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.. "examples/post_process/plot_correlations.py"
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.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_examples_post_process_plot_correlations.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_post_process_plot_correlations.py:


Correlations
============

In this example, we illustrate the use of the :class:`.Correlations` plot
on the Sobieski's SSBJ problem.

.. GENERATED FROM PYTHON SOURCE LINES 28-36

.. code-block:: default

    from __future__ import annotations

    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








.. GENERATED FROM PYTHON SOURCE LINES 37-41

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

.. GENERATED FROM PYTHON SOURCE LINES 41-44

.. code-block:: default


    configure_logger()





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <RootLogger root (INFO)>



.. GENERATED FROM PYTHON SOURCE LINES 45-64

Description
-----------

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

.. math::

   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.

.. GENERATED FROM PYTHON SOURCE LINES 66-70

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

.. GENERATED FROM PYTHON SOURCE LINES 70-79

.. code-block:: default

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








.. GENERATED FROM PYTHON SOURCE LINES 80-83

Create design space
-------------------
We also read the design space from the :class:`.SobieskiProblem`.

.. GENERATED FROM PYTHON SOURCE LINES 83-85

.. code-block:: default

    design_space = SobieskiProblem().design_space








.. GENERATED FROM PYTHON SOURCE LINES 86-93

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.

.. GENERATED FROM PYTHON SOURCE LINES 93-105

.. code-block:: default

    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})





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

        INFO - 14:43:22:  
        INFO - 14:43:22: *** Start MDOScenario execution ***
        INFO - 14:43:22: MDOScenario
        INFO - 14:43:22:    Disciplines: SobieskiAerodynamics SobieskiMission SobieskiPropulsion SobieskiStructure
        INFO - 14:43:22:    MDO formulation: MDF
        INFO - 14:43:22: Optimization problem:
        INFO - 14:43:22:    minimize -y_4(x_shared, x_1, x_2, x_3)
        INFO - 14:43:22:    with respect to x_1, x_2, x_3, x_shared
        INFO - 14:43:22:    subject to constraints:
        INFO - 14:43:22:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 14:43:22:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 14:43:22:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 14:43:22:    over the design space:
        INFO - 14:43:22:    +-------------+-------------+-------+-------------+-------+
        INFO - 14:43:22:    | name        | lower_bound | value | upper_bound | type  |
        INFO - 14:43:22:    +-------------+-------------+-------+-------------+-------+
        INFO - 14:43:22:    | x_shared[0] |     0.01    |  0.05 |     0.09    | float |
        INFO - 14:43:22:    | x_shared[1] |    30000    | 45000 |    60000    | float |
        INFO - 14:43:22:    | x_shared[2] |     1.4     |  1.6  |     1.8     | float |
        INFO - 14:43:22:    | x_shared[3] |     2.5     |  5.5  |     8.5     | float |
        INFO - 14:43:22:    | x_shared[4] |      40     |   55  |      70     | float |
        INFO - 14:43:22:    | x_shared[5] |     500     |  1000 |     1500    | float |
        INFO - 14:43:22:    | x_1[0]      |     0.1     |  0.25 |     0.4     | float |
        INFO - 14:43:22:    | x_1[1]      |     0.75    |   1   |     1.25    | float |
        INFO - 14:43:22:    | x_2         |     0.75    |   1   |     1.25    | float |
        INFO - 14:43:22:    | x_3         |     0.1     |  0.5  |      1      | float |
        INFO - 14:43:22:    +-------------+-------------+-------+-------------+-------+
        INFO - 14:43:22: Solving optimization problem with algorithm SLSQP:
        INFO - 14:43:22: ...   0%|          | 0/10 [00:00<?, ?it]
        INFO - 14:43:22: ...  20%|██        | 2/10 [00:00<00:00, 40.97 it/sec, obj=-2.12e+3]
     WARNING - 14:43:22: MDAJacobi has reached its maximum number of iterations but the normed residual 1.4486313079508508e-06 is still above the tolerance 1e-06.
        INFO - 14:43:22: ...  30%|███       | 3/10 [00:00<00:00, 23.60 it/sec, obj=-3.75e+3]
        INFO - 14:43:23: ...  40%|████      | 4/10 [00:00<00:00, 17.26 it/sec, obj=-4.01e+3]
     WARNING - 14:43:23: MDAJacobi has reached its maximum number of iterations but the normed residual 2.928004141058104e-06 is still above the tolerance 1e-06.
        INFO - 14:43:23: ...  50%|█████     | 5/10 [00:00<00:00, 13.11 it/sec, obj=-4.49e+3]
        INFO - 14:43:23: ...  60%|██████    | 6/10 [00:00<00:00, 11.09 it/sec, obj=-3.4e+3]
        INFO - 14:43:23: ...  80%|████████  | 8/10 [00:01<00:00,  9.45 it/sec, obj=-4.76e+3]
        INFO - 14:43:23: ... 100%|██████████| 10/10 [00:01<00:00,  8.23 it/sec, obj=-4.56e+3]
        INFO - 14:43:23: ... 100%|██████████| 10/10 [00:01<00:00,  8.21 it/sec, obj=-4.56e+3]
        INFO - 14:43:23: Optimization result:
        INFO - 14:43:23:    Optimizer info:
        INFO - 14:43:23:       Status: None
        INFO - 14:43:23:       Message: Maximum number of iterations reached. GEMSEO Stopped the driver
        INFO - 14:43:23:       Number of calls to the objective function by the optimizer: 12
        INFO - 14:43:23:    Solution:
        INFO - 14:43:23:       The solution is feasible.
        INFO - 14:43:23:       Objective: -3749.8868975554387
        INFO - 14:43:23:       Standardized constraints:
        INFO - 14:43:23:          g_1 = [-0.01671296 -0.03238836 -0.04350867 -0.05123129 -0.05681738 -0.13780658
        INFO - 14:43:23:  -0.10219342]
        INFO - 14:43:23:          g_2 = -0.0004062839430756249
        INFO - 14:43:23:          g_3 = [-0.66482546 -0.33517454 -0.11023156 -0.183255  ]
        INFO - 14:43:23:       Design space: 
        INFO - 14:43:23:       +-------------+-------------+---------------------+-------------+-------+
        INFO - 14:43:23:       | name        | lower_bound |        value        | upper_bound | type  |
        INFO - 14:43:23:       +-------------+-------------+---------------------+-------------+-------+
        INFO - 14:43:23:       | x_shared[0] |     0.01    | 0.05989842901423112 |     0.09    | float |
        INFO - 14:43:23:       | x_shared[1] |    30000    |  59853.73840058666  |    60000    | float |
        INFO - 14:43:23:       | x_shared[2] |     1.4     |         1.4         |     1.8     | float |
        INFO - 14:43:23:       | x_shared[3] |     2.5     |  2.527371250092273  |     8.5     | float |
        INFO - 14:43:23:       | x_shared[4] |      40     |  69.86825198198687  |      70     | float |
        INFO - 14:43:23:       | x_shared[5] |     500     |  1495.734648986894  |     1500    | float |
        INFO - 14:43:23:       | x_1[0]      |     0.1     |         0.4         |     0.4     | float |
        INFO - 14:43:23:       | x_1[1]      |     0.75    |  0.7521124139939552 |     1.25    | float |
        INFO - 14:43:23:       | x_2         |     0.75    |  0.7520888531444992 |     1.25    | float |
        INFO - 14:43:23:       | x_3         |     0.1     |  0.1398000762238233 |      1      | float |
        INFO - 14:43:23:       +-------------+-------------+---------------------+-------------+-------+
        INFO - 14:43:23: *** End MDOScenario execution (time: 0:00:01.234222) ***

