Blurb::
Aleatory uncertain variable - discrete histogram

Description::

Histogram uncertain variables are typically used to model a set of
empirical data.  When the variables take on only discrete values or
categories, a discrete, or point histogram is used to describe their
probability mass function (one could think of this as a 
\ref variables-histogram_bin_uncertain variable with "bins" of zero width).
Dakota supports integer-, string-, and real-valued point histograms.

Point histograms are similar to \ref variables-discrete_design_set and
\ref variables-discrete_state_set, but as they are uncertain
variables, include the relative probabilities of observing the
different values within the set.

The \c histogram_point_uncertain keyword is followed by one or more of
\c integer, \c string, or \c real, each of which specify the number of
variables to be characterized as discrete histograms of that sub-type.

Each discrete histogram variable is specified by one or more
abscissa/count pairs.  The \c abscissas, are the possible values the
variable can take on ("x" coordinates of type integer, string, or
real), and must be specified in increasing order.  These are paired
with \c counts \c c which provide the frequency of the given value or
string, relative to other possible values/strings.

Thus, to fully specify a point-based histogram with \c n points, \c n
\c (x,c) pairs must be specified with the following features:

\li \c x is the point value (integer, string, or real) and \c c is the
  corresponding count for that value.

\li the \c x values must be strictly increasing (lexicographically for
  strings).

\li all \c c values must be positive.

\li a minimum of one pair must be specified for each point-based
  histogram.

Topics::	discrete_variables, aleatory_uncertain_variables

Examples::

The \c pairs_per_variable specification provides for the proper
association of multiple sets of \c (x,c) or \c (x,y) pairs with
individual histogram variables. For example, in the following
specification,

\verbatim
histogram_point_uncertain
  integer            = 2
  pairs_per_variable = 2     3 
  abscissas          = 3 4   100 200 300
  counts             = 1 1   1   2   1
\endverbatim

\c pairs_per_variable associates the \c (x,c) pairs {(3,1),(4,1)} with
one point-based histogram variable (where the values 3 and 4 are
equally probable) and associates the \c (x,c) pairs
{(100,1),(200,2),(300,1)} with a second point-based histogram variable
(where the value 200 is twice as probable as either 100 or 300).

Theory::

Faq::
<b>Difference between bin and point histograms:</b> A (continuous) bin
histogram specifies bins of non-zero width, whereas a (discrete) point
histogram specifies individual point values, which can be thought of
as bins with zero width. In the terminology of LHS \cite Wyss1998, 
the bin pairs specification defines a
"continuous linear" distribution and the point pairs specification
defines a "discrete histogram" distribution (although the points are
real-valued, the number of possible values is finite).

See_Also::	variables-histogram_bin_uncertain
