Dakota Reference Manual: kde

| Dakota Reference Manual Version 6.15 Explore and Predict with Confidence

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kde

Calculate the Kernel Density Estimate of the posterior distribution

## Specification

Alias: none

Argument(s): none

## Description

A kernel density estimate (KDE) is a non-parametric, smooth approximation of the probability density function of a random variable. It is calculated using a set of samples of the random variable. If is a univariate random variable with unknown density and independent and identically distributed samples , the KDE is given by

The kernel is a non-negative function which integrates to one. Although the kernel can take many forms, such as uniform or triangular, Dakota uses a normal kernel. The bandwidth is a smoothing parameter that should be optimized. Choosing a large value of yields a wide KDE with large variance, while choosing a small value of yields a choppy KDE with large bias. Dakota approximates the bandwidth using Silverman's rule of thumb,

where is the standard deviation of the sample set .

For multivariate cases, the random variables are treated as independent, and a separate KDE is calculated for each.

Expected Output

If `kde` is specified, calculated values of will be output to the file `kde_posterior.dat`. Example output is given below.

## Examples

Below is a `method` block of a Dakota input file that indicates the calculation of the KDE

    method,
        bayes_calibration queso
          dram
          seed = 34785
          chain_samples = 1000
          posterior_stats kde

The calculated KDE values are output to the file `kde_posterior.dat`, as shown below

    uuv_1  KDE PDF estimate
    0.406479    61.2326
    0.406338    64.0245
    0.402613    114.468
    0.402613    114.468
    0.40249    114.409
    0.40282    114.162
    0.398899    65.2361
    0.400093    84.9285
    0.401264    104.105
    0.400917    98.7803