.. _ExamScreeningMorris:

Example: Using the Morris method
==================================

In the page we provide an example of applying the Morris screening
method (:ref:`ProcMorris<ProcMorris>`) on a simple synthetic example.

Suppose the simulator function is deterministic and described by the
formula :math:`f(x) = x_1 + x_2^2 + x_2 \times sin(x_3) + 0 \times x_4`.
We will use the standard Morris method to discover the three relevant
variables from a total set of four input variables.

We use :math:`R=4` trajectories, four levels for each input
:math:`p=4` and :math:`\Delta=p/2(p-1) = 0.66`. Given these
parameters, the total number of simulator runs is :math:`(k+1)R =
20`. The experimental design used is shown below. The values have been
rounded to two decimal places.

Morris Experimental Design

=========== =========== =========== =========== ============
:math:`x_1` :math:`x_2` :math:`x_3` :math:`x_4` :math:`f(x)`
=========== =========== =========== =========== ============
0.00        0.33        0.33        0.00        0.22
0.67        0.33        0.33        0.00        0.89
0.67        1.00        0.33        0.00        1.99
0.67        1.00        0.33        0.67        1.99
0.67        1.00        1.00        0.67        2.51
0.33        0.00        0.33        0.33        0.33
0.33        0.00        1.00        0.33        0.33
1.00        0.00        1.00        0.33        1.00
1.00        0.67        1.00        0.33        2.01
1.00        0.67        1.00        1.00        2.01
0.00        0.00        0.00        0.33        0.00
0.00        0.00        0.67        0.33        0.00
0.67        0.00        0.67        0.33        0.67
0.67        0.67        0.67        0.33        1.52
0.67        0.67        0.67        1.00        1.52
0.33        0.33        0.00        0.00        0.44
0.33        0.33        0.67        0.00        0.65
1.00        0.33        0.67        0.00        1.32
1.00        0.33        0.67        0.67        1.32
1.00        1.00        0.67        0.67        2.62
=========== =========== =========== =========== ============

Using this design we sampling statistics of the elementary effects for
each factor are computed:

Morris Method Indexes

=========== =========== ============= ==============
Factor      :math:`\mu` :math:`\mu_*` :math:`\sigma`
=========== =========== ============= ==============
:math:`x_1` 1           1             2e-16
:math:`x_2` 1.6         1.6           0.28
:math:`x_3` 0.27        0.27          0.36
:math:`x_4` 0           0             0
=========== =========== ============= ==============

The :math:`\mu` and :math:`\sigma` values are plotted in Figure 1 below. As
can be seen the Morris method effectively and clearly identifies the
relevant inputs. For factor :math:`x_1` we note the high :math:`\mu` and low
:math:`\sigma` values signify a linear effect. For factors :math:`x_2`,
:math:`x_3` the large :math:`\sigma` value demonstrates the
non-linear/interaction effects. Factor :math:`x_4` has zero value for both
metrics as expected for an irrelevant factor. Lastly the agreement of
:math:`\mu` to :math:`\mu_*` for all factors shows a lack of cancellation
effects, due to the monotonic nature of the input-output response in
this simple example. In general models this will not be the case,
particularly those with non-linear responses.

.. figure:: ExamScreeningMorris/morrisToolkitExample.png
   :width: 560px
   :align: center

   **Figure 1:** Summary statistics for Morris method.

References
----------

A multitude of screening and sensitivity analysis methods including the
Morris method are implemented in the sensitivity package available for
the R statistical software system:

Gilles Pujol and Bertrand Iooss (2008). Sensitivity Analysis package: A
collection of functions for factor screening and global sensitivity
analysis of model output.
`http://cran.r-project.org/web/packages/sensitivity/index.html <http://cran.r-project.org/web/packages/sensitivity/index.html>`__

R Development Core Team (2005). R: A language and environment for
statistical computing. R Foundation for Statistical Computing, Vienna,
Austria. ISBN 3-900051-07-0, `http://www.R-project.org <http://www.R-project.org>`__.