Base Polynomial Chaos Expansion

base

Base class for Polynomial Chaos Expansion (PCE) models.

This module provides the base class for Polynomial Chaos Expansion (PCE) models. The class provides the basic functionality for constructing and evaluating PCE models.

Classes:

Name	Description
BasePCE	Base class for Polynomial Chaos Expansion (PCE) models.

multichaos.base.BasePCE

Base class for Polynomial Chaos Expansion (PCE) models.

This class provides the basic functionality for constructing and evaluating PCE models. The class is designed to be inherited by other PCE models.

Attributes:

Name	Type	Description
dist	ot.Distribution	Input probability distribution.
index_set	np.ndarray	Index set of the PCE.
n_basis	int	Number of basis functions in the PCE.
risk	float	Quasi-optimality probability.
coefficients	np.ndarray	Coefficients of the PCE.
input_dim	int	Input dimension.
output_dim	int	Output dimension.
tf	callable	Transformation function.
tf_inv	callable	Inverse transformation function.

__init__

__init__(dist, index_set=None, risk=0)

Initialize Base PCE object.

Parameters:

Name	Type	Description	Default
dist	ot.Distribution	Input probability distribution.	required
index_set	np.ndarray	Index set of the PCE.	None
risk	float	Quasi-optimality probability.	0

set_transformation

set_transformation()

Set the transformation functions.

This method sets up the necessary components for transforming the input distribution to the underlying measure of the PCE.

__call__

__call__(data_in)

Evaluate the PCE at the given input data.

Parameters:

Name	Type	Description	Default
data_in	np.ndarray	Input data points.	required

compute_mean

compute_mean()

Compute the mean of the PCE.

This returns the mean of the PCE \(P = \sum_{\lambda \in \Lambda} c_\lambda P_\lambda\), which, due to orthonormality, is given by \(\mathbb{E}[P] = c_\mathbf{0}\).

Returns:

Type	Description
float	Mean of the PCE.

compute_variance

compute_variance()

Compute the variance of the PCE.

This returns the variance of the PCE \(P = \sum_{\lambda \in \Lambda} c_\lambda P_\lambda\), which, due to orthonormality, is given by \(\mathbb{V}[P] = \sum_{\lambda \in \Lambda \setminus \{\mathbf{0}\}} c_\lambda^2\).

Returns:

Type	Description
float	Variance of the PCE.

l2_error

l2_error(data_in, data_out)

Compute the \(L^2\) error of the PCE.

This method computes the root mean squared error as an estimate to the \(L^2\) error of the PCE with respect to the given data, i.e.

\[ \begin{equation} \|Q - \hat{Q}\|_{L^2} \approx \left( \frac{1}{N} \sum_{i=1}^N (Q_i - \hat{Q}(\omega_i))^2 \right)^{1/2}, \end{equation} \]

where \(\{(\omega_i, Q_i)\}_{i=1}^N\) denotes the data and \(\hat{Q}\) the PCE model.

Parameters:

Name	Type	Description	Default
data_in	np.ndarray	Input data points of shape (n_samples, input_dim).	required
data_out	np.ndarray	Output data points of shape (n_samples, output_dim).	required

Returns:

Type	Description
float	Root mean squared error.