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:
and the coefficients \((w_1, w_2, \ldots, w_n)\) are estimated by least squares minimization.
Dependence¶
The RBF model relies on the Rbf class of the scipy library.
- class gemseo.mlearning.regression.rbf.RBFRegressor(data, transformer=mappingproxy({}), input_names=None, output_names=None, function=Function.MULTIQUADRIC, der_function=None, epsilon=None, smooth=0.0, norm='euclidean')[source]
Bases:
MLRegressionAlgoRegression based on radial basis functions (RBFs).
This model relies on the SciPy class
scipy.interpolate.Rbf.- Parameters:
data (IODataset) – The learning dataset.
transformer (TransformerType) –
The strategies to transform the variables. The values are instances of
Transformerwhile 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, theTransformerwill be applied to all the variables of this group. IfIDENTITY, 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.function (Function | Callable[[float, float], float]) –
The radial basis function taking a radius \(r\) as input, representing a distance between two points. If it is a string, then it must be one of the following:
"multiquadric"for \(\sqrt{(r/\epsilon)^2 + 1}\),"inverse"for \(1/\sqrt{(r/\epsilon)^2 + 1}\),"gaussian"for \(\exp(-(r/\epsilon)^2)\),"linear"for \(r\),"cubic"for \(r^3\),"quintic"for \(r^5\),"thin_plate"for \(r^2\log(r)\).
If it is a callable, then it must take the two arguments
selfandras inputs, e.g.lambda self, r: sqrt((r/self.epsilon)**2 + 1)for the multiquadric function. The epsilon parameter will be available asself.epsilon. Other keyword arguments passed in will be available as well.By default it is set to “multiquadric”.
der_function (Callable[[ndarray], ndarray] | None) – The derivative of the radial basis function, only to be provided if
functionis a callable and if the use of the model with its derivative is required. IfNoneand iffunctionis a callable, an error will be raised. IfNoneand iffunctionis a string, the class will look for its internal implementation and will raise an error if it is missing. Theder_functionshall take three arguments (input_data,norm_input_data,eps). For an RBF of the form function(\(r\)), der_function(\(x\), \(|x|\), \(\epsilon\)) shall return \(\epsilon^{-1} x/|x| f'(|x|/\epsilon)\).epsilon (float | None) – An adjustable constant for Gaussian or multiquadric functions. If
None, use the average distance between input data.smooth (float) –
The degree of smoothness,
0involving an interpolation of the learning points.By default it is set to 0.0.
norm (str | Callable[[ndarray, ndarray], float]) –
The distance metric to be used, either a distance function name known by SciPy or a function that computes the distance between two points.
By default it is set to “euclidean”.
- Raises:
ValueError – When both the variable and the group it belongs to have a transformer.
- class Function(value)[source]
Bases:
StrEnumThe radial basis functions.
- CUBIC = 'cubic'
- GAUSSIAN = 'gaussian'
- INVERSE_MULTIQUADRIC = 'inverse_multiquadric'
- LINEAR = 'linear'
- MULTIQUADRIC = 'multiquadric'
- QUINTIC = 'quintic'
- THIN_PLATE = 'thin_plate'
- class RBFDerivatives[source]
Bases:
objectDerivatives of functions used in
RBFRegressor.For an 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 \(r\).
- classmethod der_cubic(input_data, norm_input_data, eps)[source]
Compute derivative w.r.t. \(x\) of the function \(f(r) = r^3\).
- classmethod der_gaussian(input_data, norm_input_data, eps)[source]
Compute derivative of \(f(r)=\exp(-r^2)\) w.r.t. \(x\).
- classmethod der_inverse_multiquadric(input_data, norm_input_data, eps)[source]
Compute derivative of \(f(r)=1/\sqrt{r^2 + 1}\) w.r.t. \(x\).
- classmethod der_linear(input_data, norm_input_data, eps)[source]
Compute derivative of \(f(r)=r\) w.r.t. \(x\).
If \(x=0\), return 0 (determined up to a tolerance).
- classmethod der_multiquadric(input_data, norm_input_data, eps)[source]
Compute derivative of \(f(r) = \sqrt{r^2 + 1}\) w.r.t. \(x\).
- classmethod der_quintic(input_data, norm_input_data, eps)[source]
Compute derivative w.r.t. \(x\) of the function \(f(r) = r^5\).
- classmethod der_thin_plate(input_data, norm_input_data, eps)[source]
Compute derivative of \(f(r) = r^2\log(r)\) w.r.t. \(x\).
If \(x=0\), return 0 (determined up to a tolerance).
- TOL = 2.220446049250313e-16
- EUCLIDEAN: Final[str] = 'euclidean'
- LIBRARY: Final[str] = 'SciPy'
The name of the library of the wrapped machine learning algorithm.
- SHORT_ALGO_NAME: ClassVar[str] = 'RBF'
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}"orf"{algo.SHORT_ALGO_NAME}_{discipline.name}".
- algo: Any
The interfaced machine learning algorithm.
- der_function: Callable[[ndarray], ndarray]
The derivative of the radial basis function.
- property function: str
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’
- learning_set: Dataset
The learning dataset.
- 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)whereresampleris aResampler,ml_algosis the list of the associated machine learning algorithms built during the resampling stage andpredictionsare the predictions obtained with the latter.resampling_resultsstores 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
Transformerwhile 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, theTransformerwill be applied to all the variables of this group.
- y_average: ndarray
The mean of the learning output data.
