Dakota Reference Manual: response_levels

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response_levels

Values at which to estimate desired statistics for each response

## Specification

Alias: none

Argument(s): REALLIST

Default: No CDF/CCDF probabilities/reliabilities to compute

Child Keywords:

| 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

## 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.