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 gemseo.api import configure_logger
from gemseo.api import create_design_space
from gemseo.api import create_discipline
from gemseo.api import create_scenario
from gemseo.mlearning.api import create_regression_model
from numpy import array

configure_logger()

Out:

<RootLogger root (INFO)>

Create the discipline to learn

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

expressions = {
    "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=expressions
)

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.

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

Out:

    INFO - 10:06:52:
    INFO - 10:06:52: *** Start DOEScenario execution ***
    INFO - 10:06:52: DOEScenario
    INFO - 10:06:52:    Disciplines: func
    INFO - 10:06:52:    MDO formulation: DisciplinaryOpt
    INFO - 10:06:52: Optimization problem:
    INFO - 10:06:52:    minimize y_1(x_1, x_2)
    INFO - 10:06:52:    with respect to x_1, x_2
    INFO - 10:06:52:    over the design space:
    INFO - 10:06:52:    +------+-------------+-------+-------------+-------+
    INFO - 10:06:52:    | name | lower_bound | value | upper_bound | type  |
    INFO - 10:06:52:    +------+-------------+-------+-------------+-------+
    INFO - 10:06:52:    | x_1  |      0      |  None |      1      | float |
    INFO - 10:06:52:    | x_2  |      0      |  None |      1      | float |
    INFO - 10:06:52:    +------+-------------+-------+-------------+-------+
    INFO - 10:06:52: Solving optimization problem with algorithm fullfact:
    INFO - 10:06:52: Full factorial design required. Number of samples along each direction for a design vector of size 2 with 9 samples: 3
    INFO - 10:06:52: Final number of samples for DOE = 9 vs 9 requested
    INFO - 10:06:52: ...   0%|          | 0/9 [00:00<?, ?it]
    INFO - 10:06:52: ... 100%|██████████| 9/9 [00:00<00:00, 1431.77 it/sec, obj=7]
    INFO - 10:06:52: Optimization result:
    INFO - 10:06:52:    Optimizer info:
    INFO - 10:06:52:       Status: None
    INFO - 10:06:52:       Message: None
    INFO - 10:06:52:       Number of calls to the objective function by the optimizer: 9
    INFO - 10:06:52:    Solution:
    INFO - 10:06:52:       Objective: 1.0
    INFO - 10:06:52:       Design space:
    INFO - 10:06:52:       +------+-------------+-------+-------------+-------+
    INFO - 10:06:52:       | name | lower_bound | value | upper_bound | type  |
    INFO - 10:06:52:       +------+-------------+-------+-------------+-------+
    INFO - 10:06:52:       | x_1  |      0      |   0   |      1      | float |
    INFO - 10:06:52:       | x_2  |      0      |   0   |      1      | float |
    INFO - 10:06:52:       +------+-------------+-------+-------------+-------+
    INFO - 10:06:52: *** End DOEScenario execution (time: 0:00:00.014775) ***

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

Create the regression model

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

dataset = scenario.export_to_dataset(opt_naming=False)
model = create_regression_model(
    "PolynomialRegressor", data=dataset, degree=2, transformer=None
)
model.learn()
print(model)

Out:

PolynomialRegressor(degree=2, fit_intercept=True, l2_penalty_ratio=1.0, penalty_level=0.0)
   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.])}

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:

/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/4.0.1/lib/python3.9/site-packages/sklearn/utils/deprecation.py:103: FutureWarning: The attribute `n_input_features_` was deprecated in version 1.0 and will be removed in 1.2.
  warnings.warn(msg, category=FutureWarning)
{'y_1': {'x_1': array([[4.]]), 'x_2': array([[3.]])}}

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.]
{'y_1': [1.0]}

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 -6.22449391e-16
  -5.07973866e-16]]