probability_levels
Specify probability levels at which to estimate the corresponding response value
Specification
Alias: None
Arguments: REALLIST
Default: No CDF/CCDF response levels to compute
Child Keywords:
Required/Optional |
Description of Group |
Dakota Keyword |
Dakota Keyword Description |
|---|---|---|---|
Optional |
Specify which |
||
Description
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.
Examples
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 5.1384774972e-12
3.4221494996e+11 4.0634975300e+11 5.1454122311e-12
4.0634975300e+11 5.4196114379e+11 2.4334239039e-12
PDF for response_fn_2:
Bin Lower Bin Upper Density Value
--------- --------- -------------
4.6431154744e+04 5.6511827775e+04 1.9839945149e-05
5.6511827775e+04 6.1603813790e+04 5.8916108390e-05
6.1603813790e+04 7.8702465755e+04 2.9242071306e-05
PDF for response_fn_3:
Bin Lower Bin Upper Density Value
--------- --------- -------------
2.3796737090e+05 3.6997214153e+05 5.3028386523e-06
3.6997214153e+05 3.8100966235e+05 9.0600055634e-06
3.8100966235e+05 4.4111498127e+05 3.3274925348e-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
-------------- ----------------- ----------------- -----------------
2.7604749078e+11 1.0000000000e+00
3.4221494996e+11 6.6000000000e-01
4.0634975300e+11 3.3000000000e-01
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 8.0000000000e-01
6.1603813790e+04 5.0000000000e-01
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 3.0000000000e-01
3.8100966235e+05 2.0000000000e-01
4.4111498127e+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.
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 evidence-based epistemic case, instead of the one result for each level mapping in the aleatory case).
