cov_exp_quad: Covariance Functions

In markjrieke/riekelib: A collection of personal functions

Covariance Functions

Description

Generate a N \times N covariance matrix, \Sigma, by passing a distance vector of length N, x, to one of the following covariance functions:

• Exponentiated Quadtratic

  \Sigma_{ij} = \sigma^2 \exp \left( \frac{|x_i - x_j|^2}{-2 \ \rho^2} \right)

• Rational Quadratic

  \Sigma_{ij} = \sigma^2 \left(1 + \frac{|x_i - x_j|^2}{2 \alpha \rho^2} \right)^{-\alpha}

• Periodic

  \Sigma_{ij} = \sigma^2 \exp \left(\frac{-2 \sin^2(\pi |x_i - x_j|/p)}{\rho^2} \right)

• Linear

  \Sigma_{ij} = \beta_0 + \beta (x_i - c)(x_j - c)

• White Noise

  \Sigma_{ij} = \sigma^2 I

where \sigma^2 is the amplitude and \rho is the length_scale at which the covariance function can operate.

In the rational quadratic kernel, \alpha is the scale-mixture As \alpha \rightarrow \infty the rational quadratic converges to the exponentiated quadratic.

In the periodic kernel, p is the period over which the function repeats.

In the linear kernel, \beta_0 and \beta are intercept and slope parameters, respectively. c is a constant that offsets the linear covariance from the origin.

Finally, in the white noise kernel, I is the identity matrix.

Usage

cov_exp_quad(x, amplitude = 1, length_scale = 1, delta = 1e-09)

cov_rational(x, amplitude = 1, length_scale = 1, mixture = 1, delta = 1e-09)

cov_periodic(x, amplitude = 1, length_scale = 1, period = 1, delta = 1e-09)

cov_linear(x, slope = 0, intercept = 0, constant = 0, delta = 1e-09)

cov_noise(x, amplitude = 1)

Arguments

x	A vector containing positions.
amplitude	Vertical scale of the covariance function
length_scale	Horizontal scale of the covariance function
delta	A small offset along the diagonal of the resulting covariance matrix to ensure the function returns a positive-semidefinite matrix. Can also be used as a white noise kernel to allow for increased variation at individual positions along the vector x.
mixture	Scale-mixture for the rational quadratic covariance function
period	Period of repetition for the periodic covariance function
slope	Slope of the linear covariance function
intercept	Intercept of the linear covariance function
constant	A constant that offsets the linear covariance function along the x-axis from the origin.

Value

A N \times N symmetric, positive-semidefinite covariance matrix

Examples

x <- seq(from = -2, to = 2, length.out = 50)

# heatmap of covariance - higher values = greater covariance
cov_exp_quad(x) |> heatmap(Rowv = NA, Colv = NA)

# decreasing the length scale decreases the range over which values covary
cov_exp_quad(x, length_scale = 0.5) |> heatmap(Rowv = NA, Colv = NA)

# rational quadratic includes a mixture parameter
cov_rational(x, mixture = 0.1) |> heatmap(Rowv = NA, Colv = NA)

# periodic repeats over a distance
cov_periodic(x, period = 1, length_scale = 2) |> heatmap(Rowv = NA, Colv = NA)

# linear covariance increases/decreases linearly
cov_linear(x, slope = 1) |> heatmap(Rowv = NA, Colv = NA)

# white noise is applied only along the diagonal
cov_noise(x) |> heatmap(Rowv = NA, Colv = NA)

markjrieke/riekelib documentation built on Dec. 20, 2024, 4:57 a.m.