Kernel Smoothing

Kernel smoothing is a group of powerful smoothing algorithms that consists in applying a function known as the kernel to each data point in the time-series. Kernel Smoothing belongs to the class of weighted moving averages.

This means in practice that all the points in the time-series are weighted using as weights the results of the computation of the kernel function.

Every kernel function has several properties:

All kernel values are positive or zero.
The kernel functions are normally symmetric.
Kernel function values decrease to zero from a central (maximum) value.

We are offering two versions of each kernel smoothing method: the first version computes the kernel without modifications to the original signal and the second one computes the kernel extending the original signal by linear extrapolation, simple extension of last value or linear prediction the signal. These extended methods allow better behavior of the kernel smoothing at the right edge of the time-series.

See the Excel workbook with Kernel Smoothing methods for an example of the application of these extrapolation methods and their effects on the kernels. The kernel functions supported by IPredict’s library are the following:

Gaussian

$K(t) = e^{-\lambda t^2}$

Gaussian Kernel

Hilbert

$K(t) = { {-\pi} \over t }$

Hilbert Kernel

Triangle

$K(T) = 1 - |t|$

Triangle Kernel

Epanechnicov

$K(t) = {3 \over 4} (1 - t^2)$

Epanechnicov Kernel

Quartic

$K(t) = {15 \over 16} (1 - t^2)^2$

Quartic Kernel

Triweight

$K(t) = {35 \over 32} (1 - t^2)^3$

Triweight Kernel

Cosine

$K(t) = - { {\pi \over 4} } \cos( {\pi \over 2} t)$

Cosine Kernel

References:

Kernel Smoothing, M.P. Wand and M.C. Jones White Noise Theory of Prediction, Filtering and Smoothing, G. Kallianpur and Rajeeva Karandikar

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