Kernel {KSPM}

Create a Kernel Object

Description

Create a kernel object, to use as variable in a model formula.

Usage

Kernel(x, kernel.function, scale = TRUE, rho = NULL, gamma = NULL,
  d = NULL)

Arguments

x

a formula, a vector or a matrix of variables grouped in the same kernel. It could also be a symetric matrix representing the Gram matrix, associated to a kernel function, already computed by the user.

kernel.function

type of kernel. Possible values are "gaussian", "linear", "polynomial", "sigmoid", "inverse.quadratic" or "equality". See details below. If x is a Gram matrix, associated to a kernel function, already computed by the user, kernel.function should be equal to "gram.matrix".

scale

boolean indicating if variables should be scaled before computing the kernel.

rho, gamma, d

kernel function hyperparameters. See details below.

Details

To use inside kspm() function. 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 Kernel object including all parameters needed in computation of the model

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.