. Wilson
Keyword(s):  
Probability Density ◽  
Density Estimation ◽  
Kernel Density Estimation ◽  
Probabilistic Models ◽  
Kernel Density ◽  
Random Variable ◽  
Ensemble Prediction ◽  
Prediction System ◽  
Ensemble Prediction System ◽  
Density Estimators

Abstract Kernel density estimation is employed to fit smooth probabilistic models to precipitation forecasts of the Canadian ensemble prediction system. An intuitive nonparametric technique, kernel density estimation has become a powerful tool widely used in the approximation of probability density functions. The density estimators were constructed using the gamma kernels prescribed by S.-X. Chen, confined as they are to the nonnegative real axis, which constitutes the support of the random variable representing precipitation accumulation. Performance of kernel density estimators for several different smoothing bandwidths is compared with the discrete probabilistic model obtained as the fraction of member forecasts predicting the events, which for this study consisted of threshold exceedances. A propitious choice of the smoothing bandwidth yields smooth forecasts comparable, or sometimes superior, to the discrete probabilistic forecast, depending on the character of the raw ensemble forecasts. At the same time more realistic models of the probability density are achieved, particularly in the tail of the distribution, yielding forecasts that can be optimally calibrated for extreme events.

scholarly journals Numerical Procedures for Random Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-23 ◽  
Author(s):  
Mohamed Ben Said ◽  
Lahcen Azrar ◽  
Driss Sarsri
Keyword(s):  
Differential Equations ◽  
Polynomial Chaos ◽  
Methodological Approach ◽  
Random Parameter ◽  
Random Coefficients ◽  
Superposition Method ◽  
Random Parameters ◽  
Matrix Formulation ◽  
Random Differential Equations ◽  
Linear Differential

Some methodological approaches based on generalized polynomial chaos for linear differential equations with random parameters following various types of distribution laws are proposed. Mainly, an internal random coefficients method ‘IRCM’ is elaborated for a large number of random parameters. A procedure to build a new polynomial chaos basis and a connection between the one-dimensional and multidimensional polynomials are developed. This allows handling easily random parameters with various laws. A compact matrix formulation is given and the required matrices and scalar products are explicitly presented. For random excitations with an arbitrary number of uncertain variables, the IRCM is couplet to the superposition method leading to successive random differential equations with the same main random operator and right-hand sides depending only on one random parameter. This methodological approach leads to equations with a reduced number of random variables and thus to a large reduction of CPU time and memory required for the numerical solution. The conditional expectation method is also elaborated for reference solutions as well as the Monte-Carlo procedure. The applicability and effectiveness of the developed methods are demonstrated by some numerical examples.

scholarly journals Kernel Density Estimation: Theory and Application in Discriminant Analysis

2016 ◽  
Vol 33 (3) ◽  
pp. 267-279 ◽  
Author(s):  
Thomas Ledl
Keyword(s):  
Discriminant Analysis ◽  
Density Function ◽  
Density Estimation ◽  
Kernel Density Estimation ◽  
Kernel Density ◽  
Random Variable ◽  
Kernel Density Estimator ◽  
Density Estimator ◽  
Quadratic Discriminant Analysis ◽  
Bandwidth Parameter

Nowadays, one can find a huge set of methods to estimate the density function of a random variable nonparametrically. Since the first version of the most elementary nonparametric density estimator (the histogram) researchers produced a vast amount of ideas especially corresponding to the issue of choosing the bandwidth parameter in a kernel density estimator model. To focus not only on a descriptive application, the model seems to be quite suitable for application in discriminant analysis, where (multivariate) class densities are the basis for the assignment of a vector to a given class. Thisarticle gives insight to most popular bandwidth parameter selectors as well as to the performance of the kernel density estimator as a classification method compared to the classical linear and quadratic discriminant analysis, respectively. Both a direct estimation in a multivariate space as well as an application of the concept to marginal normalizations of the single variables will be taken into consideration. From this report the gap between theory and application is going to be pointed out.

From Gaussian kernel density estimation to kernel methods

2012 ◽  
Vol 4 (2) ◽  
pp. 119-137 ◽  
Author(s):  
Shitong Wang ◽  
Zhaohong Deng ◽  
Fu-lai Chung ◽  
Wenjun Hu
Keyword(s):  
Density Estimation ◽  
Kernel Density Estimation ◽  
Kernel Methods ◽  
Kernel Density ◽  
Gaussian Kernel
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