.. only:: html

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

        Click :ref:`here <sphx_glr_download_auto_reliability_sensitivity_design_of_experiments_plot_optimal_lhs.py>`     to download the full example code
    .. rst-class:: sphx-glr-example-title

    .. _sphx_glr_auto_reliability_sensitivity_design_of_experiments_plot_optimal_lhs.py:


Optimal LHS designs
===================

This examples show how to generate optimized LHS experiments according to the different criteria.


.. code-block:: default

    from __future__ import print_function
    import openturns as ot
    import math as m
    import openturns.viewer as viewer
    from matplotlib import pylab as plt
    ot.Log.Show(ot.Log.NONE)








**LHS and space filling**


.. code-block:: default

    N = 100
    # Considering independent Uniform distributions of dimension 3
    # Bounds are (-1,1), (0,2) and (0, 0.5)
    distribution = ot.ComposedDistribution([ot.Uniform(-1.0, 1.0), ot.Uniform(0.0, 2.0), ot.Uniform(0.0, 0.5)])
    # Random LHS
    lhs = ot.LHSExperiment(distribution, N)
    lhs.setAlwaysShuffle(True) # randomized
    design = lhs.generate()
    # C2
    c2 = ot.SpaceFillingC2().evaluate(design)
    # PhiP with default p
    phip = ot.SpaceFillingPhiP().evaluate(design)
    # mindist
    mindist = ot.SpaceFillingMinDist().evaluate(design)
    # For p->infinity
    phip_inf = ot.SpaceFillingPhiP(100).evaluate(design)
    print(phip, mindist, phip_inf)





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

 Out:

 .. code-block:: none

    15.722897751576225 0.042216452730087735 15.720769617188896




**Optimized LHS using Monte Carlo**

As with Monte Carlo, user decides of a fixed number of iterations,
but this time this number is part of the temperature profile.

Two profiles are currently provided:
- Linear profile: :math:`T_i = T_0 (1-\frac{1}{n_{iter}})`
- Geometric profile: :math:`T_i = T_O c^i, 0<c<1`

Starting from an LHS design, a new design is built by permuting a random coordinate of two randomly chosen sample points; this new design is also an LHS. but not necessary a more efficient design.

A comparison of criteria of the two designs is done, and the new LHS is accepted with probability

.. math::
   min\left(exp{\left[-\frac{\Phi(LHS_{new}) - \Phi(LHS)}{T_i}\right]}, 1\right)


Considering independent Uniform(0,1) distributions of dimension 3


.. code-block:: default

    distribution = ot.ComposedDistribution([ot.Uniform(0.0, 1.0)] * 3)
    # Random LHS
    lhs = ot.LHSExperiment(distribution, N)
    lhs.setAlwaysShuffle(True) # randomized
    algo = ot.SimulatedAnnealingLHS(lhs)
    design = algo.generate()








One could also fix the criterion, the temperature profile and get more results.


.. code-block:: default


    # Considering independent Uniform distributions of dimension 3
    # Bounds are (-1,1), (0,2) and (0, 0.5)
    distribution = ot.ComposedDistribution([ot.Uniform(-1.0, 1.0), ot.Uniform(0.0, 2.0), ot.Uniform(0.0, 0.5)])
    # Random LHS
    lhs = ot.LHSExperiment(distribution, N)
    lhs.setAlwaysShuffle(True) # randomized
    # Fixing C2 crit
    space_filling = ot.SpaceFillingC2()
    # Defining a temperature profile
    # A geometric profile seems accurate with default parameters
    # e.g. T0=10, c=0.95, iMax=2000
    temperatureProfile = ot.GeometricProfile()
    algo = ot.SimulatedAnnealingLHS(lhs, temperatureProfile, space_filling)
    # optimal design
    design = algo.generate()
    result = algo.getResult()
    # Criteria for the optimal design
    crit_c2 = result.getC2()
    crit_phip = result.getPhiP()
    crit_mindist = result.getMinDist()
    # History of the criterion used for optimization
    history = result.getAlgoHistory()
    criterion_hist = history[:, 0]
    # Additional results
    temperature_hist = history[:, 1]
    probability_hist = history[:, 2]








It is also possible to chain several iterations of the whole process with different starting points.


.. code-block:: default

    N= 10

    # Considering independent Uniform distributions of dimension 3
    # Bounds are (-1,1), (0,2) and (0, 0.5)
    distribution = ot.ComposedDistribution([ot.Uniform(-1.0, 1.0), ot.Uniform(0.0, 2.0), ot.Uniform(0.0, 0.5)])
    # Random LHS
    lhs = ot.LHSExperiment(distribution, N)
    lhs.setAlwaysShuffle(True) # randomized
    # Fixing PhiP crit
    space_filling = ot.SpaceFillingPhiP()
    # Defining a temperature profile
    # T0=10, iMax=3000
    temperatureProfile = ot.LinearProfile(10.0, 3000)
    algo = ot.SimulatedAnnealingLHS(lhs, temperatureProfile, space_filling)
    restart = 50
    design = algo.generateWithRestart(restart)
    # Retrieve all optimal designs
    result = algo.getResult()
    designs = [result.getOptimalDesign(i) for i in range(restart)]








Finally, we could start the optimization process of LHS using a precomputed LHS design.


.. code-block:: default


    # Considering independent Uniform distributions of dimension 3
    # Bounds are (0,1)^3
    distribution = ot.ComposedDistribution([ot.Uniform(0.0, 1.0)] * 3)
    # Random LHS
    lhs = ot.LHSExperiment(distribution, N)
    lhs.setAlwaysShuffle(True) # randomized
    # Fixing C2 crit for example
    space_filling = ot.SpaceFillingC2()
    # Defining a temperature profile
    # T0=10, iMax=3000
    temperatureProfile = ot.LinearProfile(10.0, 3000)
    algo = ot.SimulatedAnnealingLHS(lhs, temperatureProfile, space_filling)
    design = algo.generate()
    result = algo.getResult()
    # check history ==> draw criterion
    graph = result.drawHistoryCriterion()
    view = viewer.View(graph)




.. image:: /auto_reliability_sensitivity/design_of_experiments/images/sphx_glr_plot_optimal_lhs_001.png
    :alt: C2 criterion history of optimal design
    :class: sphx-glr-single-img





Convergence needs to be performed
New algo starting from this design


.. code-block:: default

    algo = ot.SimulatedAnnealingLHS(design, distribution, temperatureProfile, space_filling)
    design = algo.generate()
    plt.show()








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

   **Total running time of the script:** ( 0 minutes  0.493 seconds)


.. _sphx_glr_download_auto_reliability_sensitivity_design_of_experiments_plot_optimal_lhs.py:


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