{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,21]],"date-time":"2026-09-21T09:35:47Z","timestamp":1789983347286,"version":"4.0.1"},"reference-count":142,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,4,15]],"date-time":"2015-04-15T00:00:00Z","timestamp":1429056000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Entropy is a measure of uncertainty and has been commonly used for various applications, including probability inferences in hydrology. Copula has been widely used for constructing joint distributions to model the dependence structure of multivariate hydrological random variables. Integration of entropy and copula theories provides new insights in hydrologic modeling and analysis, for which the development and application are still in infancy. Two broad branches of integration of the two concepts, entropy copula and copula entropy, are introduced in this study. On the one hand, the entropy theory can be used to derive new families of copulas based on information content matching. On the other hand, the copula entropy provides attractive alternatives in the nonlinear dependence measurement even in higher dimensions. We introduce in this study the integration of entropy and copula theories in the dependence modeling and analysis to illustrate the potential applications in hydrology and water resources.<\/jats:p>","DOI":"10.3390\/e17042253","type":"journal-article","created":{"date-parts":[[2015,4,15]],"date-time":"2015-04-15T10:51:24Z","timestamp":1429095084000},"page":"2253-2280","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":47,"title":["Integrating Entropy and Copula Theories for Hydrologic Modeling and Analysis"],"prefix":"10.3390","volume":"17","author":[{"given":"Zengchao","family":"Hao","sequence":"first","affiliation":[{"name":"Green Development Institute and School of Environmental Science, Beijing Normal University,  Beijing 100875, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vijay","family":"Singh","sequence":"additional","affiliation":[{"name":"Department of Biological and Agricultural Engineering and Zachry Department of Civil Engineering , Texas A&M University, College Station, TX 77843-2117, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,4,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"780","DOI":"10.1061\/(ASCE)HE.1943-5584.0000621","article-title":"Entropy-based method for bivariate drought analysis","volume":"18","author":"Hao","year":"2013","journal-title":"J. Hydrol. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Favre, A.C., El Adlouni, S., Perreault, L., Thi\u00e9monge, N., and Bob\u00e9e, B. (2004). Multivariate hydrological frequency analysis using copulas. Water Resour. Res., 40, doi: 01110.01029\/02003WR002456.","DOI":"10.1029\/2003WR002456"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.jhydrol.2006.06.033","article-title":"Bivariate rainfall frequency distributions using archimedean copulas","volume":"332","author":"Zhang","year":"2007","journal-title":"J. Hydrol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1061\/(ASCE)1084-0699(2007)12:4(431)","article-title":"Trivariate flood frequency analysis using the gumbel-hougaard copula","volume":"12","author":"Zhang","year":"2007","journal-title":"J. Hydrol. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/S0022-1694(99)00168-7","article-title":"The gumbel mixed model for flood frequency analysis","volume":"226","author":"Yue","year":"1999","journal-title":"J. Hydrol."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Singh, V.P. (2013). Entropy Theory and Its Application in Environmental and Water Engineering, Wiley.","DOI":"10.1002\/9781118428306"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1061\/(ASCE)1084-0699(2007)12:4(347)","article-title":"Everything you always wanted to know about copula modeling but were afraid to ask","volume":"12","author":"Genest","year":"2007","journal-title":"J. Hydrol. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1061\/(ASCE)HE.1943-5584.0000332","article-title":"Hydrologic synthesis using entropy theory: Review","volume":"16","author":"Singh","year":"2011","journal-title":"J. Hydrol. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1002\/(SICI)1099-1085(199705)11:6<587::AID-HYP479>3.0.CO;2-P","article-title":"The use of entropy in hydrology and water resources","volume":"11","author":"Singh","year":"1997","journal-title":"Hydrol. Process."},{"key":"ref_10","unstructured":"Singh, V.P. (2015). Introduction to Entropy Theory in Hydrologic Science and Engineering, McGraw-Hill Education."},{"key":"ref_11","unstructured":"Nelsen, R.B. (2006). An Introduction to Copulas, Springer."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"761","DOI":"10.5194\/npg-15-761-2008","article-title":"Multivariate non-normally distributed random variables in climate research\u2013introduction to the copula approach","volume":"15","author":"Schoelzel","year":"2008","journal-title":"Nonlinear Process. Geophys."