The polynomial model for regression.

Polynomial regression class is a particular case of the linear regression, where the input data is transformed before the regression is applied. This transform consists of creating a matrix of monomials (Vandermonde) by raising the input data to different powers up to a certain degree $$D$$. In the case where there is only one input variable, the input data $$(x_i)_{i=1, \dots, n}\in\mathbb{R}^n$$ is transformed into the Vandermonde matrix

$\begin{split}\begin{pmatrix} x_1^1 & x_1^2 & \cdots & x_1^D\\ x_2^1 & x_2^2 & \cdots & x_2^D\\ \vdots & \vdots & \ddots & \vdots\\ x_n^1 & x_n^2 & \cdots & x_n^D\\ \end{pmatrix} = (x_i^d)_{i=1, \dots, n;\ d=1, \dots, D}.\end{split}$

The output is expressed as a weighted sum of monomials:

$y = w_0 + w_1 x^1 + w_2 x^2 + ... + w_D x^D,$

where the coefficients $$(w_1, w_2, ..., w_d)$$ and the intercept $$w_0$$ are estimated by least square regression.

In the case of a multidimensional input, i.e. $$X = (x_{ij})_{i=1,\dots,n; j=1,\dots,m}$$, where $$n$$ is the number of samples and $$m$$ is the number of input variables, the Vandermonde matrix is expressed through different combinations of monomials of degree $$d, (1 \leq d \leq D)$$; e.g. for three variables $$(x, y, z)$$ and degree $$D=3$$, the different terms are $$x$$, $$y$$, $$z$$, $$x^2$$, $$xy$$, $$xz$$, $$y^2$$, $$yz$$, $$z^2$$, $$x^3$$, $$x^2y$$ etc. More generally, for m input variables, the total number of monomials of degree $$1 \leq d \leq D$$ is given by $$P = \binom{m+D}{m} = \frac{(m+D)!}{m!D!}$$. In the case of 3 input variables given above, the total number of monomial combinations of degree lesser than or equal to three is thus $$P = \binom{6}{3} = 20$$. The linear regression has to identify the coefficients $$(w_1, \dots, w_P)$$, in addition to the intercept $$w_0$$.

This concept is implemented through the PolynomialRegression class which inherits from the MLRegressionAlgo class.

# Dependence¶

The polynomial regression model relies on the LinearRegression class of the LinearRegression and PolynomialFeatures classes of the scikit-learn library.

Classes:

 PolynomialRegression(data, degree[, …]) Polynomial regression.
class gemseo.mlearning.regression.polyreg.PolynomialRegression(data, degree, transformer=None, input_names=None, output_names=None, fit_intercept=True, penalty_level=0.0, l2_penalty_ratio=1.0, **parameters)[source]

Polynomial regression.

Attributes
• learning_set (Dataset) – The learning dataset.

• parameters (Dict[str,MLAlgoParameterType]) – The parameters of the machine learning algorithm.

• transformer (Dict[str,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.

• algo (Any) – The interfaced machine learning algorithm.

• learning_set (Dataset) – The learning dataset.

• parameters (Dict[str,MLAlgoParameterType]) – The parameters of the machine learning algorithm.

• transformer (Dict[str,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.

• algo (Any) – The interfaced machine learning algorithm.

• input_names (List[str]) – The names of the input variables.

• output_names (List[str]) – The names of the output variables.

• input_space_center (Dict[str,ndarray]) – The center of the input space.

• learning_set (Dataset) – The learning dataset.

• parameters (Dict[str,MLAlgoParameterType]) – The parameters of the machine learning algorithm.

• transformer (Dict[str,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.

• algo (Any) – The interfaced machine learning algorithm.

• learning_set (Dataset) – The learning dataset.

• parameters (Dict[str,MLAlgoParameterType]) – The parameters of the machine learning algorithm.

• transformer (Dict[str,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.

• algo (Any) – The interfaced machine learning algorithm.

• input_names (List[str]) – The names of the input variables.

• output_names (List[str]) – The names of the output variables.

• input_space_center (Dict[str,ndarray]) – The center of the input space.

• learning_set (Dataset) – The learning dataset.

• parameters (Dict[str,MLAlgoParameterType]) – The parameters of the machine learning algorithm.

• transformer (Dict[str,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.

• algo (Any) – The interfaced machine learning algorithm.

