PoweredExponential#

class gpjax.kernels.PoweredExponential(active_dims=None, lengthscale=1.0, variance=1.0, power=1.0, n_dims=None, compute_engine=<gpjax.kernels.computations.dense.DenseKernelComputation object>)[source]#

Bases: StationaryKernel

The powered exponential family of kernels.

Computes the covariance for pairs of inputs \((x, y)\) with length-scale parameter \(\ell\), variance \(\sigma^2\) and power \(\kappa\).

\[ k(x, y)=\sigma^2\exp\Bigg(-\Big(\frac{\lVert x-y\rVert_2}{\ell}\Big)^\kappa\Bigg) \]

This also equivalent to the symmetric generalized normal distribution. See Diggle and Ribeiro (2007) - “Model-based Geostatistics”. and https://en.wikipedia.org/wiki/Generalized_normal_distribution#Symmetric_version

The kernel is positive definite in every dimension only for \(0 < \kappa \le 2\). A float power in \((0, 2)\) becomes a trainable parameter that the optimiser keeps inside \((0, 2)\). power=2.0 gives the RBF kernel and is held fixed, because a bounded parameter cannot sit on its bound; to learn the power, start it inside the interval, for example at 1.9. A parameter that you pass yourself must keep the power in \((0, 2]\).

Parameters: