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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 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", "y_2": "-1-2*x_1-3*x_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})
INFO - 15:37:55:
INFO - 15:37:55: *** Start DOEScenario execution ***
INFO - 15:37:55: DOEScenario
INFO - 15:37:55: Disciplines: func
INFO - 15:37:55: MDO formulation: DisciplinaryOpt
INFO - 15:37:55: Optimization problem:
INFO - 15:37:55: minimize y_1(x_1, x_2)
INFO - 15:37:55: with respect to x_1, x_2
INFO - 15:37:55: over the design space:
INFO - 15:37:55: +------+-------------+-------+-------------+-------+
INFO - 15:37:55: | name | lower_bound | value | upper_bound | type |
INFO - 15:37:55: +------+-------------+-------+-------------+-------+
INFO - 15:37:55: | x_1 | 0 | None | 1 | float |
INFO - 15:37:55: | x_2 | 0 | None | 1 | float |
INFO - 15:37:55: +------+-------------+-------+-------------+-------+
INFO - 15:37:55: Solving optimization problem with algorithm fullfact:
INFO - 15:37:55: ... 0%| | 0/9 [00:00<?, ?it]
INFO - 15:37:55: ... 11%|█ | 1/9 [00:00<00:00, 333.12 it/sec, obj=1]
INFO - 15:37:55: ... 22%|██▏ | 2/9 [00:00<00:00, 532.51 it/sec, obj=2]
INFO - 15:37:55: ... 33%|███▎ | 3/9 [00:00<00:00, 675.01 it/sec, obj=3]
INFO - 15:37:55: ... 44%|████▍ | 4/9 [00:00<00:00, 782.96 it/sec, obj=2.5]
INFO - 15:37:55: ... 56%|█████▌ | 5/9 [00:00<00:00, 866.99 it/sec, obj=3.5]
INFO - 15:37:55: ... 67%|██████▋ | 6/9 [00:00<00:00, 934.07 it/sec, obj=4.5]
INFO - 15:37:55: ... 78%|███████▊ | 7/9 [00:00<00:00, 988.42 it/sec, obj=4]
INFO - 15:37:55: ... 89%|████████▉ | 8/9 [00:00<00:00, 1027.13 it/sec, obj=5]
INFO - 15:37:55: ... 100%|██████████| 9/9 [00:00<00:00, 1062.42 it/sec, obj=6]
INFO - 15:37:55: Optimization result:
INFO - 15:37:55: Optimizer info:
INFO - 15:37:55: Status: None
INFO - 15:37:55: Message: None
INFO - 15:37:55: Number of calls to the objective function by the optimizer: 9
INFO - 15:37:55: Solution:
INFO - 15:37:55: Objective: 1.0
INFO - 15:37:55: Design space:
INFO - 15:37:55: +------+-------------+-------+-------------+-------+
INFO - 15:37:55: | name | lower_bound | value | upper_bound | type |
INFO - 15:37:55: +------+-------------+-------+-------------+-------+
INFO - 15:37:55: | x_1 | 0 | 0 | 1 | float |
INFO - 15:37:55: | x_2 | 0 | 0 | 1 | float |
INFO - 15:37:55: +------+-------------+-------+-------------+-------+
INFO - 15:37:55: *** End DOEScenario execution (time: 0:00:00.019675) ***
{'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("LinearRegressor", data=dataset, transformer=None)
model.learn()
model
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([9.])}
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([[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.
model.intercept, model.get_intercept()
(array([1.]), {'y_1': [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.
model.coefficients, model.get_coefficients()
(array([[2., 3.]]), {'y_1': [{'x_1': [2.000000000000001], 'x_2': [3.0000000000000018]}]})
Total running time of the script: ( 0 minutes 0.058 seconds)