Moment-based Hermite model for asymptotically small non-Gaussianity

Denoël, Vincent

doi:10.1016/j.apm.2025.116061

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Moment-based Hermite model for asymptotically small non-Gaussianity

Denoël, Vincent

2025 • In Applied Mathematical Modelling, 144, p. 116061

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https://hdl.handle.net/2268/329293

DOI
10.1016/j.apm.2025.116061

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Keywords :

Gram-Charlier series expansion; Edgeworth expansion; Monotonic region; Wind pressure; Reliability; Fatigue analysis

Abstract :

[en] The third degree Moment-based Hermite model, which expresses a random variable as a cubic transformation of a standard normal variable, oﬀers versatility in engineering applications. While its probability density function is not directly tractable, it is more complex to compute than the Gram-Charlier series, which, despite its simplicity, suﬀers from limitations such as positivity and unimodality issues, restricting its range of applicability. This paper presents two asymptotic analyses of the cubic Moment-based Hermite model for slight non-Gaussianity (i.e. small skewness and excess coeﬃcients, “small” being understood in the sense of perturbation methods), showing that it asymptotically converges to the fourth cumulant Gram-Charlier model, while oﬀering a slightly broader domain of applicability with minimal additional computational cost. Additionally, the paper derives, mathematically, a non empirical expression for the monotone limit of the original cubic translation model, and validates the theoretical findings through numerical simulations.

Disciplines :

Civil engineering

Author, co-author :

Denoël, Vincent ; Université de Liège - ULiège > Département ArGEnCo > Analyse sous actions aléatoires en génie civil

Language :

English

Title :

Moment-based Hermite model for asymptotically small non-Gaussianity

Publication date :

August 2025

Journal title :

Applied Mathematical Modelling

ISSN :

0307-904X

eISSN :

1872-8480

Publisher :

Elsevier

Volume :

144

Pages :

116061

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S0307904X25001362?httpAccept=text/xml

Funders :

SU - Stanford University

Funding text :

UPS visiting professor grant (Stanford University)

Available on ORBi :

