1Blurb::
2Aleatory uncertain variable - continuous histogram
3
4Description::
5
6Histogram uncertain variables are typically used to model a set of
7empirical data.  The bin histogram (contrast:
8\ref variables-histogram_point_uncertain) is a continuous aleatory
9distribution characterized by bins of non-zero width where the
10uncertain variable may lie, together with the relative frequencies of
11each bin.  Hence it can be used to specify a marginal probability
12density function arising from data.
13
14The \c histogram_bin_uncertain keyword specifies the number of
15variables to be characterized as continuous histograms.  The required
16sub-keywords are: \ref variables-histogram_bin_uncertain-abscissas
17(ranges of values the variable can take on) and either
18\ref variables-histogram_bin_uncertain-ordinates or
19\ref variables-histogram_bin_uncertain-counts (characterizing each
20variable's frequency information).  When using histogram bin
21variables, each variable must be defined by at least one bin (with two
22bounding value pairs).  When more than one histogram bin variable is
23active, \ref variables-histogram_bin_uncertain-pairs_per_variable can
24be used to specify unequal apportionment of provided bin pairs among
25the variables.
26
27The \c abscissas specification defines abscissa values ("x"
28coordinates) for the probability density function of each histogram
29variable.  When paired with \c counts, the specifications provide sets
30of \c (x,c) pairs for each histogram variable where \c c defines a
31count (i.e., a frequency or relative probability) associated with a
32bin. If using bins of unequal width and specification of probability
33densities is more natural, then the \c counts specification can be
34replaced with an \c ordinates specification ("y" coordinates) in order
35to support interpretation of the input as \c (x,y) pairs defining the
36profile of a "skyline" probability density function.
37
38Conversion between the two specifications is straightforward: a
39count/frequency is a cumulative probability quantity defined from the
40product of the ordinate density value and the \c x bin width. Thus,
41in the cases of bins of equal width, ordinate and count specifications
42are equivalent. In addition, ordinates and counts may be relative
43values; it is not necessary to scale them as all user inputs will be
44normalized.
45
46To fully specify a bin-based histogram with \c n bins (potentially of
47unequal width), \c n+1 \c (x,c) or \c (x,y) pairs must be specified
48with the following features:
49
50\li \c x is the parameter value for the left boundary of a histogram
51 bin and \c c is the corresponding count for that bin.  Alternatively,
52 \c y defines the ordinate density value for this bin within a skyline
53 probability density function. The right boundary of the bin is
54 defined by the left boundary of the next pair.
55
56\li the final pair specifies the right end of the last bin and must
57 have a \c c or \c y value of zero.
58
59\li the \c x values must be strictly increasing.
60
61\li all \c c or \c y values must be positive, except for the last
62 which must be zero.
63
64\li a minimum of two pairs must be specified for each bin-based
65 histogram variable.
66
67Topics::	continuous_variables, aleatory_uncertain_variables
68
69Examples::
70The \c pairs_per_variable specification provides for the proper
71association of multiple sets of \c (x,c) or \c (x,y) pairs with
72individual histogram variables. For example, in this input snippet
73
74\verbatim
75histogram_bin_uncertain = 2
76  pairs_per_variable = 3           4
77  abscissas          = 5  8  10    .1 .2 .3 .4
78  counts             = 17 21 0     12 24 12 0
79  descriptors        = 'hbu_1'     'hbu_2'
80\endverbatim
81
82\c pairs_per_variable associates the first 3 \c (x,c) pairs from \c
83abscissas and \c counts \c {(5,17),(8,21),(10,0)} with one
84bin-based histogram variable, where one bin is defined between 5 and 8
85with a count of 17 and another bin is defined between 8 and 10 with a
86count of 21. The following set of 4 \c (x,c) pairs \c
87{(.1,12),(.2,24),(.3,12),(.4,0)} defines a second bin-based histogram
88variable containing three equal-width bins with counts 12, 24, and 12
89(middle bin is twice as probable as the other two).
90
91Theory::
92
93Faq::
94<b>Difference between bin and point histograms:</b> A (continuous) bin
95histogram specifies bins of non-zero width, whereas a (discrete) point
96histogram specifies individual point values, which can be thought of
97as bins with zero width. In the terminology of LHS \cite Wyss1998,
98the bin pairs specification defines a
99"continuous linear" distribution and the point pairs specification
100defines a "discrete histogram" distribution (although the points are
101real-valued, the number of possible values is finite).
102
103See_Also::	variables-histogram_point_uncertain
104