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:
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:
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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.
- Parameters
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.
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.
degree (int) – The polynomial degree of the PCE.
n_quad (Optional[int]) – 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.
stieltjes (bool) – Use the Stieltjes method.
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.
data (Dataset) –
transformer (Optional[TransformerType]) –
input_names (Optional[Iterable[str]]) –
output_names (Optional[Iterable[str]]) –
- 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:
The first Sobol’ indices.
The input data matrix.
The dimension of the input variables before applying the transformers.
Return whether the algorithm is trained.
The output data matrix.
The dimension of the output variables before applying the transformers.
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, Dict[str, numpy.ndarray]]], Union[numpy.ndarray, Dict[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, Dict[str, numpy.ndarray]]], Union[numpy.ndarray, Dict[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, Dict[str, numpy.ndarray]]], Union[numpy.ndarray, Dict[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) – If True, apply the transformers to the input variables.
transform_outputs (bool) – If True, apply the transformers to the output variables.
- 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[List[int]]) – The indices of the learning samples. If None, use the whole learning dataset.
- Raises
NotImplementedError – If an output transformer modifies both the input and the output variables, e.g.
PLS.- Return type
None
- load_algo(directory)
Load a machine learning algorithm from a directory.
- Parameters
directory (str) – 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, Dict[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, Dict[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.
path (str) – The path to parent directory where to create the directory.
save_learning_set (bool) – If False, do not save the learning set to lighten the saved files.
- Returns
The path to the directory where the algorithm is saved.
- Return type
str
- property total_sobol_indices
The total Sobol’ indices.