Correlations

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

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

configure_logger()

Out:

<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 read the design space from the SobieskiProblem.

design_space = SobieskiProblem().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.add_constraint(constraint, "ineq")
scenario.execute({"algo": "SLSQP", "max_iter": 10})

Out:

    INFO - 15:00:38:
    INFO - 15:00:38: *** Start MDOScenario execution ***
    INFO - 15:00:38: MDOScenario
    INFO - 15:00:38:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
    INFO - 15:00:38:    MDO formulation: MDF
    INFO - 15:00:38: Optimization problem:
    INFO - 15:00:38:    minimize -y_4(x_shared, x_1, x_2, x_3)
    INFO - 15:00:38:    with respect to x_1, x_2, x_3, x_shared
    INFO - 15:00:38:    subject to constraints:
    INFO - 15:00:38:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 15:00:38:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 15:00:38:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 15:00:38:    over the design space:
    INFO - 15:00:38:    +----------+-------------+-------+-------------+-------+
    INFO - 15:00:38:    | name     | lower_bound | value | upper_bound | type  |
    INFO - 15:00:38:    +----------+-------------+-------+-------------+-------+
    INFO - 15:00:38:    | x_shared |     0.01    |  0.05 |     0.09    | float |
    INFO - 15:00:38:    | x_shared |    30000    | 45000 |    60000    | float |
    INFO - 15:00:38:    | x_shared |     1.4     |  1.6  |     1.8     | float |
    INFO - 15:00:38:    | x_shared |     2.5     |  5.5  |     8.5     | float |
    INFO - 15:00:38:    | x_shared |      40     |   55  |      70     | float |
    INFO - 15:00:38:    | x_shared |     500     |  1000 |     1500    | float |
    INFO - 15:00:38:    | x_1      |     0.1     |  0.25 |     0.4     | float |
    INFO - 15:00:38:    | x_1      |     0.75    |   1   |     1.25    | float |
    INFO - 15:00:38:    | x_2      |     0.75    |   1   |     1.25    | float |
    INFO - 15:00:38:    | x_3      |     0.1     |  0.5  |      1      | float |
    INFO - 15:00:38:    +----------+-------------+-------+-------------+-------+
    INFO - 15:00:38: Solving optimization problem with algorithm SLSQP:
    INFO - 15:00:38: ...   0%|          | 0/10 [00:00<?, ?it]
    INFO - 15:00:38: ...  20%|██        | 2/10 [00:00<00:00, 41.41 it/sec, obj=-2.12e+3]
 WARNING - 15:00:39: MDAJacobi has reached its maximum number of iterations but the normed residual 9.482098563532576e-06 is still above the tolerance 1e-06.
    INFO - 15:00:39: ...  30%|███       | 3/10 [00:00<00:00, 23.65 it/sec, obj=-3.76e+3]
    INFO - 15:00:39: ...  40%|████      | 4/10 [00:00<00:00, 16.99 it/sec, obj=-3.96e+3]
    INFO - 15:00:39: ...  50%|█████     | 5/10 [00:00<00:00, 13.34 it/sec, obj=-3.96e+3]
    INFO - 15:00:39: ...  70%|███████   | 7/10 [00:00<00:00, 10.89 it/sec, obj=-3.96e+3]
    INFO - 15:00:39: ...  90%|█████████ | 9/10 [00:01<00:00,  9.19 it/sec, obj=-3.96e+3]
 WARNING - 15:00:39: Optimization found no feasible point !  The least infeasible point is selected.
    INFO - 15:00:39: ... 100%|██████████| 10/10 [00:01<00:00,  8.50 it/sec, obj=-3.96e+3]
    INFO - 15:00:39: Optimization result:
    INFO - 15:00:39:    Optimizer info:
    INFO - 15:00:39:       Status: None
    INFO - 15:00:39:       Message: Maximum number of iterations reached. GEMSEO Stopped the driver
    INFO - 15:00:39:       Number of calls to the objective function by the optimizer: 12
    INFO - 15:00:39:    Solution:
 WARNING - 15:00:39:       The solution is not feasible.
    INFO - 15:00:39:       Objective: -3963.793570013662
    INFO - 15:00:39:       Standardized constraints:
    INFO - 15:00:39:          g_1 = [-0.01810854 -0.03338059 -0.04427602 -0.0518563  -0.05734441 -0.13720865
    INFO - 15:00:39:  -0.10279135]
    INFO - 15:00:39:          g_2 = 1.9000415800052295e-05
    INFO - 15:00:39:          g_3 = [-7.67474555e-01 -2.32525445e-01  4.34281075e-04 -1.83255000e-01]
    INFO - 15:00:39:       Design space:
    INFO - 15:00:39:       +----------+-------------+---------------------+-------------+-------+
    INFO - 15:00:39:       | name     | lower_bound |        value        | upper_bound | type  |
    INFO - 15:00:39:       +----------+-------------+---------------------+-------------+-------+
    INFO - 15:00:39:       | x_shared |     0.01    | 0.06000475010395003 |     0.09    | float |
    INFO - 15:00:39:       | x_shared |    30000    |        60000        |    60000    | float |
    INFO - 15:00:39:       | x_shared |     1.4     |         1.4         |     1.8     | float |
    INFO - 15:00:39:       | x_shared |     2.5     |         2.5         |     8.5     | float |
    INFO - 15:00:39:       | x_shared |      40     |          70         |      70     | float |
    INFO - 15:00:39:       | x_shared |     500     |         1500        |     1500    | float |
    INFO - 15:00:39:       | x_1      |     0.1     |         0.4         |     0.4     | float |
    INFO - 15:00:39:       | x_1      |     0.75    |         0.75        |     1.25    | float |
    INFO - 15:00:39:       | x_2      |     0.75    |         0.75        |     1.25    | float |
    INFO - 15:00:39:       | x_3      |     0.1     |  0.1563125997824079 |      1      | float |
    INFO - 15:00:39:       +----------+-------------+---------------------+-------------+-------+
    INFO - 15:00:39: *** End MDOScenario execution (time: 0:00:01.190024) ***

{'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=False)
# Workaround for HTML rendering, instead of ``show=True``
plt.show()
  • R=0.99054, R=0.97697, R=0.99701, R=0.96535, R=0.99203, R=0.99880, R=0.95602, R=0.98723, R=0.99659, R=0.99943, R=-0.96535, R=-0.97615, R=-0.98221, R=0.96535, R=0.97615, R=0.98221, R=0.96295, R=0.96894, R=0.97171, R=-0.98611, R=0.98611, R=0.97210, R=-0.96057, R=-0.96057, R=0.99975
  • R=-0.96000, R=0.99975, R=0.98230, R=-0.98230, R=-0.99975, R=0.96000, R=-0.99975, R=0.99975, R=-0.96000, R=0.99975, R=0.99975, R=-0.96000, R=0.99975, R=-0.96864, R=0.96864, R=-0.99804, R=-0.99975, R=0.96000, R=-0.99975, R=-0.99975, R=0.96000, R=-0.99975, R=-0.98740, R=-0.98740, R=-0.98653
  • R=0.98653, R=-0.98653, R=-0.98653, R=0.98653, R=0.98653

Out:

INFO - 15:00:39: Detected 55 correlations > 0.95

Total running time of the script: ( 0 minutes 4.002 seconds)

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