kernels: Gaussian process kernels

In greta-dev/gretaGP: Gaussian Process Modelling in 'greta'

Gaussian process kernels

Description

Create and combine Gaussian process kernels (covariance functions) for use in Gaussian process models.

Usage

bias(variance)

constant(variance)

white(variance)

iid(variance, columns = 1)

rbf(lengthscales, variance, columns = seq_along(lengthscales))

rational_quadratic(
  lengthscales,
  variance,
  alpha,
  columns = seq_along(lengthscales)
)

linear(variances, columns = seq_along(variances))

polynomial(variances, offset, degree, columns = seq_along(variances))

expo(lengthscales, variance, columns = seq_along(lengthscales))

mat12(lengthscales, variance, columns = seq_along(lengthscales))

mat32(lengthscales, variance, columns = seq_along(lengthscales))

mat52(lengthscales, variance, columns = seq_along(lengthscales))

cosine(lengthscales, variance, columns = seq_along(lengthscales))

periodic(period, lengthscale, variance)

Arguments

variance, variances	(scalar/vector) the variance of a Gaussian process prior in all dimensions (variance) or in each dimension (variances)
columns	(scalar/vector integer, not a greta array) the columns of the data matrix on which this kernel acts. Must have the same dimensions as lengthscale parameters.
alpha	(scalar) additional parameter in rational quadratic kernel
offset	(scalar) offset in polynomial kernel
degree	(scalar) degree of polynomial kernel
period	(scalar) the period of the Gaussian process
lengthscale, lengthscales	(scalar/vector) the correlation decay distance along all dimensions (lengthscale) or each dimension ((lengthscales)) of the Gaussian process

Details

The kernel constructor functions each return a function (of class greta_kernel) which can be executed on greta arrays to compute the covariance matrix between points in the space of the Gaussian process. The + and * operators can be used to combine kernel functions to create new kernel functions.

Note that bias and constant are identical names for the same underlying kernel.

iid is equivalent to bias where all entries in columns match (where the absolute euclidean distance is less than 1e-12), and white where they don't; i.e. an independent Gaussian random effect.

Value

greta kernel with class "greta_kernel"

Examples

## Not run: 
# create a radial basis function kernel on two dimensions
k1 <- rbf(lengthscales = c(0.1, 0.2), variance = 0.6)

# evaluate it on a greta array to get the variance-covariance matrix
x <- greta_array(rnorm(8), dim = c(4, 2))
k1(x)

# non-symmetric covariance between two sets of points
x2 <- greta_array(rnorm(10), dim = c(5, 2))
k1(x, x2)

# create a bias kernel, with the variance as a variable
k2 <- bias(variance = lognormal(0, 1))

# combine two kernels and evaluate
K <- k1 + k2
K(x, x2)

# other kernels
constant(variance = lognormal(0, 1))
white(variance = lognormal(0, 1))
iid(variance = lognormal(0, 1))
rational_quadratic(lengthscales = c(0.1, 0.2), alpha = 0.5, variance = 0.6)
linear(variances = 0.1)
polynomial(variances = 0.6, offset = 0.8, degree = 2)
expo(lengthscales = 0.6, variance = 0.9)
mat12(lengthscales = 0.5, variance = 0.7)
mat32(lengthscales = 0.4, variance = 0.8)
mat52(lengthscales = 0.3, variance = 0.9)
cosine(lengthscales = 0.68, variance = 0.8)
periodic(period = 0.71, lengthscale = 0.59, variance = 0.2)

## End(Not run)

greta-dev/gretaGP documentation built on Nov. 17, 2024, 12:48 p.m.