• learning_set (Dataset) – The learning dataset.

• parameters (Dict[str,MLAlgoParameterType]) – The parameters of the machine learning algorithm.

• transformer (Dict[str,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.

• algo (Any) – The interfaced machine learning algorithm.

• input_names (List[str]) – The names of the input variables.

• output_names (List[str]) – The names of the output variables.

• input_space_center (Dict[str,ndarray]) – The center of the input space.

• learning_set (Dataset) – The learning dataset.

• parameters (Dict[str,MLAlgoParameterType]) – The parameters of the machine learning algorithm.

• transformer (Dict[str,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.

• algo (Any) – The interfaced machine learning algorithm.

• learning_set (Dataset) – The learning dataset.

• parameters (Dict[str,MLAlgoParameterType]) – The parameters of the machine learning algorithm.

• transformer (Dict[str,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.

• algo (Any) – The interfaced machine learning algorithm.

• input_names (List[str]) – The names of the input variables.

• output_names (List[str]) – The names of the output variables.

• input_space_center (Dict[str,ndarray]) – The center of the input space.

Parameters
• data (Dataset) –

• degree (int) –

• transformer (Optional[TransformerType]) –

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

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

• fit_intercept (bool) –

• penalty_level (float) –

• l2_penalty_ratio (float) –

• parameters (Optional[Union[float,int,str,bool]]) –

Return type

None

Parameters: input_names: The names of the input variables.

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

output_names: The names of the output variables.

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

fit_intercept: If True, fit intercept. penalty_level: The penalty level greater or equal to 0.

If 0, there is no penalty.

l2_penalty_ratio: The penalty ratio related to the l2 regularization.

If 1, use the Ridge penalty. If 0, use the Lasso penalty. Between 0 and 1, use the ElasticNet penalty.

**parameters: The parameters of the machine learning algorithm.

degree: The polynomial degree. fit_intercept: If True, fit intercept. penalty_level: The penalty level greater or equal to 0.

If 0, there is no penalty.

l2_penalty_ratio: The penalty ratio

related to the l2 regularization. If 1, the penalty is the Ridge penalty. If 0, this is the Lasso penalty. Between 0 and 1, the penalty is the ElasticNet penalty.

Raises

ValueError – If the degree is lower than one.

Parameters
• data (Dataset) –

• degree (int) –

• transformer (Optional[TransformerType]) –

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

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

• fit_intercept (bool) –

• penalty_level (float) –

• l2_penalty_ratio (float) –

• parameters (Optional[Union[float,int,str,bool]]) –

Return type

None

Classes:

 Machine learning regression model decorators.

Attributes:

 coefficients The regression coefficients of the linear model. input_data The input data matrix. input_shape The dimension of the input variables before applying the transformers. intercept The regression intercepts of the linear model. is_trained Return whether the algorithm is trained. output_data The output data matrix. output_shape The dimension of the output variables before applying the transformers.

Methods:

 get_coefficients([as_dict]) Return the regression coefficients of the linear model. get_intercept([as_dict]) Return the regression intercepts of the linear model. 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 coefficients

The regression coefficients of the linear model.

get_coefficients(as_dict=False)[source]

Return the regression coefficients of the linear model.

Parameters
• as_dict (bool) – If True, return the coefficients as a dictionary. Otherwise, return the coefficients as a numpy.array

• as_dict – If True, return the coefficients as a dictionary of Numpy arrays indexed by the names of the coefficients. Otherwise, return the coefficients as a Numpy array. For now the only valid value is False.

Returns

The regression coefficients of the linear model. The regression coefficients of the linear model.

Raises
• ValueError – If the coefficients are required as a dictionary even though the transformers change the variables dimensions.

• NotImplementedError – If the coefficients are required as a dictionary.

Return type

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

get_intercept(as_dict=True)

Return the regression intercepts of the linear model.

Parameters

as_dict (bool) – If True, return the intercepts as a dictionary. Otherwise, return the intercepts as a numpy.array

Returns

The regression intercepts of the linear model.

Raises

ValueError – If the coefficients are required as a dictionary even though the transformers change the variables dimensions.

Return type

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

property input_data

The input data matrix.

property input_shape

The dimension of the input variables before applying the transformers.

property intercept

The regression intercepts of the linear model.

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 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