kernel.function {KSPM} | R Documentation |
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 |
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 betweenx
andy
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.