rbf module¶

The RBF network for regression.

The radial basis function surrogate discipline expresses the model output as a weighted sum of kernel functions centered on the learning input data:

\[y = w_1K(\|x-x_1\|;\epsilon) + w_2K(\|x-x_2\|;\epsilon) + ... + w_nK(\|x-x_n\|;\epsilon)\]

and the coefficients \((w_1, w_2, ..., w_n)\) are estimated by least squares minimization.

Dependence¶

The RBF model relies on the Rbf class of the scipy library.

Classes:

RBFRegression(data[, transformer, …])

Regression based on radial basis functions.

class gemseo.mlearning.regression.rbf.RBFRegression(data, transformer=None, input_names=None, output_names=None, function='multiquadric', der_function=None, epsilon=None, **parameters)[source]¶

Bases: gemseo.mlearning.regression.regression.MLRegressionAlgo

Regression based on radial basis functions.

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.
der_function (Callable[[ndarray],ndarray]) – The derivative of the radial basis function.
y_average (ndarray) – The mean of the learning output data.

Parameters

function (str) – The radial basis function.
der_function (Optional[Callable[[ndarray],ndarray]]) – The derivative of the radial basis function, only to be provided if function is a callable and if the use of the model with its derivative is required. If None and if function is a callable, an error will be raised. If None and if function is a string, the class will look for its internal implementation and will raise an error if it is missing. The der_function shall take three arguments (input_data, norm_input_data, eps). For a RBF of the form function(\(r\)), der_function(\(x\), \(|x|\), \(\epsilon\)) shall return \(\epsilon^{-1} x/|x| f'(|x|/\epsilon)\).
epsilon (Optional[float]) – An adjustable constant for Gaussian or multiquadrics functions. If None, use the average distance between input data.
data (Dataset) –
transformer (Optional[TransformerType]) –
input_names (Optional[Iterable[str]]) –
output_names (Optional[Iterable[str]]) –
parameters (Optional[MLAlgoParameterType]) –

Return type

None

Attributes:

`ABBR`
`AVAILABLE_FUNCTIONS`
`CUBIC`
`DEFAULT_TRANSFORMER`
`EUCLIDEAN`
`FILENAME`
`GAUSSIAN`
`INVERSE_MULTIQUADRIC`
`LIBRARY`
`LINEAR`
`MULTIQUADRIC`
`QUINTIC`
`THIN_PLATE`
`function`	The name of the kernel function.
`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.
`output_data`	The output data matrix.
`output_shape`	The dimension of the output variables before applying the transformers.

Classes:

`DataFormatters`()	Machine learning regression model decorators.
`RBFDerivatives`()	Derivatives of functions used in `RBFRegression`.

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.

ABBR = 'RBF'¶

AVAILABLE_FUNCTIONS = ['multiquadric', 'inverse_multiquadric', 'gaussian', 'linear', 'cubic', 'quintic', 'thin_plate']¶

CUBIC = 'cubic'¶

DEFAULT_TRANSFORMER = {'inputs': <gemseo.mlearning.transform.scaler.min_max_scaler.MinMaxScaler object>, 'outputs': <gemseo.mlearning.transform.scaler.min_max_scaler.MinMaxScaler object>}¶

class DataFormatters¶

Bases: gemseo.mlearning.core.supervised.MLSupervisedAlgo.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]

EUCLIDEAN = 'euclidean'¶

FILENAME = 'ml_algo.pkl'¶

GAUSSIAN = 'gaussian'¶

INVERSE_MULTIQUADRIC = 'inverse_multiquadric'¶

LIBRARY = 'scipy'¶

LINEAR = 'linear'¶

MULTIQUADRIC = 'multiquadric'¶

QUINTIC = 'quintic'¶

class RBFDerivatives[source]¶

Bases: object

Derivatives of functions used in RBFRegression.

For a RBF of the form \(f(r)\), \(r\) scalar, the derivative functions are defined by \(d(f(r))/dx\), with \(r=|x|/\epsilon\). The functions are thus defined by \(df/dx = \epsilon^{-1} x/|x| f'(|x|/\epsilon)\). This convention is chosen to avoid division by \(|x|\) when the terms may be cancelled out, as \(f'(r)\) often has a term in \(r\).

