# PCE 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_parameter_space
from gemseo.api import create_scenario
from gemseo.mlearning.api import create_regression_model
from gemseo.mlearning.api import import_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", "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()


## 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 - 10:06:51:
INFO - 10:06:51: *** Start DOEScenario execution ***
INFO - 10:06:51: DOEScenario
INFO - 10:06:51:    Disciplines: func
INFO - 10:06:51:    MDO formulation: DisciplinaryOpt
INFO - 10:06:51: Optimization problem:
INFO - 10:06:51:    minimize y_1(x_1, x_2)
INFO - 10:06:51:    with respect to x_1, x_2
INFO - 10:06:51:    over the design space:
INFO - 10:06:51:    +------+-------------+-------+-------------+-------+
INFO - 10:06:51:    | name | lower_bound | value | upper_bound | type  |
INFO - 10:06:51:    +------+-------------+-------+-------------+-------+
INFO - 10:06:51:    | x_1  |      0      |  None |      1      | float |
INFO - 10:06:51:    | x_2  |      0      |  None |      1      | float |
INFO - 10:06:51:    +------+-------------+-------+-------------+-------+
INFO - 10:06:51: Solving optimization problem with algorithm fullfact:
INFO - 10:06:51: Full factorial design required. Number of samples along each direction for a design vector of size 2 with 9 samples: 3
INFO - 10:06:51: Final number of samples for DOE = 9 vs 9 requested
INFO - 10:06:51: ...   0%|          | 0/9 [00:00<?, ?it]
INFO - 10:06:51: ... 100%|██████████| 9/9 [00:00<00:00, 1439.09 it/sec, obj=6]
INFO - 10:06:51: Optimization result:
INFO - 10:06:51:    Optimizer info:
INFO - 10:06:51:       Status: None
INFO - 10:06:51:       Message: None
INFO - 10:06:51:       Number of calls to the objective function by the optimizer: 9
INFO - 10:06:51:    Solution:
INFO - 10:06:51:       Objective: 1.0
INFO - 10:06:51:       Design space:
INFO - 10:06:51:       +------+-------------+-------+-------------+-------+
INFO - 10:06:51:       | name | lower_bound | value | upper_bound | type  |
INFO - 10:06:51:       +------+-------------+-------+-------------+-------+
INFO - 10:06:51:       | x_1  |      0      |   0   |      1      | float |
INFO - 10:06:51:       | x_2  |      0      |   0   |      1      | float |
INFO - 10:06:51:       +------+-------------+-------+-------------+-------+
INFO - 10:06:51: *** End DOEScenario execution (time: 0:00:00.014625) ***

{'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.

prob_space = create_parameter_space()
dataset = scenario.export_to_dataset(opt_naming=False)
model = create_regression_model(
"PCERegressor", data=dataset, probability_space=prob_space, transformer=None
)
model.learn()
print(model)


Out:

PCERegressor(degree=2, n_quad=None, probability_space=<gemseo.algos.parameter_space.ParameterSpace object at 0x7fdbf80f37c0>, sparse_param=None, stieltjes=True, strategy='LS')
based on the OpenTURNS 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.])}


## Save the regression model¶

Lastly, we save the model.

directory = model.save()


In an other study, we could load this model.

loaded_model = import_regression_model(directory)


Out:

PCERegressor(degree=2, n_quad=None, probability_space=<gemseo.algos.parameter_space.ParameterSpace object at 0x7fdbe18cd3a0>, sparse_param=None, stieltjes=True, strategy='LS')
based on the OpenTURNS library
built from 0 learning samples


## Use the loaded regression model¶

And use it!

print(loaded_model.predict(input_value))


Out:

{'y_1': array([9.])}


Total running time of the script: ( 0 minutes 0.154 seconds)

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