KRLS (version 1.0-0)

gausskernel: Gaussian Kernel Distance Computation

Description

Given a N by D numeric data matrix, this function computes the N by N distance matrix with the pairwise distances between the rows of the data matrix as measured by a Gaussian Kernel.

Usage

gausskernel(X = NULL, sigma = NULL)

Arguments

X

N by N numeric data matrix.

sigma

Positive scalar that specifies the bandwidth of the Gaussian kernel (see details).

Value

An N by N numeric distance matrix that contains the pairwise distances between the rows in X.

Details

Given two D dimensional vectors \(x_i\) and \(x_j\). The Gaussian kernel is defined as

$$k(x_i,x_j)=exp(\frac{-|| x_i - x_j ||^2}{\sigma^2})$$

where \(||x_i - x_j||\) is the Euclidean distance given by

$$||x_i - x_j||=((x_i1-x_j1)^2 + (x_i2-x_j2)^2 + ... + (x_iD-x_jD)^2)^.5$$

and \(\sigma^2\) is the bandwidth of the kernel.

Note that the Gaussian kernel is a measure of similarity between \(x_i\) and \(x_j\). It evalues to 1 if the \(x_i\) and \(x_j\) are identical, and approaches 0 as \(x_i\) and \(x_j\) move further apart.

The function relies on the dist function in the stats package for an initial estimate of the euclidean distance.

See Also

dist function in the stats package.

Examples

# NOT RUN {
X <- matrix(rnorm(6),ncol=2)
gausskernel(X=X,sigma=1)
# }