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6.13 Gaussian Process Covariance Functions

The Gaussian process covariance functions compute the covariance between observations in an input data set or the cross-covariance between two input data sets.

For one dimensional GPs, the input data sets are arrays of scalars. The covariance matrix is given by $K_{ij} = k(x_i, x_j)$ (where $x_i$ is the $i^{th}$ element of the array $x$) and the cross-covariance is given by $K_{ij} = k(x_i, y_j)$.

For multi-dimensional GPs, the input data sets are arrays of vectors. The covariance matrix is given by $K_{ij} = k(\mathbf{x}_i, \mathbf{x}_j)$ (where $\mathbf{x}_i$ is the $i^{th}$ vector in the array $x$) and the cross-covariance is given by $K_{ij} = k(\mathbf{x}_i, \mathbf{y}_j)$.

6.13.1 Exponentiated quadratic kernel

With magnitude $\sigma$ and length scale $l$, the exponentiated quadratic kernel is:

$$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \exp \left( -\frac{|\mathbf{x}_i - \mathbf{x}_j|^2}{2l^2} \right)$$

matrix gp_exp_quad_cov(array[] real x, real sigma, real length_scale)

Gaussian process covariance with exponentiated quadratic kernel in one dimension.
Available since 2.20

matrix gp_exp_quad_cov(array[] real x1, array[] real x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponentiated quadratic kernel in one dimension.
Available since 2.20

matrix gp_exp_quad_cov(vectors x, real sigma, real length_scale)

Gaussian process covariance with exponentiated quadratic kernel in multiple dimensions.
Available since 2.20

matrix gp_exp_quad_cov(vectors x, real sigma, array[] real length_scale)

Gaussian process covariance with exponentiated quadratic kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20

matrix gp_exp_quad_cov(vectors x1, vectors x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponentiated quadratic kernel in multiple dimensions.
Available since 2.20

matrix gp_exp_quad_cov(vectors x1, vectors x2, real sigma, array[] real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponentiated quadratic kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20

6.13.2 Dot product kernel

With bias $\sigma_0$ the dot product kernel is:

$$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma_0^2 + \mathbf{x}_i^T \mathbf{x}_j$$

matrix gp_dot_prod_cov(array[] real x, real sigma)

Gaussian process covariance with dot product kernel in one dimension.
Available since 2.20

matrix gp_dot_prod_cov(array[] real x1, array[] real x2, real sigma)

Gaussian process cross-covariance of x1 and x2 with dot product kernel in one dimension.
Available since 2.20

matrix gp_dot_prod_cov(vectors x, real sigma)

Gaussian process covariance with dot product kernel in multiple dimensions.
Available since 2.20

matrix gp_dot_prod_cov(vectors x1, vectors x2, real sigma)

Gaussian process cross-covariance of x1 and x2 with dot product kernel in multiple dimensions.
Available since 2.20

6.13.3 Exponential kernel

With magnitude $\sigma$ and length scale $l$, the exponential kernel is:

$$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \exp \left( -\frac{|\mathbf{x}_i - \mathbf{x}_j|}{l} \right)$$

matrix gp_exponential_cov(array[] real x, real sigma, real length_scale)

Gaussian process covariance with exponential kernel in one dimension.
Available since 2.20

matrix gp_exponential_cov(array[] real x1, array[] real x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponential kernel in one dimension.
Available since 2.20

matrix gp_exponential_cov(vectors x, real sigma, real length_scale)

Gaussian process covariance with exponential kernel in multiple dimensions.
Available since 2.20

matrix gp_exponential_cov(vectors x, real sigma, array[] real length_scale)

Gaussian process covariance with exponential kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20

matrix gp_exponential_cov(vectors x1, vectors x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponential kernel in multiple dimensions.
Available since 2.20

matrix gp_exponential_cov(vectors x1, vectors x2, real sigma, array[] real length_scale)

