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:

$y = \mu + w_1\kappa(\|x-x_1\|;\epsilon) + w_2\kappa(\|x-x_2\|;\epsilon) + ... + w_N\kappa(\|x-x_N\|;\epsilon)$

# 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:

$\hat{f}(x) = \hat{\mu} + \hat{w}^T k(x)$

and its covariance:

$\hat{c}(x,x') = \hat{\sigma}^2 - k(x)^T K^{-1} k(x')$

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 $$\hat{f}$$:

$\hat{s}(x):=\sqrt{\hat{c}(x,x)}$

# 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=None, input_names=None, output_names=None, kernel=None, bounds=None, alpha=1e-10, optimizer='fmin_l_bfgs_b', n_restarts_optimizer=10, random_state=None)[source]

Gaussian process regression model.

Parameters
• data (Dataset) – The learning dataset.

• transformer (Mapping[str, TransformerType] | None) –

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 (Iterable[str] | None) –

The names of the input variables. If None, consider all the input variables of the learning dataset.

By default it is set to None.

• output_names (Iterable[str] | None) –

The names of the output variables. If None, consider all the output variables of the learning dataset.

By default it is set to None.

• kernel (Kernel | None) –

The kernel specifying the covariance model. If None, use a Matérn(2.5).

By default it is set to None.

• bounds (__Bounds | Mapping[str, __Bounds] | None) –

The lower and upper bounds of the parameter length scales when kernel is None. Either a unique lower-upper pair common to all the inputs or lower-upper pairs for some of them. When bounds is None or when an input has no pair, the lower bound is 0.01 and the upper bound is 100.

By default it is set to None.

• 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 seed used to initialize the centers. If None, the random number generator is the RandomState instance used by numpy.random.

By default it is set to None.

Raises

ValueError – When both the variable and the group it belongs to have a transformer.

class DataFormatters

Machine learning regression model decorators.

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_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]

learn(samples=None, fit_transformers=True)

Train the machine learning algorithm from the learning dataset.

Parameters
• samples (Sequence[int] | None) –

The indices of the learning samples. If None, use the whole learning dataset.

By default it is set to None.

• fit_transformers (bool) –

Whether to fit the variable transformers.

By default it is set to True.

Return type

None

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 ‘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_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.

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 (Union[numpy.ndarray, Mapping[str, numpy.ndarray]]) – The input data.

Returns

The standard deviation at the query points.

Return type

numpy.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.

save(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.

By default it is set to None.

• 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

str

property input_data: numpy.ndarray

The input data matrix.

property input_dimension: int

The input space dimension.

property is_trained: bool

Return whether the algorithm is trained.

property kernel

The kernel used for prediction.

property learning_samples_indices: Sequence[int]

The indices of the learning samples used for the training.

property output_data: numpy.ndarray

The output data matrix.

property output_dimension: int

The output space dimension.