# A Comparison of Surrogate Modeling Techniques for Global Sensitivity Analysis in Hybrid Simulation

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## Abstract

**:**

## 1. Introduction

## 2. Surrogate Modeling Methods

#### 2.1. Polynomial Chaos Expansion

#### 2.1.1. Polynomial Basis

#### 2.1.2. Truncation Schemes

#### 2.1.3. PCE Coefficient Calculation

#### 2.2. Kriging

#### 2.2.1. Trend Families

#### 2.2.2. Autocorrelation Functions

#### 2.2.3. Hyperparameter Estimation

#### 2.2.4. Optimization Methods

#### 2.3. Polynomial Chaos Kriging

#### 2.4. Leave-One-Out Cross-Validation Error

## 3. Global Sensitivity Analysis with Sobol’ Indices

#### 3.1. Sobol’–Hoeffding Decomposition

#### 3.2. Sobol’ Indices

#### 3.3. Sobol’ Indices Estimation

#### 3.3.1. Monte Carlo-Based Estimation

#### 3.3.2. Sobol’ Indices from Polynomial Chaos Expansion

## 4. Case Study

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Tsokanas, N.; Wagg, D.; Stojadinović, B. Robust Model Predictive Control for Dynamics Compensation in Real-Time Hybrid Simulation. Front. Built Environ.
**2020**, 6, 127. [Google Scholar] [CrossRef] - Tsokanas, N.; Pastorino, R.; Stojadinovic, B. Adaptive model predictive control for actuation dynamics compensation in real-time hybrid simulation. engrXiv
**2021**. [Google Scholar] [CrossRef] - Li, H.; Maghareh, A.; Montoya, H.; Condori Uribe, J.W.; Dyke, S.J.; Xu, Z. Sliding mode control design for the benchmark problem in real-time hybrid simulation. Mech. Syst. Signal Process.
**2021**, 151, 107364. [Google Scholar] [CrossRef] - Simpson, T.; Dertimanis, V.K.; Chatzi, E.N. Towards Data-Driven Real-Time Hybrid Simulation: Adaptive Modeling of Control Plants. Front. Built Environ.
**2020**, 6, 158. [Google Scholar] [CrossRef] - Tsokanas, N.; Simpson, T.; Pastorino, R.; Chatzi, E.; Stojadinovic, B. Model Order Reduction for Real-Time Hybrid Simulation: Comparing Polynomial Chaos Expansion and Neural Network methods. engrXiv
**2021**. [Google Scholar] [CrossRef] - Miraglia, G.; Petrovic, M.; Abbiati, G.; Mojsilovic, N.; Stojadinovic, B. A model-order reduction framework for hybrid simulation based on component-mode synthesis. Earthq. Eng. Struct. Dyn.
**2020**, 49, 737–753. [Google Scholar] [CrossRef] - Schellenberg, A.H.; Mahin, S.A.; Fenves, G.L. Advanced Implementation of Hybrid Simulation; Technical Report PEER 2009/104; Pacific Earthquake Engineering Research Center, University of California: Berkeley, CA, USA, 2009. [Google Scholar]
- Tsokanas, N. Real-Time and Stochastic Hybrid Simulation. Ph.D. Thesis, ETH Zurich, Zurich, Switzerland, 2021. [Google Scholar] [CrossRef]
- Abbiati, G.; Lanese, I.; Cazzador, E.; Bursi, O.S.; Pavese, A. A computational framework for fast-time hybrid simulation based on partitioned time integration and state-space modeling. Struct. Control Health Monit.
**2019**, 26, e2419. [Google Scholar] [CrossRef] - Abbiati, G.; Covi, P.; Tondini, N.; Bursi, O.S.; Stojadinović, B. A Real-Time Hybrid Fire Simulation Method Based on Dynamic Relaxation and Partitioned Time Integration. J. Eng. Mech.
**2020**, 146, 04020104. [Google Scholar] [CrossRef] - Song, W.; Sun, C.; Zuo, Y.; Jahangiri, V.; Lu, Y.; Han, Q. Conceptual Study of a Real-Time Hybrid Simulation Framework for Monopile Offshore Wind Turbines Under Wind and Wave Loads. Front. Built Environ.
**2020**, 6, 129. [Google Scholar] [CrossRef] - Idinyang, S.; Franza, A.; Heron, C.M.; Marshall, A.M. Real-time data coupling for hybrid testing in a geotechnical centrifuge. Int. J. Phys. Model. Geotech.
