Mathematical approaches and software to solve the problem of sensitivity analysis by factors of tabularly specified mathematical models using remodeling concept

Abstract


Sensitivity Analysis of mathematical models involves a large number of approaches, among which there are local methods (studying the influence of a factor on the response in the case of its isolated variation) and global methods (involving the study of simultaneous fluctuations in groups of factors). Classification of methods is also based on the applied mathematical tools. However, the known methods are approximate or allow using surrogate models approximating the original function, which is a source of error.Previously, the authors proposed an analytical method for analyzing the sensitivity by factors of mathematical models based on Analysis of Finite Fluctuations. In this case, the well-known Lagrange mean value theorem is used to study fluctuations of function responce. However, in some situations the process of finding partial derivatives can be a computationally labor-intensive task, and in some cases the function is given tabularly. In this case, it is possible to use numerical differentiation with further restoration of the analytical representation of the function. For this purpose it is suggested to use the approach of mathematical remodeling and to apply linear regression models with interaction effects as a remodeling class. This assumption is natural, as it models the presence of a linear relationship between the model factors.The paper presents a numerical example - the analysis of the Rosenbrock function, performed in two ways: analytical method and using remodeling to recover partial derivatives. The results obtained show the high quality of the obtained sensitivity estimates, which testifies to the validity of the remodeling approach in such problems. Promising aspects of the presented approach are: application of a wider set of classes of remodeling models (fully connected neural networks, approximating polynomials) and optimal choice of the numerical differentiation step.

Full Text

1

About the authors

A. S Sysoev

Lipetsk State Technical University

A. I Miroshnikov

Lipetsk State Technical University

P. V Saraev

Lipetsk State Technical University

References

  1. Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models / A. Saltelli, S. Tarantola, F. Campolongo, M. Ratto. – Hoboken, John Wiley & Sons Ltd., 2004 – 232 p.
  2. Global Sensitivity Analysis: The Primer / A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, S. Tarantola. – John Wiley & Sons: Chichester, UK, 2008. – 305 p.
  3. Sobol, I.M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates / I.M. Sobol // Math. Comput. Simul. – 2001. – Vol. 55. – P. 271–280.
  4. Lamboni, M. Multivariate sensitivity analysis and derivative-based global sensitivity measures with dependent variables / M. Lamboni, S. Kucherenko // Reliab. Eng. Syst. Saf. – 2021. – Vol. 212. – P. 107519.
  5. Borgonovo, E. Model emulation and moment-independent sensitivity analysis: An application to environmental modelling / E. Borgonovo, W. Castaings, S. Tarantola // Environ. Model. Softw. – 2012. – Vol. 34. – P. 105–115.
  6. Rana, S. An efficient assisted history matching and uncertainty quantification workflow using Gaussian processes proxy models and variogram based sensitivity analysis: GP-VARS / S. Rana, T. Ertekin, G.R. King // Comput. Geosci. – 2018. – Vol. 114. – P. 73–83.
  7. Pujol, G. Simplex-based screening designs for estimating metamodels / G. Pujol // Reliab. Eng. Syst. Saf. – 2009. – Vol. 94. – P. 1156–1160.
  8. Hamby, D.M. A comparison of sensitivity analysis techniques / D. M. Hamby // Health Phys. – 1995. – Vol. 68. – P. 195–204.
  9. Box, G.E. An analysis for unreplicated fractional factorials / G. E. Box, R. D. Meyer // Technometrics. – 1986. – Vol. 28. – P. 11–18.
  10. Dean, A. Screening: Methods for Experimentation in Industry, Drug Discovery, and Genetics / A. Dean, S. Lewis. – Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006. – 234 p.
  11. Kurowicka, D. Uncertainty analysis with high dimensional dependence modeling / D. Kurowicka, D.M. Cooke. – John Wiley & Sons. – 2006. – 304 p.
  12. Christensen, R. Linear models for multivariate, time series, and spatial data / R. Christensen. – New York: Springer-Verlag, 1991. – Vol. 1. – P. 354–355.
  13. Cacuci, D.G. Sensitivity and uncertainty analysis, volume II: applications to large-scale systems / D.G. Cacuci, M. Ionescu-Bujor, I.M. Navon. – CRC press, 2005. – 368 p.
  14. Sobol, I.M. On sensitivity estimation for nonlinear mathematical models / I.M. Sobol // Matematicheskoe modelirovanie, 1990. – Vol. 2, iss. 1. – P. 112–118.
  15. Efron, B. The jacknife estimate of variance / B. Efron, C. Stein // Ann. Stat. – 1981. – Vol. 9. – P. 586–596.
  16. Блюмин, С.Л. Экономический факторный анализ: Монография / С.Л. Блюмин, В.Ф. Суханов, С.В. Чеботарев. – Липецк: ЛЭГИ, 2004. – 148 с.
  17. Analysis of finite fluctuations for solving big data management problems / S.L. Blyumin, G.S. Borovkova, K.V. Serova, A.S. Sysoev // 2015 9th International Conference on Application of Information and Communication Technologies (AICT). – IEEE, 2015. – P. 48–51. doi: 10.1109/ICAICT.2015.7338514
  18. Sensitivity analysis of neural network models: Applying methods of analysis of finite fluctuations / A. Sysoev, A. Ciurlia, R. Sheglevatych, S. Blyumin // Periodica polytechnica Electrical engineering and computer science. – 2019. – Vol. 63(4). – P. 306–311.
  19. Применение несимметричного ортогонального планирования при исследовании процессов прокатки / С.Л. Блюмин, В.Г. Барышев, С.Л. Коцарь, Б.А. Поляков // Применение ЭВМ в металлургии: I Всесоюзн. науч.-техн. конф. – М.: МИСиС, 1973. – С. 118–119.
  20. Surrogate Modeling of Buckling Analysis in Support of Composite Structure Optimization / S. Grihon, S. Alestra, E. Burnaev, P. Prikhodko // 1st International Conference on Composite Dynamics, Arcachon, France, 2012.
  21. Zhao, D. A multi-surrogate approximation method for metamodeling / D. Zhao, D. Xue // Engineering with Computers, 2011. – Vol. 27(2).
  22. Co-simulation: State of the Art. [arXiv:1702.00686] / C. Gomes, C. Thule, D. Broman, P. Larsen, H. Vangheluwe. – 2017.
  23. Benureau, F. Re-run, Repeat, Reproduce, Reuse, Replicate: Transforming Code into Scientific Contributions. [arXiv:1708.08205] / F. Benureau, N. Rougier. – 2017.

Statistics

Views

Abstract - 42

PDF (Russian) - 31

Refbacks

  • There are currently no refbacks.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies