cherab.phix.inversion.GCV

class cherab.phix.inversion.GCV(*args, **kwargs)Source

Bases: _SVDBase

Generalized Cross-Validation (GCV) criterion optimization for regularization parameter.

GCV criterion function is defined as follows:

\[GCV(\lambda) := \frac{||Ax_\lambda - b||^2}{\left[1 - \sum_{i=1}^N w_i(\lambda)\right]^2}\]

The optimal regularization parameter corresponds to the minimum value of GCV function.

Parameters:

• s (vector_like) – singular values of \(A\) like \(\sigma = (\sigma_1, \sigma_2, ...) \in \mathbb{R}^r\)

• u (array_like) – left singular vectors of \(A\) like \(U = (u_1, u_2, ...) \in \mathbb{R}^{m\times r}\)

• basis (array_like) – inverted solution basis \(\tilde{V} \in \mathbb{R}^{n\times r}\). Here, \(\tilde{V} = L^{-1}V\), where \(V\in\mathbb{R}^{n\times r}\) is the right singular vectors of \(A\) and \(L^{-1}\) is the inverse of regularization operator \(L \in \mathbb{R}^{n\times n}\).

• **kwargs (_SVDBase properties, optional) – kwargs are used to specify properties like a data

Methods

eta(beta)	Calculate squared regularization norm: \(\eta = ||Lx_\lambda||^2\)
eta_diff(beta)	Calculate differential of eta: \(\eta' = \frac{d\eta}{d\lambda}\)
gcv(beta)	Calculate of GCV criterion function.
inverted_solution(beta)	Calculate the inverted solution using SVD components at given regularization parameter.
optimize([itemax, bounds])	Optimize the regularization parameter using GCV criterion.
plot_gcv([fig, axes, bounds, n_beta])	Plotting GCV vs regularization parameters in log-log scale.
regularization_norm(beta)	Return the residual norm: \(\sqrt{\eta} = ||L x_\lambda||\)
residual_norm(beta)	Return the residual norm: \(\sqrt{\rho} = ||Ax_\lambda - b||\)
rho(beta)	Calculate squared residual norm: \(\rho = ||Ax_\lambda - b||^2\).
solve([bounds, stepsize])	Solve the ill-posed inversion equation.
w(beta)	Calculate window function using regularization parameter \(\lambda\).

Attributes

basis	The inverted solution basis \(\tilde{V} \in \mathbb{R}^{n\times r}\).
data	Given data for inversion calculation.
lambda_opt	Optimal regularization parameter defined after solve is executed.
s	Singular values of \(A\)
u	Left singular vectors of \(A\).