Squared-exponential covariance kernel

Computes the squared-exponential (radial basis function) kernel between rows of x and y. A scalar length scale gives an isotropic kernel; one length scale per input dimension gives automatic relevance determination (ARD).

Usage

kernel_rbf(x, y = NULL, variance = 1, length_scale = 1)

Arguments

x

Numeric vector or matrix. Matrix rows are observations and columns are input dimensions.

y

Optional numeric vector or matrix with the same number of input dimensions as x. If NULL, computes the covariance of x with itself.

variance

Positive marginal variance.

length_scale

Positive scalar or numeric vector with one value per input dimension. A vector enables ARD.

Value

A numeric covariance matrix with nrow(x) rows and nrow(y) columns. Numeric vectors are treated as one-dimensional observations.

Stability

Stable: from version 1.0.0 this interface changes incompatibly only in a major release, after a deprecation period. Results and options that concern an experimental model class, kernel, or argument follow that interface's tier. See gaussianprocesses-package for the policy.

See also

rbf_kernel() creates the same kernel as a reusable specification, which can be combined, optimized, and used in models.

Examples

x <- seq(0, 2, by = 0.5)
kernel_rbf(x, variance = 2, length_scale = 0.7)
#>            [,1]      [,2]      [,3]      [,4]       [,5]
#> [1,] 2.00000000 1.5496749 0.7208956 0.2013378 0.03375977
#> [2,] 1.54967486 2.0000000 1.5496749 0.7208956 0.20133780
#> [3,] 0.72089558 1.5496749 2.0000000 1.5496749 0.72089558
#> [4,] 0.20133780 0.7208956 1.5496749 2.0000000 1.54967486
#> [5,] 0.03375977 0.2013378 0.7208956 1.5496749 2.00000000

# ARD: one length scale per input dimension.
x2 <- cbind(c(0, 1, 2), c(0, 0, 1))
kernel_rbf(x2, length_scale = c(0.5, 3))
#>             [,1]      [,2]        [,3]
#> [1,] 1.000000000 0.1353353 0.000317334
#> [2,] 0.135335283 1.0000000 0.128021693
#> [3,] 0.000317334 0.1280217 1.000000000

# Cross-covariance between two sets of inputs.
kernel_rbf(x, c(0.25, 1.75))
#>           [,1]      [,2]
#> [1,] 0.9692332 0.2162652
#> [2,] 0.9692332 0.4578334
#> [3,] 0.7548396 0.7548396
#> [4,] 0.4578334 0.9692332
#> [5,] 0.2162652 0.9692332