pce module¶

Polynomial chaos expansion model.

The polynomial chaos expansion (PCE) model expresses an output variable as a weighted sum of polynomial functions which are orthonormal in the stochastic input space spanned by the random input variables:

\[Y = w_0 + w_1\phi_1(X) + w_2\phi_2(X) + ... + w_K\phi_K(X)\]

where \(\phi_i(x)=\psi_{\tau_1(i),1}(x_1)\times\ldots\times \psi_{\tau_d(i),d}(x_d)\) and \(d\) is the number of input variables.

Enumeration strategy¶

The choice of the function \(\tau=(\tau_1,\ldots,\tau_d)\) is an enumeration strategy and \(\tau_j(i)\) is the polynomial degree of \(\psi_{\tau_j(i),j}\).

Distributions¶

PCE models depend on random input variables and are often used to deal with uncertainty quantification problems.

If \(X_j\) is a Gaussian random variable, \((\psi_{ij})_{i\geq 0}\) is the Legendre basis. If \(X_j\) is a uniform random variable, \((\psi_{ij})_{i\geq 0}\) is the Hermite basis.

When the problem is deterministic, we can still use PCE models under the assumption that the input variables are independent uniform random variables. Then, the orthonormal function basis is the Hermite one.

Degree¶

The degree \(P\) of a PCE model is defined in such a way that \(\max_i \text{degree}(\phi_i)=\sum_{j=1}^d\tau_j(i)=P\).

Estimation¶

The coefficients \((w_1, w_2, ..., w_K)\) and the intercept \(w_0\) are estimated either by least-squares regression or a quadrature rule. In the case of least-squares regression, a sparse strategy can be considered with the LARS algorithm and in both cases, the CleaningStrategy can also remove the non-significant coefficients.

Dependence¶

The PCE model relies on the OpenTURNS class FunctionalChaosAlgorithm.

class gemseo.mlearning.regression.pce.CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001)[source]

Bases: object

The options of the CleaningStrategy.

Parameters:

max_considered_terms (int) –

By default it is set to 100.
most_significant (int) –

By default it is set to 20.
significance_factor (float) –

By default it is set to 0.0001.

max_considered_terms: int = 100: The maximum number of coefficients of the polynomial basis to be considered.

most_significant: int = 20: The maximum number of efficient coefficients of the polynomial basis to be kept.

significance_factor: float = 0.0001: The threshold to select the efficient coefficients of the polynomial basis.

class gemseo.mlearning.regression.pce.PCERegressor(data, probability_space, transformer=mappingproxy({}), input_names=None, output_names=None, degree=2, discipline=None, use_quadrature=False, use_lars=False, use_cleaning=False, hyperbolic_parameter=1.0, n_quadrature_points=0, cleaning_options=None)[source]

Bases: MLRegressionAlgo

Polynomial chaos expansion model.

See Also: API documentation of the OpenTURNS class FunctionalChaosAlgorithm.

Parameters:

data (IODataset | None) – The learning dataset required in the case of the least-squares regression or when discipline is None in the case of quadrature.
probability_space (ParameterSpace) – The set of random input variables defined by OTDistribution instances.
transformer (TransformerType) –
The strategies to transform the variables. The values are instances of Transformer while the keys are the names of either the variables or the groups of variables, e.g. "inputs" or "outputs" in the case of the regression algorithms. If a group is specified, the Transformer will be applied to all the variables of this group. If IDENTITY, do not transform the variables.

By default it is set to {}.
input_names (Iterable[str] | None) – The names of the input variables. If None, consider all the input variables of the learning dataset.
output_names (Iterable[str] | None) – The names of the output variables. If None, consider all the output variables of the learning dataset.
degree (int) –
The polynomial degree of the PCE.

By default it is set to 2.
discipline (MDODiscipline | None) – The discipline to be sampled if use_quadrature is True and data is None.
use_quadrature (bool) –
Whether to estimate the coefficients of the PCE by a quadrature rule; if so, use the quadrature points stored in data or sample discipline. otherwise, estimate the coefficients by least-squares regression.

By default it is set to False.
use_lars (bool) –
Whether to use the LARS algorithm in the case of the least-squares regression.

