covfunc

This module provides covariance functions for the nodes and weights of the GPRN.

covFunction(*args)

Base class for covariance functions (kernels) used for nodes and weights in the GPRN.

Sum

Bases: _operator

Sum of two covariance functions

Multiplication

Bases: _operator

Product of two covariance functions

Constant(c)

Bases: covFunction

This kernel returns the square of its constant argument c $$ K_{ij} = c^2 $$

Parameters:

Name	Type	Description	Default
c	float	Constant value	required

WhiteNoise(w)

Bases: covFunction

White noise (diagonal) kernel $$ K_{ij} = w^2 \, \delta_{ij} $$

Parameters:

Name	Type	Description	Default
w	float	White noise amplitude	required

SquaredExponential(theta, ell)

Bases: covFunction

Squared Exponential kernel, also known as radial basis function $$ K_{ij} = \theta^2 \, \exp \left[ - \frac{(t_i - t_j)^2}{2 \ell^2} \right] $$

Parameters:

Name	Type	Description	Default
theta	float	Amplitude	required
ell	float	Length-scale	required

Periodic(theta, P, ell)

Bases: covFunction

Periodic kernel, also known as the exponential sine squared $$ K_{ij} = \theta^2 \, \exp \left[ - \frac{2 \sin^2 \left( \frac{\pi (t_i - t_j)}{P} \right)}{\ell^2} \right] $$

Parameters:

Name	Type	Description	Default
theta	float	Amplitude	required
P	float	Period	required
ell	float	Lenght scale	required

Parameterization

Note that the periodic kernel is sometimes parameterized differently, namely using \(\Gamma = 2/\ell^2\).

QuasiPeriodic(theta, elle, P, ellp)

Bases: covFunction

This kernel is the product between the periodic kernel and the squared exponential kernel, and is commonly known as the quasi-periodic kernel $$ K_{ij} = \theta^2 \, \exp \left[ - \frac{(t_i - t_j)^2}{2 \ell_e^2} - \frac{2 \sin^2 \left( \frac{\pi (t_i - t_j)}{P} \right)}{\ell_p^2} \right] $$

Parameters:

Name	Type	Description	Default
theta	float	Amplitude	required
elle	float	Evolutionary length scale	required
P	float	Kernel periodicity	required
ellp	float	Length scale of the periodic component	required

Info

The QuasiPeriodic kernel is implemented on its own for convenience, but it is exactly equivalent to the product of a SquaredExponential and a Periodic kernel with the same parameters.

RationalQuadratic(theta, alpha, ell)

Bases: covFunction

The rational quadratic kernel $$ $$

Parameters:

Name	Type	Description	Default
theta	float	Amplitude of the kernel	required
alpha	float	Amplitude of large and small scale variations	required
ell	float	Characteristic lenght scale to define the kernel "smoothness"	required

RQP(theta, alpha, elle, P, ellp)

Bases: covFunction

Product between the periodic kernel and the rational quadratic kernel that we call RQP kernel.

Parameters:

Name	Type	Description	Default
theta	float	Amplitude	required
alpha	float	Alpha of the rational quadratic kernel	required
elle	float	Aperiodic length scale	required
P	float	Periodic repetitions of the kernel	required
ellp	float	Periodic length scale	required

Cosine(theta, P)

Bases: covFunction

the cosine kernel

Parameters:

Name	Type	Description	Default
theta	float	Amplitude	required
P	float	Period	required

Exponential(theta, ell)

Bases: covFunction

The exponential kernel $$ K_{ij} = \theta^2 \, \exp \left( - \frac{|t_i - t_j|}{\ell} \right) $$

Parameters:

Name	Type	Description	Default
theta	float	Amplitude	required
ell	float	Characteristic lenght scale	required

Matern32(theta, ell)

Bases: covFunction

the Matern 3/2 kernel. This kernel arise when setting v=3/2 in the matern family of kernels

Parameters:

Name	Type	Description	Default
theta	float	Amplitude	required
ell	float	Characteristic lenght scale	required

Matern52(theta, ell)