**2018**, 19, 208–220. [Google Scholar] [CrossRef] - Tsokanas, N.; Abbiati, G.; Kanellopoulos, K.; Stojadinovic, B. Multi-Axial Hybrid Fire Testing based on Dynamic Relaxation. Fire Saf. J.
**2021**, 126, 103468. [Google Scholar] [CrossRef] - Tsokanas, N.; Stojadinovic, B. A stochastic real-time hybrid simulation of the seismic response of a magnetorheological damper. In Proceedings of the 17th World Conference on Earthquake Engineering (17WCEE 2020), Sendai, Japan, 13–18 September 2020. [Google Scholar] [CrossRef]
- Tsokanas, N.; Stojadinovic, B. Design of an Actuation Controller for Physical Substructures in Stochastic Real-Time Hybrid Simulations. In Model Validation and Uncertainty Quantification; Mao, Z., Ed.; Society for Experimental Mechanics Series; Springer International Publishing: Cham, Switzerland, 2020; Volume 3, pp. 69–82. [Google Scholar] [CrossRef]
- Abbiati, G.; Marelli, S.; Tsokanas, N.; Sudret, B.; Stojadinović, B. A global sensitivity analysis framework for hybrid simulation. Mech. Syst. Signal Process.
**2021**, 146, 106997. [Google Scholar] [CrossRef] - Tsokanas, N.; Zhu, X.; Abbiati, G.; Marelli, S.; Sudret, B.; Stojadinović, B. A Global Sensitivity Analysis Framework for Hybrid Simulation with Stochastic Substructures. Front. Built Environ.
**2021**, 7, 154. [Google Scholar] [CrossRef] - Saltelli, A.; Ratto, M.; Andres, T.; Campolongo, F.; Cariboni, J.; Gatelli, D.; Saisana, M.; Tarantola, S. Global Sensitivity Analysis. The Primer; John Wiley & Sons, Ltd.: Chichester, UK, 2007; Volume 76. [Google Scholar]
- Sudret, B. Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf.
**2008**, 93, 964–979. [Google Scholar] [CrossRef] - Le Gratiet, L.; Marelli, S.; Sudret, B. Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes. In Handbook of Uncertainty Quantification; Ghanem, R., Higdon, D., Owhadi, H., Eds.; Springer International Publishing: Cham, Switzerland, 2017; pp. 1289–1325. [Google Scholar]
- Ghanem, R.; Spanos, P.D. Polynomial Chaos in Stochastic Finite Elements. J. Appl. Mech.
**1990**, 57, 197–202. [Google Scholar] [CrossRef] - Xiu, D.; Karniadakis, G.E. The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations. SIAM J. Sci. Comput.
**2002**, 24, 619–644. [Google Scholar] [CrossRef] - Spiridonakos, M.D.; Chatzi, E.N. Metamodeling of dynamic nonlinear structural systems through polynomial chaos NARX models. Comput. Struct.
**2015**, 157, 99–113. [Google Scholar] [CrossRef] - Torre, E.; Marelli, S.; Embrechts, P.; Sudret, B. Data-driven polynomial chaos expansion for machine learning regression. J. Comput. Phys.
**2019**, 388, 601–623. [Google Scholar] [CrossRef][Green Version] - Marelli, S.; Sudret, B. UQLab User Manual—Polynomial Chaos Expansions; Technical Report # UQLab-V1.3-104; Risk, Safety and Uncertainty Quantification, ETH Zurich: Zurich, Switzerland, 2019. [Google Scholar]
- Blatman, G.; Sudret, B. Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys.
**2011**, 230, 2345–2367. [Google Scholar] [CrossRef] - Berveiller, M.; Sudret, B.; Lemaire, M. Stochastic finite element: A non intrusive approach by regression. Eur. J. Comput. Mech.
**2006**, 15, 81–92. [Google Scholar] [CrossRef][Green Version] - Eldred, M.; Webster, C.; Constantine, P. Evaluation of Non-Intrusive Approaches for Wiener-Askey Generalized Polynomial Chaos. In Proceedings of the 49th AIAA Structures, Structural Dynamics, and Materials Conference, Schaumburg, IL, USA, 7–10 April 2008. [Google Scholar]
- Zhang, J.; Yue, X.; Qiu, J.; Zhuo, L.; Zhu, J. Sparse polynomial chaos expansion based on Bregman-iterative greedy coordinate descent for global sensitivity analysis. Mech. Syst. Signal Process.