Attributes:

TOL

Methods:

`der_cubic`(input_data, norm_input_data, eps)	Compute derivative w.r.t.
`der_gaussian`(input_data, norm_input_data, eps)	Compute derivative w.r.t.
`der_inverse_multiquadric`(input_data, …)	Compute derivative w.r.t.
`der_linear`(input_data, norm_input_data, eps)	Compute derivative w.r.t.
`der_multiquadric`(input_data, …)	Compute derivative of \(f(r) = \sqrt{r^2 + 1}\) wrt \(x\).
`der_quintic`(input_data, norm_input_data, eps)	Compute derivative w.r.t.
`der_thin_plate`(input_data, norm_input_data, eps)	Compute derivative w.r.t.

TOL = 2.220446049250313e-16¶

classmethod der_cubic(input_data, norm_input_data, eps)[source]¶

Compute derivative w.r.t. \(x\) of the function \(f(r) = r^3\).

Parameters

input_data (numpy.ndarray) – The 1D input data.
norm_input_data (float) – The norm of the input variable.
eps (float) – The correlation length.

Returns

The derivative of the function.

Return type

numpy.ndarray

classmethod der_gaussian(input_data, norm_input_data, eps)[source]¶

Compute derivative w.r.t. \(x\) of the function \(f(r) = \exp(-r^2)\).

Parameters

input_data (numpy.ndarray) – The 1D input data.
norm_input_data (float) – The norm of the input variable.
eps (float) – The correlation length.

Returns

The derivative of the function.

Return type

numpy.ndarray

classmethod der_inverse_multiquadric(input_data, norm_input_data, eps)[source]¶

Compute derivative w.r.t. \(x\) of the function \(f(r) = 1/\sqrt{r^2 + 1}\).

Parameters

input_data (numpy.ndarray) – The 1D input data.
norm_input_data (float) – The norm of the input variable.
eps (float) – The correlation length.

Returns

The derivative of the function.

Return type

numpy.ndarray

classmethod der_linear(input_data, norm_input_data, eps)[source]¶

Compute derivative w.r.t. \(x\) of the function \(f(r) = r\). If \(x=0\), return 0 (determined up to a tolerance).

Parameters

input_data (numpy.ndarray) – The 1D input data.
norm_input_data (float) – The norm of the input variable.
eps (float) – The correlation length.

Returns

The derivative of the function.

Return type

numpy.ndarray

classmethod der_multiquadric(input_data, norm_input_data, eps)[source]¶

Compute derivative of \(f(r) = \sqrt{r^2 + 1}\) wrt \(x\).

Parameters

input_data (numpy.ndarray) – The 1D input data.
norm_input_data (float) – The norm of the input variable.
eps (float) – The correlation length.

Returns

The derivative of the function.

Return type

numpy.ndarray

classmethod der_quintic(input_data, norm_input_data, eps)[source]¶

Compute derivative w.r.t. \(x\) of the function \(f(r) = r^5\).

Parameters

input_data (numpy.ndarray) – The 1D input data.
norm_input_data (float) – The norm of the input variable.
eps (float) – The correlation length.

Returns

The derivative of the function.

Return type

numpy.ndarray

classmethod der_thin_plate(input_data, norm_input_data, eps)[source]¶

Compute derivative w.r.t. \(x\) of the function \(f(r) = r^2 \log(r)\). If \(x=0\), return 0 (determined up to a tolerance).

Parameters

input_data (numpy.ndarray) – The 1D input data.
norm_input_data (float) – The norm of the input variable.
eps (float) – The correlation length.

Returns

The derivative of the function.

Return type

numpy.ndarray

THIN_PLATE = 'thin_plate'¶

property function¶

The name of the kernel function.

The name is possibly different from self.parameters[‘function’], as it is mapped (scipy). Examples:

‘inverse’ -> ‘inverse_multiquadric’ ‘InverSE MULtiQuadRIC’ -> ‘inverse_multiquadric’

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)[source]¶

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