# Mixture of experts¶

The mixture of experts for regression.

The mixture of experts (MoE) regression model expresses the output as a weighted sum of local surrogate models, where the weights are indicating the class of the input.

Inputs are grouped into clusters by a classification model that is trained on a training set where the output labels are determined through a clustering algorithm. The outputs may be preprocessed through a sensor or a dimension reduction algorithm.

The classification may either be hard, in which case only one of the weights is equal to one, and the rest equal to zero:

$y = \sum_{k=1}^K i_{C_k}(x) f_k(x),$

or soft, in which case the weights express the probabilities of belonging to each class:

$y = \sum_{k=1}^K \mathbb{P}(x\in C_k) f_k(x),$

where $$x$$ is the input, $$y$$ is the output, $$K$$ is the number of classes, $$(C_k)_{k=1,\cdots,K}$$ are the input spaces associated to the classes, $$i_{C_k}(x)$$ is the indicator of class $$k$$, $$\mathbb{P}(x\in C_k)$$ is the probability of class $$k$$ given $$x$$ and $$f_k(x)$$ is the local surrogate model on class $$k$$.

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

Classes:

 MixtureOfExperts(data[, transformer, ...]) Mixture of experts regression.
class gemseo.mlearning.regression.moe.MixtureOfExperts(data, transformer=None, input_names=None, output_names=None, hard=True)[source]

Mixture of experts regression.

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]

hard

Whether clustering/classification should be hard or soft.

Type

bool

cluster_algo

The name of the clustering algorithm.

Type

str

classif_algo

The name of the classification algorithm.

Type

str

regress_algo

The name of the regression algorithm.

Type

str

cluster_params

The parameters of the clustering algorithm.

Type

Optional[MLAlgoParameterType]

classif_params

The parameters of the classification algorithm.

Type

Optional[MLAlgoParameterType]

regress_params

The parameters of the regression algorithm.

Type

Optional[MLAlgoParameterType]

cluster_measure

The quality measure for the clustering algorithms.

Type

Dict[str,Union[str,EvalOptionType]]

classif_measure

The quality measure for the classification algorithms.

Type

Dict[str,Union[str,EvalOptionType]]

regress_measure

The quality measure for the regression algorithms.

Type

Dict[str,Union[str,EvalOptionType]]

cluster_cands

The clustering algorithm candidates.

Type

List[MLAlgoType]

classif_cands

The classification algorithm candidates.

Type

List[MLAlgoType]

regress_cands

The regression algorithm candidates.

Type

List[MLAlgoType]

clusterer

The clustering algorithm.

Type

MLClusteringAlgo

classifier

The classification algorithm.

Type

MLClassificationAlgo

regress_models

The regression algorithms.

Type

List(MLRegressionAlgo)

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

Parameters
• data (Dataset) – The learning dataset.

• 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.

• hard (bool) –

Whether clustering/classification should be hard or soft.

By default it is set to True.

Return type

None

Classes:

 Machine learning regression model decorators.

Methods:

 add_classifier_candidate(name[, ...]) Add a candidate for classification. add_clusterer_candidate(name[, calib_space, ...]) Add a candidate for clustering. add_regressor_candidate(name[, calib_space, ...]) Add a candidate for regression. 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_class(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_local_model(input_data, *args, **kwargs) Evaluate 'predict' with either array or dictionary-based input data. predict_raw(input_data) Predict output data from input data. save([directory, path, save_learning_set]) Save the machine learning algorithm. set_classification_measure(measure, ...) Set the quality measure for the classification algorithms. set_classifier(classif_algo, **classif_params) Set the classification algorithm. set_clusterer(cluster_algo, **cluster_params) Set the clustering algorithm. set_clustering_measure(measure, **eval_options) Set the quality measure for the clustering algorithms. set_regression_measure(measure, **eval_options) Set the quality measure for the regression algorithms. set_regressor(regress_algo, **regress_params) Set the regression algorithm.

Attributes:

 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. labels The cluster labels. learning_samples_indices The indices of the learning samples used for the training. n_clusters The number of clusters. output_data The output data matrix. output_shape The dimension of the output variables before applying the transformers.
class DataFormatters[source]

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_predict_class_dict(predict) Make an array-based function be called with a dictionary of NumPy arrays. 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_predict_class_dict(predict)[source]

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_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]

Parameters
• name (str) – The name of a classification algorithm.

• calib_space (Optional[gemseo.algos.design_space.DesignSpace]) –

The space defining the calibration variables.

By default it is set to None.

• calib_algo (Optional[Dict[str, Union[str, int]]]) –

The name and options of the DOE or optimization algorithm, e.g. {“algo”: “fullfact”, “n_samples”: 10}). If None, do not perform calibration.

By default it is set to None.

• option_lists (***) – Parameters for the clustering algorithm candidate. Each parameter has to be enclosed within a list. The list may contain different values to try out for the given parameter, or only one.

• **option_lists (Optional[List[Optional[Any]]]) –

Return type

None

Parameters
• name (str) – The name of a clustering algorithm.

• calib_space (Optional[gemseo.algos.design_space.DesignSpace]) –

The space defining the calibration variables.

By default it is set to None.

• calib_algo (Optional[Dict[str, Union[str, int]]]) –

The name and options of the DOE or optimization algorithm, e.g. {“algo”: “fullfact”, “n_samples”: 10}). If None, do not perform calibration.

By default it is set to None.

• option_lists (***) – Parameters for the clustering algorithm candidate. Each parameter has to be enclosed within a list. The list may contain different values to try out for the given parameter, or only one.

• **option_lists (Optional[List[Optional[Any]]]) –

Return type

None

Parameters
• name (str) – The name of a regression algorithm.

• calib_space (Optional[gemseo.algos.design_space.DesignSpace]) –

The space defining the calibration variables.

By default it is set to None.

• calib_algo (Optional[Dict[str, Union[str, int]]]) –

The name and options of the DOE or optimization algorithm, e.g. {“algo”: “fullfact”, “n_samples”: 10}). If None, do not perform calibration.

By default it is set to None.

• option_lists (***) – Parameters for the clustering algorithm candidate. Each parameter has to be enclosed within a list. The list may contain different values to try out for the given parameter, or only one.

• **option_lists (Optional[List[Optional[Any]]]) –

Return type

None

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.

property labels

The cluster labels.

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 n_clusters

The number of clusters.

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_class(input_data, *args, **kwargs)[source]

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_local_model(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_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

set_classification_measure(measure, **eval_options)[source]

Set the quality measure for the classification algorithms.

Parameters
Return type

None

set_classifier(classif_algo, **classif_params)[source]

Set the classification algorithm.

Parameters
• classif_algo (str) – The name of the classification algorithm.

• **classif_params (Optional[Any]) – The parameters of the classification algorithm.

Return type

None

set_clusterer(cluster_algo, **cluster_params)[source]

Set the clustering algorithm.

Parameters
• cluster_algo (str) – The name of the clustering algorithm.

• **cluster_params (Optional[Any]) – The parameters of the clustering algorithm.

Return type

None

set_clustering_measure(measure, **eval_options)[source]

Set the quality measure for the clustering algorithms.

Parameters
Return type

None

set_regression_measure(measure, **eval_options)[source]

Set the quality measure for the regression algorithms.

Parameters
Return type

None

set_regressor(regress_algo, **regress_params)[source]

Set the regression algorithm.

Parameters
• regress_algo (str) – The name of the regression algorithm.

• **regress_params (Optional[Any]) – The parameters of the regression algorithm.

Return type

None