kernel.function {KSPM}

Kernel Functions

Description

These functions transform a n \times p matrix into a n \times n kernel matrix.

Usage

kernel.gaussian(x, rho = ncol(x))

kernel.linear(x)

kernel.polynomial(x, rho = 1, gamma = 0, d = 1)

kernel.sigmoid(x, rho = 1, gamma = 1)

kernel.inverse.quadratic(x, gamma = 1)

kernel.equality(x)

Arguments

x

a n \times p matrix

gamma, rho, d

kernel hyperparameters (see details)

Details

Given two p-dimensional vectors x and y,

• the Gaussian kernel is defined as k(x,y) = exp\left(-\frac{\parallel x-y \parallel^2}{\rho}\right) where \parallel x-y \parallel is the Euclidean distance between x and y and \rho > 0 is the bandwidth of the kernel,

• the linear kernel is defined as k(x,y) = x^Ty,

• the polynomial kernel is defined as k(x,y) = (\rho x^Ty + \gamma)^d with \rho > 0, d is the polynomial order. Of note, a linear kernel is a polynomial kernel with \rho = d = 1 and \gamma = 0,

• the sigmoid kernel is defined as k(x,y) = tanh(\rho x^Ty + \gamma) which is similar to the sigmoid function in logistic regression,

• the inverse quadratic function defined as k(x,y) = \frac{1}{\sqrt{\parallel x-y \parallel^2 + \gamma}} with \gamma > 0,

• the equality kernel defined as k(x,y) = \left\lbrace \begin{array}{ll} 1 & if x = y \\ 0 & otherwise \end{array}\right..

Of note, Gaussian, inverse quadratic and equality kernels are measures of similarity resulting to a matrix containing 1 along the diagonal.

Value

A n \times n matrix.

Author(s)

Catherine Schramm, Aurelie Labbe, Celia Greenwood

References

Liu, D., Lin, X., and Ghosh, D. (2007). Semiparametric regression of multidimensional genetic pathway data: least squares kernel machines and linear mixed models. Biometrics, 63(4), 1079:1088.