gemseo / mlearning / regression

# linreg module¶

Linear regression model.

The linear regression model expresses the output variables as a weighted sum of the input ones:

$y = w_0 + w_1x_1 + w_2x_2 + ... + w_dx_d + \alpha \left( \lambda \|w\|_2 + (1-\lambda) \|w\|_1 \right),$

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

The penalty level $$\alpha$$ is a non-negative parameter intended to prevent overfitting, while the penalty ratio $$\lambda\in [0, 1]$$ expresses the ratio between $$\ell_2$$- and $$\ell_1$$-regularization. When $$\lambda=1$$, there is no $$\ell_1$$-regularization, and a Ridge regression is performed. When $$\lambda=0$$, there is no $$\ell_2$$-regularization, and a Lasso regression is performed. For $$\lambda$$ between 0 and 1, an Elastic Net regression is performed.

One may also choose not to penalize the regression at all, by setting $$\alpha=0$$. In this case, a simple least squares regression is performed.

## Dependence¶

The linear model relies on the LinearRegression, Ridge, Lasso and ElasticNet classes of the scikit-learn library.

class gemseo.mlearning.regression.linreg.LinearRegressor(data, transformer=mappingproxy({}), input_names=None, output_names=None, fit_intercept=True, penalty_level=0.0, l2_penalty_ratio=1.0, random_state=0, **parameters)[source]

Linear regression model.

Parameters:
• data (IODataset) – The learning dataset.

• 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.

• fit_intercept (bool) –

Whether to fit the intercept.

By default it is set to True.

• penalty_level (float) –

The penalty level greater or equal to 0. If 0, there is no penalty.

By default it is set to 0.0.

• l2_penalty_ratio (float) –

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.

By default it is set to 1.0.

• random_state (int | None) –

The random state passed to the random number generator when there is a penalty. Use an integer for reproducible results.

By default it is set to 0.

• **parameters (float | int | str | bool | None) – The parameters of the machine learning algorithm.

Raises:

ValueError – When both the variable and the group it belongs to have a transformer.

get_coefficients(as_dict=True)[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

By default it is set to True.

Returns:

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.

Return type:

DataType

get_intercept(as_dict=True)[source]

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

By default it is set to True.

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:

DataType

LIBRARY: Final[str] = 'scikit-learn'

The name of the library of the wrapped machine learning algorithm.

SHORT_ALGO_NAME: ClassVar[str] = 'LinReg'

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 coefficients: ndarray

The regression coefficients of the linear model.

input_names: list[str]

The names of the input variables.

input_space_center: dict[str, ndarray]

The center of the input space.

property intercept: ndarray

The regression intercepts of the linear model.

learning_set: Dataset

The learning dataset.

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").

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.

## Examples using LinearRegressor¶

Cross-validation

Cross-validation

Leave-one-out

Leave-one-out

MSE for regression models

MSE for regression models

R2 for regression models

R2 for regression models

RMSE for regression models

RMSE for regression models

API

API

Linear regression

Linear regression