Vai ai contenuti. | Spostati sulla navigazione | Spostati sulla ricerca | Vai al menu | Contatti | Accessibilità

| Crea un account

Manzi, Maddalena (2011) New construction methods for copulas and the multivariate case. [Tesi di dottorato]

Full text disponibile come:

[img]
Anteprima
Documento PDF - Versione accettata
1034Kb

Abstract (inglese)

Aggregation functions are mathematical objects that have the function of reducing a set of numbers into a unique representative number, combining several degrees of membership into one aggregated value. Particular kinds of aggregation functions are copulas which permit
to represent joint distribution functions by splitting the marginal behaviour, embedded in the marginal distributions, from the dependence captured by the copula itself.
The concept of copula can be extended to n dimensions, but multivariate extensions are generally not easily to
be done.
This thesis addresses and develops a new unified approach to copula-based modelling and characterizations
of aggregation functions in the multivariate case.
To cope with
this problem, we have to understand the algebraic structure of lattice and supermodularity on
a general lattice, because supermodularity is strictly connected to 2-increasingness and in the bivariate case copulas are a subclass of supermodular aggregation functions.

Abstract (italiano)

Le funzioni di aggregazione sono strumenti matematici importanti che riducono un insieme di numeri in un unico numero rappresentativo, combinando i vari gradi di appartenenza nel valore aggregato. Importanti funzioni di aggregazione sono le copule, che permettono di rappresentare funzioni di distribuzione marginali con la funzione di distribuzione congiunta, che cattura proprio nella copula la dipendenza tra le marginali.
Il concetto di copula può essere esteso al caso multidimensionale, ma costruire copule multivariate non è un problema semplice.
Questa tesi, partendo dalla generalizzazione dell'assioma tipico delle copule nel caso bivariato, apre la strada alle costruzioni multivariate. Si generalizza l'assioma di supermodularità con quello di ultramodularità e l'investigazione affronta il problema con un approccio unificato, studiando le copule proprio come particolari tipi di funzioni d'aggregazione.

Statistiche Download - Aggiungi a RefWorks
Tipo di EPrint:Tesi di dottorato
Relatore:Cardin , Marta
Correlatore:Mesiar, Radko
Dottorato (corsi e scuole):Ciclo 22 > Scuole per il 22simo ciclo > SCIENZE MATEMATICHE > MATEMATICA COMPUTAZIONALE
Data di deposito della tesi:NON SPECIFICATO
Anno di Pubblicazione:25 Gennaio 2011
Parole chiave (italiano / inglese):aggregation functions, copulas, supermodularity, ultramodularity, monotonicity, n-monotonicity, strong n-monotonicity, n-increasingness
Settori scientifico-disciplinari MIUR:Area 01 - Scienze matematiche e informatiche > MAT/06 Probabilità e statistica matematica
Area 13 - Scienze economiche e statistiche > SECS-S/06 Metodi matematici dell'economia e delle scienze attuariali e finanziarie
Struttura di riferimento:Dipartimenti > Dipartimento di Matematica Pura e Applicata
Codice ID:3438
Depositato il:29 Lug 2011 16:11
Simple Metadata
Full Metadata
EndNote Format

Bibliografia

I riferimenti della bibliografia possono essere cercati con Cerca la citazione di AIRE, copiando il titolo dell'articolo (o del libro) e la rivista (se presente) nei campi appositi di "Cerca la Citazione di AIRE".
Le url contenute in alcuni riferimenti sono raggiungibili cliccando sul link alla fine della citazione (Vai!) e tramite Google (Ricerca con Google). Il risultato dipende dalla formattazione della citazione.

