离散copula和quasi-copula的研究
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摘要
Copula理论在统计分析中有着很重要的作用。quasi-copula和copula具有许多共性,是copula概念的推广,因而对quasi-copula的研究具有一定的理论价值。离散copula和离散quasi-copula分别作为copula和quasi-copula的一种离散化,对离散copula理论的研究,有助于进一步丰富和完善copula理论。
本文首先从离散copula和离散quasi-copula的基本理论入手,然后利用组合数学理论、矩阵理论和格理论对离散copula和离散quasi-copula进行了研究。具体内容包括如下三个方面：第一,利用组合数学的方法,从不可约离散copula的幂等元的角度研究了不可约离散copula的计数问题。此外,由于置换矩阵是特殊的布尔矩阵,所以还讨论了置换矩阵间的三种布尔运算是否仍为置换矩阵的问题。第二,利用一般的双随机矩阵(GBM)和离散quasi-copula的关系,从矩阵的角度,研究了离散quasi-copula的延拓问题。具体来说,对于一个给定的GBM,给出了GBMs序列的一种构造方式,进而得到了离散quasi-copulas序列的构造方式,且证明了离散quasi-copulas序列满足一致性条件,由于满足一致性条件的离散quasi-copulas序列的极限是一个quasi-copula,所以给出离散quasi-copula的延拓方法。第三,考虑多元quasi-copula的情况,利用格理论,着重研究了多元quasi-copulas集合的格结构问题。
最后对前面所做的内容作了总结,并且对下一步将要研究的工作做了展望。
Copula theory was proven to be important in statistical analysis. Quasi-copula, a more general concept, share many properties with copulas, so the study of quasi-copulas has theoretical value. The discrete copulas and discrete quasi-copulas can be regarded as the discreting of copulas and quasi-copulas, so the study of discrete copulas and discrete quasi-copulas can improve the copula theory.
First of all, we introduce the basic knowledge of discrete copula and discrete quasi-copula in this paper, and then the discrete copulas and quasi-copulas are studied further by using the theory of combinatorial mathematics, matrix and lattice. The specific content include three aspects as the following:Firstly, the numbers of irreducible discrete copulas are studied from the standpoint of its idempotent elements by using combinatorial mathematics. In addition, because the permutation matrix is a special Boolean matrix, the question that the result of three basic Boolean operations between permutation matrices is still or not a permutation matrix is also discussed. Secondly, according to the relationship between GBM and discrete copula, the extension of discrete quasi-copula is studied from the standpoint of matrix. Specifically, for a given GBM, we get a way to construct a sequence of GBMs, and then correspondingly the sequence of discrete quasi-copulas. Furthermore, the sequence of discrete quasi-copulas that corresponding to the sequence of GBMs satisfy consistence condition is proved. Because the limit of the sequence of discrete quasi-copulas is a quasi-copula, we get a way to extend the discrete quasi-copula. Thirdly, the multivariate quasi-copulas are considered, and the lattice-theoretic structure of the sets of multivariate quasi-copulas is studied especially by applying the theory of lattice.
At last to summary the content of this paper mentioned before and point out the direction of research in the next step.
本文首先从离散copula和离散quasi-copula的基本理论入手,然后利用组合数学理论、矩阵理论和格理论对离散copula和离散quasi-copula进行了研究。具体内容包括如下三个方面：第一,利用组合数学的方法,从不可约离散copula的幂等元的角度研究了不可约离散copula的计数问题。此外,由于置换矩阵是特殊的布尔矩阵,所以还讨论了置换矩阵间的三种布尔运算是否仍为置换矩阵的问题。第二,利用一般的双随机矩阵(GBM)和离散quasi-copula的关系,从矩阵的角度,研究了离散quasi-copula的延拓问题。具体来说,对于一个给定的GBM,给出了GBMs序列的一种构造方式,进而得到了离散quasi-copulas序列的构造方式,且证明了离散quasi-copulas序列满足一致性条件,由于满足一致性条件的离散quasi-copulas序列的极限是一个quasi-copula,所以给出离散quasi-copula的延拓方法。第三,考虑多元quasi-copula的情况,利用格理论,着重研究了多元quasi-copulas集合的格结构问题。
最后对前面所做的内容作了总结,并且对下一步将要研究的工作做了展望。
Copula theory was proven to be important in statistical analysis. Quasi-copula, a more general concept, share many properties with copulas, so the study of quasi-copulas has theoretical value. The discrete copulas and discrete quasi-copulas can be regarded as the discreting of copulas and quasi-copulas, so the study of discrete copulas and discrete quasi-copulas can improve the copula theory.
