# Variables influence¶

In this example, we illustrate the use of the VariableInfluence plot on the Sobieski’s SSBJ problem.

from __future__ import division, unicode_literals

from matplotlib import pyplot as plt


## Import¶

The first step is to import some functions from the API and a method to get the design space.

from gemseo.api import configure_logger, create_discipline, create_scenario
from gemseo.problems.sobieski.core import SobieskiProblem

configure_logger()


Out:

<RootLogger root (INFO)>


## Description¶

The VariableInfluence post-processing performs first-order variable influence analysis.

The method computes $$\frac{d f}{d x_i} \cdot \left(x_{i_*} - x_{i_0}\right)$$, where $$x_{i_0}$$ is the initial value of the variable and $$x_{i_*}$$ is the optimal value of the variable.

## Create disciplines¶

At this point, we instantiate the disciplines of Sobieski’s SSBJ problem: Propulsion, Aerodynamics, Structure and Mission

disciplines = create_discipline(
[
"SobieskiPropulsion",
"SobieskiAerodynamics",
"SobieskiStructure",
"SobieskiMission",
]
)


## Create design space¶

We also read the design space from the SobieskiProblem.

design_space = SobieskiProblem().read_design_space()


## Create and execute scenario¶

The next step is to build an MDO scenario in order to maximize the range, encoded ‘y_4’, with respect to the design parameters, while satisfying the inequality constraints ‘g_1’, ‘g_2’ and ‘g_3’. We can use the MDF formulation, the SLSQP optimization algorithm and a maximum number of iterations equal to 100.

scenario = create_scenario(
disciplines,
formulation="MDF",
objective_name="y_4",
maximize_objective=True,
design_space=design_space,
)
scenario.set_differentiation_method("user")
for constraint in ["g_1", "g_2", "g_3"]:
scenario.execute({"algo": "SLSQP", "max_iter": 10})


Out:

    INFO - 14:42:38:
INFO - 14:42:38: *** Start MDO Scenario execution ***
INFO - 14:42:38: MDOScenario
INFO - 14:42:38:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
INFO - 14:42:38:    MDOFormulation: MDF
INFO - 14:42:38:    Algorithm: SLSQP
INFO - 14:42:38: Optimization problem:
INFO - 14:42:38:    Minimize: -y_4(x_shared, x_1, x_2, x_3)
INFO - 14:42:38:    With respect to: x_shared, x_1, x_2, x_3
INFO - 14:42:38:    Subject to constraints:
INFO - 14:42:38:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 14:42:38:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 14:42:38:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
INFO - 14:42:38: Design space:
INFO - 14:42:38: +----------+-------------+-------+-------------+-------+
INFO - 14:42:38: | name     | lower_bound | value | upper_bound | type  |
INFO - 14:42:38: +----------+-------------+-------+-------------+-------+
INFO - 14:42:38: | x_shared |     0.01    |  0.05 |     0.09    | float |
INFO - 14:42:38: | x_shared |    30000    | 45000 |    60000    | float |
INFO - 14:42:38: | x_shared |     1.4     |  1.6  |     1.8     | float |
INFO - 14:42:38: | x_shared |     2.5     |  5.5  |     8.5     | float |
INFO - 14:42:38: | x_shared |      40     |   55  |      70     | float |
INFO - 14:42:38: | x_shared |     500     |  1000 |     1500    | float |
INFO - 14:42:38: | x_1      |     0.1     |  0.25 |     0.4     | float |
INFO - 14:42:38: | x_1      |     0.75    |   1   |     1.25    | float |
INFO - 14:42:38: | x_2      |     0.75    |   1   |     1.25    | float |
INFO - 14:42:38: | x_3      |     0.1     |  0.5  |      1      | float |
INFO - 14:42:38: +----------+-------------+-------+-------------+-------+
INFO - 14:42:38: Optimization:   0%|          | 0/10 [00:00<?, ?it]
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.2.2/lib/python3.8/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:407: SparseEfficiencyWarning: splu requires CSC matrix format
warn('splu requires CSC matrix format', SparseEfficiencyWarning)
INFO - 14:42:38: Optimization:  20%|██        | 2/10 [00:00<00:00, 53.01 it/sec, obj=2.12e+3]
INFO - 14:42:38: Optimization:  40%|████      | 4/10 [00:00<00:00, 21.20 it/sec, obj=3.97e+3]
INFO - 14:42:39: Optimization:  50%|█████     | 5/10 [00:00<00:00, 16.55 it/sec, obj=3.96e+3]
INFO - 14:42:39: Optimization:  60%|██████    | 6/10 [00:00<00:00, 13.56 it/sec, obj=3.96e+3]
INFO - 14:42:39: Optimization:  70%|███████   | 7/10 [00:00<00:00, 11.50 it/sec, obj=3.96e+3]
INFO - 14:42:39: Optimization:  90%|█████████ | 9/10 [00:01<00:00,  9.80 it/sec, obj=3.96e+3]
INFO - 14:42:39: Optimization: 100%|██████████| 10/10 [00:01<00:00,  9.11 it/sec, obj=3.96e+3]
INFO - 14:42:39: Optimization result:
INFO - 14:42:39: Objective value = 3963.595455433326
INFO - 14:42:39: The result is feasible.
INFO - 14:42:39: Status: None
INFO - 14:42:39: Optimizer message: Maximum number of iterations reached. GEMSEO Stopped the driver
INFO - 14:42:39: Number of calls to the objective function by the optimizer: 12
INFO - 14:42:39: Constraints values:
INFO - 14:42:39:    g_1 = [-0.01814919 -0.03340982 -0.04429875 -0.05187486 -0.05736009 -0.13720854
INFO - 14:42:39:  -0.10279146]
INFO - 14:42:39:    g_2 = 3.236261671801799e-05
INFO - 14:42:39:    g_3 = [-7.67067574e-01 -2.32932426e-01 -9.19662628e-05 -1.83255000e-01]
INFO - 14:42:39: Design space:
INFO - 14:42:39: +----------+-------------+--------------------+-------------+-------+
INFO - 14:42:39: | name     | lower_bound |       value        | upper_bound | type  |
INFO - 14:42:39: +----------+-------------+--------------------+-------------+-------+
INFO - 14:42:39: | x_shared |     0.01    | 0.0600080906541795 |     0.09    | float |
INFO - 14:42:39: | x_shared |    30000    |       60000        |    60000    | float |
INFO - 14:42:39: | x_shared |     1.4     |        1.4         |     1.8     | float |
INFO - 14:42:39: | x_shared |     2.5     |        2.5         |     8.5     | float |
INFO - 14:42:39: | x_shared |      40     |         70         |      70     | float |
INFO - 14:42:39: | x_shared |     500     |        1500        |     1500    | float |
INFO - 14:42:39: | x_1      |     0.1     | 0.3999993439500847 |     0.4     | float |
INFO - 14:42:39: | x_1      |     0.75    |        0.75        |     1.25    | float |
INFO - 14:42:39: | x_2      |     0.75    |        0.75        |     1.25    | float |
INFO - 14:42:39: | x_3      |     0.1     | 0.156230376400943  |      1      | float |
INFO - 14:42:39: +----------+-------------+--------------------+-------------+-------+
INFO - 14:42:39: *** MDO Scenario run terminated in 0:00:01.107482 ***