[1] C. Alsina, M. J. Frank, and B. Schweizer. Associative functions: triangular norms and Cerca con Google

copulas. World Scientific Publishing Co. Pte. Ltd., Singapore, 2006. Cerca con Google

[2] C. Alsina, E. Trillas, and L. Valverde. On non-distributive logical connectives for fuzzy Cerca con Google

sets theory. Busefal, 3:18–29, 1980. Cerca con Google

[3] C. Amblard and S. Girard. Symmetry and dependence properties within a semiparametric Cerca con Google

family of bivariate copulas. J. Nonparametr. Stat., 14(6):715–727, 2002. Cerca con Google

[4] G. Barbieri and H. Weber. A representation theorem and a Lyapunov theorem for Tsmeasures: Cerca con Google

The solution of two problems of Butnariu and Klement. J. Math. Anal. Appl., Cerca con Google

244:408–424, 2000. Cerca con Google

[5] G. Beliakov, R. Mesiar, and L. Valášková. Fitting generated aggregation operators to Cerca con Google

empirical data. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 12(2):219– Cerca con Google

236, 2004. Cerca con Google

[6] G. Beliakov, A. Pradera, and T. Calvo. Aggregation Functions: A Guide for Practitioners, Cerca con Google

volume 221 of Studies in Fuzziness and Soft Computing. Springer, Heidelberg, Cerca con Google

2007. Cerca con Google

[7] S. Benferhat, D. Dubois, and H. Prade. From semantic to syntactic approaches to information Cerca con Google

combination in possibilistic logic. In B. Bouchon-Meunier, editor, Aggregation Cerca con Google

and Fusion of Imperfect Information, pages 141–161. Physica-Verlag, Heidelberg, 1997. Cerca con Google

[8] E. Berkson and T. A. Gillespie. Absolutely continuous-functions of 2 variables and Cerca con Google

well-bounded operators. Journal of the London Mathematical Society, 30(2):305–321, Cerca con Google

1984. Cerca con Google

[9] G. Birkhoff. Lattice theory. Revised edition. American Mathematical Society Colloquium Cerca con Google

Publications, Vol. XXV. American Mathematical Society, New York City, 1948. Cerca con Google

[10] G. Birkhoff. Lattice Theory. 3rd edition. American Mathematical Society Colloquium Cerca con Google

Publications, Vol. XXV. American Mathematical Society, Providence, 1967. Cerca con Google

[11] H. W. Block, W. S. Griffith, and T. H. Savits. L-superadditive structure functions. Adv. Cerca con Google

in Appl. Probab., 21(4):919–929, 1989. Cerca con Google

122 BIBLIOGRAPHY Cerca con Google

[12] S. Bodjanova. TL and SL evaluators. In Proceedings of AGOP 2007, pages 165–172. Cerca con Google

Academia Press, Ghent, 2007. Cerca con Google

[13] S. Bodjanova. TL and SL evaluators: aggregation and modification. Acta Univ. M. Belii Cerca con Google

Ser. Math., (14):5–17, 2007. Cerca con Google

[14] S. Bodjanova and M. Kalina. T-evaluators and S-evaluators. Fuzzy Sets and Systems, Cerca con Google

160(14):1965 – 1983, 2009. Cerca con Google

[15] S. Bodjanova and M. Kalina. Fuzzy integral-based T-evaluators and S-evaluators. Part Cerca con Google

I: Sugeno integral. Integration: Mathematical Theory and Applications, accepted. Cerca con Google

[16] S. Bodjanova and M. Kalina. Fuzzy integral-based T-evaluators and S-evaluators. Part Cerca con Google

II: Shilkret and choquet integrals. Integration: Mathematical Theory and Applications, Cerca con Google

submitted. Cerca con Google

[17] A. G. Bronevich. On the closure of families of fuzzy measures under eventwise aggregations. Cerca con Google

Fuzzy Sets and Systems, 153:45–70, 2005. Cerca con Google

[18] D. Butnariu. Additive fuzzy measures and integrals. I. J. Math. Anal. Appl., 93:436–452, Cerca con Google

1983. Cerca con Google

[19] D. Butnariu. Values and cores of fuzzy games with infinitely many players. Internat. J. Cerca con Google

Game Theory, 16:43–68, 1987. Cerca con Google

[20] D. Butnariu and E. P. Klement. Triangular norm-based measures and their Markov Cerca con Google

kernel representation. J. Math. Anal. Appl., 162:111–143, 1991. Cerca con Google

[21] D. Butnariu and E. P. Klement. Triangular Norm-Based Measures and Games with Cerca con Google

Fuzzy Coalitions, volume 10 of Theory and Decision Library, Series C: Game Theory, Cerca con Google

Mathematical Programming and Operations Research. Kluwer, Dordrecht, 1993. Cerca con Google

