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

    .. note::
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        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 30-34

.. code-block:: default

    from __future__ import division, unicode_literals

    from matplotlib import pyplot as plt








.. GENERATED FROM PYTHON SOURCE LINES 35-39

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 39-44

.. code-block:: default

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

    configure_logger()





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

 Out:

 .. code-block:: none


    <RootLogger root (INFO)>



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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().read_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

 Out:

 .. code-block:: none

        INFO - 14:41:51:  
        INFO - 14:41:51: *** Start MDO Scenario execution ***
        INFO - 14:41:51: MDOScenario
        INFO - 14:41:51:    Disciplines: SobieskiPropulsion SobieskiAerodynamics SobieskiStructure SobieskiMission
        INFO - 14:41:51:    MDOFormulation: MDF
        INFO - 14:41:51:    Algorithm: SLSQP
        INFO - 14:41:51: Optimization problem:
        INFO - 14:41:51:    Minimize: -y_4(x_shared, x_1, x_2, x_3)
        INFO - 14:41:51:    With respect to: x_shared, x_1, x_2, x_3
        INFO - 14:41:51:    Subject to constraints:
        INFO - 14:41:51:       g_1(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 14:41:51:       g_2(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 14:41:51:       g_3(x_shared, x_1, x_2, x_3) <= 0.0
        INFO - 14:41:51: Design space:
        INFO - 14:41:51: +----------+-------------+-------+-------------+-------+
        INFO - 14:41:51: | name     | lower_bound | value | upper_bound | type  |
        INFO - 14:41:51: +----------+-------------+-------+-------------+-------+
        INFO - 14:41:51: | x_shared |     0.01    |  0.05 |     0.09    | float |
        INFO - 14:41:51: | x_shared |    30000    | 45000 |    60000    | float |
        INFO - 14:41:51: | x_shared |     1.4     |  1.6  |     1.8     | float |
        INFO - 14:41:51: | x_shared |     2.5     |  5.5  |     8.5     | float |
        INFO - 14:41:51: | x_shared |      40     |   55  |      70     | float |
        INFO - 14:41:51: | x_shared |     500     |  1000 |     1500    | float |
        INFO - 14:41:51: | x_1      |     0.1     |  0.25 |     0.4     | float |
        INFO - 14:41:51: | x_1      |     0.75    |   1   |     1.25    | float |
        INFO - 14:41:51: | x_2      |     0.75    |   1   |     1.25    | float |
        INFO - 14:41:51: | x_3      |     0.1     |  0.5  |      1      | float |
        INFO - 14:41:51: +----------+-------------+-------+-------------+-------+
        INFO - 14:41:51: 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 - 14:41:52: Optimization:  20%|██        | 2/10 [00:00<00:00, 52.86 it/sec, obj=2.12e+3]
        INFO - 14:41:52: Optimization:  40%|████      | 4/10 [00:00<00:00, 21.21 it/sec, obj=3.97e+3]
        INFO - 14:41:52: Optimization:  50%|█████     | 5/10 [00:00<00:00, 16.58 it/sec, obj=3.96e+3]
        INFO - 14:41:52: Optimization:  60%|██████    | 6/10 [00:00<00:00, 13.58 it/sec, obj=3.96e+3]
        INFO - 14:41:52: Optimization:  70%|███████   | 7/10 [00:00<00:00, 11.52 it/sec, obj=3.96e+3]
        INFO - 14:41:52: Optimization:  90%|█████████ | 9/10 [00:01<00:00,  9.83 it/sec, obj=3.96e+3]
        INFO - 14:41:52: Optimization: 100%|██████████| 10/10 [00:01<00:00,  9.14 it/sec, obj=3.96e+3]
        INFO - 14:41:52: Optimization result:
        INFO - 14:41:52: Objective value = 3963.595455433326
        INFO - 14:41:52: The result is feasible.
        INFO - 14:41:52: Status: None
        INFO - 14:41:52: Optimizer message: Maximum number of iterations reached. GEMSEO Stopped the driver
        INFO - 14:41:52: Number of calls to the objective function by the optimizer: 12
        INFO - 14:41:52: Constraints values:
        INFO - 14:41:52:    g_1 = [-0.01814919 -0.03340982 -0.04429875 -0.05187486 -0.05736009 -0.13720854
        INFO - 14:41:52:  -0.10279146]
        INFO - 14:41:52:    g_2 = 3.236261671801799e-05
        INFO - 14:41:52:    g_3 = [-7.67067574e-01 -2.32932426e-01 -9.19662628e-05 -1.83255000e-01]
        INFO - 14:41:52: Design space:
        INFO - 14:41:52: +----------+-------------+--------------------+-------------+-------+
        INFO - 14:41:52: | name     | lower_bound |       value        | upper_bound | type  |
        INFO - 14:41:52: +----------+-------------+--------------------+-------------+-------+
        INFO - 14:41:52: | x_shared |     0.01    | 0.0600080906541795 |     0.09    | float |
        INFO - 14:41:52: | x_shared |    30000    |       60000        |    60000    | float |
        INFO - 14:41:52: | x_shared |     1.4     |        1.4         |     1.8     | float |
        INFO - 14:41:52: | x_shared |     2.5     |        2.5         |     8.5     | float |
        INFO - 14:41:52: | x_shared |      40     |         70         |      70     | float |
        INFO - 14:41:52: | x_shared |     500     |        1500        |     1500    | float |
        INFO - 14:41:52: | x_1      |     0.1     | 0.3999993439500847 |     0.4     | float |
        INFO - 14:41:52: | x_1      |     0.75    |        0.75        |     1.25    | float |
        INFO - 14:41:52: | x_2      |     0.75    |        0.75        |     1.25    | float |
        INFO - 14:41:52: | x_3      |     0.1     | 0.156230376400943  |      1      | float |
        INFO - 14:41:52: +----------+-------------+--------------------+-------------+-------+
        INFO - 14:41:52: *** MDO Scenario run terminated in 0:00:01.104148 ***

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



.. 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:: /examples/post_process/images/sphx_glr_plot_correlations_001.png
          :alt: 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
          :class: sphx-glr-multi-img

    *

      .. image:: /examples/post_process/images/sphx_glr_plot_correlations_002.png
          :alt: 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
          :class: sphx-glr-multi-img

    *

      .. image:: /examples/post_process/images/sphx_glr_plot_correlations_003.png
          :alt: R=-0.98694, R=0.98694, R=-0.98694, R=-0.98694, R=0.98694, R=0.98694
          :class: sphx-glr-multi-img


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

 Out:

 .. code-block:: none

        INFO - 14:41:52: Detected 56 correlations > 0.95





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

   **Total running time of the script:** ( 0 minutes  3.974 seconds)


.. _sphx_glr_download_examples_post_process_plot_correlations.py:


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