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 $X$ is a univariate random variable with unknown density $f$ and independent and identically distributed samples $x_{1}, x_{2}, \ldots, x_{n}$, the KDE is given by

\[ \hat{f} = \frac{1}{nh} \sum_{i = 1}^{n} K \left( \frac{x - x_{i}}{h} \right). \]

The kernel $K$ 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 $h$ is a smoothing parameter that should be optimized. Choosing a large value of $h$ yields a wide KDE with large variance, while choosing a small value of $h$ yields a choppy KDE with large bias. Dakota approximates the bandwidth using Silverman's rule of thumb,

\[ h = \hat{\sigma} \left( \frac{4}{3n} \right)^{1/5}, \]

where $\hat{\sigma}$ is the standard deviation of the sample set $\left\{ x_{i} \right\}$.

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 $\hat{f}$ 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