gpr module¶
Gaussian process regression model.
Overview¶
The Gaussian process regression (GPR) 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 surrogate model of \(f\).
Error measure¶
The standard deviation \(\hat{s}\) is a local error measure of \(\hat{f}\):
Interpolation or regression¶
The GPR 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.
- class gemseo.mlearning.regression.gpr.GaussianProcessRegressor(data, transformer=mappingproxy({}), input_names=None, output_names=None, kernel=None, bounds=None, alpha=1e-10, optimizer='fmin_l_bfgs_b', n_restarts_optimizer=10, random_state=0)[source]¶
Bases:
MLRegressionAlgo
Gaussian process 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, theTransformer
will 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.kernel (Kernel | None) – The kernel specifying the covariance model. If
None
, use a Matérn(2.5).bounds (__Bounds | Mapping[str, __Bounds] | None) – The lower and upper bounds of the parameter length scales when
kernel
isNone
. Either a unique lower-upper pair common to all the inputs or lower-upper pairs for some of them. Whenbounds
isNone
or when an input has no pair, the lower bound is 0.01 and the upper bound is 100.alpha (float | ndarray) –
The nugget effect to regularize the model.
By default it is set to 1e-10.
optimizer (str | Callable) –
The optimization algorithm to find the parameter length scales.
By default it is set to “fmin_l_bfgs_b”.
n_restarts_optimizer (int) –
The number of restarts of the optimizer.
By default it is set to 10.
random_state (int | None) –
The random state passed to the random number generator. Use an integer for reproducible results.
By default it is set to 0.
- Raises:
ValueError – When both the variable and the group it belongs to have a transformer.
- DataFormatters¶
alias of
RegressionDataFormatters
- learn(samples=None, fit_transformers=True)¶
Train the machine learning algorithm from the learning dataset.
- load_algo(directory)¶
Load a machine learning algorithm from a directory.
- Parameters:
directory (str | Path) – The path to the directory where the machine learning algorithm is saved.
- Return type:
None
- predict(input_data, *args, **kwargs)¶
Evaluate
func
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
func
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:
algo (MLSupervisedAlgo) – The supervised learning algorithm.
input_data (DataType) – The input data.
*args (Any) – The positional arguments of the function
func
.**kwargs (Any) – The keyword arguments of the function
func
.
- Returns:
The output data with the same type as the input one.
- Return type:
- predict_jacobian(input_data, *args, **kwargs)¶
Evaluate
func
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
func
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:
algo (MLRegressionAlgo) – The regression algorithm.
input_data (DataType) – The input data.
*args (Any) – The positional arguments of the function
func
.**kwargs (Any) – The keyword arguments of the function
func
.
- Returns:
The output data with the same type as the input one.
- Return type:
- predict_raw(input_data)¶
Predict output data from input data.
- Parameters:
input_data (ndarray) – The input data with shape (n_samples, n_inputs).
- Returns:
The predicted output data with shape (n_samples, n_outputs).
- Return type:
ndarray
- predict_std(input_data)[source]¶
Predict the standard deviation from input data.
The user can specify these input data either as a NumPy array, e.g.
array([1., 2., 3.])
or as a dictionary of NumPy arrays, e.g.{'a': array([1.]), 'b': array([2., 3.])}
.If the NumPy arrays are of dimension 2, their i-th rows represent the input data of the i-th sample; while if the NumPy arrays are of dimension 1, there is a single sample.
- Parameters:
input_data (DataType) – The input data.
- Returns:
The standard deviation at the query points.
- Return type:
ndarray
Warning
If the output variables are transformed before the training stage, then the standard deviation is related to this transformed output space unlike
predict()
which returns values in the original output space.
- to_pickle(directory=None, path='.', save_learning_set=False)¶
Save the machine learning algorithm.
- Parameters:
directory (str | None) – The name of the directory to save the algorithm.
path (str | 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:
- DEFAULT_TRANSFORMER: DefaultTransformerType = mappingproxy({'inputs': <gemseo.mlearning.transformers.scaler.min_max_scaler.MinMaxScaler object>, 'outputs': <gemseo.mlearning.transformers.scaler.min_max_scaler.MinMaxScaler object>})¶
The default transformer for the input and output data, if any.
- IDENTITY: Final[DefaultTransformerType] = mappingproxy({})¶
A transformer leaving the input and output variables as they are.
- LIBRARY: Final[str] = 'scikit-learn'¶
The name of the library of the wrapped machine learning algorithm.
- SHORT_ALGO_NAME: ClassVar[str] = 'GPR'¶
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.
- property kernel¶
The kernel used for prediction.
- property learning_samples_indices: Sequence[int]¶
The indices of the learning samples used for the training.
- 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)
whereresampler
is aResampler
,ml_algos
is the list of the associated machine learning algorithms built during the resampling stage andpredictions
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, theTransformer
will be applied to all the variables of this group.