SmoothingKernels.jl

These kernels are designed for use in smoothing algorithms such as kernel regression and kernel density estimation. They are implemented in both unnormalized and normalized form.

Mathematical Form of Implemented Kernels

Currently, the kernels implemented are those found in the Wikipedia article on kernels in statistics.

In normalized form, the kernels are:

• Uniform: $K(u) = \frac{1}{2} I(|u| \leq 1)$
• Triangular: $K(u) = (1 - |u|) I(|u| \leq 1)$
• Epanechnikov: $K(u) = \frac{3}{4} (1 - |u|^2) I(|u| \leq 1)$
• Biweight (Quartic): $K(u) = \frac{15}{16} (1 - |u|^2)^2 I(|u| \leq 1)$
• Triweight: $K(u) = \frac{35}{32} (1 - |u|^2)^3 I(|u| \leq 1)$
• Tricube: $K(u) = \frac{70}{81} (1 - |u|^3)^3 I(|u| \leq 1)$
• Gaussian: $K(u) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}u^2}$
• Cosine: $K(u) = \frac{\pi}{4} \cos(\frac{\pi}{2} u) I(|u| \leq 1)$
• Logistic: $K(u) = \frac{1}{e^u + 2 + e^{-u}}$

Usage Example

using SmoothingKernels, StatsBase

x = randn(100)

h = StatsBase.bandwidth(x)

λ = 1 / h

kval = λ * SmoothingKernels.kernels[:uniform](λ * (x - 0))
kval = λ * SmoothingKernels.unnormalized_kernels[:uniform](λ * (x - 0))

kval = λ * SmoothingKernels.kernels[:gaussian](λ * (x - 0))
kval = λ * SmoothingKernels.unnormalized_kernels[:gaussian](λ * (x - 0))

To Do

Extend these kernels to work with data points in $\mathbb{R}^k$ and not just $\mathbb{R}$.