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Dakota Reference Manual
Version 6.15
Explore and Predict with Confidence
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Calculate the Kernel Density Estimate of the posterior distribution
Alias: none
Argument(s): none
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.
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