kernel {iprior}

Reproducing kernels for the I-prior package

Description

The kernel functions used in this package are:

• The (canonical) linear kernel

• The fractional Brownian motion (fBm) kernel with Hurst index \gamma

• The Pearson kernel

• The (scaled) d-degree polynomial kernel with offset c

• The squared exponential (SE) kernel with lengthscale l

Usage

kern_canonical(x, y = NULL, centre = TRUE)

kern_linear(x, y = NULL, centre = TRUE)

kern_pearson(x, y = NULL)

kern_fbm(x, y = NULL, gamma = 0.5, centre = TRUE)

kern_se(x, y = NULL, l = 1, centre = FALSE)

kern_poly(x, y = NULL, c = 0, d = 2, lam.poly = 1, centre = TRUE)

Arguments

x

A vector, matrix or data frame.

y

(Optional) vector, matrix or data frame. x and y must have identical column sizes.

centre

Logical. Whether to centre the data (default) or not.

gamma

The Hurst coefficient for the fBm kernel.

l

The lengthscale for the SE kernel.

c

The offset for the polynomial kernel. This is a value greater than zero.

d

The degree for the polynomial kernel. This is an integer value greater than oe equal to two.

lam.poly

The scale parameter for the polynomial kernel.

Details

The Pearson kernel is used for nominal-type variables, and thus factor-type variables are treated with the Pearson kernel automatically when fitting I-prior models. The other kernels are for continuous variables, and each emits different properties of functions.

The linear kernel is used for "straight-line" functions. In addition, if squared, cubic, or higher order terms are to be modelled, then the polynomial kernel is suitable for this purpose. For smoothing models, the fBm kernel is preferred, although the SE kernel may be used as well.

Value

A matrix whose [i, j] entries are given by h(\code{x[i]}, \code{y[j]}), with h being the appropriate kernel function. The matrix has dimensions m by n according to the lengths of y and x respectively. When a single argument x is supplied, then y is taken to be equal to x, and a symmetric n by n matrix is returned.

The matrix has a "kernel" attribute indicating which type of kernel function was called.

References

https://phd.haziqj.ml/intro/

Examples

kern_linear(1:3)
kern_fbm(1:5, 1:3, gamma = 0.7)