gallifrey: JAX-based Gaussian process structure learning for astronomical time series

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Volume 699 (July 2025)

A&A, 699 (2025) A42

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Table A.1

Common kernels in GP regression.

Name	Equation	# of Parameters	Parameter	Stationary
Periodic	$σ^{2} \exp (- \frac{\sin^{2} (π \| x - x^{'} \| / p)}{2 l^{2}})$ $\[\sigma^2 \exp \left(-\frac{\sin ^2\left(\pi\left\|x-x^{\prime}\right\| / p\right)}{2 l^2}\right)\]$	3	lengthscale : l variance : σ² period : p	Yes
Powered Exponential	$σ^{2} \exp (- {(\frac{\| x - x^{'} \|}{l})}^{k})$ $\[\sigma^2 \exp \left(-\left(\frac{\left\|x-x^{\prime}\right\|}{l}\right)^k\right)\]$	3	lengthscale : l variance : σ² power : κ	Yes
Squared Exponential/Radial Basis Function	$σ^{2} \exp (- \frac{{(x - x^{'})}^{2}}{2 l^{2}})$ $\[\sigma^2 \exp \left(-\frac{\left(x-x^{\prime}\right)^2}{2 l^2}\right)\]$	2	lengthscale : l variance : σ²	Yes
Matérn ( $v = \frac{1}{2}$ $\[v=\frac{1}{2}\]$ )/Ornstein-Uhlenbeck	$σ^{2} \exp (- \frac{\| x - x^{'} \|}{l})$ $\[\sigma^2 \exp \left(-\frac{\left\|x-x^{\prime}\right\|}{l}\right)\]$	2	lengthscale : l variance : σ²	Yes
Matérn ( $v = \frac{3}{2}$ $\[v=\frac{3}{2}\]$ )	$σ^{2} (1 + \frac{\sqrt{3} \| x - x^{'} \|}{l}) \exp (- \frac{\sqrt{3} \| x - x^{'} \|}{l})$ $\[\sigma^{2}\left(1+\frac{\sqrt{3}\left\|x-x^{\prime}\right\|}{l}\right) \exp \left(-\frac{\sqrt{3}\left\|x-x^{\prime}\right\|}{l}\right)\]$	2	lengthscale : l variance : σ²	Yes
Matérn ( $v = \frac{5}{2}$ $\[v=\frac{5}{2}\]$ )	$σ^{2} (1 + \frac{\sqrt{5} \| x - x^{'} \|}{l} + \frac{5 {(x - x^{'})}^{2}}{3 l^{2}}) \exp (- \frac{\sqrt{5} \| x - x^{'} \|}{l})$ $\[\sigma^{2}\left(1+\frac{\sqrt{5}\left\|x-x^{\prime}\right\|}{l}+\frac{5\left(x-x^{\prime}\right)^{2}}{3 l^{2}}\right) \exp \left(-\frac{\sqrt{5}\left\|x-x^{\prime}\right\|}{l}\right)\]$	2	lengthscale : l variance : σ²	Yes
Rational Quadratic	$σ^{2} {(1 + \frac{{(x - x^{'})}^{2}}{2 α l^{2}})}^{- α}$ $\[\sigma^{2}\left(1+\frac{\left(x-x^{\prime}\right)^{2}}{2 \alpha l^{2}}\right)^{-\alpha}\]$	3	lengthscale : l variance : σ² shape : α	Yes
White	σ² δ(x − x′)	1	variance : σ²	Yes
Linear	σ² (x · x′)	1	variance : σ²	No
Linear with Shift	b + σ²((x − c) · (x′ − c))	3	bias : b variance : σ² shift : c	No

Notes. Each of the listed kernels is implemented as an atomic kernel in gallifrey.

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