Correlations

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

from __future__ import annotations

from gemseo import configure_logger
from gemseo import create_discipline
from gemseo import create_scenario
from gemseo.problems.sobieski.core.design_space import SobieskiDesignSpace

Import

The first step is to import some high-level functions and a method to get the design space.

configure_logger()

<RootLogger root (INFO)>

Description

A correlation coefficient indicates whether there is a linear relationship between 2 quantities \(x\) and \(y\), in which case it equals 1 or -1. It is the normalized covariance between the two quantities:

\[R_{xy}=\frac {\sum \limits _{i=1}^n(x_i-{\bar{x}})(y_i-{\bar{y}})}{ns_{x}s_{y}} =\frac {\sum \limits _{i=1}^n(x_i-{\bar{x}})(y_i-{\bar{y}})}{\sqrt {\sum \limits _{i=1}^n(x_i-{\bar{x}})^{2}\sum \limits _{i=1}^n(y_i-{\bar{y}})^{2}}}\]

The Correlations post-processing builds scatter plots of correlated variables among design variables, output functions, and constraints.

The plot method considers all variable correlations greater than 95%. A different threshold value and/or a sublist of variable names can be passed as options.

Create disciplines

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

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

Create design space

We also create the SobieskiDesignSpace.

design_space = SobieskiDesignSpace()

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,
    "MDF",
    "y_4",
    design_space,
    maximize_objective=True,
)
scenario.set_differentiation_method()
for constraint in ["g_1", "g_2", "g_3"]:
    scenario.add_constraint(constraint, constraint_type="ineq")
scenario.execute({"algo": "SLSQP", "max_iter": 10})

    INFO - 13:12:37:
    INFO - 13:12:37: *** Start MDOScenario execution ***
    INFO - 13:12:37: MDOScenario
    INFO - 13:12:37:    Disciplines: SobieskiAerodynamics SobieskiMission SobieskiPropulsion SobieskiStructure
    INFO - 13:12:37:    MDO formulation: MDF
    INFO - 13:12:37: Optimization problem:
    INFO - 13:12:37:    minimize -y_4(x_shared, x_1, x_2, x_3)
    INFO - 13:12:37:    with respect to x_1, x_2, x_3, x_shared
    INFO - 13:12:37:    subject to constraints:
    INFO - 13:12:37:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 13:12:37:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 13:12:37:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 13:12:37:    over the design space:
    INFO - 13:12:37:       +-------------+-------------+-------+-------------+-------+
    INFO - 13:12:37:       | Name        | Lower bound | Value | Upper bound | Type  |
    INFO - 13:12:37:       +-------------+-------------+-------+-------------+-------+
    INFO - 13:12:37:       | x_shared[0] |     0.01    |  0.05 |     0.09    | float |
    INFO - 13:12:37:       | x_shared[1] |    30000    | 45000 |    60000    | float |
    INFO - 13:12:37:       | x_shared[2] |     1.4     |  1.6  |     1.8     | float |
    INFO - 13:12:37:       | x_shared[3] |     2.5     |  5.5  |     8.5     | float |
    INFO - 13:12:37:       | x_shared[4] |      40     |   55  |      70     | float |
    INFO - 13:12:37:       | x_shared[5] |     500     |  1000 |     1500    | float |
    INFO - 13:12:37:       | x_1[0]      |     0.1     |  0.25 |     0.4     | float |
    INFO - 13:12:37:       | x_1[1]      |     0.75    |   1   |     1.25    | float |
    INFO - 13:12:37:       | x_2         |     0.75    |   1   |     1.25    | float |
    INFO - 13:12:37:       | x_3         |     0.1     |  0.5  |      1      | float |
    INFO - 13:12:37:       +-------------+-------------+-------+-------------+-------+
    INFO - 13:12:37: Solving optimization problem with algorithm SLSQP:
    INFO - 13:12:37:     10%|█         | 1/10 [00:00<00:01,  8.80 it/sec, obj=-536]
    INFO - 13:12:37:     20%|██        | 2/10 [00:00<00:01,  6.24 it/sec, obj=-2.12e+3]
 WARNING - 13:12:38: MDAJacobi has reached its maximum number of iterations but the normed residual 2.338273970736908e-06 is still above the tolerance 1e-06.
    INFO - 13:12:38:     30%|███       | 3/10 [00:00<00:01,  5.22 it/sec, obj=-3.56e+3]
    INFO - 13:12:38:     40%|████      | 4/10 [00:00<00:01,  4.97 it/sec, obj=-3.96e+3]
    INFO - 13:12:38:     50%|█████     | 5/10 [00:01<00:01,  4.83 it/sec, obj=-3.96e+3]
    INFO - 13:12:38: Optimization result:
    INFO - 13:12:38:    Optimizer info:
    INFO - 13:12:38:       Status: 8
    INFO - 13:12:38:       Message: Positive directional derivative for linesearch
    INFO - 13:12:38:       Number of calls to the objective function by the optimizer: 6
    INFO - 13:12:38:    Solution:
    INFO - 13:12:38:       The solution is feasible.
    INFO - 13:12:38:       Objective: -3963.403105287515
    INFO - 13:12:38:       Standardized constraints:
    INFO - 13:12:38:          g_1 = [-0.01806054 -0.03334606 -0.04424918 -0.05183437 -0.05732588 -0.13720864
    INFO - 13:12:38:  -0.10279136]
    INFO - 13:12:38:          g_2 = 3.1658077606078194e-06
    INFO - 13:12:38:          g_3 = [-7.67177346e-01 -2.32822654e-01 -5.57051011e-06 -1.83255000e-01]
    INFO - 13:12:38:       Design space:
    INFO - 13:12:38:          +-------------+-------------+---------------------+-------------+-------+
    INFO - 13:12:38:          | Name        | Lower bound |        Value        | Upper bound | Type  |
    INFO - 13:12:38:          +-------------+-------------+---------------------+-------------+-------+
    INFO - 13:12:38:          | x_shared[0] |     0.01    | 0.06000079145194018 |     0.09    | float |
    INFO - 13:12:38:          | x_shared[1] |    30000    |        60000        |    60000    | float |
    INFO - 13:12:38:          | x_shared[2] |     1.4     |         1.4         |     1.8     | float |
    INFO - 13:12:38:          | x_shared[3] |     2.5     |         2.5         |     8.5     | float |
    INFO - 13:12:38:          | x_shared[4] |      40     |          70         |      70     | float |
    INFO - 13:12:38:          | x_shared[5] |     500     |         1500        |     1500    | float |
    INFO - 13:12:38:          | x_1[0]      |     0.1     |  0.3999999322608766 |     0.4     | float |
    INFO - 13:12:38:          | x_1[1]      |     0.75    |         0.75        |     1.25    | float |
    INFO - 13:12:38:          | x_2         |     0.75    |         0.75        |     1.25    | float |
    INFO - 13:12:38:          | x_3         |     0.1     |  0.1562438752833519 |      1      | float |
    INFO - 13:12:38:          +-------------+-------------+---------------------+-------------+-------+
    INFO - 13:12:38: *** End MDOScenario execution (time: 0:00:01.181721) ***

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

Post-process scenario

Lastly, we post-process the scenario by means of the Correlations plot which provides scatter plots of correlated variables among design variables, outputs functions and constraints any of the constraint or objective functions w.r.t. optimization iterations or sampling snapshots. This method requires the list of functions names to 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: Post-processing algorithms.

scenario.post_process("Correlations", save=False, show=True)

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    INFO - 13:12:38: Detected 40 correlations > 0.95

<gemseo.post.correlations.Correlations object at 0x7f8b96f4cc10>