Gaussian Process Regression (GPR)-based missing data imputation and its uses for bridge structural health monitoring

Abstract

Structural health monitoring (SHM) apparatuses rely on continuous measurement and analysis to assess the safety condition of a target system. However, in field applications, the SHM framework is often hampered by practical issues. Among them, missing data in recorded time series is arguably the most common and most disruptive challenge that can arise. Therefore, imputing missing values is necessary to maintain the integrity and utility of the SHM data. This research work investigates the use of Gaussian Process Regression (GPR) for imputing missing data in ordered time series. In particular, this approach is here proposed and tested for Vibration-Based Monitoring (VBM) and ambient monitoring, with applications to modal parameters and air temperature. Both punctual missing-at-random (MAR) and prolonged missing-not-at-random (MNAR) gaps in the time histories of recorded natural frequencies are analysed. The performance of the proposed GPR-based approach is evaluated on real-life data from field tests on a well-known case study, the KW51 rail bridge. The method is first tested to actual missing values in the dataset. Then, the accuracy is tested using artificially removed data, and the imputed values are compared to the ground truth (i.e., the actual measured data). In the first case, the results show that the complete time series are deemed qualitatively similar to what would be expected by an expert user. The outcomes of the second part quantitatively demonstrate that GPR can accurately impute missing data in modal parameter time series, preserving the statistical properties of the data.

References

[1] 1994–2024 The MathWorks, I. (n.d.). Gaussian Process Regression Models. https://It.Mathworks.Com/Help/Stats/Gaussian-Process-Regression-Models.Html.

[2] Al-GhalibAA, MahmoudSM. Structural Condition Classification of Railway Bridge KW51 Before, During, and After Retrofitting. Ce/papers, 2023, 6(5): 840-847

[3] Anastasopoulos, D. (2020). Structural health monitoring based on operational modal analysis from long gauge dynamic strain measurements. KU Leuven.

[4] Anastasopoulos D, Maes K, De Roeck G, Lombaert G, Reynders EPB (2023) Structural health monitoring of the KW51 bridge based on detailed strain mode shapes: Environmental influences versus simulated damage. In Life-Cycle of Structures and Infrastructure Systems (pp. 391–398). CRC Press. https://doi.org/10.1201/9781003323020-45

[5] AssolieA. Advanced modeling techniques using hierarchical gaussian process regression in civil engineering. Asian Journal of Civil Engineering, 2024, 25(7): 5599-5612

[6] Rasmussen C E, & Nickisch H, (2020, July 14). Documentation for GPML Matlab Code version 4.2. https://Gaussianprocess.Org/Gpml/Code/Matlab/Doc/-Last. Access 15/09/2024

[7] CiveraM, BoscatoG, Zanotti FragonaraL. Treed gaussian process for manufacturing imperfection identification of pultruded GFRP thin-walled profile. Compos Struct, 2020, 254, ArticleID: 112882

[8] CiveraM, SuraceC. A Comparative Analysis of Signal Decomposition Techniques for Structural Health Monitoring on an Experimental Benchmark. Sensors, 2021, 21(5): 1825

[9] Civera M, Surace C, Worden K (2017) Detection of Cracks in Beams Using Treed Gaussian Processes (pp. 85–97). https://doi.org/10.1007/978-3-319-54109-9_10

[10] CrossEJ, MansonG, WordenK, PierceSG. Features for damage detection with insensitivity to environmental and operational variations. Proceedings of the Royal Society a: Mathematical, Physical and Engineering Sciences, 2012, 468(2148): 4098-4122

[11] CrossEJ, WordenK, ChenQ. Cointegration: a novel approach for the removal of environmental trends in structural health monitoring data. Proceedings of the Royal Society a: Mathematical, Physical and Engineering Sciences, 2011, 467(2133): 2712-2732

[12] DanielsMJ, HoganJW. Missing Data in Longitudinal Studies. Chapman and Hall/CRC, 2008

[13] DingY, LiA. Temperature-induced variations of measured modal frequencies of steel box girder for a long-span suspension bridge. International Journal of Steel Structures, 2011, 11(2): 145-155

[14] Emmanuel T, Maupong T, Mpoeleng D, Semong T, Mphago B, Tabona O (2021) A survey on missing data in machine learning. J. Big Data, 8(1). https://doi.org/10.1186/s40537-021-00516-9

[15] ErlerNS, RizopoulosD, van RosmalenJ, JaddoeVWV, FrancoOH, LesaffreEMEH. Dealing with missing covariates in epidemiologic studies: a comparison between multiple imputation and a full Bayesian approach. Stat Med, 2016, 35(17): 2955-2974

[16] Farrar CR, Baker WE, Bell TM, Cone KM, Darling TW, Duffey TA, Eklund A, Migliori A (1994) Dynamic Characterization and Damage Detection in the I-40 Bridge Over the Rio Grande.

[17] Faruqi MA, Roy S, Salem A (2012) Elevated temperature deflection behavior of concrete members reinforced with FRP bars. https://doi.org/10.1177/1042391512447045, 22(3), 183–196. https://doi.org/10.1177/1042391512447045

[18] HoangN-D, PhamA-D, NguyenQ-L, PhamQ-N. Estimating Compressive Strength of High Performance Concrete with Gaussian Process Regression Model. Advances in Civil Engineering, 2016, 2016: 1-8

[19] IbrahimJG, ChenM-H, LipsitzSR, HerringAH. Missing-Data Methods for Generalized Linear Models. J Am Stat Assoc, 2005, 100(469): 332-346

[20] Jie Wang (2024) An Intuitive Tutorial to Gaussian Process Regression. Computing in Science & Engineering.

[21] Kim D, Kim D-Y, Ko T, Hwan Lee S (2024) Physics-informed Gaussian process regression model for predicting the fatigue life of welded joints. International Journal of Fatigue, 108644. https://doi.org/10.1016/j.ijfatigue.2024.108644

[22] LinWC, TsaiCF. Missing value imputation: a review and analysis of the literature (2006–2017). Artif Intell Rev, 2020, 53(2): 1487-1509

[23] Luo Y (2022) Evaluating the state of the art in missing data imputation for clinical data. Brief. Bioinforma., 23(1), bbab489. https://doi.org/10.1093/bib/bbab489

[24] MaZ, ChenG. Bayesian methods for dealing with missing data problems. Journal of the Korean Statistical Society, 2018, 47(3): 297-313

[25] Maes K, Lombaert G (2021) Monitoring Railway Bridge KW51 Before, During, and After Retrofitting. Journal of Bridge Engineering, 26(3). https://doi.org/10.1061/(ASCE)BE.1943-5592.0001668

[26] MaesK, Van MeerbeeckL, ReyndersEPB, LombaertG. Validation of vibration-based structural health monitoring on retrofitted railway bridge KW51. Mech Syst Signal Process, 2022, 165, ArticleID: 108380

[27] Mason A, Best N, Plewis I, Richardson S (2010a) Insights into the use of Bayesian models for informative missing data. In Technical report. Imperial College.

[28] Mason A, Richardson S, Plewis I, Best N (2010b) Strategy for modelling non-random missing data mechanisms in observational studies using Bayesian methods. Technical Report.

[29] OmidinasabF, Sahraei MoghadamA, DowlatshahiMB. Predictive model for shear strength estimation in reinforced concrete beams with recycled aggregates using Gaussian process regression. Neural Comput Appl, 2023, 35(11): 8487-8503

[30] PalM, DeswalS. Modelling pile capacity using Gaussian process regression. Comput Geotech, 2010, 37(7–8): 942-947

[31] PozziM, WangQ. Gaussian Process Regression and Classification for Probabilistic Damage Assessment of Spatially Distributed Systems. KSCE J Civ Eng, 2018, 22(3): 1016-1026

[32] Rasmussen CE (2004) Gaussian Processes in Machine Learning (pp. 63–71). https://doi.org/10.1007/978-3-540-28650-9_4

[33] Rasmussen CE, De HM (2010) Gaussian Processes for Machine Learning (GPML) Toolbox Hannes Nickisch. Journal of Machine Learning Research, 11, 3011–3015. http://www.kyb.tuebingen.mpg.de/bs/people/carl/code/minimize/.

[34] RoyS, UnobeI, SorensenAD. Vehicle-Impact Damage of Reinforced Concrete Bridge Piers: A State-of-the Art Review. J Perform Constr Facil, 2021, 35(5): 03121001

[35] SaidaT, RashidM, NishioM. System fragility analysis of highway bridge using multi-output Gaussian process regression surrogate model. Adv Struct Eng, 2024, 27(16): 2803-2822

[36] SamuiP, JaganJ. Determination of effective stress parameter of unsaturated soils: A Gaussian process regression approach. Front Struct Civ Eng, 2013, 7(2): 133-136

[37] Shadbahr T, Roberts M, Stanczuk J, Gilbey J, Teare P, Dittmer S, Thorpe M, Torné RV, Sala E, Lió P, Patel M, Preller J, Selby I, Breger A, Weir-McCall JR, Gkrania-Klotsas E, Korhonen A, Jefferson E, Langs G, … Schönlieb C-B (2023) The impact of imputation quality on machine learning classifiers for datasets with missing values. Communications Medicine 2023 3:1, 3(1), 1–15. https://doi.org/10.1038/s43856-023-00356-z

[38] SohnH. Effects of environmental and operational variability on structural health monitoring. Philosophical Transactions of the Royal Society a: Mathematical, Physical and Engineering Sciences, 2007, 365(1851): 539-560

[39] TsudaY, TomizawaY, YoshidaI, WadaM, SuemasaN, OtakeY. Estimation of engineering bedrock layer utilizing ground surface elevation in Gaussian process regression. Comput Geotech, 2023, 160, ArticleID: 105548

[40] van de Velde M, Maes K, Lombaert G (2023) Modelling the Nonlinear Behavior of the Pot Bearings of Railway Bridge KW51. Journal of Bridge Engineering, 28(6). https://doi.org/10.1061/JBENF2.BEENG-6144

[41] WordenK, BeckerWE, RogersTJ, CrossEJ. On the confidence bounds of Gaussian process NARX models and their higher-order frequency response functions. Mech Syst Signal Process, 2018, 104: 188-223

[42] WordenK, SuraceC, BeckerW. Uncertainty Bounds on Higher-Order FRFs from Gaussian Process NARX Models. Procedia Engineering, 2017, 199: 1994-2000

[43] WuX, YangF, HuangS. Predicting CBR values using gaussian process regression and meta-heuristic algorithms in geotechnical engineering. Multiscale and Multidisciplinary Modeling, Experiments and Design, 2024, 7(4): 3799-3813

Funding

-(This work is part of the research activity developed by the authors within the framework of the “PNRR”: SPOKE 7 “CCAM)

RIGHTS & PERMISSIONS

The Author(s)