Blurb::
Aleatory uncertain variable - continuous histogram

Description::
Histogram uncertain variables are typically used to model a set of
empirical data.  The bin histogram (contrast: :dakkw:`variables-histogram_point_uncertain`) is a continuous aleatory
distribution characterized by bins of non-zero width where the
uncertain variable may lie, together with the relative frequencies of
each bin.  Hence it can be used to specify a marginal probability
density function arising from data.

The ``histogram_bin_uncertain`` keyword specifies the number of
variables to be characterized as continuous histograms.  The required
sub-keywords are: :dakkw:`variables-histogram_bin_uncertain-abscissas`
(ranges of values the variable can take on) and either
:dakkw:`variables-histogram_bin_uncertain-ordinates` or
:dakkw:`variables-histogram_bin_uncertain-counts` (characterizing each
variable's frequency information).  When using histogram bin
variables, each variable must be defined by at least one bin (with two
bounding value pairs).  When more than one histogram bin variable is
active, :dakkw:`variables-histogram_bin_uncertain-pairs_per_variable` can
be used to specify unequal apportionment of provided bin pairs among
the variables.

The ``abscissas`` specification defines abscissa values (:math:`x`
coordinates) for the probability density function of each histogram
variable.  When paired with ``counts``, the specifications provide sets
of :math:`(x,c)` pairs for each histogram variable where :math:`c` defines a
count (i.e., a frequency or relative probability) associated with a
bin. If using bins of unequal width and specification of probability
densities is more natural, then the ``counts`` specification can be
replaced with an ``ordinates`` specification (:math:`y` coordinates) in order
to support interpretation of the input as :math:`(x,y)` pairs defining the
profile of a "skyline" probability density function.

Conversion between the two specifications is straightforward: a
count/frequency is a cumulative probability quantity defined from the
product of the ordinate density value and the :math:`x` bin width. Thus,
in the cases of bins of equal width, ordinate and count specifications
are equivalent. In addition, ordinates and counts may be relative
values; it is not necessary to scale them as all user inputs will be
normalized.

To fully specify a bin-based histogram with :math:`n` bins (potentially of
unequal width), :math:`n+1` :math:`(x,c)` or :math:`(x,y)` pairs must be specified
with the following features:

- :math:`x` is the parameter value for the left boundary of a histogram  bin and :math:`c` is the corresponding count for that bin.  Alternatively,  :math:`y` defines the ordinate density value for this bin within a skyline  probability density function. The right boundary of the bin is defined by the left boundary of the next pair.
- the final pair specifies the right end of the last bin and must  have a :math:`c` or :math:`y` value of zero.
- the :math:`x` values must be strictly increasing.
- all :math:`c` or :math:`y` values must be positive, except for the last  which must be zero.
- a minimum of two pairs must be specified for each bin-based  histogram variable.

Topics::
continuous_variables, aleatory_uncertain_variables

Examples::
The ``pairs_per_variable`` specification provides for the proper
association of multiple sets of :math:`(x,c)` or :math:`(x,y)` pairs with
individual histogram variables. For example, in this input snippet


.. code-block::

    histogram_bin_uncertain = 2
      pairs_per_variable = 3           4
      abscissas          = 5  8  10    .1 .2 .3 .4
      counts             = 17 21 0     12 24 12 0
      descriptors        = 'hbu_1'     'hbu_2'


``pairs_per_variable`` associates the first 3 :math:`(x,c)` pairs from ``abscissas`` and ``counts`` {(5,17),(8,21),(10,0)} with one bin-based histogram variable,
where one bin is defined between 5 and 8 with a count of 17 and another bin is defined between 8 and 10 with a count of 21. The following set of 4 :math:`(x,c)`
pairs {(.1,12),(.2,24),(.3,12),(.4,0)} defines a second bin-based histogram variable containing three equal-width bins with counts 12, 24, and 12 (middle
bin is twice as probable as the other two).

Theory::

Faq::

*Difference between bin and point histograms:* 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:p:`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::
