kernel.function: Kernel Functions
In KSPM: Kernel Semi-Parametric Models

Description Usage Arguments Details Value Author(s) References

These functions transform a n x p matrix into a n x n kernel matrix.

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)

`x`	a n x p matrix
`gamma, rho, d`	kernel hyperparameters (see details)

Given two p-dimensional vectors x and y,

the Gaussian kernel is defined as k(x,y) = exp(-||x-y||^2 / rho) where ||x-y|| 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) = t(x).y,
the polynomial kernel is defined as k(x,y) = (rho.t(x).y + 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.t(x).y + gamma) which is similar to the sigmoid function in logistic regression,
the inverse quadratic function defined as k(x,y) = 1 / sqrt( ||x-y||^2 + gamma) with gamma > 0,
the equality kernel defined as k(x,y) = 1 if x = y, 0 otherwise.

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

A n x n matrix.

Catherine Schramm, Aurelie Labbe, Celia Greenwood

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.

KSPM documentation built on Aug. 10, 2020, 5:07 p.m.