[22] T. Calvo, A. Kolesárová, M. Komorníková, and R. Mesiar. Aggregation operators: Cerca con Google

properties, classes and construction methods. In Aggregation Operators. New Trends Cerca con Google

and Applications, volume 97 of Studies in Fuzziness and Soft Computing, pages 3–104. Cerca con Google

Physica-Verlag, Heidelberg, 2002. Cerca con Google

[23] T. Calvo and R. Mesiar. Continuous generated associative aggregation operators. Fuzzy Cerca con Google

Sets and Systems, 126:191–197, 2002. Cerca con Google

[24] T. Calvo and R. Mesiar. Aggregation operators: ordering and bounds. Fuzzy Sets and Cerca con Google

Systems, 139:685–697, 2003. Cerca con Google

[25] T. Calvo and A. Pradera. Double aggregation operators. Fuzzy Sets and Systems, Cerca con Google

142:15–33, 2004. Cerca con Google

[26] G. Choquet. Theory of capacities. Ann. Inst. Fourier, Grenoble, 5:131–295 (1955), Cerca con Google

1953–1954. Cerca con Google

[27] P. Civitanovic. Group Theory: Birdtracks, Lie’s and Exceptional Groups. Princeton Cerca con Google

University Press, Princeton, NJ, 2008. Cerca con Google

[28] A. H. Clifford. Naturally totally ordered commutative semigroups. Amer. J. Math., Cerca con Google

76:631–646, 1954. Cerca con Google

[29] B. De Baets, H. De Meyer, J. Kalická, and R. Mesiar. Flipping and cyclic shifting of Cerca con Google

binary aggregation functions. Fuzzy Sets and Systems, 160:752–765, 2009. Cerca con Google

[30] G. de Cooman, M. C. M. Troffaes, and E. Miranda. n-monotone exact functionals. J. Cerca con Google

Math. Anal. Appl., 347(1):143–156, 2008. Cerca con Google

[31] D. Denneberg. Non-additive measure and integral, volume 27 of Theory and Decision Cerca con Google

Library. Series B: Mathematical and Statistical Methods. Kluwer Academic Publishers Cerca con Google

Group, Dordrecht, 1994. Cerca con Google

[32] D. Denneberg and M. Grabisch. Measure and integral with purely ordinal scales. J. Cerca con Google

Math. Psych., 48(1):15–27, 2004. Cerca con Google

[33] A. Dolati and M. Úbeda-Flores. Some new parametric families of multivariate copulas. Cerca con Google

Int. Math. Forum, 1(1-4):17–25, 2006. Cerca con Google

[34] J. Dombi. A general class of fuzzy operators, the De Morgan class of fuzzy operators Cerca con Google

and fuzziness measures induced by fuzzy operators. Fuzzy Sets and Systems, 8:149–163, Cerca con Google

1982. Cerca con Google

[35] F. Durante. Generalized composition of aggregation operators. Internat. J. Uncertain. Cerca con Google

Fuzziness Knowledge-Based Systems, 13:567–577, 2005. Cerca con Google

[36] F. Durante and P. Jaworski. Invariant dependence structure under univariate truncation. Cerca con Google

Submitted for publication. Cerca con Google

[37] F. Durante, R. Mesiar, P. L. Papini, and C. Sempi. 2-increasing binary aggregation Cerca con Google

operators. Inform. Sci., 177(1):111–129, 2007. Cerca con Google

[38] F. Durante, R. Mesiar, and C. Sempi. On a family of copulas constructed from the Cerca con Google

diagonal section. Soft Comput., 10(6):490–494, 2006. Cerca con Google

[39] F. Durante, S. Saminger-Platz, and P. Sarkoci. On representations of 2-increasing binary Cerca con Google

aggregation functions. Inform. Sci., 178(23):4534–4541, 2008. Cerca con Google

[40] F. Durante, S. Saminger-Platz, and P. Sarkoci. Rectangular patchwork for bivariate Cerca con Google

copulas and tail dependence. Comm. Statist. Theory and Method, 38:2515–2527, 2009. Cerca con Google

[41] F. Durante and C. Sempi. On the characterization of a class of binary operations on Cerca con Google

bivariate distribution functions. Publ. Math. Debrecen, 69:47–63, 2006. Cerca con Google

