Note
Click here to download the full example code
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()
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:
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)