Dakota Reference Manual
Version 6.4
LargeScale Engineering Optimization and Uncertainty Analysis

Values at which to estimate desired statistics for each response
Alias: none
Argument(s): REALLIST
Default: No CDF/CCDF probabilities/reliabilities to compute
Required/Optional  Description of Group  Dakota Keyword  Dakota Keyword Description  

Optional  num_response_levels  Number of values at which to estimate desired statistics for each response  
Optional  compute  Selection of statistics to compute at each response level 
The response_levels
specification provides the target response values for which to compute probabilities, reliabilities, or generalized reliabilities (forward mapping).
Default Behavior
If response_levels
are not specified, no statistics will be computed. If they are, probabilities will be computed by default.
Expected Outputs
If response_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.
Usage Tips
The num_response_levels
is used to specify which arguments of the response_level
correspond to which response.
For example, specifying a response_level
of 52.3 followed with compute
probabilities
will result in the calculation of the probability that the response value is less than or equal to 52.3, given the uncertain distributions on the inputs.
For an example with multiple responses, the following specification
response_levels = 1. 2. .1 .2 .3 .4 10. 20. 30. num_response_levels = 2 4 3
would assign the first two response levels (1., 2.) to response function 1, the next four response levels (.1, .2, .3, .4) to response function 2, and the final three response levels (10., 20., 30.) to response function 3. If the num_response_levels
key were omitted from this example, then the response levels would be evenly distributed among the response functions (three levels each in this case).
The Dakota input file below specifies a sampling method with response levels of interest.
method, sampling, samples = 100 seed = 1 complementary distribution response_levels = 3.6e+11 4.0e+11 4.4e+11 6.0e+04 6.5e+04 7.0e+04 3.5e+05 4.0e+05 4.5e+05 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 specified in the input file. The probability levels corresponding to those response values are 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 3.6000000000e+11 5.3601733194e12 3.6000000000e+11 4.0000000000e+11 4.2500000000e12 4.0000000000e+11 4.4000000000e+11 3.7500000000e12 4.4000000000e+11 5.4196114379e+11 2.2557612778e12 PDF for response_fn_2: Bin Lower Bin Upper Density Value    4.6431154744e+04 6.0000000000e+04 2.8742313192e05 6.0000000000e+04 6.5000000000e+04 6.4000000000e05 6.5000000000e+04 7.0000000000e+04 4.0000000000e05 7.0000000000e+04 7.8702465755e+04 1.0341896485e05 PDF for response_fn_3: Bin Lower Bin Upper Density Value    2.3796737090e+05 3.5000000000e+05 4.2844660868e06 3.5000000000e+05 4.0000000000e+05 8.6000000000e06 4.0000000000e+05 4.5000000000e+05 1.8000000000e06 Level mappings for each response function: Complementary Cumulative Distribution Function (CCDF) for response_fn_1: Response Level Probability Level Reliability Index General Rel Index     3.6000000000e+11 5.5000000000e01 4.0000000000e+11 3.8000000000e01 4.4000000000e+11 2.3000000000e01 Complementary Cumulative Distribution Function (CCDF) for response_fn_2: Response Level Probability Level Reliability Index General Rel Index     6.0000000000e+04 6.1000000000e01 6.5000000000e+04 2.9000000000e01 7.0000000000e+04 9.0000000000e02 Complementary Cumulative Distribution Function (CCDF) for response_fn_3: Response Level Probability Level Reliability Index General Rel Index     3.5000000000e+05 5.2000000000e01 4.0000000000e+05 9.0000000000e02 4.5000000000e+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.
A forward mapping involves computing the belief and plausibility probability level for a specified response level.