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Kernel Density Estimation

Kernel Density Estimation (KDE) is a non-parametric method for estimating the probability density function of a random variable. Given \(N\) samples \(\{x_1, \ldots, x_N\}\), the KDE estimate is:

\[\hat{p}(x) = \frac{1}{Nh} \sum_{i=1}^{N} K\left(\frac{x - x_i}{h}\right)\]

where \(K(\cdot)\) is a kernel function (typically Gaussian) and \(h > 0\) is the bandwidth parameter.

Bandwidth Selection

The bandwidth \(h\) controls the smoothness of the estimate:

  • Too small: overfitting, noisy estimate
  • Too large: oversmoothing, loss of detail

Common selection methods include Scott's rule (\(h = 1.06 \hat{\sigma} N^{-1/5}\)) and Silverman's rule.

Role in RBIG

In the RBIG pipeline, KDE is one option for estimating the marginal CDF \(F_d(x_d)\) during the uniformization step. See also PDF Estimation for alternative approaches.

Resources

Built-In