},{"key":"ref_13","first-page":"530","article-title":"Copula Modeling for Extremes","volume":"2","author":"Piegorsch","year":"2012","journal-title":"Encyclopedia of Environmetrics"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1061\/(ASCE)1084-0699(2007)12:4(369)","article-title":"On the use of copulas in hydrology: Theory and practice","volume":"12","author":"Salvadori","year":"2007","journal-title":"J. Hydrol. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hao, Z., and Singh, V.P. (2015). Drought characterization from a multivariate perspective: A review. J. Hydrol., submitted.","DOI":"10.1016\/j.jhydrol.2015.05.031"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"68003","DOI":"10.1209\/0295-5075\/88\/68003","article-title":"An information-theoretic approach to statistical dependence: Copula information","volume":"88","author":"Calsaverini","year":"2009","journal-title":"Europhys. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1824","DOI":"10.1016\/j.jbankfin.2010.12.011","article-title":"Recovering copulas from limited information and an application to asset allocation","volume":"35","author":"Chu","year":"2011","journal-title":"J. Bank. Financ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1007\/s10463-013-0422-0","article-title":"On the construction of minimum information bivariate copula families","volume":"66","author":"Bedford","year":"2014","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"W06604","DOI":"10.1029\/2011WR011419","article-title":"Entropy-copula method for single-site monthly streamflow simulation","volume":"48","author":"Hao","year":"2012","journal-title":"Water Resour. Res."},{"key":"ref_20","unstructured":"Hao, Z. (2012). Application of Entropy Theory in Hydrologic Analysis and Simulation. [Ph.D Thesis, Texas A&M University, College Station]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"747","DOI":"10.3390\/e16020747","article-title":"Modelling and simulation of seasonal rainfall using the principle of maximum entropy","volume":"16","author":"Borwein","year":"2014","journal-title":"Entropy"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Hao, Z., and Singh, V.P. (2013). Modeling multi-site streamflow dependence with maximum entropy copula. Water Resour. Res., 49.","DOI":"10.1002\/wrcr.20523"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"833","DOI":"10.1007\/s00477-014-0978-0","article-title":"Maximum entropy-gumbel-hougaard copula method for simulation of monthly streamflow in Xiangxi River, China","volume":"29","author":"Kong","year":"2014","journal-title":"Stoch. Env. Res. Risk Assess"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1080\/09715010.2013.804697","article-title":"Development of an entropy-copula-based stochastic simulation model for generation of monthly inflows into the Hirakud Dam","volume":"19","author":"Zachariah","year":"2013","journal-title":"ISH J. Hyd. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Singh, V.P. (1998). Entropy-Based Parameter Estimation in Hydrology, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-017-1431-0"},{"key":"ref_26","unstructured":"Kapur, J. (1989). Maximum-Entropy Models in Science and Engineering, Wiley."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Frank, S.A., and Smith, E. (2010). A simple derivation and classification of common probability distributions based on information symmetry and measurement scale. J. Evol. Biol., 469\u2013484.","DOI":"10.1111\/j.1420-9101.2010.02204.x"},{"key":"ref_28","first-page":"263","article-title":"Exponential series estimation of empirical copulas with application to financial returns","volume":"25","author":"Chui","year":"2009","journal-title":"Adv. Econ."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2258","DOI":"10.1016\/j.patrec.2010.05.001","article-title":"Link between copula and tomography","volume":"31","author":"Pougaza","year":"2010","journal-title":"Pattern Recogn. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"305","DOI":"10.3934\/jimo.2007.3.305","article-title":"Matching the grade correlation coefficient using a copula with maximum disorder","volume":"3","author":"Piantadosi","year":"2007","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/s11590-010-0254-2","article-title":"Copulas with maximum entropy","volume":"6","author":"Piantadosi","year":"2012","journal-title":"Optim. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1080\/00949659708811806","article-title":"Minimally informative distributions with given rank correlation for use in uncertainty analysis","volume":"57","author":"Meeuwissen","year":"1997","journal-title":"J. Stat. Comput. Sim."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1142\/S0219024907004391","article-title":"Empirical copulas for cdo tranche pricing using relative entropy","volume":"10","author":"Dempster","year":"2007","journal-title":"Int. J. Theor. Appl. Financ."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"W09528","DOI":"10.1029\/2010WR010208","article-title":"Single-site monthly streamflow simulation using entropy theory","volume":"47","author":"Hao","year":"2011","journal-title":"Water Resour. Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S1007-0214(11)70008-6","article-title":"Mutual information is copula entropy","volume":"16","author":"Ma","year":"2011","journal-title":"Tsinghua Sci. Technol."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Alfonso, L., Lobbrecht, A., and Price, R. (2010). Information theory\u2013based approach for location of monitoring water level gauges in polders. Water Resour. Res., 46.","DOI":"10.1029\/2009WR008101"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advwatres.2011.11.007","article-title":"Entropy based derivation of probability distributions: A case study to daily rainfall","volume":"45","author":"Papalexiou","year":"2012","journal-title":"Adv. Water Resour"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1029\/2011JD017394","article-title":"Entropy-based method for extreme rainfall analysis in texas","volume":"118","author":"Hao","year":"2013","journal-title":"J. Geophys. Res.-Atmos."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1007\/s00477-008-0286-7","article-title":"Entropy-based parameter estimation for extended burr xii distribution","volume":"23","author":"Hao","year":"2009","journal-title":"Stoch. Env. Res. Risk Assess"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2847","DOI":"10.1029\/93WR00923","article-title":"Minimum relative entropy: Forward probabilistic modeling","volume":"29","author":"Woodbury","year":"1993","journal-title":"Water Resour. Res."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"W12425","DOI":"10.1029\/2005WR004082","article-title":"On minimum relative entropy concepts and prior compatibility issues in vadose zone inverse and forward modeling","volume":"41","author":"Hou","year":"2005","journal-title":"Water Resour. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2671","DOI":"10.1029\/95WR03818","article-title":"Minimum relative entropy inversion: Theory and application to recovering the release history of a groundwater contaminant","volume":"32","author":"Woodbury","year":"1996","journal-title":"Water Resour. Res."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.jhydrol.2009.03.006","article-title":"An entropy-based investigation into the variability of precipitation","volume":"370","author":"Mishra","year":"2009","journal-title":"J. Hydrol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1061\/(ASCE)HE.1943-5584.0000150","article-title":"Association between uncertainties in meteorological variables and water-resources planning for the state of texas","volume":"16","author":"Mishra","year":"2010","journal-title":"J. Hydrol. Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"870","DOI":"10.1061\/(ASCE)HE.1943-5584.0000683","article-title":"Regionalization of drought characteristics using an entropy approach","volume":"18","author":"Rajsekhar","year":"2012","journal-title":"J. Hydrol. Eng."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Li, C., Singh, V.P., and Mishra, A.K. (2012). Entropy theory-based criterion for hydrometric network evaluation and design: Maximum information minimum redundancy. Water Resour. Res., 48.","DOI":"10.1029\/2011WR011251"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"05014003","DOI":"10.1061\/(ASCE)HE.1943-5584.0000971","article-title":"Variability in canadian seasonal streamflow information and its implication for hydrometric network design","volume":"19","author":"Mishra","year":"2014","journal-title":"J. Hydrol. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1016\/j.jhydrol.2009.11.015","article-title":"Hydrometric network evaluation for canadian watersheds","volume":"380","author":"Mishra","year":"2010","journal-title":"J. Hydrol."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communications","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J"},{"key":"ref_50","unstructured":"Shannon, C., and Weaver, W. (1949). The Mathematical Theory of Communication, University Illinois Press."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Singh, V.P. (2014). Entropy Theory in Hydraulic Engineering: An Introduction, ASCE Press.","DOI":"10.1061\/9780784412725"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1103\/PhysRev.108.171","article-title":"Information theory and statistical mechanics. II","volume":"108","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_54","unstructured":"Kesavan, H., and Kapur, J. (1992). Entropy Optimization Principles with Applications, Academic Press."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"2404","DOI":"10.1063\/1.526446","article-title":"Maximum entropy in the problem of moments","volume":"25","author":"Mead","year":"1984","journal-title":"J. Math. Phys."},{"key":"ref_56","unstructured":"Cover, T., and Thomas, J. (1991). Elementary of Information Theory, Wiley."