Dakota Reference Manual
Version 6.4
LargeScale Engineering Optimization and Uncertainty Analysis

Specify probability levels at which to estimate the corresponding response value
Alias: none
Argument(s): REALLIST
Default: No CDF/CCDF response levels to compute
Required/Optional  Description of Group  Dakota Keyword  Dakota Keyword Description  

Optional  num_probability_levels  Specify which probability_levels correspond to which response 
Response levels are calculated for specified CDF/CCDF probabilities by indexing into a sorted samples array (the response levels computed are not interpolated and will correspond to one of the sampled values).
Expected Output
If probability_levels
are specified, Dakota will create two tables in the standard output: a Probability Density function (PDF) histogram and a Cumulative Distribution Function (CDF) table. The PDF histogram has the lower and upper endpoints of each bin and the corresponding density of that bin. Note that the PDF histogram has bins defined by the probability_levels
and/or response_levels
in the Dakota input file. If there are not very many levels, the histogram will be coarse. Dakota does not do anything to optimize the bin size or spacing. The CDF table has the list of response levels and the corresponding probability that the response value is less than or equal to each response level threshold.
The Dakota input file below specifies a sampling method with probability levels of interest.
method, sampling, samples = 100 seed = 1 complementary distribution probability_levels = 1. .66 .33 0. 1. .8 .5 0. 1. .3 .2 0. variables, normal_uncertain = 2 means = 248.89, 593.33 std_deviations = 12.4, 29.7 descriptors = 'TF1n' 'TF2n' uniform_uncertain = 2 lower_bounds = 199.3, 474.63 upper_bounds = 298.5, 712. descriptors = 'TF1u' 'TF2u' weibull_uncertain = 2 alphas = 12., 30. betas = 250., 590. descriptors = 'TF1w' 'TF2w' histogram_bin_uncertain = 2 num_pairs = 3 4 abscissas = 5 8 10 .1 .2 .3 .4 counts = 17 21 0 12 24 12 0 descriptors = 'TF1h' 'TF2h' histogram_point_uncertain real = 1 num_pairs = 2 abscissas = 3 4 counts = 1 1 descriptors = 'TF3h' interface, system asynch evaluation_concurrency = 5 analysis_driver = 'text_book' responses, response_functions = 3 no_gradients no_hessians
Given the above Dakota input file, the following excerpt from the output shows the PDF and CCDF generated. Note that the bounds on the bins of the PDF are the response values that correspond the probability levels specified in the input file. Those response values are also shown in the CCDF.
Probability Density Function (PDF) histograms for each response function: PDF for response_fn_1: Bin Lower Bin Upper Density Value    2.7604749078e+11 3.4221494996e+11 5.1384774972e12 3.4221494996e+11 4.0634975300e+11 5.1454122311e12 4.0634975300e+11 5.4196114379e+11 2.4334239039e12 PDF for response_fn_2: Bin Lower Bin Upper Density Value    4.6431154744e+04 5.6511827775e+04 1.9839945149e05 5.6511827775e+04 6.1603813790e+04 5.8916108390e05 6.1603813790e+04 7.8702465755e+04 2.9242071306e05 PDF for response_fn_3: Bin Lower Bin Upper Density Value    2.3796737090e+05 3.6997214153e+05 5.3028386523e06 3.6997214153e+05 3.8100966235e+05 9.0600055634e06 3.8100966235e+05 4.4111498127e+05 3.3274925348e06 Level mappings for each response function: Complementary Cumulative Distribution Function (CCDF) for response_fn_1: Response Level Probability Level Reliability Index General Rel Index     2.7604749078e+11 1.0000000000e+00 3.4221494996e+11 6.6000000000e01 4.0634975300e+11 3.3000000000e01 5.4196114379e+11 0.0000000000e+00 Complementary Cumulative Distribution Function (CCDF) for response_fn_2: Response Level Probability Level Reliability Index General Rel Index     4.6431154744e+04 1.0000000000e+00 5.6511827775e+04 8.0000000000e01 6.1603813790e+04 5.0000000000e01 7.8702465755e+04 0.0000000000e+00 Complementary Cumulative Distribution Function (CCDF) for response_fn_3: Response Level Probability Level Reliability Index General Rel Index     2.3796737090e+05 1.0000000000e+00 3.6997214153e+05 3.0000000000e01 3.8100966235e+05 2.0000000000e01 4.4111498127e+05 0.0000000000e+00
Sets of responseprobability pairs computed with the forward/inverse mappings define either a cumulative distribution function (CDF) or a complementary cumulative distribution function (CCDF) for each response function.
In the case of evidencebased epistemic methods, this is generalized to define either cumulative belief and plausibility functions (CBF and CPF) or complementary cumulative belief and plausibility functions (CCBF and CCPF) for each response function.
An inverse mapping involves computing the belief and plausibility response level for either a specified probability level or a specified generalized reliability level (two results for each level mapping in the evidencebased epistemic case, instead of the one result for each level mapping in the aleatory case).