**2021**, 157, 107727. [Google Scholar] [CrossRef] - Santner, T.J.; Williams, B.J.; Notz, W.I. The Design and Analysis of Computer Experiments; Springer Series in Statistics; Springer: New York, NY, USA, 2003. [Google Scholar]
- Rasmussen, C.E.; Williams, C.K.I. Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning); The MIT Press: Cambridge, MA, USA, 2005. [Google Scholar]
- Lataniotis, C.; Wicaksono, D.; Marelli, S.; Sudret, B. UQLab User Manual —Kriging (Gaussian Process Modeling); Technical Report # UQLab-V1.3-105; Risk, Safety and Uncertainty Quantification, ETH Zurich: Zurich, Switzerland, 2019. [Google Scholar]
- Dubourg, V. Adaptive Surrogate Models for Reliability Analysis and Reliability-Based Design Optimization. Ph.D. Thesis, Université Blaise Pascal-Clermont-Ferrand II, Clermont-Ferrand, France, 2011. [Google Scholar]
- Goldberg, D.E.; Holland, J.H. Genetic Algorithms and Machine Learning. Mach. Learn.
**1988**, 3, 95–99. [Google Scholar] [CrossRef] - Schobi, R.; Sudret, B.; Wiart, J. Polynomial-chaos-based Kriging. Int. J. Uncertain. Quantif.
**2015**, 5, 171–193. [Google Scholar] [CrossRef] - Schöbi, R.; Sudret, B.; Marelli, S. Rare Event Estimation Using Polynomial-Chaos Kriging. ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civ. Eng.
**2017**, 3, D4016002. [Google Scholar] [CrossRef][Green Version] - Kersaudy, P.; Sudret, B.; Varsier, N.; Picon, O.; Wiart, J. A new surrogate modeling technique combining Kriging and polynomial chaos expansions—Application to uncertainty analysis in computational dosimetry. J. Comput. Phys.
**2015**, 286, 103–117. [Google Scholar] [CrossRef][Green Version] - Schöbi, R.; Marelli, S.; Sudret, B. UQLab User Manual—Polynomial Chaos Kriging; Technical Report # UQLab-V1.3-109; Risk, Safety and Uncertainty Quantification, ETH Zurich: Zurich, Switzerland, 2019. [Google Scholar]
- Sobol’, I.M. Sensitivity analysis for non-linear mathematical models. Math. Model. Comput. Exp.
**1993**, 1, 407–414. [Google Scholar] - Marelli, S.; Lamas, C.; Konakli, K.; Mylonas, C.; Wiederkehr, P.; Sudret, B. UQLab User Manual—Sensitivity Analysis; Technical Report # UQLab-V1.3-106; Risk, Safety and Uncertainty Quantification, ETH Zurich: Zurich, Switzerland, 2019. [Google Scholar]
- Pinheiro, S.M. Motorcycle Modeling for eCVT-in-the-Loop Real-Time Hybrid Testing. Master’s Thesis, University of Porto, Porto, Portugal, 2020. [Google Scholar]
- Kimishima, T.; Nakamura, T.; Suzuki, T. The Effects on Motorcycle Behavior of the Moment of Inertia of the Crankshaft. SAE Trans.
**1997**, 106, 1993–2003. [Google Scholar] - Tanelli, M. Modelling, Simulation and Control of Two-Wheeled Vehicles; John Wiley & Sons: New, York, NY, USA, 2014. [Google Scholar]
- Sharp, R.; Evangelou, S.; Limebeer, D. Advances in the Modelling of Motorcycle Dynamics. Multibody Syst. Dyn.
**2004**, 12, 251–283. [Google Scholar] [CrossRef][Green Version] - Jia, S.; Li, Q. Friction-induced vibration and noise on a brake system. In Proceedings of the 2013 IEEE International Conference on Information and Automation (ICIA), Yinchuan, China, 26–28 August 2013; pp. 489–492. [Google Scholar] [CrossRef]
- Mckay, M.D.; Beckman, R.J.; Conover, W.J. A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code. Technometrics
**2000**, 42, 55–61. [Google Scholar] [CrossRef] - Marelli, S.; Sudret, B. UQLab: A Framework for Uncertainty Quantification in Matlab. In Proceedings of the 2nd International Conference on Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool, UK, 13–16 July 2014; pp. 2554–2563. [Google Scholar] [CrossRef][Green Version]