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1175\/JHM-D-13-053.1","article-title":"Comparing evaporative sources of terrestrial precipitation and their extremes in merra using relative entropy","volume":"15","author":"Dirmeyer","year":"2014","journal-title":"J. Hydrometeorol."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"2057","DOI":"10.1175\/1520-0469(2002)059<2057:MDPUUR>2.0.CO;2","article-title":"Measuring dynamical prediction utility using relative entropy","volume":"59","author":"Kleeman","year":"2002","journal-title":"J. Atmos. Sci."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.jhydrol.2010.02.016","article-title":"A multiscale information theory approach to assess spatial\u2013temporal variability of daily precipitation","volume":"385","author":"Brunsell","year":"2010","journal-title":"J. Hydrol."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"425","DOI":"10.4310\/MAA.2002.v9.n3.a8","article-title":"A mathematical framework for quantifying predictability through relative entropy","volume":"9","author":"Majda","year":"2002","journal-title":"Methods Appl. Anal."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"DelSole, T., and Tippett, M.K. (2007). Predictability: Recent insights from information theory. Rev. Geophys., 45.","DOI":"10.1029\/2006RG000202"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"3368","DOI":"10.1175\/JAS3522.1","article-title":"Predictability and information theory. Part II: Imperfect forecasts","volume":"62","author":"DelSole","year":"2005","journal-title":"J. Atmos. Sci."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"2425","DOI":"10.1175\/1520-0469(2004)061<2425:PAITPI>2.0.CO;2","article-title":"Predictability and information theory. Part I: Measures of predictability","volume":"61","author":"DelSole","year":"2004","journal-title":"J. Atmos. Sci."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1038\/nclimate1907","article-title":"Changes in rainfall seasonality in the tropics","volume":"3","author":"Feng","year":"2013","journal-title":"Nat. Clim. Chang."},{"key":"ref_65","unstructured":"Pascale, S., Lucarini, V., Feng, X., and Porporato, A. (2014). Projected changes of rainfall seasonality and dry spells in a high concentration pathway 21st century scenario, arXiv, 1410.3116. arXiv preprint."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_67","unstructured":"Kullback, S. (1959). Statistics and Information Theory, Wiley."},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Salvadori, G., and de Michele, C. (2010). Multivariate multiparameter extreme value models and return periods: A copula approach. Water Resour. Res., 46.","DOI":"10.1029\/2009WR009040"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1061\/(ASCE)HE.1943-5584.0000637","article-title":"Water deficit duration and severity analysis based on runoff derived from noah land surface model","volume":"18","author":"Khedun","year":"2012","journal-title":"J. Hydrol. Eng."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"W07412","DOI":"10.1029\/2007WR006115","article-title":"Geostatistical interpolation using copulas","volume":"44","author":"Bardossy","year":"2008","journal-title":"Water Resour. Res."},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Piani, C., and Haerter, J. (2012). Two dimensional bias correction of temperature and precipitation copulas in climate models. Geophys. Res. Lett., 39.","DOI":"10.1029\/2012GL053839"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"W11516","DOI":"10.1029\/2009WR008490","article-title":"Reducing uncertainty in estimates of frequency distribution parameters using composite likelihood approach and copula-based bivariate distributions","volume":"46","author":"Chowdhary","year":"2010","journal-title":"Water Resour. Res."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"2401","DOI":"10.5194\/hess-15-2401-2011","article-title":"Copula-based statistical refinement of precipitation in rcm simulations over complex terrain","volume":"15","author":"Laux","year":"2011","journal-title":"Hydrol. Earth Syst. Sci."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1002\/2013WR013763","article-title":"A copula-based precipitation forecasting model: Investigating the interdecadal modulation of enso\u2019s impacts on monthly precipitation","volume":"50","author":"Khedun","year":"2014","journal-title":"Water Resour. Res."},{"key":"ref_75","unstructured":"Jaworski, P., Durante, F., H\u00e4rdle, W.K., and Rychlik, T. (2010). Lecture Notes in Statistics\u2014Proceedings, Springer."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.jmva.2012.02.021","article-title":"A review of copula models for economic time series","volume":"110","author":"Patton","year":"2012","journal-title":"J. Multivar. Anal."},{"key":"ref_77","first-page":"541","article-title":"Copulas and Copula Models","volume":"2","author":"Piegorsch","year":"2012","journal-title":"Encyclopedia of Environmetrics"},{"key":"ref_78","doi-asserted-by":"crossref","unstructured":"Joe, H. (1997). Multivariate Models and Dependence Concepts, Chapman & Hall.","DOI":"10.1201\/9780367803896"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"W09401","DOI":"10.1029\/2006WR005275","article-title":"Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data","volume":"43","author":"Genest","year":"2007","journal-title":"Water Resour. Res."