# Linear 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 $$[0,1]\times[0,1]$$.

## Import¶

from __future__ import division, unicode_literals

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()


Out:

<RootLogger root (INFO)>


## 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", "y_2": "-1-2*x_1-3*x_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()


## 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:

    INFO - 09:23:17:
INFO - 09:23:17: *** Start DOE Scenario execution ***
INFO - 09:23:17: DOEScenario
INFO - 09:23:17:    Disciplines: func
INFO - 09:23:17:    MDOFormulation: DisciplinaryOpt
INFO - 09:23:17:    Algorithm: fullfact
INFO - 09:23:17: Optimization problem:
INFO - 09:23:17:    Minimize: y_1(x_1, x_2)
INFO - 09:23:17:    With respect to: x_1, x_2
INFO - 09:23:17: Full factorial design required. Number of samples along each direction for a design vector of size 2 with 9 samples: 3
INFO - 09:23:17: Final number of samples for DOE = 9 vs 9 requested
INFO - 09:23:17: DOE sampling:   0%|          | 0/9 [00:00<?, ?it]
INFO - 09:23:17: DOE sampling: 100%|██████████| 9/9 [00:00<00:00, 421.75 it/sec, obj=6]
INFO - 09:23:17: Optimization result:
INFO - 09:23:17: Objective value = 1.0
INFO - 09:23:17: The result is feasible.
INFO - 09:23:17: Status: None
INFO - 09:23:17: Optimizer message: None
INFO - 09:23:17: Number of calls to the objective function by the optimizer: 9
INFO - 09:23:17: Design Space:
INFO - 09:23:17: +------+-------------+-------+-------------+-------+
INFO - 09:23:17: | name | lower_bound | value | upper_bound | type  |
INFO - 09:23:17: +------+-------------+-------+-------------+-------+
INFO - 09:23:17: | x_1  |      0      |   0   |      1      | float |
INFO - 09:23:17: | x_2  |      0      |   0   |      1      | float |
INFO - 09:23:17: +------+-------------+-------+-------------+-------+
INFO - 09:23:17: *** DOE Scenario run terminated ***

{'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("LinearRegression", data=dataset, transformer=None)
model.learn()
print(model)


Out:

LinearRegression(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([9.]), 'y_2': array([-9.])}


## 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([[2.]]), 'x_2': array([[3.]])}, 'y_2': {'x_1': array([[-2.]]), '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. -1.]
{'y_1': [0.9999999999999987], '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)
print(model.get_coefficients())


Out:

[[ 2.  3.]
[-2. -3.]]
{'y_1': [{'x_1': [2.000000000000001], 'x_2': [3.0000000000000018]}], 'y_2': [{'x_1': [-2.000000000000001], 'x_2': [-3.0000000000000018]}]}


Total running time of the script: ( 0 minutes 0.105 seconds)

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