Blurb::
Activates global sensitivity analysis based on decomposition of
response variance into contributions from variables
Description::
Dakota can calculate sensitivity indices through variance based
decomposition using the keyword ``variance_based_decomp``.  These
indicate how important the uncertainty in each input variable is in
contributing to the output variance.

*Default Behavior*

Variance based decomposition is turned off as a default.

*Note that specifying this keyword with the default* ``vbd_sampling_method``
*of* ``pick_and_freeze`` *selected will increase the number of function 
evaluations above the number requested with the* ``samples`` *keyword 
since replicated sets of sample values are evaluated.* See documentation
for ``pick_and_freeze`` on the increased sample size.

*Expected Output*

When ``variance_based_decomp`` is specified, Sobol' sensitivity indices 
will be reported. Sobol' indices represent the percent of the variance in 
the model response that can be attributed to each individual variable. 

*Usage Tips*

To obtain sensitivity indices that are reasonably accurate, we
recommend that N, the number of samples, be at least one hundred and
preferably several hundred or thousands.
Topics::

Examples::

.. code-block::

    method,
      sampling
        sample_type lhs
        samples = 100
        variance_based_decomp

Theory::

In this context, we take sensitivity analysis to be global, not local
as when calculating derivatives of output variables with respect to
input variables. Our definition is similar to that of :cite:p:`Sal04`:
"The study of how uncertainty in the output of a model can be
apportioned to different sources of uncertainty in the model input."

Variance based decomposition is a way of using sets of samples to
understand how the variance of the output behaves, with respect to
each input variable. 

Two types of variance-based sensitivity indices are commonly 
computed: main effects and total effects indices. Main effects (roughly) 
represent the percent contribution of each individual
variable to the variance in the model response. Total effects
represent the percent contribution of each individual variable in
combination with all other variables to the variance in the model
response. For the variable :math:`X_i`, the main and total effect
indices are denoted here :math:`S_i` and :math:`T_i`, respectively,
and their formulas are

.. math::
   S_i = \frac{V(E_{\mathbf{X}_{\sim i}}(f(\mathbf{X})|X_i))}{V(f)} \\
   T_i = \frac{E(V_{X_{i}}(f(\mathbf{X})|\mathbf{X}_{\sim i}))}{V(f)},

where :math:`\mathbf{X}_{\sim i}` denotes all variables except 
:math:`X_i`.

A larger sensitivity index means that the uncertainty in the input
variable :math:`X_i`` has a larger effect on the variance of the 
output. More details on the calculations and interpretation of the 
sensitivity indices can be found in :cite:p:`Sal04` and :cite:p:`Weirs10`.

Two methods to compute variance-based sensitivity indices are 
implemented in Dakota. The default "pick-and-freeze" method is based on
the algorithm discussed in :cite:p:`Weirs10`. Pick-and-freeze methods
are currently the most popular approach for varianced-based sensitivity
index computation, but they incur significant computational cost. These
approaches rely on structured sampling wherein two independent random
sample sets of the input variables are generated, then the random samples
of the variable whose sensitivity index is being computed are substituted
from one sample set into the other. Specifically, if the user specified a 
number of samples, N, and a number of nondeterministic variables, M, 
pick-and-freeze variance-based decomposition requires the evaluation of 
N*(M+2) samples.

The "binned" method uses the N samples generated during the sampling study
directly to computed variance-based sensitivity indices, but it can only
compute main effect indices, so the user will not be provided with 
information from the total effect indices about the percent variance of 
the model response arising from interactions between each variable and 
all other variables. However, the main effect indices can be computed with 
no further model evaluations beyond the N samples already specified for the
sampling study. The binned approach is detailed in :cite:p:`Li16`.

Faq::

See_Also::