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s00477-009-0331-1","article-title":"Meta-elliptical copulas for drought frequency analysis of periodic hydrologic data","volume":"24","author":"Song","year":"2010","journal-title":"Stoch. Env. Res. Risk Assess"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1061\/(ASCE)1084-0699(2007)12:6(651)","article-title":"Idf curves using the frank archimedean copula","volume":"12","author":"Singh","year":"2007","journal-title":"J. Hydrol. Eng."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"W02415","DOI":"10.1029\/2007WR006261","article-title":"Trivariate statistical analysis of extreme rainfall events via the plackett family of copulas","volume":"44","author":"Kao","year":"2008","journal-title":"Water Resour. Res."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1007\/s00477-010-0364-5","article-title":"Frequency analysis of droughts using the plackett copula and parameter estimation by genetic algorithm","volume":"24","author":"Song","year":"2010","journal-title":"Stoch. Env. Res. Risk Assess"},{"key":"ref_84","doi-asserted-by":"crossref","unstructured":"Salvadori, G., de Michele, C., Kottegoda, N., and Rosso, R. (2007). Extremes in Nature: An Approach Using Copulas, Springer.","DOI":"10.1007\/1-4020-4415-1"},{"key":"ref_85","doi-asserted-by":"crossref","unstructured":"Kurowicka, D., and Joe, H. (2011). Dependence Modeling: Vine Copula Handbook, World Scientific.","DOI":"10.1142\/9789814299886"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1023\/A:1016725902970","article-title":"Probability density decomposition for conditionally dependent random variables modeled by vines","volume":"32","author":"Bedford","year":"2001","journal-title":"Ann. Math. Artif. Intel."},{"key":"ref_87","first-page":"W07535","article-title":"Copula-based daily rainfall disaggregation model","volume":"47","year":"2011","journal-title":"Water Resour. Res."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.jmva.2009.08.002","article-title":"Tail dependence functions and vine copulas","volume":"101","author":"Joe","year":"2010","journal-title":"J. Multivar. Anal."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.insmatheco.2007.02.001","article-title":"Pair-copula constructions of multiple dependence","volume":"44","author":"Aas","year":"2009","journal-title":"Insur. Math. Econ."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1093\/biomet\/82.3.543","article-title":"A semiparametric estimation procedure of dependence parameters in multivariate families of distributions","volume":"82","author":"Genest","year":"1995","journal-title":"Biometrika"},{"key":"ref_91","first-page":"5","article-title":"Copula parameter estimation using blomqvist\u2019s beta","volume":"154","author":"Genest","year":"2013","journal-title":"J. Soc. Fr. Stat."},{"key":"ref_92","unstructured":"Available online: http:\/\/www.math.wsu.edu\/math\/faculty\/genz\/homepage."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1109\/TFUZZ.2004.840129","article-title":"Copula-like operations on finite settings","volume":"13","author":"Mayor","year":"2005","journal-title":"IEEE Trans. Fuzzy Syst"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1109\/TFUZZ.2006.880003","article-title":"Discrete copulas","volume":"14","author":"Mesiar","year":"2006","journal-title":"IEEE Trans. Fuzzy Syst"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1006\/jmaa.1998.6056","article-title":"Strong approximation of copulas","volume":"225","author":"Li","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"233","DOI":"10.3934\/naco.2012.2.233","article-title":"Maximum entropy methods for generating simulated rainfall","volume":"2","author":"Piantadosi","year":"2012","journal-title":"Numer. Algebra. Contr. Optim."},{"key":"ref_97","doi-asserted-by":"crossref","unstructured":"Bene\u0161, V., and \u0160t\u011bp\u00e1n, J. (1997). Distributions with Given Marginals and Moment Problems, Springer.","DOI":"10.1007\/978-94-011-5532-8"},{"key":"ref_98","doi-asserted-by":"crossref","unstructured":"Lenard, A. (1973). Statistical Mechanics and Mathematical Problems, Springer.","DOI":"10.1007\/BFb0112755"},{"key":"ref_99","unstructured":"Borwein, J.M., and Lewis, A.S. (2010). Convex Analysis and Nonlinear Optimization: Theory and Examples, Springer."},{"key":"ref_100","unstructured":"Available online: http:\/\/docserver.carma.newcastle.edu.au\/1453\/1\/ecr.pdf."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"066138","DOI":"10.1103\/PhysRevE.69.066138","article-title":"Estimating mutual information","volume":"69","author":"Kraskov","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1080\/01621459.1989.10478751","article-title":"Relative entropy measures of multivariate dependence","volume":"84","author":"Joe","year":"1989","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1109\/TIT.1968.1054142","article-title":"Approximating discrete probability distributions with dependence trees","volume":"14","author":"Chow","year":"1968","journal-title":"IEEE Trans. Inf. Theory."