The Gaussian process algorithm for regression.
Overview¶
The Gaussian process regression (GPR) surrogate model expresses the model output as a weighted sum of kernel functions centered on the learning input data:
Details¶
The GPR model relies on the assumption that the original model \(f\) to replace is an instance of a Gaussian process (GP) with mean \(\mu\) and covariance \(\sigma^2\kappa(\|x-x'\|;\epsilon)\).
Then, the GP conditioned by the learning set \((x_i,y_i)_{1\leq i \leq N}\) is entirely defined by its expectation:
and its covariance:
where \([\hat{\mu};\hat{w}]=([1_N~K]^T[1_N~K])^{-1}[1_N~K]^TY\) with \(K_{ij}=\kappa(\|x_i-x_j\|;\hat{\epsilon})\), \(k_i(x)=\kappa(\|x-x_i\|;\hat{\epsilon})\) and \(Y_i=y_i\).
The correlation length vector \(\epsilon\) is estimated by numerical non-linear optimization.
Surrogate model¶
The expectation \(\hat{f}\) is the GPR surrogate model of \(f\).
Error measure¶
The standard deviation \(\hat{s}\) is a local error measure of \(\hat{f}\):
Interpolation or regression¶
The GPR surrogate model can be regressive or interpolative according to the value of the nugget effect \(\\alpha\geq 0\) which is a regularization term applied to the correlation matrix \(K\). When \(\alpha = 0\), the surrogate model interpolates the learning data.
Dependence¶
The GPR model relies on the GaussianProcessRegressor class of the scikit-learn library.
Classes:
|
Gaussian process regression. |
- class gemseo.mlearning.regression.gpr.GaussianProcessRegression(data, transformer=None, input_names=None, output_names=None, kernel=None, alpha=1e-10, optimizer='fmin_l_bfgs_b', n_restarts_optimizer=10, random_state=None)[source]
Gaussian process regression.
- Parameters
kernel (Optional[openturns.CovarianceModel]) – The kernel function. If None, use a
Matern(2.5)
.alpha (Union[float,ndarray]) – The nugget effect to regularize the model.
optimizer (Union[str,Callable]) – The optimization algorithm to find the hyperparameters.
n_restarts_optimizer (int) – The number of restarts of the optimizer.
random_state (Optional[int]) – The seed used to initialize the centers. If None, the random number generator is the RandomState instance used by numpy.random.
data (Dataset) –
transformer (Optional[TransformerType]) –
input_names (Optional[Iterable[str]]) –
output_names (Optional[Iterable[str]]) –
- Return type
None
Classes:
Machine learning regression model decorators.
Attributes:
The input data matrix.
The dimension of the input variables before applying the transformers.
Return whether the algorithm is trained.
The output data matrix.
The dimension of the output variables before applying the transformers.
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.
predict_std
(input_data)Predict the standard deviation 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 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)
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
- predict_std(input_data)[source]
Predict the standard deviation from input data.
- Parameters
input_data (Union[numpy.ndarray, Dict[str, numpy.ndarray]]) – The input data with shape (n_samples, n_inputs).
- Returns
The output data with shape (n_samples, n_outputs).
- Return type
output_data
- 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