Correlations

In this example, we illustrate the use of the Correlations 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

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

Out:

    INFO - 14:41:31:
    INFO - 14:41:31: *** Start MDO Scenario execution ***
    INFO - 14:41:31: MDOScenario
    INFO - 14:41:31:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
    INFO - 14:41:31:    MDOFormulation: MDF
    INFO - 14:41:31:    Algorithm: SLSQP
    INFO - 14:41:31: Optimization problem:
    INFO - 14:41:31:    Minimize: -y_4(x_shared, x_1, x_2, x_3)
    INFO - 14:41:31:    With respect to: x_shared, x_1, x_2, x_3
    INFO - 14:41:31:    Subject to constraints:
    INFO - 14:41:31:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 14:41:31:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 14:41:31:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
    INFO - 14:41:31: Design space:
    INFO - 14:41:31: +----------+-------------+-------+-------------+-------+
    INFO - 14:41:31: | name     | lower_bound | value | upper_bound | type  |
    INFO - 14:41:31: +----------+-------------+-------+-------------+-------+
    INFO - 14:41:31: | x_shared |     0.01    |  0.05 |     0.09    | float |
    INFO - 14:41:31: | x_shared |    30000    | 45000 |    60000    | float |
    INFO - 14:41:31: | x_shared |     1.4     |  1.6  |     1.8     | float |
    INFO - 14:41:31: | x_shared |     2.5     |  5.5  |     8.5     | float |
    INFO - 14:41:31: | x_shared |      40     |   55  |      70     | float |
    INFO - 14:41:31: | x_shared |     500     |  1000 |     1500    | float |
    INFO - 14:41:31: | x_1      |     0.1     |  0.25 |     0.4     | float |
    INFO - 14:41:31: | x_1      |     0.75    |   1   |     1.25    | float |
    INFO - 14:41:31: | x_2      |     0.75    |   1   |     1.25    | float |
    INFO - 14:41:31: | x_3      |     0.1     |  0.5  |      1      | float |
    INFO - 14:41:31: +----------+-------------+-------+-------------+-------+
    INFO - 14:41:31: Optimization:   0%|          | 0/10 [00:00<?, ?it]
/home/docs/checkouts/readthedocs.org/user_builds/gemseo/conda/stable/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 - 14:41:31: Optimization:  20%|██        | 2/10 [00:00<00:00, 52.97 it/sec, obj=2.12e+3]
    INFO - 14:41:32: Optimization:  40%|████      | 4/10 [00:00<00:00, 21.25 it/sec, obj=3.97e+3]
    INFO - 14:41:32: Optimization:  50%|█████     | 5/10 [00:00<00:00, 16.58 it/sec, obj=3.96e+3]
    INFO - 14:41:32: Optimization:  60%|██████    | 6/10 [00:00<00:00, 13.61 it/sec, obj=3.96e+3]
    INFO - 14:41:32: Optimization:  70%|███████   | 7/10 [00:00<00:00, 11.51 it/sec, obj=3.96e+3]
    INFO - 14:41:32: Optimization:  90%|█████████ | 9/10 [00:01<00:00,  9.82 it/sec, obj=3.96e+3]
    INFO - 14:41:32: Optimization: 100%|██████████| 10/10 [00:01<00:00,  9.13 it/sec, obj=3.96e+3]
    INFO - 14:41:32: Optimization result:
    INFO - 14:41:32: Objective value = 3963.595455433326
    INFO - 14:41:32: The result is feasible.
    INFO - 14:41:32: Status: None
    INFO - 14:41:32: Optimizer message: Maximum number of iterations reached. GEMSEO Stopped the driver
    INFO - 14:41:32: Number of calls to the objective function by the optimizer: 12
    INFO - 14:41:32: Constraints values:
    INFO - 14:41:32:    g_1 = [-0.01814919 -0.03340982 -0.04429875 -0.05187486 -0.05736009 -0.13720854
    INFO - 14:41:32:  -0.10279146]
    INFO - 14:41:32:    g_2 = 3.236261671801799e-05
    INFO - 14:41:32:    g_3 = [-7.67067574e-01 -2.32932426e-01 -9.19662628e-05 -1.83255000e-01]
    INFO - 14:41:32: Design space:
    INFO - 14:41:32: +----------+-------------+--------------------+-------------+-------+
    INFO - 14:41:32: | name     | lower_bound |       value        | upper_bound | type  |
    INFO - 14:41:32: +----------+-------------+--------------------+-------------+-------+
    INFO - 14:41:32: | x_shared |     0.01    | 0.0600080906541795 |     0.09    | float |
    INFO - 14:41:32: | x_shared |    30000    |       60000        |    60000    | float |
    INFO - 14:41:32: | x_shared |     1.4     |        1.4         |     1.8     | float |
    INFO - 14:41:32: | x_shared |     2.5     |        2.5         |     8.5     | float |
    INFO - 14:41:32: | x_shared |      40     |         70         |      70     | float |
    INFO - 14:41:32: | x_shared |     500     |        1500        |     1500    | float |
    INFO - 14:41:32: | x_1      |     0.1     | 0.3999993439500847 |     0.4     | float |
    INFO - 14:41:32: | x_1      |     0.75    |        0.75        |     1.25    | float |
    INFO - 14:41:32: | x_2      |     0.75    |        0.75        |     1.25    | float |
    INFO - 14:41:32: | x_3      |     0.1     | 0.156230376400943  |      1      | float |
    INFO - 14:41:32: +----------+-------------+--------------------+-------------+-------+
    INFO - 14:41:32: *** MDO Scenario run terminated in 0:00:01.104073 ***

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

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: Options for Post-processing algorithms.

scenario.post_process("Correlations", save=False, show=False)
# Workaround for HTML rendering, instead of ``show=True``
plt.show()
  • R=0.99049, R=0.97686, R=0.99700, R=0.96518, R=0.99200, R=0.99880, R=0.95582, R=0.98718, R=0.99657, R=0.99943, R=-0.96500, R=-0.97578, R=-0.98183, R=0.96500, R=0.97578, R=0.98183, R=0.95083, R=0.96497, R=0.97093, R=0.97367, R=-0.98786, R=0.98786, R=0.97242, R=-0.96043, R=-0.96043
  • R=0.99996, R=-0.95998, R=0.99996, R=0.98231, R=-0.98231, R=-0.99996, R=0.95998, R=-0.99996, R=0.99996, R=-0.95998, R=0.99996, R=0.99996, R=-0.95998, R=0.99996, R=-0.96868, R=0.96868, R=-0.99804, R=-0.99996, R=0.95998, R=-0.99996, R=-0.99996, R=0.95998, R=-0.99996, R=-0.98719, R=-0.98719
  • R=-0.98694, R=0.98694, R=-0.98694, R=-0.98694, R=0.98694, R=0.98694

Out:

INFO - 14:41:32: Detected 56 correlations > 0.95

Total running time of the script: ( 0 minutes 3.978 seconds)

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