.. _method-polynomial_chaos-cubature_integrand:

""""""""""""""""""
cubature_integrand
""""""""""""""""""


Cubature using Stroud rules and their extensions


.. toctree::
   :hidden:
   :maxdepth: 1



**Specification**

- *Alias:* None

- *Arguments:* INTEGER


**Description**


Multi-dimensional integration by Stroud cubature rules
:cite:p:`stroud` and extensions
:cite:p:`xiu_cubature`, as specified with ``cubature_integrand``.
A total-order
expansion is used, where the isotropic order *p* of the
expansion is half of the integrand order, rounded down. The
total number of terms *N* for an isotropic total-order expansion
of order *p* over *n* variables is given by
   
.. math:: N~=~1 + P ~=~1 + \sum_{s=1}^{p} {\frac{1}{s!}}
   \prod_{r=0}^{s-1} (n + r) ~=~\frac{(n+p)!}{n!p!}

Since the maximum integrand order is currently five for normal
and uniform and two for all other types, at most second- and
first-order expansions, respectively, will be used. As a result,
cubature is primarily useful for global sensitivity analysis,
where the Sobol' indices will provide main effects and, at most,
two-way interactions. In addition, the random variable set must
be independent and identically distributed ( *iid*), so the use
of ``askey`` or ``wiener`` transformations may be required to
create *iid* variable sets in the transformed space (as well as
to allow usage of the higher order cubature rules for normal and
uniform). Note that global sensitivity analysis often assumes
uniform bounded regions, rather than precise probability
distributions, so the *iid* restriction would not be problematic
in that case.