response_levels
Values at which to estimate desired statistics for each response
Specification
Alias: None
Arguments: REALLIST
Default: No CDF/CCDF probabilities/reliabilities to compute
Child Keywords:
Required/Optional |
Description of Group |
Dakota Keyword |
Dakota Keyword Description |
|---|---|---|---|
Optional |
Number of values at which to estimate desired statistics for each response |
||
Optional |
Selection of statistics to compute at each response level |
||
Description
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.
Examples
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 5.3601733194e-12
3.6000000000e+11 4.0000000000e+11 4.2500000000e-12
4.0000000000e+11 4.4000000000e+11 3.7500000000e-12
4.4000000000e+11 5.4196114379e+11 2.2557612778e-12
PDF for response_fn_2:
Bin Lower Bin Upper Density Value
--------- --------- -------------
4.6431154744e+04 6.0000000000e+04 2.8742313192e-05
6.0000000000e+04 6.5000000000e+04 6.4000000000e-05
6.5000000000e+04 7.0000000000e+04 4.0000000000e-05
7.0000000000e+04 7.8702465755e+04 1.0341896485e-05
PDF for response_fn_3:
Bin Lower Bin Upper Density Value
--------- --------- -------------
2.3796737090e+05 3.5000000000e+05 4.2844660868e-06
3.5000000000e+05 4.0000000000e+05 8.6000000000e-06
4.0000000000e+05 4.5000000000e+05 1.8000000000e-06
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 5.5000000000e-01
4.0000000000e+11 3.8000000000e-01
4.4000000000e+11 2.3000000000e-01
Complementary Cumulative Distribution Function (CCDF) for response_fn_2:
Response Level Probability Level Reliability Index General Rel Index
-------------- ----------------- ----------------- -----------------
6.0000000000e+04 6.1000000000e-01
6.5000000000e+04 2.9000000000e-01
7.0000000000e+04 9.0000000000e-02
Complementary Cumulative Distribution Function (CCDF) for response_fn_3:
Response Level Probability Level Reliability Index General Rel Index
-------------- ----------------- ----------------- -----------------
3.5000000000e+05 5.2000000000e-01
4.0000000000e+05 9.0000000000e-02
4.5000000000e+05 0.0000000000e+00
Theory
Sets of response-probability 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 evidence-based 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.
