Polynomial regression

We want to approximate a discipline with two inputs and two outputs:

• \(y_1=1+2x_1+3x_2\)

• \(y_2=-1-2x_1-3x_2\)

over the unit hypercube \([0,1]\times[0,1]\).

from __future__ import absolute_import, division, print_function, unicode_literals

from future import standard_library
from numpy import array

from gemseo.api import (
    configure_logger,
    create_design_space,
    create_discipline,
    create_scenario,
)
from gemseo.mlearning.api import create_regression_model

configure_logger()

standard_library.install_aliases()

Create the discipline to learn

We can implement this analytic discipline by means of the AnalyticDiscipline class.

expressions_dict = {
    "y_1": "1 + 2*x_1 + 3*x_2 + x_1**2",
    "y_2": "-1 - 2*x_1 + x_1*x_2 - 3*x_2**2",
}
discipline = create_discipline(
    "AnalyticDiscipline", name="func", expressions_dict=expressions_dict
)

Create the input sampling space

We create the input sampling space by adding the variables one by one.

design_space = create_design_space()
design_space.add_variable("x_1", l_b=0.0, u_b=1.0)
design_space.add_variable("x_2", l_b=0.0, u_b=1.0)

Create the learning set

We can build a learning set by means of a DOEScenario with a full factorial design of experiments. The number of samples can be equal to 9 for example.

discipline.set_cache_policy(discipline.MEMORY_FULL_CACHE)
scenario = create_scenario(
    [discipline], "DisciplinaryOpt", "y_1", design_space, scenario_type="DOE"
)
scenario.execute({"algo": "fullfact", "n_samples": 9})

Out:

{'eval_jac': False, 'algo': 'fullfact', 'n_samples': 9}

Create the regression model

Then, we build the linear regression model from the discipline cache and displays this model.

dataset = discipline.cache.export_to_dataset()
model = create_regression_model(
    "PolynomialRegression", data=dataset, degree=2, transformer=None
)
model.learn()
print(model)

Out:

PolynomialRegression(fit_intercept=True, penalty_level=0.0, l2_penalty_ratio=1.0, degree=2)
| based on the scikit-learn library
| built from 9 learning samples

Predict output

Once it is built, we can use it for prediction.

input_value = {"x_1": array([1.0]), "x_2": array([2.0])}
output_value = model.predict(input_value)
print(output_value)

Out:

{'y_1': array([10.]), 'y_2': array([-13.])}

Predict Jacobian

We can also use it to predict the jacobian of the discipline.

jacobian_value = model.predict_jacobian(input_value)
print(jacobian_value)

Out:

{'y_1': {'x_1': array([[4.]]), 'x_2': array([[3.]])}, 'y_2': {'x_1': array([[1.33226763e-15]]), 'x_2': array([[-11.]])}}

Get intercept

In addition, it is possible to access the intercept of the model, either directly or by means of a method returning either a dictionary (default option) or an array.

print(model.intercept)
print(model.get_intercept())

Out:

[ 1. -1.]
{'y_1': [1.0], 'y_2': [-0.9999999999999987]}

Get coefficients

In addition, it is possible to access the coefficients of the model, either directly or by means of a method returning either a dictionary (default option) or an array.

print(model.coefficients)

Out:

[[ 2.00000000e+00  3.00000000e+00  1.00000000e+00 -2.22044605e-16
  -8.88178420e-16]
 [-2.00000000e+00 -3.21964677e-15 -2.77555756e-16  1.00000000e+00
  -3.00000000e+00]]