Gibbs#

class gpjax.kernels.Gibbs(base_kernel, lengthscale, compute_engine=<gpjax.kernels.computations.dense.DenseKernelComputation object>)[source]#

Bases: AbstractKernel

A base kernel whose lengthscale changes with location.

The Gibbs kernel (Gibbs, 1997), also known as the Paciorek–Schervish kernel (Paciorek & Schervish, 2006). A location function \(g\) gives \(\log\ell(x)\), and \(\ell(x)\) multiplies the lengthscale of an isotropic base kernel \(k_0\) with correlation \(\rho\) and variance \(\sigma^2\):

\[ k(x, y) = \sigma^2 \left(\frac{2\,\ell(x)\,\ell(y)}{\ell(x)^2 + \ell(y)^2}\right)^{d/2} \rho\!\left(\sqrt{\frac{2}{\ell(x)^2 + \ell(y)^2}}\, \lVert x - y\rVert\right), \qquad \ell(x) = \exp g(x). \]

Here \(d\) is the number of columns over which the base kernel measures distance, and \(\lVert\cdot\rVert\) uses the base kernel’s lengthscales, so an ARD base kernel keeps its shape and \(\ell(x)\) scales it. Correlation decays faster where \(\ell(x)\) is small, for example over mountains, and more slowly where it is large. The marginal variance is \(\sigma^2\) at every location.

The kernel is positive definite when \(\rho\) is positive definite in every dimension. Only base kernels with isotropic_radial = True meet this condition: RBF, the Matérn kernels, RationalQuadratic and PoweredExponential.

The base kernel selects the columns over which it measures distance with its own active_dims, and the location function selects its covariate columns. The wrapper itself always receives every column.

Parameters: