Dakota Reference Manual  Version 6.4 Large-Scale Engineering Optimization and Uncertainty Analysis
probability_levels

Specify probability levels at which to estimate the corresponding response value

## Specification

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

## 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_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).