[42] J. Fodor and M. Roubens. Fuzzy preference modelling and multicriteria decision support. Cerca con Google

Theory and Decision Library. Series D: Systems Theory, Knowledge Engineering Cerca con Google

and Problem Solving. 14. Dordrecht: Kluwer Academic Publishers., 1994. Cerca con Google

[43] M. J. Frank. On the simoultaneous associativity of f (x;y) and x+Y- f (x;y). Aequationes Cerca con Google

Math., 19:194–226, 1979. Cerca con Google

[44] C. Genest, J. J. Quesada Molina, and J. A. Rodríguez Lallena. De l’impossibilité de Cerca con Google

construire des lois à marges multidimensionnelles données à partir de copules. C. R. Cerca con Google

Acad. Sci. Paris Sér. I Math., (320):723–726, 1995. Cerca con Google

[45] M. Grabisch. Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems, Cerca con Google

69(3):279–298, 1995. Cerca con Google

[46] M. Grabisch, T. Murofushi, and M. Sugeno, editors. Fuzzy Measures and Integrals. Theory Cerca con Google

and Applications, volume 40 of Studies in Fuzziness and Soft Computing. Physica- Cerca con Google

Verlag, Heidelberg, 2000. Cerca con Google

[47] H. Hamacher. Über logische Aggregationen nicht-binär explizierter Entscheidungskriterien. Cerca con Google

Rita G. Fischer Verlag, Frankfurt, 1978. Cerca con Google

[48] M. K. Jensen. Monotone comparative statics in ordered vector spaces. The B. E. Journal Cerca con Google

of Theoretical Economics, 7(4):Issue 1, Article 35, 2007. Cerca con Google

[49] H. Joe. Multivariate Models and Dependence Concepts. Chapman & Hall, London, Cerca con Google

1997. Cerca con Google

[50] A. Khoudraji. Contributions à l’étude des copules et à la modélisation des valeurs Cerca con Google

extrêmes bivariées. PhD thesis, Université Laval, Québec, 1995. Cerca con Google

[51] E. P. Klement. Characterization of finite fuzzy measures using Markoff-kernels. J. Math. Cerca con Google

Anal. Appl., 75:330–339, 1980. Cerca con Google

[52] E. P. Klement. Characterization of fuzzy measures constructed by means of triangular Cerca con Google

norms. J. Math. Anal. Appl., 86:345–358, 1982. Cerca con Google

[53] E. P. Klement. Construction of fuzzy s-algebras using triangular norms. J. Math. Anal. Cerca con Google

Appl., 85:543–565, 1982. Cerca con Google

[54] E. P. Klement, A. Kolesárová, R. Mesiar, and C. Sempi. Copulas constructed from Cerca con Google

horizontal sections. Comm. Statist. Theory Methods, 36:2901–2911, 2007. Cerca con Google

[55] E. P. Klement, A. Kolesárová, R. Mesiar, and A. Stupˇnanová. Lipschitz continuity of discrete Cerca con Google

universal integrals based on copulas. Internat. J. Uncertain. Fuzziness Knowledge- Cerca con Google

Based Systems, 18:39–52, 2010. Cerca con Google

[56] E. P. Klement, M. Manzi, and R. Mesiar. Ultramodular aggregation functions and a new Cerca con Google

construction method for copulas. Submitted for publication. Cerca con Google

[57] E. P. Klement, M. Manzi, and R. Mesiar. Aggregation functions with stronger types Cerca con Google

of monotonicity. In Computational Intelligence for Knowledge-Based Systems Design, Cerca con Google

Proceedings of 13th International Conference on Information Processing and Management Cerca con Google

of Uncertainty (IPMU 2010), pages 418–424. Eyke Hüllermeier, Rudolf Kruse, Cerca con Google

and Frank Hoffmann (Eds.), Springer LNAI 6178, Springer-Verlag Berlin Heidelberg, Cerca con Google

2010. Cerca con Google

[58] E. P. Klement, R. Mesiar, and E. Pap. Quasi- and pseudo-inverses of monotone functions, Cerca con Google

and the construction of t-norms. Fuzzy Sets and Systems, 104(1):3–13, 1999. Cerca con Google

