# 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 annotations

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

configure_logger()

<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()


## 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})

    INFO - 13:53:55:
INFO - 13:53:55: *** Start DOEScenario execution ***
INFO - 13:53:55: DOEScenario
INFO - 13:53:55:    Disciplines: func
INFO - 13:53:55:    MDO formulation: DisciplinaryOpt
INFO - 13:53:55: Optimization problem:
INFO - 13:53:55:    minimize y_1(x_1, x_2)
INFO - 13:53:55:    with respect to x_1, x_2
INFO - 13:53:55:    over the design space:
INFO - 13:53:55:    +------+-------------+-------+-------------+-------+
INFO - 13:53:55:    | name | lower_bound | value | upper_bound | type  |
INFO - 13:53:55:    +------+-------------+-------+-------------+-------+
INFO - 13:53:55:    | x_1  |      0      |  None |      1      | float |
INFO - 13:53:55:    | x_2  |      0      |  None |      1      | float |
INFO - 13:53:55:    +------+-------------+-------+-------------+-------+
INFO - 13:53:55: Solving optimization problem with algorithm fullfact:
INFO - 13:53:55: ...   0%|          | 0/9 [00:00<?, ?it]
INFO - 13:53:55: ...  11%|█         | 1/9 [00:00<00:00, 272.96 it/sec, obj=1]
INFO - 13:53:55: ...  22%|██▏       | 2/9 [00:00<00:00, 429.81 it/sec, obj=2.25]
INFO - 13:53:55: ...  33%|███▎      | 3/9 [00:00<00:00, 541.13 it/sec, obj=4]
INFO - 13:53:55: ...  44%|████▍     | 4/9 [00:00<00:00, 634.54 it/sec, obj=2.5]
INFO - 13:53:55: ...  56%|█████▌    | 5/9 [00:00<00:00, 710.25 it/sec, obj=3.75]
INFO - 13:53:55: ...  67%|██████▋   | 6/9 [00:00<00:00, 769.71 it/sec, obj=5.5]
INFO - 13:53:55: ...  78%|███████▊  | 7/9 [00:00<00:00, 821.01 it/sec, obj=4]
INFO - 13:53:55: ...  89%|████████▉ | 8/9 [00:00<00:00, 860.77 it/sec, obj=5.25]
INFO - 13:53:55: ... 100%|██████████| 9/9 [00:00<00:00, 897.75 it/sec, obj=7]
INFO - 13:53:55: Optimization result:
INFO - 13:53:55:    Optimizer info:
INFO - 13:53:55:       Status: None
INFO - 13:53:55:       Message: None
INFO - 13:53:55:       Number of calls to the objective function by the optimizer: 9
INFO - 13:53:55:    Solution:
INFO - 13:53:55:       Objective: 1.0
INFO - 13:53:55:       Design space:
INFO - 13:53:55:       +------+-------------+-------+-------------+-------+
INFO - 13:53:55:       | name | lower_bound | value | upper_bound | type  |
INFO - 13:53:55:       +------+-------------+-------+-------------+-------+
INFO - 13:53:55:       | x_1  |      0      |   0   |      1      | float |
INFO - 13:53:55:       | x_2  |      0      |   0   |      1      | float |
INFO - 13:53:55:       +------+-------------+-------+-------------+-------+
INFO - 13:53:55: *** End DOEScenario execution (time: 0:00:00.022026) ***

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


## Create the regression model¶

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

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

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)
output_value

{'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)
jacobian_value

{'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.

model.intercept, model.get_intercept()

(array([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.

model.coefficients

array([[ 2.00000000e+00,  3.00000000e+00,  1.00000000e+00,
-6.22449391e-16, -5.07973866e-16]])


Total running time of the script: (0 minutes 0.076 seconds)

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