gpLib.SpaceTime1d

class gpLib.SpaceTime1d(**kwargs)

One-dimensional space–time covariance (Bolin & Wallin 2016).

Inputs are 2-vectors \((x,t)\). With parameters \(\phi,a,a',c,\alpha,\delta,\alpha',\gamma,\varepsilon \ge 0\)

\[k\bigl((x,t),(x',t')\bigr) \;=\; \underbrace{\frac{\phi} {\bigl(1+a\,(t-t')^{2}\bigr)^{\alpha}} \exp\!\Bigl[ -\frac{c\;\|x-x'-\varepsilon(t-t')\|^{2\gamma}} {\bigl(1+a\,(t-t')^{2}\bigr)^{\alpha\gamma}} \Bigr]}_{\text{spatio-temporal interaction}} \;+\; \underbrace{\frac{\delta\,\mathbf 1_{x=x'}} {\bigl(1+a'(t-t')^{2}\bigr)^{\alpha'}}}_{\text{pure nugget}}\]

Parameters:

batch_shape – Optional batch shape.

Variables:

• epsilon (torch.Tensor) – Space–time advection parameter.

• gamma (torch.Tensor a, a_prime, c, phi, alpha, delta, alpha_prime,) – Positive parameters with constraints shown in code.