[59] E. P. Klement, R. Mesiar, and E. Pap. Triangular Norms, volume 8 of Trends in Logic. Cerca con Google

Studia Logica Library. Kluwer, Dordrecht, 2000. Cerca con Google

[60] E. P. Klement, R. Mesiar, and E. Pap. Transformations of copulas. Kybernetika Cerca con Google

(Prague), 41(4):425–434, 2005. Cerca con Google

[61] G. J. Klir and T. A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Cerca con Google

Englewood Cliff, 1988. Cerca con Google

[62] H. König. New facts around the choquet integral. preprint n. 62, 2002. Cerca con Google

[63] B. Larose and A. Krokhin. A note on supermodular sublattices in finite relatively complemented Cerca con Google

lattices. 7(4):Issue 1, Article 35, 2007. Cerca con Google

[64] E. Liebscher. Construction of asymmetric multivariate copulas. J. Multivariate Anal., Cerca con Google

99:2234–2250, 2008. Cerca con Google

[65] S. Maaß. Exact functionals and their core. Statist. Papers, 43(1):75–93, 2002. Choquet Cerca con Google

integral and applications. Cerca con Google

[66] J-L. Marichal. An axiomatic approach of the discrete choquet integral as a tool to aggregate Cerca con Google

interacting criteria. IEEE Trans. Fuzzy Systems, 8(6):800–807, 2000. Cerca con Google

[67] M. Marinacci and L. Montrucchio. Ultramodular functions. Math. Oper. Res., Cerca con Google

30(2):311–332, 2005. Cerca con Google

[68] M. Marinacci and L. Montrucchio. On concavity and supermodularity. J. Math. Anal. Cerca con Google

Appl., 344:642–654, 2008. Cerca con Google

[69] A. W. Marshall and I. Olkin. Inequalities: theory of majorization and its applications, Cerca con Google

volume 143 of Mathematics in Science and Engineering. Academic Press Inc. [Harcourt Cerca con Google

Brace Jovanovich Publishers], New York, 1979. Cerca con Google

[70] A. J. McNeil and J. Nešlehová. Multivariate Archimedean copulas, d-monotone functions Cerca con Google

and l1-norm symmetric distributions. Ann. Statist., 37:3059–3097, 2009. Cerca con Google

[71] K. Menger. Statistical metrics. Proc. Nat. Acad. Sci. U.S.A., 8:535–537, 1942. Cerca con Google

[72] R. Mesiar. Approximation of continuous t-norms by strict t-norms with smooth generators. Cerca con Google

Busefal, 75:72–79, 1998. Cerca con Google

[73] R. Mesiar. A note on moderate growth of t-conorms. Fuzzy Sets and Systems, 122:357– Cerca con Google

359, 2001. Cerca con Google

[74] R. Mesiar, V. Jágr, M. Juráˇnová, and M. Komorníková. Univariate conditioning of Cerca con Google

copulas. Kybernetika (Prague), 44:807–816, 2008. Cerca con Google

[75] R. Mesiar and S. Saminger. Domination of ordered weighted averaging operators over Cerca con Google

t-norms. Soft Computing, 8(1-2):562–570, 2004. Cerca con Google

[76] R. Mesiar and C. Sempi. Ordinal sums and idempotents of copulas. Aequationes Math., Cerca con Google

79(1–2):39–52, 2010. Cerca con Google

[77] P. M. Morillas. A characterization of absolutely monotonic (D) functions of a fixed Cerca con Google

order. Publ. Inst. Math. (Beograd) (N.S.), 78(92):93–105, 2005. Cerca con Google

[78] P. M. Morillas. A method to obtain new copulas from a given one. Metrika, 61(2):169– Cerca con Google

184, 2005. Cerca con Google

[79] R. Moynihan. On tT semigroups of probability distribution functions II. Aequationes Cerca con Google

Math., 17:19–40, 1978. Cerca con Google

[80] M. Navara. Characterization of measures based on strict triangular norms. J. Math. Cerca con Google

Anal. Appl., 236:370–383, 1999. Cerca con Google

[81] R. B. Nelsen. An Introduction to Copulas, volume 139 of Lecture Notes in Statistics. Cerca con Google

