# 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 __future__ import annotations

from gemseo import configure_logger
from gemseo import create_design_space
from gemseo import create_discipline
from gemseo import create_parameter_space
from gemseo import create_scenario
from gemseo.mlearning import create_regression_model
from gemseo.mlearning import import_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()


## 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 - 16:28:54:
INFO - 16:28:54: *** Start DOEScenario execution ***
INFO - 16:28:54: DOEScenario
INFO - 16:28:54:    Disciplines: func
INFO - 16:28:54:    MDO formulation: DisciplinaryOpt
INFO - 16:28:54: Optimization problem:
INFO - 16:28:54:    minimize y_1(x_1, x_2)
INFO - 16:28:54:    with respect to x_1, x_2
INFO - 16:28:54:    over the design space:
INFO - 16:28:54:    +------+-------------+-------+-------------+-------+
INFO - 16:28:54:    | name | lower_bound | value | upper_bound | type  |
INFO - 16:28:54:    +------+-------------+-------+-------------+-------+
INFO - 16:28:54:    | x_1  |      0      |  None |      1      | float |
INFO - 16:28:54:    | x_2  |      0      |  None |      1      | float |
INFO - 16:28:54:    +------+-------------+-------+-------------+-------+
INFO - 16:28:54: Solving optimization problem with algorithm fullfact:
INFO - 16:28:54: ...   0%|          | 0/9 [00:00<?, ?it]
INFO - 16:28:54: ...  11%|█         | 1/9 [00:00<00:00, 301.84 it/sec, obj=1]
INFO - 16:28:54: ...  22%|██▏       | 2/9 [00:00<00:00, 495.90 it/sec, obj=2]
INFO - 16:28:54: ...  33%|███▎      | 3/9 [00:00<00:00, 640.22 it/sec, obj=3]
INFO - 16:28:54: ...  44%|████▍     | 4/9 [00:00<00:00, 751.84 it/sec, obj=2.5]
INFO - 16:28:54: ...  56%|█████▌    | 5/9 [00:00<00:00, 833.20 it/sec, obj=3.5]
INFO - 16:28:54: ...  67%|██████▋   | 6/9 [00:00<00:00, 904.07 it/sec, obj=4.5]
INFO - 16:28:54: ...  78%|███████▊  | 7/9 [00:00<00:00, 960.33 it/sec, obj=4]
INFO - 16:28:54: ...  89%|████████▉ | 8/9 [00:00<00:00, 1009.73 it/sec, obj=5]
INFO - 16:28:54: ... 100%|██████████| 9/9 [00:00<00:00, 1052.02 it/sec, obj=6]
INFO - 16:28:54: Optimization result:
INFO - 16:28:54:    Optimizer info:
INFO - 16:28:54:       Status: None
INFO - 16:28:54:       Message: None
INFO - 16:28:54:       Number of calls to the objective function by the optimizer: 9
INFO - 16:28:54:    Solution:
INFO - 16:28:54:       Objective: 1.0
INFO - 16:28:54:       Design space:
INFO - 16:28:54:       +------+-------------+-------+-------------+-------+
INFO - 16:28:54:       | name | lower_bound | value | upper_bound | type  |
INFO - 16:28:54:       +------+-------------+-------+-------------+-------+
INFO - 16:28:54:       | x_1  |      0      |   0   |      1      | float |
INFO - 16:28:54:       | x_2  |      0      |   0   |      1      | float |
INFO - 16:28:54:       +------+-------------+-------+-------------+-------+
INFO - 16:28:54: *** End DOEScenario execution (time: 0:00:00.020524) ***

{'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.

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

PCERegressor(cleaning_options=CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001), degree=2, hyperbolic_parameter=1.0, n_quadrature_points=0, probability_space=+----------------------------------------------------------------------------------+
|                                 Parameter space                                  |
+------+-------------+-------+-------------+-------+-------------------------------+
| name | lower_bound | value | upper_bound | type  |      Initial distribution     |
+------+-------------+-------+-------------+-------+-------------------------------+
| x_1  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |
| x_2  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |
+------+-------------+-------+-------------+-------+-------------------------------+, use_cleaning=False, use_lars=False)
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)

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


## Save the regression model¶

Lastly, we save the model.

directory = model.to_pickle()

/home/docs/checkouts/readthedocs.org/user_builds/gemseo/envs/5.0.0/lib/python3.9/site-packages/gemseo/mlearning/core/ml_algo.py:327: UserWarning: Pandas doesn't allow columns to be created via a new attribute name - see https://pandas.pydata.org/pandas-docs/stable/indexing.html#attribute-access
self.learning_set.data = {}


In an other study, we could load this model.

loaded_model = import_regression_model(directory)

PCERegressor(cleaning_options=CleaningOptions(max_considered_terms=100, most_significant=20, significance_factor=0.0001), degree=2, hyperbolic_parameter=1.0, n_quadrature_points=0, probability_space=+----------------------------------------------------------------------------------+
|                                 Parameter space                                  |
+------+-------------+-------+-------------+-------+-------------------------------+
| name | lower_bound | value | upper_bound | type  |      Initial distribution     |
+------+-------------+-------+-------------+-------+-------------------------------+
| x_1  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |
| x_2  |      0      |  0.5  |      1      | float | Uniform(lower=0.0, upper=1.0) |
+------+-------------+-------+-------------+-------+-------------------------------+, use_cleaning=False, use_lars=False)
based on the OpenTURNS library
built from 9 learning samples


## Use the loaded regression model¶

And use it!

print(loaded_model.predict(input_value))

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


Total running time of the script: ( 0 minutes 0.202 seconds)

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