{'algo': 'SLSQP', 'max_iter': 10}


## Post-process scenario¶

Lastly, we post-process the scenario by means of the BasicHistory plot.

Tip

Each post-processing method requires different inputs and offers a variety of customization options. Use the API function get_post_processing_options_schema() to print a table with the options for any post-processing algorithm. Or refer to our dedicated page: Options for Post-processing algorithms.

scenario.post_process("VariableInfluence", save=False, show=False, fig_size=(15, 12))
# Workaround for HTML rendering, instead of show=True
plt.show()


Out:

    INFO - 14:42:39: VariableInfluence for function -y_4
INFO - 14:42:39: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:39: [1 4 3 2 5 9 7 8 0]
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/3.2.2/lib/python3.8/site-packages/gemseo/post/variable_influence.py:223: UserWarning: FixedFormatter should only be used together with FixedLocator
axe.set_xticklabels(x_labels, fontsize=12, rotation=90)
INFO - 14:42:39: VariableInfluence for function g_1_0
INFO - 14:42:39: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:39: [0 7 3 5 6]
INFO - 14:42:39: VariableInfluence for function g_1_1
INFO - 14:42:39: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:39: [7 0 3 5 6]
INFO - 14:42:39: VariableInfluence for function g_1_2
INFO - 14:42:39: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:39: [7 0 3 5 6]
INFO - 14:42:39: VariableInfluence for function g_1_3
INFO - 14:42:39: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:39: [7 0 3 5 6]
INFO - 14:42:39: VariableInfluence for function g_1_4
INFO - 14:42:39: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:39: [7 0 3 5 6]
INFO - 14:42:40: VariableInfluence for function g_1_5
INFO - 14:42:40: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:40: [3 7 5 6]
INFO - 14:42:40: VariableInfluence for function g_1_6
INFO - 14:42:40: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:40: [3 7 5 6]
INFO - 14:42:40: VariableInfluence for function g_2
INFO - 14:42:40: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:40: [0]
INFO - 14:42:40: VariableInfluence for function g_3_0
INFO - 14:42:40: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:40: [1 9 5 2 4 0 8]
INFO - 14:42:40: VariableInfluence for function g_3_1
INFO - 14:42:40: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:40: [1 9 5 2 4 0 8]
INFO - 14:42:40: VariableInfluence for function g_3_2
INFO - 14:42:40: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:40: [1 9 2]
INFO - 14:42:40: VariableInfluence for function g_3_3
INFO - 14:42:40: Most influent variables indices to explain % of the function variation : 99
INFO - 14:42:40: [9 1 2]


Total running time of the script: ( 0 minutes 2.150 seconds)

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