{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T03:48:53Z","timestamp":1771472933362,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,18]],"date-time":"2020-08-18T00:00:00Z","timestamp":1597708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["18-13212S"],"award-info":[{"award-number":["18-13212S"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001823","name":"Ministerstvo \u0160kolstv\u00ed, Ml\u00e1de\u017ee a T\u011blov\u00fdchovy","doi-asserted-by":"publisher","award":["FAST-J-20-6417"],"award-info":[{"award-number":["FAST-J-20-6417"]}],"id":[{"id":"10.13039\/501100001823","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The paper is focused on Taylor series expansion for statistical analysis of functions of random variables with special attention to correlated input random variables. It is shown that the standard approach leads to significant deviations in estimated variance of non-linear functions. Moreover, input random variables are often correlated in industrial applications; thus, it is crucial to obtain accurate estimations of partial derivatives by a numerical differencing scheme. Therefore, a novel methodology for construction of Taylor series expansion of increasing complexity of differencing schemes is proposed and applied on several analytical examples. The methodology is adapted for engineering applications by proposed asymmetric difference quotients in combination with a specific step-size parameter. It is shown that proposed differencing schemes are suitable for functions of correlated random variables. Finally, the accuracy, efficiency, and limitations of the proposed methodology are discussed.<\/jats:p>","DOI":"10.3390\/sym12081379","type":"journal-article","created":{"date-parts":[[2020,8,19]],"date-time":"2020-08-19T21:37:40Z","timestamp":1597873060000},"page":"1379","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["On Taylor Series Expansion for Statistical Moments of Functions of Correlated Random Variables"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9387-2745","authenticated-orcid":false,"given":"Luk\u00e1\u0161","family":"Nov\u00e1k","sequence":"first","affiliation":[{"name":"Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Drahom\u00edr","family":"Nov\u00e1k","sequence":"additional","affiliation":[{"name":"Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,18]]},"reference":[{"key":"ref_1","first-page":"974","article-title":"A probability based structural code","volume":"66","author":"Cornell","year":"1969","journal-title":"J. Am. Concr. Inst."},{"key":"ref_2","unstructured":"Comit\u00e9 Europ\u00e9en de Normalisation (CEN) (2002). EN 1990: Eurocode: Basis of Structural Design, Comit\u00e9 Europ\u00e9en de Normalisation."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1061\/JMCEA3.0001848","article-title":"Exact and Invariant Second-moment Code Format","volume":"100","author":"Hasofer","year":"1974","journal-title":"J. Eng. Mech. Div."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1214\/aoms\/1177729394","article-title":"Remarks on a multivariate transformation","volume":"23","author":"Rosenblatt","year":"1952","journal-title":"Ann. Math. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.engstruct.2014.04.005","article-title":"Safety examination of existing concrete structures using the global resistance safety factor concept","volume":"70","author":"Pimentel","year":"2014","journal-title":"Eng. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1002\/suco.201200022","article-title":"Reliability-based non-linear analysis according to fib Model Code 2010","volume":"14","author":"Cervenka","year":"2013","journal-title":"Struct. Concr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/S0167-4730(96)00025-2","article-title":"Reliability evaluation in nonlinear analysis of reinforced concrete structures","volume":"19","author":"Val","year":"1997","journal-title":"Struct. Saf."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1002\/best.200810117","article-title":"Global Safety Format for Nonlinear Calculation of Reinforced Concrete","volume":"103","author":"Cervenka","year":"2008","journal-title":"Beton Und Stahlbetonbau"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.1016\/j.engstruct.2011.04.008","article-title":"Safety formats for nonlinear analysis tested on concrete beams subjected to shear forces and bending moments","volume":"33","author":"Schlune","year":"2011","journal-title":"Eng. Struct."},{"key":"ref_10","first-page":"31","article-title":"Safety format for the nonlinear analysis of concrete structures","volume":"25","author":"Bertagnoli","year":"2004","journal-title":"Studi e Ricerche- Politecnico di Milano. Scuola di Specializzazione in Costruzioni in Cemento Armato"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"McKay, M.D. (1992, January 13\u201316). Latin Hypercube Sampling as a Tool in Uncertainty Analysis of Computer Models. Proceedings of the 24th Conference on Winter Simulation, Arlington, VA, USA.","DOI":"10.1145\/167293.167637"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1749","DOI":"10.1080\/03610928008827996","article-title":"Small sample sensitivity analysis techniques for computer models with an application to risk assessment","volume":"9","author":"Iman","year":"1980","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1016\/j.ress.2007.04.002","article-title":"Global sensitivity analysis using polynomial chaos expansions","volume":"93","author":"Sudret","year":"2008","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.probengmech.2009.10.003","article-title":"An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis","volume":"25","author":"Blatman","year":"2010","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.strusafe.2011.01.002","article-title":"AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation","volume":"33","author":"Echard","year":"2011","journal-title":"Struct. Saf."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1007\/s00521-016-2485-3","article-title":"Reliability calculation of time-consuming problems using a small-sample artificial neural network-based response surface method","volume":"28","author":"Lehky","year":"2016","journal-title":"Neural Comput. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Melchers, R.E., and Beck, A.T. (2017). Second-Moment and Transformation Methods. Structural Reliability Analysis and Prediction, John Wiley & Sons, Ltd.. Chapter 4.","DOI":"10.1002\/9781119266105.ch4"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Paudel, A., Thapa, M., Gupta, S., Mulani, S.B., and Walters, R.W. (2020). Higher-Order Taylor Series Expansion with Efficient Sensitivity Estimation for Uncertainty Analysis. AIAA Aviation 2020 Forum, American Institute of Aeronautics and Astronautics.","DOI":"10.2514\/6.2020-3164"},{"key":"ref_19","unstructured":"Sykora, M., Cervenka, J., Cervenka, V., Mlcoch, J., Novak, D., and Novak, L. (2019, January 27\u201329). Pilot comparison of safety formats for reliability assessment of RC structures. Proceedings of the fib Symposium 2019: Concrete\u2014Innovations in Materials, Design and Structures, Krak\u00f3w, Poland."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Nov\u00e1k, L., Nov\u00e1k, D., and Pukl, R. (2019). Probabilistic and semi-probabilistic design of large concrete beams failing in shear. Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications, Taylor and Francis Group CRC Press.","DOI":"10.1201\/9780429426506-231"},{"key":"ref_21","unstructured":"Nov\u00e1k, D., Nov\u00e1k, L., Slowik, O., and Strauss, A. (2018, January 28\u201331). Prestressed concrete roof girders: Part III\u2014Semi-probabilistic design. Proceedings of the Sixth International Symposium on Life-Cycle Civil Engineering (IALCCE 2018), Ghent, Belgium."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.probengmech.2008.08.001","article-title":"An innovating analysis of the Nataf transformation from the copula viewpoint","volume":"24","author":"Lebrun","year":"2009","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/j.probengmech.2009.04.006","article-title":"Do Rosenblatt and Nataf isoprobabilistic transformations really differ?","volume":"24","author":"Lebrun","year":"2009","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_24","unstructured":"Comit\u00e9 Europ\u00e9en de Normalisation (CEN) (2004). EN 1992: Eurocode 2: Design of Concrete Structures, Comit\u00e9 Europ\u00e9en de Normalisation."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1379\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:02:46Z","timestamp":1760176966000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1379"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,18]]},"references-count":24,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["sym12081379"],"URL":"https:\/\/doi.org\/10.3390\/sym12081379","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,18]]}}}