`ipynb source code <linearPOD.ipynb>`_

Linear model POD
================

.. code:: python

    # import relevant module
    import openturns as ot
    import otpod
    # enable display figure in notebook
    %matplotlib inline

Generate data
-------------

.. code:: python

    N = 100
    ot.RandomGenerator.SetSeed(123456)
    defectDist = ot.Uniform(0.1, 0.6)
    # normal epsilon distribution
    epsilon = ot.Normal(0, 1.9)
    defects = defectDist.getSample(N)
    signalsInvBoxCox = defects * 43. + epsilon.getSample(N) + 2.5
    # Inverse Box Cox transformation
    invBoxCox = ot.InverseBoxCoxTransform(0.3)
    signals = invBoxCox(signalsInvBoxCox)

Build POD using previous linear analysis
----------------------------------------

.. code:: python

    # run the analysis with Gaussian hypothesis of the residuals (default case)
    analysis = otpod.UnivariateLinearModelAnalysis(defects, signals, boxCox=True)

.. code:: python

    # signal detection threshold
    detection = 200.
    # Use the analysis to build the POD with Gaussian hypothesis
    # keyword arguments must be given
    PODGauss = otpod.UnivariateLinearModelPOD(analysis=analysis, detection=detection)
    PODGauss.run()

Build POD with Gaussian hypothesis
----------------------------------

.. code:: python

    # The previous POD is equivalent to the following POD
    PODGauss = otpod.UnivariateLinearModelPOD(defects, signals, detection,
                                         resDistFact=ot.NormalFactory(),
                                         boxCox=True)
    PODGauss.run()

Get the R2 value of the regression
----------------------------------

.. code:: python

    print('R2 : {:0.3f}'.format(PODGauss.getR2()))


.. parsed-literal::

    R2 : 0.895


Compute detection size
----------------------

.. code:: python

    # Detection size at probability level 0.9
    # and confidence level 0.95
    print(PODGauss.computeDetectionSize(0.9, 0.95))
    
    # probability level 0.95 with confidence level 0.99
    print(PODGauss.computeDetectionSize(0.95, 0.99))


.. parsed-literal::

    [a90 : 0.303982, a90/95 : 0.317157]
    [a95 : 0.323048, a95/99 : 0.343536]


get POD NumericalMathFunction
-----------------------------

.. code:: python

    # get the POD model
    PODmodel = PODGauss.getPODModel()
    # get the POD model at the given confidence level
    PODmodelCl95 = PODGauss.getPODCLModel(0.95)
    
    # compute the probability of detection for a given defect value
    print('POD : {:0.3f}'.format(PODmodel([0.3])[0]))
    print('POD at level 0.95 : {:0.3f}'.format(PODmodelCl95([0.3])[0]))


.. parsed-literal::

    POD : 0.886
    POD at level 0.95 : 0.834


Show POD graphs
---------------

Only the mean POD
~~~~~~~~~~~~~~~~~

.. code:: python

    fig, ax = PODGauss.drawPOD()
    fig.show()


.. parsed-literal::

    /home/dumas/anaconda2/lib/python2.7/site-packages/matplotlib/figure.py:397: UserWarning: matplotlib is currently using a non-GUI backend, so cannot show the figure
      "matplotlib is currently using a non-GUI backend, "



.. image:: linearPOD_files/linearPOD_17_1.png


Mean POD with the detection size for a given probability level
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: python

    fig, ax = PODGauss.drawPOD(probabilityLevel=0.9)
    fig.show()



.. image:: linearPOD_files/linearPOD_19_0.png


Mean POD with POD at confidence level
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: python

    fig, ax = PODGauss.drawPOD(confidenceLevel=0.95)
    fig.show()



.. image:: linearPOD_files/linearPOD_21_0.png


Mean POD and POD at confidence level with the detection size for a given probability level
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code:: python

    fig, ax = PODGauss.drawPOD(probabilityLevel=0.9, confidenceLevel=0.95,
                          name='figure/PODGauss.png')
    # The figure is saved in PODGauss.png
    fig.show()



.. image:: linearPOD_files/linearPOD_23_0.png


Build POD with no hypothesis on the residuals
---------------------------------------------

This corresponds with the Berens Binomial method.

.. code:: python

    PODBinomial = otpod.UnivariateLinearModelPOD(defects, signals, detection, boxCox=True)
    PODBinomial.run()

.. code:: python

    # Detection size at probability level 0.9
    # and confidence level 0.95
    print(PODBinomial.computeDetectionSize(0.9, 0.95))


.. parsed-literal::

    [a90 : 0.298739, a90/95 : 0.329606]


.. code:: python

    fig, ax = PODBinomial.drawPOD(0.9, 0.95)
    fig.show()



.. image:: linearPOD_files/linearPOD_27_0.png


Build POD with kernel smoothing on the residuals
------------------------------------------------

The POD at the given confidence level is built using bootstrap. It may
take few seconds. A progress bar if displayed is in this case. It can be
remove using setVerbose(False)

.. code:: python

    PODks = otpod.UnivariateLinearModelPOD(defects, signals, detection,
                                                 resDistFact=ot.KernelSmoothing(),
                                                 boxCox=True)
    PODks.run()


.. parsed-literal::

    Computing POD (bootstrap): [==================================================] 100.00% Done


.. code:: python

    # Detection size at probability level 0.9
    # and confidence level 0.95
    print(PODks.computeDetectionSize(0.9, 0.95))


.. parsed-literal::

    [a90 : 0.308381, a90/95 : 0.331118]


.. code:: python

    fig, ax = PODks.drawPOD(0.9, 0.95)
    fig.show()



.. image:: linearPOD_files/linearPOD_31_0.png