In this repo, I implement algorithm 5.1 from Plischke, Rabitti and Borgonovo (2020). The implementation differs on whether model inputs are dependent or not. I translate the algorithms from MATLAB to Python and implement corresponding test cases, which are taken from Iooss and Prieur (2019) and Plischke, Rabitti and Borgonovo (2020).
The implementations of algorithm 5.1 in the case of independent and dependent inputs are located in the modules
shapley_moebius_independent.py and shapley_moebius_dependent.py. Test programmes can be found in the scripts starting with
test_. I chose this structure because it is simple. auxiliary.py contains functions needed for test programmes.
For the tests, I calculated expected values with the help of the original MATLAB implementations. I do not upload
those scripts but I provide the expected values that were generated by the MATLAB code. Expected results can be found in
the directory \data.
Warning: Not all tests pass. For an additive model, the implementation of the algorithm with dependent inputs fails the tests. Even the MATLAB implementation yielded wrong results. Expected values were taken from Iooss and Prieur (2019). I checked the analytical expected results from Iooss and Prieur (2019) by another algorithm for the computation of Shapley effects.
Shapley effects were introduced as a sensitivity measure by Owen (2014). They yield information about the input-output relationship for models which are considered a black box. For instance, a model that can be solved by numerical methods only, has no analytical solution. Thus, it is not clear, how the output depends on the inputs. Sensitivity analysis tries to shed light on this relationship. Shapley effects give the contribution to the reduction of output variance for each input. Shapley effects incorporate effects due to interactions and input dependence.
I thank Dr. Elmar Plischke for providing me with the MATLAB implementation of algorithm 5.1 for both, dependent and independent inputs.
Iooss, Betrand and Clémentine Prieur. 2019. Shapley effects for sensitivity analysis with correlated inputs: comparisons with Sobol’ indices, numerical estimation and applications. hal-01556303v6
Owen, Art B. 2014. Sobol’ indices and shapley value. SIAM/ASA Journal on Uncertainty Quantification, 2(1):245–251.
Plischke, Elmar, Giovanni Rabitti, Emanuele Borgonovo. 2020. Computing Shapley Effects for Sensitivity Analysis. arXiv.