since 12 March 2025

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Bibliography

Anders, H., The early history of the cumulants and the Gram-Charlier series. Int. Stat. Rev. 68:2 (2000), 137–153.
Blinnikov, S., Moessner, R., Expansions for nearly Gaussian distributions. Astron. Astrophys. Suppl. Ser. 130:1 (1998), 193–205.
Draper, N.R., Tierney, D.E., Regions of positive and unimodal series expansion of the Edgeworth and Gram-Charlier approximations. Biometrika 59:2 (1972), 463–465.
Gullo, I., Muscolino, G., Vasta, M., Non-Gaussian probability density function of SDOF linear structures under wind actions. J. Wind Eng. Ind. Aerodyn. 74–76 (1998), 1123–1134.
Choi, M., Sweetman, B., The Hermite moment model for highly skewed response with application to tension leg platforms. J. Offshore Mech. Arct. Eng., 132(2), 2010, 021602.
Mahmoudi, L., Wang, W., Ikegaya, N., Comparing annual extreme winds in Iran predicted by numerical weather forecasting and Gram-Charlier statistical model with meteorological observation data. Build. Environ., 261, 2024.
Pender, J., Gram Charlier expansion for time varying multiserver queues with abandonment. SIAM J. Appl. Math. 74:4 (2014), 1238–1265.
Ñíguez, T.-M., Perote, J., Moments expansion densities for quantifying financial risk. N. Am. J. Econ. Finance 42 (2017), 53–69.
Zoia, M.G., Biffi, P., Nicolussi, F., Value at risk and expected shortfall based on Gram-Charlier-like expansions. J. Bank. Finance 93 (2018), 92–104.
Winterstein, S.R., Moment-based Hermite models of random vibration. Afdelingen for bærende konstruktioner, 1987, Danmarks tekniske højskole.
Wu, F., Liu, M., Yang, Q., Peng, L., Estimation of extremes of non-Gaussian wind pressure on building roof: Sampling error in moment-based translation process model with no monotonic limit. Adv. Struct. Eng. 23:4 (2020), 810–826.
Winterstein, S.R., Nonlinear vibration models for extremes and fatigue. J. Eng. Mech. 114:10 (1988), 1772–1790.
Winterstein, S.R., Kashef, T., Moment-based load and response models with wind engineering applications. J. Sol. Energy Eng. 122:3 (2000), 122–128.
Winterstein, S.R., MacKenzie, C.A., Extremes of nonlinear vibration: Comparing models based on moments, L-moments, and maximum entropy. J. Offshore Mech. Arct. Eng., 135(2), 2013, 021602.
Gioffrè, M., Gusella, V., Grigoriu, M., Non-Gaussian wind pressure on prismatic buildings. II: Numerical simulation. J. Struct. Eng. 127:9 (2001), 990–995.
Yang, L., Gurley, K.R., Prevatt, D.O., Probabilistic modeling of wind pressure on low-rise buildings. J. Wind Eng. Ind. Aerodyn. 114 (2013), 18–26.
Gurley, K.R., Tognarelli, M.A., Kareem, A., Analysis and simulation tools for wind engineering. Probab. Eng. Mech. 12:1 (1997), 9–31.
Gioffrè, M., Gusella, V., Grigoriu, M., Simulation of non-Gaussian field applied to wind pressure fluctuations. Probab. Eng. Mech. 15:4 (2000), 339–345.
Li, Y., Xu, J., An efficient methodology for simulating multivariate non-Gaussian stochastic processes. Mech. Syst. Signal Process., 209, 2024.
Zhang, X.-Y., Zhao, Y.-G., Lu, Z.-H., Unified Hermite polynomial model and its application in estimating non-Gaussian processes. J. Eng. Mech., 145(3), 2019.
Denoël, V., Bastin, T., Pomaranzi, G., Zasso, A., A bivariate statistical model of large pressure peaks measured in wind tunnel tests. Conference of the Italian Association for Wind Engineering, 2022, Springer, 288–297.
Rigo, F., Andrianne, T., Denoël, V., On the use of the cubic translation to model bimodal wind pressures. Math. Modell. Civ. Eng., 15(2), 2019.
Wang, J., Zheng, X., He, Q., Artificial intelligence applied to extreme value prediction of non-Gaussian processes with bandwidth effect and non-monotonicity. 2021 IEEE International Conference on Artificial Intelligence and Computer Applications (ICAICA), 2021, IEEE, 721–727.
Dang, C., Xu, J., Novel algorithm for reconstruction of a distribution by fitting its first-four statistical moments. Appl. Math. Model. 71 (2019), 505–524.
Zhao, Y.-G., Wang, T., Ji, X., Huang, G., Quartic Hermite polynomial model-based translation method for extreme wind load estimation. J. Wind Eng. Ind. Aerodyn., 245, 2024.
Li, Y., Xu, J., Novel Hermite polynomial model based on probability-weighted moments for simulating non-Gaussian stochastic processes. J. Eng. Mech., 150(4), 2024.
Chen, W., Tian, Y., A simplified analytical formula for the coefficient of 3rd-order Hermite moment model and its application. J. Wind Eng. Ind. Aerodyn., 254, 2024.
Calif, R., PDF models and synthetic model for the wind speed fluctuations based on the resolution of Langevin equation. Appl. Energy 99 (2012), 173–182.
Zhang, L.-W., An improved fourth-order moment reliability method for strongly skewed distributions. Struct. Multidiscip. Optim. 62:3 (2020), 1213–1225.
Wu, F., Huang, G., Liu, M., Simulation and peak value estimation of non-Gaussian wind pressures based on Johnson transformation model. J. Eng. Mech., 146(1), 2020.
Barton, D.E., Dennis, K.E., The conditions under which Gram-Charlier and Edgeworth curves are positive definite and unimodal. Biometrika 39:3/4 (1952), 425–427.
Wei, W., Koki, S., Naoki, I., Modelling probability density functions based on the Gram-Charlier series with higher-order statistics: Theoretical derivation and application. J. Wind Eng. Ind. Aerodyn., 231, 2022, 105227.
Lin, W., Zhang, J.E., The valid regions of Gram–Charlier densities with high-order cumulants. J. Comput. Appl. Math., 407, 2022, 113945.
Lin, W., Shen, K., Zhang, J.E., Further exploration into the valid regions of Gram–Charlier densities. J. Comput. Appl. Math., 429, 2023.
Kwon, O.K., Analytic expressions for the positive definite and unimodal regions of Gram-Charlier series. Commun. Stat., Theory Methods 51:15 (2022), 5064–5084.
Peng, X., Yang, L., Gavanski, E., Gurley, K., Prevatt, D.O., A comparison of methods to estimate peak wind loads on buildings. J. Wind Eng. Ind. Aerodyn. 126 (2014), 11–23.
Papoulis, A., Pillai, S.U., Probability, Random Variables, and Stochastic Processes. fourth edition, 2002, McGraw Hill, Boston.
Bender, C.M., Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers. 1978, McGraw-Hill.
Blaise, N., Andrianne, T., Denoël, V., Assessment of extreme value overestimations with equivalent static wind loads. J. Wind Eng. Ind. Aerodyn. 168 (2017), 123–133.