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