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In statistics, kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable. The objective is to find a non-linear relation between a pair of random variables X and Y .
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 ...
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.
In nonparametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable.
[3] [4] [5] Geostatistical approaches to multivariate modeling are mostly formulated around the linear model of coregionalization (LMC), a generative approach for developing valid covariance functions that has been used for multivariate regression and in statistics for computer emulation of expensive multivariate computer codes. The ...
Unsupervised multiple kernel learning algorithms have also been proposed by Zhuang et al. The problem is defined as follows. Let = be a set of unlabeled data. The kernel definition is the linear combined kernel ′ = =. In this problem, the data needs to be "clustered" into groups based on the kernel distances.
The linear regression model turns out to be a special case of this setting when the kernel function is chosen to be the linear kernel. In general, under the kernel machine setting, the vector of covariates is first mapped into a high-dimensional (potentially infinite-dimensional ) feature space characterized by the kernel function chosen.
Since the value of the RBF kernel decreases with distance and ranges between zero (in the infinite-distance limit) and one (when x = x'), it has a ready interpretation as a similarity measure. [2] The feature space of the kernel has an infinite number of dimensions; for =, its expansion using the multinomial theorem is: [3]