Covariance Functions — cov_exp

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)