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MATLAB: A free MATLAB toolbox with implementation of kernel regression, kernel density estimation, kernel estimation of hazard function and many others is available on these pages (this toolbox is a part of the book [6]).
The kernel of a reproducing kernel Hilbert space is used in the suite of techniques known as kernel methods to perform tasks such as statistical classification, regression analysis, and cluster analysis on data in an implicit space. This usage is particularly common in machine learning.
Kernel regression estimates the continuous dependent variable from a limited set of data points by convolving the data points' locations with a kernel function—approximately speaking, the kernel function specifies how to "blur" the influence of the data points so that their values can be used to predict the value for nearby locations.
Under the linear regression model (which corresponds to choosing the kernel function as the linear kernel), this amounts to considering a spectral decomposition of the corresponding kernel matrix and then regressing the outcome vector on a selected subset of the eigenvectors of so obtained. It can be easily shown that this is the same as ...
Multivariate Kernel Smoothing and Its Applications is a comprehensive book on many topics in kernel smoothing, including density estimation. Includes ks package code snippets in R. kde2d.m A Matlab function for bivariate kernel density estimation. libagf A C++ library for multivariate, variable bandwidth kernel density estimation.
Output after kernel PCA, with a Gaussian kernel. Note in particular that the first principal component is enough to distinguish the three different groups, which is impossible using only linear PCA, because linear PCA operates only in the given (in this case two-dimensional) space, in which these concentric point clouds are not linearly separable.
GRNN can be used for regression, prediction, and classification. GRNN can also be a good solution for online dynamical systems. GRNN represents an improved technique in the neural networks based on the nonparametric regression. The idea is that every training sample will represent a mean to a radial basis neuron. [2]
Kernel average smoother example. The idea of the kernel average smoother is the following. For each data point X 0, choose a constant distance size λ (kernel radius, or window width for p = 1 dimension), and compute a weighted average for all data points that are closer than to X 0 (the closer to X 0 points get higher weights).