Derivative-based Global Sensitivity Measures

Derivative Based Global Sensitivity Measures (DGSM) is a technique used in global sensitivity analysis to identify the importance of different subsets of input variables to variation in model output. It is gaining popularity among practitioners due to its link with Sobol’ sensitivity indices and easiness of implementation. The computational time required for DGSM is usually much less than that needed for estimating Sobol’ sensitivity indices.

## Overview

The variance-based method of Sobol’ sensitivity indices [1] is one of the most efficient and popular global sensitivity analysis techniques. It provides information on the importance of different subsets of input variables to the output variance. However, the direct implementation of variance-based global sensitivity analysis measures can be time-consuming and impractical for models with high dimensionality.

DGSM is based on averaging of local derivatives using Monte Carlo or Quasi Monte Carlo sampling methods. Unlike the Morris method,[2] which uses finite differences to calculate elementary effects with increments comparable to the variable uncertainty ranges, DGSM evaluates strict local derivatives with smaller increments. Moreover, derivatives are calculated at randomly or quasi-randomly selected points within the full range of uncertainty, rather than points from a fixed grid. Consequently, DGSM is a more accurate technique than the Morris method. In was suggested as a practical tool in,[3] while the theory and the link between DGSM and Sobol' senstitivity indices was developed in,.[4][5] Other important developments were given in,[6][7],.[8]

DGSM encompasses upper and lower bounds on the values of Sobol’ total sensitivity indices,[9] offering a comprehensive understanding of the model's behavior. This approach has gained popularity due to its link with Sobol’ sensitivity indices and its ease of implementation. One of the key advantages of DGSM is its relatively lower computational time compared to estimating Sobol’ sensitivity indices, making it a practical choice for sensitivity analysis in high-dimensional models.

DGSM is closely related to the active subspace method,[10] providing a deeper understanding of the model's behavior and reducing computational costs. The relationship between the activity scores of the active subspace method and DGSM has been a subject of interest, further highlighting the utility of DGSM in practical applications.

The numerical efficiency of the DGSM method can be improved by using automatic differentiation for calculations as was shown in.[11] However, the number of required function evaluations still remains to be proportional to the number of inputs. This dependence can be greatly reduced using an approach based on algorithmic differentiation in the adjoint (reverse) mode,.[12][13] It allows estimating all derivatives at a cost at most 4-6 times of that for evaluating the original function.

## References

1. ↑ Sobol, I. (1993). Sensitivity estimates for non linear mathematical models. Mathematical Modelling and Computational Experiments, 1:407–414.
2. ↑ Morris, M. (1991). Factorial sampling plans for preliminary computational experiments. Technometrics, 33:161–174.
3. ↑ Kucherenko, S., Rodriguez-Fernandez, M., Pantelides, C., and Shah, N. (2009). Monte carlo evaluation of Bibliography derivative-based global sensitivity measures. Reliability Engineering and System Safety, 94:1135–1148.
4. ↑ Sobol, I. and Kucherenko, S. (2009). Derivative based global sensitivity measures and their links with global sensitivity indices. Mathematics and Computers in Simulation, 79:3009–3017.
5. ↑ Sobol, I. and Kucherenko, S. (2010). A new derivative based importance criterion for groups of variables and its link with the global sensitivity indices. Computer Physics Communications, 181:1212 – 1217
6. ↑ Lamboni, M., Iooss, B., Popelin, A.-L., and Gamboa, F. (2013). Derivative-based global sensitivity measures: general links with sobol’ indices and numerical tests. Mathematics and Computers in Simulation, 87:45–54.
7. ↑ Kucherenko, S. and Song, S. (2016). Derivative-based global sensitivity measures and their link with sobol’sensitivity indices. In Monte Carlo and Quasi-Monte Carlo Methods: MCQMC, Leuven, Belgium, April 2014, pages 455–469. Springer.
8. ↑ Lamboni, M. and Kucherenko, S. (2021). Multivariate sensitivity analysis and derivative-based global sensitivity measures with dependent variables. Reliability Engineering & System Safety, 212:107519.
9. ↑ Homma, T. and Saltelli, A. (1996). Importance measures in global sensitivity analysis of non linear models. Reliability Engineering and System Safety, 52:1–17.
10. ↑ Constantine, P. G. and Diaz, P. (2017). Global sensitivity metrics from active subspaces. Reliability Engineering & System Safety, 162:1–13.
11. ↑ Kiparissides, A., Kucherenko, S., Mantalaris, A., and Pistikopoulos, E. (2009). Global sensitivity analysis challenges in biological systems modeling. Journal of Industrial and Engineering Chemistry Research, 48:1135–1148.
12. ↑ Griewank, A. and Walther, A. (2008). Evaluating derivatives: Principles and techniques of automatic differentiation. SIAM Philadelphia.
13. ↑ Molkenthin, C., Scherbaum, F., Griewank, A., Leovey, H., Kucherenko, S., and Cotton, F. (2017). Derivative based global sensitivity analysis: Upper bounding of sensitivities in seismic-hazard assessment using automatic differentiation. Bulletin of the Seismological Society of America, 107(2):984–1004.

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