**Figure 4.**Sample vHS time history response of the motorcycle prototype hybrid model, using the mean (nominal) values of Table 1. The QoI are highlighted in red color.

**Figure 5.**Indicative hybrid model responses, using the mean (nominal) values of Table 1: (

**a**) angular velocity of the rear wheel, (

**b**) braking torque of the rear wheel, (

**c**) angular velocity of the transmission output shaft, (

**d**) torque of the transmission output shaft, (

**e**) torque of the engine and (

**f**) angular velocity of the engine.

**Figure 6.**Convergence of LOO error estimates of (

**a**) ${v}_{\mathrm{max}}$ and (

**b**) ${v}_{\mathrm{mean}}$ surrogates.

**Figure 7.**Comparison between QoI measurements and corresponding surrogate predictions for (

**a**) ${v}_{\mathrm{max}}$ and (

**b**) ${v}_{\mathrm{mean}}$.

**Figure 8.**Convergence of PCE-based moments estimates of (

**a**) mean, (

**b**) standard deviation and (

**c**) CV of QoI ${v}_{\mathrm{max}}$.

**Figure 9.**Convergence of PCE-based moments estimates of (

**a**) mean, (

**b**) standard deviation and (

**c**) CV of QoI ${v}_{\mathrm{mean}}$.

**Figure 10.**Comparison between histogram measurements and corresponding surrogate estimates for (

**a**) ${v}_{\mathrm{max}}$ and (

**b**) ${v}_{\mathrm{mean}}$.

**Figure 11.**First-order and total Sobol’ indices for (

**a**,

**b**) ${v}_{\mathrm{max}}$ and (

**c**,

**d**) ${v}_{\mathrm{mean}}$.

Param. | Prob. Distrib. | Mean Value | Stand. Dev. | CV (%) | Parameter Description | Units |
---|---|---|---|---|---|---|

${K}_{rt}$ | Unif. | 58,570 | 11,714 | 20 | Vertical stiffness rear tire | $\frac{\mathrm{N}}{\mathrm{m}}$ |

${Z}_{rt}$ | Unif. | 11,650 | 3495 | 30 | Vertical damping rear tire | $\frac{\mathrm{N}\mathrm{s}}{\mathrm{m}}$ |

${K}_{ft}$ | Unif. | 25,000 | 5000 | 20 | Vertical stiffness front tire | $\frac{\mathrm{N}}{\mathrm{m}}$ |

${Z}_{ft}$ | Unif. | 2134 | $640.2$ | 30 | Vertical damping front tire | $\frac{\mathrm{N}\mathrm{s}}{\mathrm{m}}$ |

${K}_{rs}$ | Unif. | 125,000 | 25,000 | 20 | Stiffness rear suspension | $\frac{\mathrm{N}}{\mathrm{m}}$ |

${Z}_{rs}$ | Unif. | 10,000 | 3000 | 30 | Damping rear suspension | $\frac{\mathrm{N}\mathrm{s}}{\mathrm{m}}$ |

${K}_{fs}$ | Unif. | 19,000 | 3800 | 20 | Stiffness front suspension | $\frac{\mathrm{N}}{\mathrm{m}}$ |

${Z}_{fs}$ | Unif. | 1250 | 375 | 30 | Damping front suspension | $\frac{\mathrm{N}\mathrm{s}}{\mathrm{m}}$ |

M | Unif. | 300 | 15 | 5 | Motorcycle mass | $\genfrac{}{}{0.0pt}{}{\mathrm{Kg}}{}$ |

J | Unif. | $0.0115$ | $0.0023$ | 20 | Engine moment of inertia | $\genfrac{}{}{0.0pt}{}{{\mathrm{Kgm}}^{2}}{}$ |

$E\mu $ | Unif. | $0.0012$ | $0.00018$ | 15 | Engine coefficient of viscous friction | $\frac{\mathrm{Nm}}{\mathrm{rev}/\mathrm{min}}$ |

$B\mu $ | Unif. | $0.13$ | $0.0195$ | 15 | Brake coefficient of viscous friction | $\frac{\mathrm{Nm}}{\mathrm{rad}/\mathrm{s}}$ |

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**MDPI and ACS Style**

Tsokanas, N.; Pastorino, R.; Stojadinović, B. A Comparison of Surrogate Modeling Techniques for Global Sensitivity Analysis in Hybrid Simulation. *Mach. Learn. Knowl. Extr.* **2022**, *4*, 1-21.
https://doi.org/10.3390/make4010001

**AMA Style**

Tsokanas N, Pastorino R, Stojadinović B. A Comparison of Surrogate Modeling Techniques for Global Sensitivity Analysis in Hybrid Simulation. *Machine Learning and Knowledge Extraction*. 2022; 4(1):1-21.
https://doi.org/10.3390/make4010001

**Chicago/Turabian Style**

Tsokanas, Nikolaos, Roland Pastorino, and Božidar Stojadinović. 2022. "A Comparison of Surrogate Modeling Techniques for Global Sensitivity Analysis in Hybrid Simulation" *Machine Learning and Knowledge Extraction* 4, no. 1: 1-21.
https://doi.org/10.3390/make4010001