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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.
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.
Kernel PCR essentially works around this problem by considering an equivalent dual formulation based on using the spectral decomposition of the associated kernel matrix. 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 ...
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.
The complexity of training is basically the cost of computing the kernel matrix plus the cost of solving the linear system which is roughly (). The computation of the kernel matrix for the linear or Gaussian kernel is (). The complexity of testing is ().
Kernel methods owe their name to the use of kernel functions, which enable them to operate in a high-dimensional, implicit feature space without ever computing the coordinates of the data in that space, but rather by simply computing the inner products between the images of all pairs of data in the feature space. This operation is often ...