Gaussian process cross-covariance of x1 and x2 with exponential kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20

6.13.4 Matern 3/2 kernel

With magnitude $\sigma$ and length scale $l$, the Matern 3/2 kernel is:

$$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \left( 1 + \frac{\sqrt{3}|\mathbf{x}_i - \mathbf{x}_j|}{l} \right) \exp \left( -\frac{\sqrt{3}|\mathbf{x}_i - \mathbf{x}_j|}{l} \right)$$

matrix gp_matern32_cov(array[] real x, real sigma, real length_scale)

Gaussian process covariance with Matern 3/2 kernel in one dimension.
Available since 2.20

matrix gp_matern32_cov(array[] real x1, array[] real x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 3/2 kernel in one dimension.
Available since 2.20

matrix gp_matern32_cov(vectors x, real sigma, real length_scale)

Gaussian process covariance with Matern 3/2 kernel in multiple dimensions.
Available since 2.20

matrix gp_matern32_cov(vectors x, real sigma, array[] real length_scale)

Gaussian process covariance with Matern 3/2 kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20

matrix gp_matern32_cov(vectors x1, vectors x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 3/2 kernel in multiple dimensions.
Available since 2.20

matrix gp_matern32_cov(vectors x1, vectors x2, real sigma, array[] real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 3/2 kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20

6.13.5 Matern 5/2 kernel

With magnitude $\sigma$ and length scale $l$, the Matern 5/2 kernel is:

$$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \left( 1 + \frac{\sqrt{5}|\mathbf{x}_i - \mathbf{x}_j|}{l} + \frac{5 |\mathbf{x}_i - \mathbf{x}_j|^2}{3l^2} \right) \exp \left( -\frac{\sqrt{5} |\mathbf{x}_i - \mathbf{x}_j|}{l} \right)$$

matrix gp_matern52_cov(array[] real x, real sigma, real length_scale)

Gaussian process covariance with Matern 5/2 kernel in one dimension.
Available since 2.20

matrix gp_matern52_cov(array[] real x1, array[] real x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 5/2 kernel in one dimension.
Available since 2.20

matrix gp_matern52_cov(vectors x, real sigma, real length_scale)

Gaussian process covariance with Matern 5/2 kernel in multiple dimensions.
Available since 2.20

matrix gp_matern52_cov(vectors x, real sigma, array[] real length_scale)

Gaussian process covariance with Matern 5/2 kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20

matrix gp_matern52_cov(vectors x1, vectors x2, real sigma, real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 5/2 kernel in multiple dimensions.
Available since 2.20

matrix gp_matern52_cov(vectors x1, vectors x2, real sigma, array[] real length_scale)

Gaussian process cross-covariance of x1 and x2 with Matern 5/2 kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20

6.13.6 Periodic kernel

With magnitude $\sigma$, length scale $l$, and period $p$, the periodic kernel is:

$$k(\mathbf{x}_i, \mathbf{x}_j) = \sigma^2 \exp \left(-\frac{2 \sin^2 \left( \pi \frac{|\mathbf{x}_i - \mathbf{x}_j|}{p} \right) }{l^2} \right)$$

matrix gp_periodic_cov(array[] real x, real sigma, real length_scale, real period)

Gaussian process covariance with periodic kernel in one dimension.
Available since 2.20

matrix gp_periodic_cov(array[] real x1, array[] real x2, real sigma, real length_scale, real period)

Gaussian process cross-covariance of x1 and x2 with periodic kernel in one dimension.
Available since 2.20

matrix gp_periodic_cov(vectors x, real sigma, real length_scale, real period)

Gaussian process covariance with periodic kernel in multiple dimensions.
Available since 2.20

matrix gp_periodic_cov(vectors x1, vectors x2, real sigma, real length_scale, real period)

Gaussian process cross-covariance of x1 and x2 with periodic kernel in multiple dimensions with a length scale for each dimension.
Available since 2.20