Sampling

sampling

Sampling routines for optimal weighted least squares.

This module provides routines for sampling from the optimal sampling distribution. Given a polynomial space \(V:= \text{span} \{P_\lambda: \lambda \in \Lambda\}\) defined by a multi-index set \(\Lambda \subset \mathbb{N}_0^d\), there exists an optimal sampling distribution \(\nu = \nu(V)\) that minimizes the number of samples required for the weighted least squares estimator onto \(V\). This module also contains routines for arcsine sampling and determining the optimal sample size for quasi-optimality of the weighted least squares estimator.

Functions:

Name	Description
arcsine	Arcsine density function.
sample_arcsine	Samples from the arcsine distribution.
sample_squared_legendre	Samples from the distribution with squared Legendre density.
optimal_sample_size	Optimal sample size for quasi-optimality.
sample_optimal_distribution	Samples from the optimal distribution.

multichaos.sampling.arcsine

arcsine(x: np.ndarray) -> np.ndarray

Arcsine density function.

This evaluates the arcsine density function

\[ \begin{equation} p_d(x) = \prod_{i=1}^d \frac{1}{\pi \sqrt{1- x_i^2}}, \quad x \in (-1, 1)^d. \end{equation} \]

Parameters:

Name	Type	Description	Default
x	np.ndarray	Input array.	required

Returns:

Type	Description
np.ndarray	Arcsine density function evaluated at x.

multichaos.sampling.sample_arcsine

sample_arcsine(size: tuple[int, ...]) -> np.ndarray

Samples from the arcsine distribution.

This is based on the fact that if \(U \sim \mathcal{U}(-\pi, \pi)\), then \(X = \sin(U) \sim p_1\) follows the arcsine distribution.

Parameters:

Name	Type	Description	Default
size	tuple	Size of the sample.	required

Returns:

Type	Description
np.ndarray	Sample from the arcsine distribution with shape size.

multichaos.sampling.sample_squared_legendre

sample_squared_legendre(n: int, size: int) -> np.ndarray

Samples from the distribution with squared Legendre density.

This function samples from the distribution with squared Legendre density \(P_n^2\), where \(P_n\) is the Legendre polynomial of degree \(n\). This is done using rejection sampling with arcsine proposals, since

\[ \begin{equation} |P_n(x)| \leq 2 p_1(x)^{1/2}, \quad x \in (-1, 1), \,\, \forall n \in \mathbb{N}_0. \end{equation} \]

Parameters:

Name	Type	Description	Default
n	int	Degree of Legendre polynomial.	required
size	int	Number of samples.	required

Returns:

Type	Description
np.ndarray	Sample from the square Legendre density with shape size.

multichaos.sampling.optimal_sample_size

optimal_sample_size(dim: float, risk: float = 0) -> int

Optimal sample size for quasi-optimality.

Determines the optimal number of samples \(N\) required to ensure quasi-optimality of a weighted least squares estimator onto a polynomial space \(V\). For \(p \in (0, 1)\) and \(\theta = \frac{1}{2}(1 - \log 2)\), the condition

\[ \begin{equation} N \geq \frac{m}{\theta} \log \left(\frac{2m}{1 - p}\right), \end{equation} \]

where \(m = \text{dim}(V)\), ensures that

\[ \begin{equation} \mathbb{P}(\Vert \mathbf{G} - \mathbf{I}\Vert \leq 1/2) \geq p, \end{equation} \]

where \(\mathbf{G} \in \mathbb{R}^{m \times m}\) denotes the Gramian least squares and \(\mathbf{I} \in \mathbb{R}^{m \times m}\) the identity matrix.

Parameters:

Name	Type	Description	Default
dim	float	Dimension of the polynomial space used for projection.	required
risk	float	Probability \(p\) for quasi-optimality to hold.	0

Returns:

Name	Type	Description
int	int	Optimal sample size.

multichaos.sampling.sample_optimal_distribution

sample_optimal_distribution(index_set: np.ndarray, size: int) -> np.ndarray

Samples from the optimal distribution.

Given a polynomial subspace \(\text{span} \{P_\lambda: \lambda \in \Lambda\} \subset L^2([-1, 1]^d)\) defined by an index set \(\Lambda \subset \mathbb{N}_0^d\), the density of the optimal distribution is defined by

\[ \begin{equation} \frac{1}{|\Lambda|} \sum_{\lambda \in \Lambda} P_\lambda^2. \end{equation} \]

Parameters:

Name	Type	Description	Default
index_set	np.ndarray	Index set defining the polynomial subspace of shape (n_basis, dim).	required
size	int	Number of samples.	required

Returns:

Type	Description
np.ndarray	Samples with shape (size, dim).