{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T14:32:37Z","timestamp":1773498757112,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,30]],"date-time":"2023-08-30T00:00:00Z","timestamp":1693353600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Stochastic characterizations of functions subject to constraints result in treating them as functions with non-independent variables. By using the distribution function or copula of the input variables that comply with such constraints, we derive two types of partial derivatives of functions with non-independent variables (i.e., actual and dependent derivatives) and argue in favor of the latter. Dependent partial derivatives of functions with non-independent variables rely on the dependent Jacobian matrix of non-independent variables, which is also used to define a tensor metric. The differential geometric framework allows us to derive the gradient, Hessian, and Taylor-type expansions of functions with non-independent variables.<\/jats:p>","DOI":"10.3390\/axioms12090845","type":"journal-article","created":{"date-parts":[[2023,8,30]],"date-time":"2023-08-30T10:30:52Z","timestamp":1693391452000},"page":"845","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Derivative Formulas and Gradient of Functions with Non-Independent Variables"],"prefix":"10.3390","volume":"12","author":[{"given":"Matieyendou","family":"Lamboni","sequence":"first","affiliation":[{"name":"Department DFR-ST, University of Guyane, 97346 Cayenne, France"},{"name":"228-UMR Espace-Dev, University of Guyane, University of R\u00e9union, IRD, University of Montpellier, 34090 Montpellier, France"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1903","DOI":"10.1214\/aop\/1022677553","article-title":"Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures","volume":"27","author":"Bobkov","year":"1999","journal-title":"Ann. Probab."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3081","DOI":"10.1214\/17-EJS1310","article-title":"Poincar\u00e9 inequalities on intervals-application to sensitivity analysis","volume":"11","author":"Roustant","year":"2017","journal-title":"Electron. J. Statist."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107519","DOI":"10.1016\/j.ress.2021.107519","article-title":"Multivariate sensitivity analysis and derivative-based global sensitivity measures with dependent variables","volume":"212","author":"Lamboni","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/j.matcom.2021.12.019","article-title":"Weak derivative-based expansion of functions: ANOVA and some inequalities","volume":"194","author":"Lamboni","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_5","unstructured":"Russi, T.M. (2010). Uncertainty Quantification with Experimental Data and Complex System Models, University of California."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1500","DOI":"10.1137\/130916138","article-title":"Active subspace methods in theory and practice: Applications to kriging surfaces","volume":"36","author":"Constantine","year":"2014","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.jmaa.2005.08.072","article-title":"A global Implicit Function Theorem without initial point and its applications to control of non-affine systems of high dimensions","volume":"313","author":"Zhang","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1016\/j.jmaa.2017.07.058","article-title":"On global implicit function theorem","volume":"456","author":"Cristea","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Jost, J.J. (2011). Riemannian Geometry and Geometric Analysis, Springer.","DOI":"10.1007\/978-3-642-21298-7"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Petersen, P. (2016). Riemannian Geometry, Springer International Publishing AG.","DOI":"10.1007\/978-3-319-26654-1"},{"key":"ref_11","unstructured":"Sommer, S., Fletcher, T., and Pennec, X. (2020). Riemannian Geometric Statistics in Medical Image Analysis, Elsevier."},{"key":"ref_12","unstructured":"MITOpenCourseWare (2007). Non-Independent Variables, MIT Institute. Open Course."},{"key":"ref_13","first-page":"645","article-title":"On a representation of random variables","volume":"21","author":"Skorohod","year":"1976","journal-title":"Theory Probab. Appl."},{"key":"ref_14","unstructured":"Lamboni, M. (2021). On dependency models and dependent generalized sensitivity indices. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.matcom.2022.04.018","article-title":"Efficient dependency models: Simulating dependent random variables","volume":"200","author":"Lamboni","year":"2022","journal-title":"Math. Comput. Simul. MATCOM"},{"key":"ref_16","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. Statist."},{"key":"ref_17","first-page":"80","article-title":"The Comparison Method for Stochastic Processes","volume":"3","year":"1975","journal-title":"Ann. Probab."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s11009-023-09993-2","article-title":"On Exact Distribution for Multivariate Weighted Distributions and Classification","volume":"25","author":"Lamboni","year":"2023","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_19","unstructured":"Robert, C.P. (2007). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, Springer."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"125898","DOI":"10.1016\/j.jmaa.2021.125898","article-title":"On the class of truncation invariant bivariate copulas under constraints","volume":"509","author":"Durante","year":"2022","journal-title":"J. Math. Anal. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1214\/aoms\/1177728190","article-title":"Remarks on some nonparametric estimates of a density function","volume":"27","author":"Rosenblatt","year":"1956","journal-title":"Ann. Math. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1214\/aoms\/1177704472","article-title":"On estimation of a probability density function and mode","volume":"33","author":"Parzen","year":"1962","journal-title":"Ann. Math. Stat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1137\/1114019","article-title":"Nonparametric estimation of a multidimensional probability density","volume":"14","author":"Epanechnikov","year":"1969","journal-title":"Theory Probab. Appl."},{"key":"ref_24","unstructured":"McNeil, A.J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management, Princeton University Press."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Durante, F., and Sempi, C. (2015). Principles of copula theory, CRC\/Chapman & Hall.","DOI":"10.1201\/b18674"},{"key":"ref_26","first-page":"394","article-title":"On the Reciprocal of the General Algebraic Matrix","volume":"26","author":"Moore","year":"1920","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_27","first-page":"97","article-title":"General analysis, Part 1","volume":"1","author":"Moore","year":"1935","journal-title":"Mem. Amer. Phil. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1017\/S0305004100030401","article-title":"A generalized inverse for matrices","volume":"51","author":"Penrose","year":"1955","journal-title":"Proc. Cambrid. Phil. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/BF02193040","article-title":"Differential-geometric and variational background of classical gauge field theories","volume":"24","author":"Rund","year":"1982","journal-title":"Aequationes Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s00010-021-00859-x","article-title":"On the extremal compatible linear connection of a generalized Berwald manifold","volume":"96","author":"Vincze","year":"2022","journal-title":"Aequationes Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"126702","DOI":"10.1016\/j.jmaa.2022.126702","article-title":"Rigid properties for gradient generalized m-quasi-Einstein manifolds and gradient shrinking Ricci solitons","volume":"518","author":"YiHua","year":"2023","journal-title":"J. Math. Anal. Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/9\/845\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:43:08Z","timestamp":1760128988000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/9\/845"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,30]]},"references-count":31,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["axioms12090845"],"URL":"https:\/\/doi.org\/10.3390\/axioms12090845","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,30]]}}}