The polynomial chaos expansion algorithm for regression.

The polynomial chaos expansion (PCE) model expresses the model output 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 $$\phi_i(x)=\psi_{\tau_1(i),1}(x_1)\times\ldots\times \psi_{\tau_d(i),d}(x_d)$$.

# Enumerating strategy¶

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

# Distributions¶

PCE are stochastic models whose inputs are random 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 an uniform random variable, $$(\psi_{ij})_{i\geq 0}$$ is the Hermite basis.

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

# Degree¶

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

# Estimation¶

The coefficients $$(w_1, w_2, ..., w_K)$$ and the intercept $$w_0$$ are estimated either by least squares regression, sparse least squares regression or quadrature.

# Dependence¶

The PCE model relies on the FunctionalChaosAlgorithm class of the openturns library.

Classes:

 PCERegression(data, probability_space[, ...]) Polynomial chaos expansion.
class gemseo.mlearning.regression.pce.PCERegression(data, probability_space, discipline=None, transformer=None, input_names=None, output_names=None, strategy='LS', degree=2, n_quad=None, stieltjes=True, sparse_param=None)[source]

Polynomial chaos expansion.

learning_set

The learning dataset.

Type

Dataset

parameters

The parameters of the machine learning algorithm.

Type

Dict[str,MLAlgoParameterType]

transformer

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 None, do not transform the variables.

Type

Dict[str,Transformer]

algo

The interfaced machine learning algorithm.

Type

Any

input_names

The names of the input variables.

Type

List[str]

output_names

The names of the output variables.

Type

List[str]

input_space_center

The center of the input space.

Type

Dict[str,ndarray]

Initialize self. See help(type(self)) for accurate signature.

Parameters
• data (Dataset) – The learning dataset.

• probability_space (ParameterSpace) – The probability space defining the probability distributions of the model inputs.

• discipline (Optional[MDODiscipline]) –

The discipline to evaluate with the quadrature strategy if the learning set does not have output data. If None, use the output data from the learning set.

By default it is set to None.

• transformer (Optional[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 None, do not transform the variables.

By default it is set to None.

• input_names (Optional[Iterable[str]]) –

The names of the input variables. If None, consider all input variables mentioned in the learning dataset.

By default it is set to None.

• output_names (Optional[Iterable[str]]) –

The names of the output variables. If None, consider all input variables mentioned in the learning dataset.

By default it is set to None.

• strategy (str) –

The strategy to compute the parameters of the PCE, either ‘LS’ for least-square, ‘Quad’ for quadrature or ‘SparseLS’ for sparse least-square.

By default it is set to LS.

• degree (int) –

The polynomial degree of the PCE.

By default it is set to 2.

The total number of quadrature points used by the quadrature strategy to compute the marginal number of points by input dimension. If None, this degree will be set equal to the polynomial degree of the PCE plus one.

By default it is set to None.

• stieltjes (bool) –

Whether to use the Stieltjes method.

By default it is set to True.

• sparse_param (Optional[Mapping[str,Union[int,float]]]) –

The parameters for the Sparse Cleaning Truncation Strategy and/or hyperbolic truncation of the initial basis:

• max_considered_terms (int) – The maximum considered terms (default: 120),

• most_significant (int) – The most Significant number to retain (default: 30),

• significance_factor (float) – Significance Factor (default: 1e-3),

• hyper_factor (float) – The factor for the hyperbolic truncation strategy (default: 1.0).

If None, use default values.

By default it is set to None.

Raises

ValueError – Either if the variables of the probability space and the input variables of the dataset are different, if transformers are specified for the inputs, or if the strategy to compute the parameters of the PCE is unknown.

Return type

None

Classes:

 Machine learning regression model decorators.

Attributes:

 first_sobol_indices The first Sobol' indices. input_data The input data matrix. input_shape The dimension of the input variables before applying the transformers. is_trained Return whether the algorithm is trained. learning_samples_indices The indices of the learning samples used for the training. output_data The output data matrix. output_shape The dimension of the output variables before applying the transformers. total_sobol_indices The total Sobol' indices.

Methods:

 learn([samples]) Train the machine learning algorithm from the learning dataset. load_algo(directory) Load a machine learning algorithm from a directory. predict(input_data, *args, **kwargs) Evaluate 'predict' with either array or dictionary-based input data. predict_jacobian(input_data, *args, **kwargs) Evaluate 'predict_jac' with either array or dictionary-based data. predict_raw(input_data) Predict output data from input data. save([directory, path, save_learning_set]) Save the machine learning algorithm.
class DataFormatters

Machine learning regression model decorators.

Methods:

 format_dict(predict) Make an array-based function be called with a dictionary of NumPy arrays. format_dict_jacobian(predict_jac) Wrap an array-based function to make it callable with a dictionary of NumPy arrays. format_input_output(predict) Make a function robust to type, array shape and data transformation. format_samples(predict) Make a 2D NumPy array-based function work with 1D NumPy array. format_transform([transform_inputs, ...]) Force a function to transform its input and/or output variables. transform_jacobian(predict_jac) Apply transformation to inputs and inverse transformation to outputs.
classmethod format_dict(predict)

Make an array-based function be called with a dictionary of NumPy arrays.

Parameters

predict (Callable[[numpy.ndarray], numpy.ndarray]) – The function to be called; it takes a NumPy array in input and returns a NumPy array.

Returns

A function making the function ‘predict’ work with either a NumPy data array or a dictionary of NumPy data arrays indexed by variables names. The evaluation will have the same type as the input data.

