Dakota Reference Manual
Version 6.4
LargeScale Engineering Optimization and Uncertainty Analysis

Aleatory uncertain variable  continuous histogram
This keyword is related to the topics:
Alias: none
Argument(s): INTEGER
Default: no histogram bin uncertain variables
Required/Optional  Description of Group  Dakota Keyword  Dakota Keyword Description  

Optional  pairs_per_variable  Number of pairs defining each histogram bin variable  
Required  abscissas  Real abscissas for a bin histogram  
Required (Choose One)  Group 1  ordinates  Ordinates specifying a "skyline" probability density function  
counts  Frequency or relative probability of each bin  
Optional  initial_point  Initial values  
Optional  descriptors  Labels for the variables 
Histogram uncertain variables are typically used to model a set of empirical data. The bin histogram (contrast: histogram_point_uncertain) is a continuous aleatory distribution characterized by bins of nonzero 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 subkeywords are: abscissas (ranges of values the variable can take on) and either ordinates or 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, pairs_per_variable can be used to specify unequal apportionment of provided bin pairs among the variables.
The abscissas
specification defines abscissa values ("x" coordinates) for the probability density function of each histogram variable. When paired with counts
, the specifications provide sets of (x,c) pairs for each histogram variable where
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 ("y" coordinates) in order to support interpretation of the input as (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 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 binbased histogram with n
bins (potentially of unequal width), n+1
(x,c) or
(x,y) pairs must be specified with the following features:
x
is the parameter value for the left boundary of a histogram bin and c
is the corresponding count for that bin. Alternatively, 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.c
or y
value of zero.x
values must be strictly increasing.c
or y
values must be positive, except for the last which must be zero.The pairs_per_variable
specification provides for the proper association of multiple sets of (x,c) or
(x,y) pairs with individual histogram variables. For example, in this input snippet
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 (x,c) pairs from
abscissas
and counts
{
(5,17),(8,21),(10,0)} with one binbased 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 (x,c) pairs
{
(.1,12),(.2,24),(.3,12),(.4,0)} defines a second binbased histogram variable containing three equalwidth bins with counts 12, 24, and 12 (middle bin is twice as probable as the other two).
These keywords may also be of interest:
Difference between bin and point histograms: A (continuous) bin histogram specifies bins of nonzero 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[89], the bin pairs specification defines a "continuous linear" distribution and the point pairs specification defines a "discrete histogram" distribution (although the points are realvalued, the number of possible values is finite).