By default it is set to False.
use_cleaning (bool) –
Whether to use the CleaningStrategy algorithm. Otherwise, use a fixed truncation strategy (FixedStrategy).

By default it is set to False.
hyperbolic_parameter (float) –
The \(q\)-quasi norm parameter of the hyperbolic and anisotropic enumerate function, defined over the interval \(]0,1]\).

By default it is set to 1.0.
n_quadrature_points (int) –
The total number of quadrature points used by the quadrature strategy to compute the marginal number of points by input dimension when discipline is not None. If 0, use \((1+P)^d\) points, where \(d\) is the dimension of the input space and \(P\) is the polynomial degree of the PCE.

By default it is set to 0.
cleaning_options (CleaningOptions | None) – The options of the CleaningStrategy. If None, use DEFAULT_CLEANING_OPTIONS.

Raises:

ValueError – When both data and discipline are missing, when both data and discipline are provided, when discipline is provided in the case of least-squares regression, when data is missing in the case of least-squares regression, when the probability space does not contain the distribution of an input variable, when an input variable has a data transformer or when a probability distribution is not an OTDistribution.

LIBRARY: Final[str] = 'OpenTURNS': The name of the library of the wrapped machine learning algorithm.

SHORT_ALGO_NAME: ClassVar[str] = 'PCE'

The short name of the machine learning algorithm, often an acronym.

Typically used for composite names, e.g. f"{algo.SHORT_ALGO_NAME}_{dataset.name}" or f"{algo.SHORT_ALGO_NAME}_{discipline.name}".

algo: Any: The interfaced machine learning algorithm.

property covariance: ndarray: The covariance matrix of the PCE model output.

Warning

This statistic is expressed in relation to the transformed output space. You can sample the predict() method to estimate it in relation to the original output space if it is different from the transformed output space.

property first_sobol_indices: list[dict[str, float]]: The first-order Sobol’ indices for the different output components.

Warning

These statistics are expressed in relation to the transformed output space. You can use a SobolAnalysis to estimate them in relation to the original output space if it is different from the transformed output space.

input_names: list[str]: The names of the input variables.

input_space_center: dict[str, ndarray]: The center of the input space.

learning_set: Dataset: The learning dataset.

property mean: ndarray: The mean vector of the PCE model output.

Warning

This statistic is expressed in relation to the transformed output space. You can sample the predict() method to estimate it in relation to the original output space if it is different from the transformed output space.

output_names: list[str]: The names of the output variables.

parameters: dict[str, MLAlgoParameterType]: The parameters of the machine learning algorithm.

resampling_results: dict[str, tuple[Resampler, list[MLAlgo], list[ndarray] | ndarray]]

The resampler class names bound to the resampling results.

A resampling result is formatted as (resampler, ml_algos, predictions) where resampler is a Resampler, ml_algos is the list of the associated machine learning algorithms built during the resampling stage and predictions are the predictions obtained with the latter.

resampling_results stores only one resampling result per resampler type (e.g., "CrossValidation", "LeaveOneOut" and "Boostrap").

property second_sobol_indices: list[dict[str, dict[str, float]]]: The second-order Sobol’ indices for the different output components.

Warning

These statistics are expressed in relation to the transformed output space. You can use a SobolAnalysis to estimate them in relation to the original output space if it is different from the transformed output space.

property standard_deviation: ndarray: The standard deviation vector of the PCE model output.

Warning

This statistic is expressed in relation to the transformed output space. You can sample the predict() method to estimate it in relation to the original output space if it is different from the transformed output space.

property total_sobol_indices: list[dict[str, float]]: The total Sobol’ indices for the different output components.

Warning

These statistics are expressed in relation to the transformed output space. You can use a SobolAnalysis to estimate them in relation to the original output space if it is different from the transformed output space.

transformer: dict[str, Transformer]

The strategies to transform the variables, if any.

The values are instances of Transformer while the keys are the names of either the variables or the groups of variables, e.g. “inputs” or “outputs” in the case of the regression algorithms. If a group is specified, the Transformer will be applied to all the variables of this group.

property variance: ndarray: The variance vector of the PCE model output.

Warning

This statistic is expressed in relation to the transformed output space. You can sample the predict() method to estimate it in relation to the original output space if it is different from the transformed output space.

Examples using PCERegressor¶

PCE regression