Return type

Callable[[Union[numpy.ndarray, Mapping[str, numpy.ndarray]]], Union[numpy.ndarray, Mapping[str, numpy.ndarray]]]

classmethod format_dict_jacobian(predict_jac)

Wrap an array-based function to make it callable with a dictionary of NumPy arrays.

Parameters

predict_jac (Callable[[numpy.ndarray], numpy.ndarray]) – The function to be called; it takes a NumPy array in input and returns a NumPy array.

Returns

The wrapped ‘predict_jac’ function, callable with either a NumPy data array or a dictionary of numpy data arrays indexed by variables names. The return value will have the same type as the input data.

Return type

Callable[[Union[numpy.ndarray, Mapping[str, numpy.ndarray]]], Union[numpy.ndarray, Mapping[str, numpy.ndarray]]]

classmethod format_input_output(predict)

Make a function robust to type, array shape and data transformation.

Parameters

predict (Callable[[numpy.ndarray], numpy.ndarray]) – The function of interest to be called.

Returns

A function calling the function of interest ‘predict’, while guaranteeing consistency in terms of data type and array shape, and applying input and/or output data transformation if required.

Return type

Callable[[Union[numpy.ndarray, Mapping[str, numpy.ndarray]]], Union[numpy.ndarray, Mapping[str, numpy.ndarray]]]

classmethod format_samples(predict)

Make a 2D NumPy array-based function work with 1D NumPy array.

Parameters

predict (Callable[[numpy.ndarray], numpy.ndarray]) – The function to be called; it takes a 2D NumPy array in input and returns a 2D NumPy array. The first dimension represents the samples while the second one represents the components of the variables.

Returns

A function making the function ‘predict’ work with either a 1D NumPy array or a 2D NumPy array. The evaluation will have the same dimension as the input data.

Return type

Callable[[numpy.ndarray], numpy.ndarray]

classmethod format_transform(transform_inputs=True, transform_outputs=True)

Force a function to transform its input and/or output variables.

Parameters
• transform_inputs (bool) –

Whether to transform the input variables.

By default it is set to True.

• transform_outputs (bool) –

Whether to transform the output variables.

By default it is set to True.

Returns

A function evaluating a function of interest, after transforming its input data and/or before transforming its output data.

Return type

Callable[[numpy.ndarray], numpy.ndarray]

classmethod transform_jacobian(predict_jac)

Apply transformation to inputs and inverse transformation to outputs.

Parameters

predict_jac (Callable[[numpy.ndarray], numpy.ndarray]) – The function of interest to be called.

Returns

A function evaluating the function ‘predict_jac’, after transforming its input data and/or before transforming its output data.

Return type

Callable[[numpy.ndarray], numpy.ndarray]

property first_sobol_indices

The first Sobol’ indices.

property input_data

The input data matrix.

property input_shape

The dimension of the input variables before applying the transformers.

property is_trained

Return whether the algorithm is trained.

learn(samples=None)

Train the machine learning algorithm from the learning dataset.

Parameters

samples (Optional[Sequence[int]]) –

The indices of the learning samples. If None, use the whole learning dataset.

By default it is set to None.

Return type

None

property learning_samples_indices

The indices of the learning samples used for the training.

Load a machine learning algorithm from a directory.

Parameters

directory (Union[str, pathlib.Path]) – The path to the directory where the machine learning algorithm is saved.

Return type

None

property output_data

The output data matrix.

property output_shape

The dimension of the output variables before applying the transformers.

predict(input_data, *args, **kwargs)

Evaluate ‘predict’ with either array or dictionary-based input data.

Firstly, the pre-processing stage converts the input data to a NumPy data array, if these data are expressed as a dictionary of NumPy data arrays.

Then, the processing evaluates the function ‘predict’ from this NumPy input data array.

Lastly, the post-processing transforms the output data to a dictionary of output NumPy data array if the input data were passed as a dictionary of NumPy data arrays.

Parameters
• input_data (Union[numpy.ndarray, Mapping[str, numpy.ndarray]]) – The input data.

• *args – The positional arguments of the function ‘predict’.

• **kwargs – The keyword arguments of the function ‘predict’.

Returns

The output data with the same type as the input one.

Return type

Union[numpy.ndarray, Mapping[str, numpy.ndarray]]

predict_jacobian(input_data, *args, **kwargs)

Evaluate ‘predict_jac’ with either array or dictionary-based data.

Firstly, the pre-processing stage converts the input data to a NumPy data array, if these data are expressed as a dictionary of NumPy data arrays.

Then, the processing evaluates the function ‘predict_jac’ from this NumPy input data array.

Lastly, the post-processing transforms the output data to a dictionary of output NumPy data array if the input data were passed as a dictionary of NumPy data arrays.

Parameters
• input_data – The input data.

• *args – The positional arguments of the function ‘predict_jac’.

• **kwargs – The keyword arguments of the function ‘predict_jac’.

Returns

The output data with the same type as the input one.

predict_raw(input_data)

Predict output data from input data.

Parameters

input_data (numpy.ndarray) – The input data with shape (n_samples, n_inputs).

Returns

The predicted output data with shape (n_samples, n_outputs).

Return type

numpy.ndarray

save(directory=None, path='.', save_learning_set=False)

Save the machine learning algorithm.

Parameters
• directory (Optional[str]) –

The name of the directory to save the algorithm.

By default it is set to None.

• path (Union[str, pathlib.Path]) –

The path to parent directory where to create the directory.

By default it is set to ..

• save_learning_set (bool) –

Whether to save the learning set or get rid of it to lighten the saved files.

By default it is set to False.

Returns

The path to the directory where the algorithm is saved.

Return type

str

property total_sobol_indices

The total Sobol’ indices.