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF02289159","article-title":"Multivariate information transmission","volume":"19","author":"McGill","year":"1954","journal-title":"Psychometrika"},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1147\/rd.41.0066","article-title":"Information theoretical analysis of multivariate correlation","volume":"4","author":"Watanabe","year":"1960","journal-title":"IBM J. Res. Dev."},{"key":"ref_106","unstructured":"Fass, D. (2007). Human Sensitivity to Mutual Information, The State University of New Jersey."},{"key":"ref_107","unstructured":"Lee, T., and Salas, J.D. Available online: www.usbr.gov\/lc\/region\/g4000\/NaturalFlow\/Final.RecordExtensionReport.2006.pdf."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1111\/1467-842X.00110","article-title":"Rank correlation\u2014An alternative measure","volume":"42","author":"Blest","year":"2000","journal-title":"Aust. N. Z. J. Stat."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"35","DOI":"10.2307\/3315902","article-title":"On blest\u2019s measure of rank correlation","volume":"31","author":"Genest","year":"2003","journal-title":"Can. J. Stat."},{"key":"ref_110","unstructured":"Stocker, T.F., Qin, D., Plattner, G.-K., Tignor, M.M.B., Allen, S.K., Boschung, J., Nauels, A., Xia, Y., Bex, V., and Midgley, P.M. (2013). Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1175\/1520-0477(2000)081<0417:OVATIE>2.3.CO;2","article-title":"Observed variability and trends in extreme climate events: A brief review","volume":"81","author":"Easterling","year":"2000","journal-title":"Bull. Am. Meteor. Soc."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10584-010-9834-5","article-title":"Statistics of extremes in climate change","volume":"100","author":"Katz","year":"2010","journal-title":"Clim. Chang."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"17905","DOI":"10.1073\/pnas.1101766108","article-title":"Increase of extreme events in a warming world","volume":"108","author":"Rahmstorf","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1038\/nclimate1452","article-title":"A decade of weather extremes","volume":"2","author":"Coumou","year":"2012","journal-title":"Nat. Clim. Chang."},{"key":"ref_115","first-page":"115","article-title":"Changes in climate and weather extremes in the 21st century","volume":"3","author":"Rummukainen","year":"2012","journal-title":"Wiley Interdiscip. Rev."},{"key":"ref_116","first-page":"D22104","article-title":"Relationships between global precipitation and surface temperature on interannual and longer timescales 1979\u20132006","volume":"113","author":"Adler","year":"2008","journal-title":"J. Geophys. Res."},{"key":"ref_117","unstructured":"Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, K.B., Averyt, M., Tignor, M., and Miller, H.L. (2007). Climate Change 2007: The Physical Science Basis: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press."},{"key":"ref_118","first-page":"L13803","article-title":"Co-variation of temperature and precipitation in cmip5 models and satellite observations","volume":"39","author":"Liu","year":"2012","journal-title":"Geophys. Res. Lett."},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"L14703","DOI":"10.1029\/2005GL022760","article-title":"Relationships between precipitation and surface temperature","volume":"32","author":"Trenberth","year":"2005","journal-title":"Geophys. Res. Lett."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1111\/j.1467-985X.2008.00545.x","article-title":"Joint projections of temperature and precipitation change from multiple climate models: A hierarchical bayesian approach","volume":"172","author":"Tebaldi","year":"2009","journal-title":"J. R. Stat. Soc. A"},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"2513","DOI":"10.1007\/s00382-011-1208-9","article-title":"Multivariate probabilistic projections using imperfect climate models part I: Outline of methodology","volume":"38","author":"Sexton","year":"2012","journal-title":"Clim. Dynam."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"211","DOI":"10.22499\/2.6104.002","article-title":"Calculation of joint pdfs for climate change with properties matching recent australian projections","volume":"61","author":"Watterson","year":"2011","journal-title":"Aust. Meteorol. Ocean. J"},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1002\/joc.3542","article-title":"Recent and future climate extremes arising from changes to the bivariate distribution of temperature and precipitation in bavaria, germany","volume":"33","author":"Estrella","year":"2012","journal-title":"Int. J. Climatol."