Springer, New York, 1999. Cerca con Google

[82] R. B. Nelsen. An Introduction to Copulas, volume 139 of Lecture Notes in Statistics. Cerca con Google

Springer, New York, second edition, 2006. Cerca con Google

[83] E. Pap. Null-Additive Set Functions. Kluwer Academic Publishers, Dordrecht, 1995. Cerca con Google

[84] H. Prade. Unions et intersections d’ensembles flous. Busefal, 3:58–62, 1980. Cerca con Google

[85] G. Puccetti and M. Scarsini. Multivariate comonotonicity. Journal of Multivariate Cerca con Google

Analysis, Volume 101(1):291–304, 2010. Cerca con Google

[86] J. J. Quesada Molina and J. A. Rodríguez Lallena. Some advances in the study of the Cerca con Google

compatibility of three bivariate copulas. J. Ital. Statist. Soc., (3):397–417, 1994. Cerca con Google

[87] A. W. Roberts and D. E. Varberg. Convex functions, volume 57. Academic Press, New Cerca con Google

York, 1973. Pure and Applied Mathematics. Cerca con Google

[88] R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, 1970. Cerca con Google

[89] H. H. Schaefer and M. P. Wolff. Topological vector spaces, volume 3 of Graduate Texts Cerca con Google

in Mathematics. Springer-Verlag, New York, second edition, 1999. Cerca con Google

[90] B. Schweizer. Thirty years of copulas. In Advances in Probability Distributions with Cerca con Google

Given Marginals. Beyond the Copulas. Lectures Presented at a Symposium Held in Cerca con Google

Rome, Italy, volume 67 of Mathematics and Its Applications, pages 13–50. Kluwer Academic Cerca con Google

Publishers, Dordrecht, 1991. Cerca con Google

[91] B. Schweizer and A. Sklar. Probabilistic metric spaces. North-Holland Series in Probability Cerca con Google

and Applied Mathematics. North-Holland Publishing Co., New York, 1983. Cerca con Google

[92] B. Schweizer and A. Sklar. Probabilistic Metric Spaces. Dover Publications, Mineola, Cerca con Google

N. Y., 2006. Cerca con Google

[93] M. Shaked and J. G. Shanthikumar. Stochastic orders. Springer Series in Statistics. Cerca con Google

Springer, New York, 2007. Cerca con Google

[94] A. Sklar. Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Cerca con Google

Univ. Paris, 8:229–231, 1959. Cerca con Google

[95] P. Struk and A. Stupˇnanová. S-measures, T-measures and distinguished classes of fuzzy Cerca con Google

measures. Kybernetika, 42(3), 2006. Cerca con Google

[96] K. Sundaresan. Monotone gradients on Banach lattices. Proc. Amer. Math. Soc., 98:448– Cerca con Google

454, 1986. Cerca con Google

[97] D. M. Topkis. Minimizing a submodular function on a lattice. Operations Res., Cerca con Google

26(2):305–321, 1978. Cerca con Google

[98] D. M. Topkis. Supermodularity and complementarity. Frontiers of Economic Research. Cerca con Google

Princeton University Press, Princeton, NJ, 1998. Cerca con Google

[99] L. Valášková and P. Struk. Classes of fuzzy measures and distortion. Kybernetika, 41(2), Cerca con Google

2005. Cerca con Google

[100] J. van Tiel. Convex analysis. John Wiley & Sons, New York, 1984. Cerca con Google

[101] Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992. Cerca con Google

[102] Z. Wang and G. J. Klir. Generalized Measure Theory. Springer, New York, 2009. Cerca con Google

[103] R. Webster. Convexity. Oxford University Press, Oxford, 1994. Cerca con Google

[104] R. R. Yager. On ordered weighted averaging aggregation operators in multicriteria decisionmaking. Cerca con Google

IEEE Trans. Systems Man Cybernet., 18(1):183–190, 1988. Cerca con Google

[105] L. A. Zadeh. Fuzzy sets. Inform. and Control, 8:338–353, 1965. Cerca con Google

[106] L. A. Zadeh. Probability measures of fuzzy events. J. Math. Anal. Appl., 23:421–427, Cerca con Google

1968 Cerca con Google

Download statistics

Solo per lo Staff dell Archivio: Modifica questo record