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



.. GENERATED FROM PYTHON SOURCE LINES 106-113

Post-process scenario
---------------------
Lastly, we post-process the scenario by means of the :class:`.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.

.. GENERATED FROM PYTHON SOURCE LINES 115-123

.. tip::

   Each post-processing method requires different inputs and offers a variety
   of customization options. Use the API function
   :meth:`~gemseo.api.get_post_processing_options_schema` to print a table with
   the options for any post-processing algorithm.
   Or refer to our dedicated page:
   :ref:`gen_post_algos`.

.. GENERATED FROM PYTHON SOURCE LINES 123-127

.. code-block:: default


    scenario.post_process("Correlations", save=False, show=False)
    # Workaround for HTML rendering, instead of ``show=True``
    plt.show()



.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /examples/post_process/images/sphx_glr_plot_correlations_001.png
         :alt: R=0.99663, R=0.99159, R=0.99886, R=0.98710, R=0.99691, R=0.99952, R=0.98339, R=0.99496, R=0.99861, R=0.99976, R=0.96409, R=0.98227, R=-0.98227, R=-0.96414, R=-1.00000, R=0.96410, R=1.00000, R=-1.00000, R=0.96414, R=1.00000, R=-1.00000, R=1.00000, R=-0.96865, R=0.96865, R=-0.99805
         :srcset: /examples/post_process/images/sphx_glr_plot_correlations_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/post_process/images/sphx_glr_plot_correlations_002.png
         :alt: R=-0.96416, R=-1.00000, R=1.00000, R=-1.00000, R=-1.00000, R=-0.96417, R=-1.00000, R=1.00000, R=-1.00000, R=-1.00000, R=1.00000
         :srcset: /examples/post_process/images/sphx_glr_plot_correlations_002.png
         :class: sphx-glr-multi-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

        INFO - 14:43:23: Detected 36 correlations > 0.95





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  3.446 seconds)


.. _sphx_glr_download_examples_post_process_plot_correlations.py:

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  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_correlations.py <plot_correlations.py>`

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      :download:`Download Jupyter notebook: plot_correlations.ipynb <plot_correlations.ipynb>`


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