},{"key":"ref_124","doi-asserted-by":"crossref","unstructured":"Rodrigo, F. (2014). On the covariability of seasonal temperature and precipitation in spain, 1956\u20132005. Int. J. Climatol., in press.","DOI":"10.1002\/joc.4214"},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"2811","DOI":"10.1175\/JCLI-D-12-00270.1","article-title":"Anatomy of an extreme event","volume":"26","author":"Hoerling","year":"2013","journal-title":"J. Clim."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1029\/2012EO470001","article-title":"Us temperature and drought: Recent anomalies and trends","volume":"93","author":"Karl","year":"2012","journal-title":"Eos Trans Am. Geophys. Union."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"1658","DOI":"10.1175\/1520-0442(1995)008<1658:ADCOTA>2.0.CO;2","article-title":"A diagnostic comparison of the 1980 and 1988 US summer heat wave-droughts","volume":"8","author":"Lyon","year":"1995","journal-title":"J. Clim."},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1080\/07055900.2000.9649654","article-title":"Temperature and precipitation trends in Canada during the 20th century","volume":"38","author":"Zhang","year":"2000","journal-title":"Atmos. Ocean."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"L07707","DOI":"10.1029\/2008GL037119","article-title":"Trends in joint quantiles of temperature and precipitation in europe since 1901 and projected for 2100","volume":"36","author":"Beniston","year":"2009","journal-title":"Geophys. Res. Lett."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"034014","DOI":"10.1088\/1748-9326\/8\/3\/034014","article-title":"Changes in concurrent monthly precipitation and temperature extremes","volume":"8","author":"Hao","year":"2013","journal-title":"Environ. Res. Lett."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1214\/lnms\/1215452614","article-title":"Families of m-variate distributions with given margins and m (m-1)\/2 bivariate dependence parameters","volume":"28","author":"Joe","year":"1996","journal-title":"Lect. Notes Monogr. Ser."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1214\/aos\/1031689016","article-title":"Vines\u2014A new graphical model for dependent random variables","volume":"30","author":"Bedford","year":"2002","journal-title":"Ann. Stat."},{"key":"ref_133","doi-asserted-by":"crossref","unstructured":"Kurowicka, D., and Cooke, R.M. (2006). Uncertainty Analysis with High Dimensional Dependence Modelling, Wiley.","DOI":"10.1002\/0470863072"},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1155\/S0161171299222594","article-title":"On approximation of copulas","volume":"22","author":"Kulpa","year":"1999","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.insmatheco.2010.11.002","article-title":"Approximation of bivariate copulas by patched bivariate fr\u00e9chet copulas","volume":"48","author":"Zheng","year":"2011","journal-title":"Insur. Math. Econ."},{"key":"ref_136","first-page":"61","article-title":"Shuffles of min","volume":"13","author":"Mikusinski","year":"1992","journal-title":"Stochastica: Revista de Matem\u00e1tica Pura y Aplicada"},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1016\/j.jmaa.2008.11.064","article-title":"Shuffles of copulas","volume":"352","author":"Durante","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1016\/j.insmatheco.2012.02.007","article-title":"Dependence modeling in non-life insurance using the bernstein copula","volume":"50","author":"Diers","year":"2012","journal-title":"Insur. Math. Econ."},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1017\/S026646660420305X","article-title":"The bernstein copula and its applications to modeling and approximations of multivariate distributions","volume":"20","author":"Sancetta","year":"2004","journal-title":"Econom. Theory."},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.csda.2011.07.016","article-title":"Copula density estimation by total variation penalized likelihood with linear equality constraints","volume":"56","author":"Qu","year":"2012","journal-title":"Comput. Stat. Data Anal"},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"3806","DOI":"10.1016\/j.csda.2008.01.002","article-title":"Linear b-spline copulas with applications to nonparametric estimation of copulas","volume":"52","author":"Shen","year":"2008","journal-title":"Comput. Stat. Data Anal"},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1007\/s10687-014-0199-4","article-title":"Using b-splines for nonparametric inference on bivariate extreme-value copulas","volume":"17","author":"Cormier","year":"2014","journal-title":"Extremes"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/4\/2253\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:44:45Z","timestamp":1760215485000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/4\/2253"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4,15]]},"references-count":142,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,4]]}},"alternative-id":["e17042253"],"URL":"https:\/\/doi.org\/10.3390\/e17042253","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,4,15]]}}}