Bases: covFunction

the Matern 5/2 kernel. This kernel arise when setting v=5/2 in the matern family of kernels

Parameters:

Name	Type	Description	Default
theta	float	Amplitude	required
ell	float	Characteristic lenght scale	required

Linear(c)

Bases: covFunction

the Linear kernel

Parameters:

Name	Type	Description	Default
c		Constant	required

GammaExp(theta, gamma, l)

Bases: covFunction

the gamma-exponential kernel

Parameters:

Name	Type	Description	Default
theta		Amplitude	required
gamma		Shape parameter ( 0 < gamma <= 2)	required
l		Lenght scale	required

Polynomial(theta, a, b, c)

Bases: covFunction

the polynomial kernel

Parameters:

Name	Type	Description	Default
theta		Amplitude ???	required
a		Real value > 0	required
b		Real value >= 0	required
c		Integer value	required
wn		White noise amplitude	required

Piecewise(eta)

Bases: covFunction

third-order piecewise polynomial kernel

Parameters:

Name	Type	Description	Default
eta		float	required

Paciorek(amplitude, ell_1, ell_2)

Bases: covFunction

Definition of the modified Paciorek's kernel (stationary version).

Parameters:

Name	Type	Description	Default
amplitude		float	required
ell_1		float	required
ell_2		float	required

NewPeriodic(amplitude, alpha2, P, l)

Bases: covFunction

Definition of a new periodic kernel derived from mapping the rational quadratic kernel to the 2D space u(x) = (cos x, sin x)

Parameters:

Name	Type	Description	Default
amplitude		float	required
alpha2		float	required
P		float	required
l		float	required

QuasiNewPeriodic(amplitude, alpha2, ell_e, P, ell_p)

Bases: covFunction

Definition of a new quasi-periodic kernel. Derived from mapping the rational quadratic kernel to the 2D space u(x) = (cos x, sin x) and multiplying it by a squared exponential kernel

Parameters:

Name	Type	Description	Default
amplitude		float	required
alpha2		float	required
ell_e		float	required
P		float	required
ell_p		float	required

NewRQP(amplitude, alpha1, alpha2, ell_e, P, ell_p)

Bases: covFunction

Definition of a new quasi-periodic kernel. Derived from mapping the rational quadratic kernel to the 2D space u(x) = (cos x, sin x) and multiplying it by a rational quadratic kernel

Parameters:

Name	Type	Description	Default
amplitude		float	required
alpha1		float	required
ell_e		float	required
P		float	required
ell_p		float	required
alpha2		float	required

HarmonicPeriodic(N, amplitude, P, ell)

Bases: covFunction

Definition of a periodic kernel that models a periodic signal with a N number of harmonics. Obtained by mapping the squared exponetial with the Lagrange trigonometric identities

Parameters:

Name	Type	Description	Default
N		int	required
amplitude		float	required
P		float	required
ell		float	required

QuasiHarmonicPeriodic(N, amplitude, ell_e, P, ell_p)

Bases: covFunction

Definition of a quasi-periodic kernel that models a periodic signals with a N number of harmonics. Comes from the multiplication of the squared exponential by the HarmonicPeriodic

Parameters:

Name	Type	Description	Default
N		int	required
amplitude		float	required
ell_e		float	required
P		float	required
ell_p		float	required

CosPeriodic(amplitude, P, ell)

Bases: covFunction

Periodic kernel derived by mapping the squared exponential kernel into thw 2D space u(t) = [cos(t + phi), sin(t + phi)]

SPOILER ALERT: If you do the math the phi terms disappear

Parameters:

Name	Type	Description	Default
amplitude		float	required
P		float	required
ell_p		float	required
phi		float	required

QuasiCosPeriodic(amplitude, ell_e, P, ell_p)

Bases: covFunction

This kernel is the product between the cosPeriodic kernel and the squared exponential kernel, it is just another the quasi-periodic kernel.

Parameters:

Name	Type	Description	Default
amplitude		float	required
ell_e		float	required
ell_p		float	required
P		float	required