First of all, we introduce the basic knowledge of discrete copula and discrete quasi-copula in this paper, and then the discrete copulas and quasi-copulas are studied further by using the theory of combinatorial mathematics, matrix and lattice. The specific content include three aspects as the following:Firstly, the numbers of irreducible discrete copulas are studied from the standpoint of its idempotent elements by using combinatorial mathematics. In addition, because the permutation matrix is a special Boolean matrix, the question that the result of three basic Boolean operations between permutation matrices is still or not a permutation matrix is also discussed. Secondly, according to the relationship between GBM and discrete copula, the extension of discrete quasi-copula is studied from the standpoint of matrix. Specifically, for a given GBM, we get a way to construct a sequence of GBMs, and then correspondingly the sequence of discrete quasi-copulas. Furthermore, the sequence of discrete quasi-copulas that corresponding to the sequence of GBMs satisfy consistence condition is proved. Because the limit of the sequence of discrete quasi-copulas is a quasi-copula, we get a way to extend the discrete quasi-copula. Thirdly, the multivariate quasi-copulas are considered, and the lattice-theoretic structure of the sets of multivariate quasi-copulas is studied especially by applying the theory of lattice.
At last to summary the content of this paper mentioned before and point out the direction of research in the next step.
引文
[1]Sklar A. Fonctions de repartition a n dimensions et leur marges[M]. Publications de l'Institut de Statistique de l'Universite de Paris,1959,8:229-231.
[2]Nelsen R B. An Introduction to Copulas.[M]. New York:Springer,1999.
[3]Alsina C. Nelsen R B, Schweiser B. On the characterization of a class of binary operations on distribution functions[J]. Statist. Probab. Lett,1993,17:85-89.
[4]Genest C, Quesada Molina J, Rodriguez Lallena J A, Sempi C. A characterization of quasi-copulas[J]. Journal of Multivariate Analysis,1999,69:193-205.
[5]Cuculescu I, Theodorescu R. Copulas:diagonals, tracks [J]. Rev. Roumaine Math. Pure Appl,46:731-742.
[6]Nelsen R B, Manuel Ubeda Flores. The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas[J]. C R Acad Sci Paris Ser.,2005,1341: 583-586.
[7]Kolesarova A, Mesiar R, MordelovaJ, Sempi C. Discrete copulas[J]. IEEE Transactions on Fuzzy System,2006,14 (5):698-705.
[8]Quesada Molina J J, Sempi C. Discrete quasi-copulas[J]. Insurance:Mathematics & Economics,2005,37 (1):27-41.
[9]Aguilo I, Suner J, and Torrens J. Matrix representation of discrete quasi-copulas[J]. Fuzzy Sets and Systems,2008,159 (13):1658-1672.
[10]Aguilo I, Suner J, and Torrens J. Matrix representation of copulas and quasi-copulas defined on non-square grids of the unit square[J]. Fuzzy Sets and Systems,2010,161 (2):254-268.
[11]Aguilo I, Suner J, and Torrens J. Representation of discrete quasi-copulas through non-square matrices[J]. in:T. Alsinet, J. Puyol-Gruart, C. Torras (Eds.), Artificial Intelligence Research and Development, Vol.184, IOS Press, Amsterdam,2008, pp.273-282.
[12]Kolesarova A, Mordelova J. Quasi-copulas and copulas on a discrete scale[J]. Soft Computing,2006,10:495-501.
[13]Mayor G, Suner J, and Torrens. Copula-like operations on finite settings[J]. IEEE Transactions on Fuzzy Systems,2005,13 (4):468-477.
[14]Mayor G, Suner J, and Torrens J. Sklar's theorem in finite settings[J]. IEEE Transactions on Fuzzy Systems,2007,15 (3):410-416.
[15]Carley H, Maximum and minimum extensions of finite sub-copulas [J]. Commun. Stat. Theory Meth,2002,31:2151-2166.
[16]Klement E P, Kolesarova A. Extension to copulas and quasi-copulas as special 1-Lipschitz aggregation operators[J]. Kybernetika,2005,41 (3):329-348.
[17]Kolesarova A.1-Lipschitz aggregation operators and quasi-copulas [J]. Kybernetika,2003,39:615-629.
[18]Mordelova J, Kolesarova A. Some results on discrete copulas[J]. in:Proc. EUSFLAT-2007, pp.145-150.
[19]Robbins D P, Rumsey H. Determinants and alternating-sign matrices [J]. Advances in Mathematics,1986,62:169-184.
[20]Nelsen R B, Quesada Molina J J, Rodriguez Lallena J A, Ubeda Flores M. Best-possible bounds on sets of bivariate distribution functions[J]. Journal of Multivariate Analysis,2004,90:348-358.
[21]Durante F, Mesiar R, Sempi C. On a family of copulas constructed from the diagonal section [J]. Soft Computing,2006,10(6):490-494.
[22]. Davey B A, Priestley H A. Introduction to Lattices and Order[M].seconded., Cambridge University Press, Cambridge,2002.
[23]Ouyang Y, Fang J. Some observations about the convex combination of continuous triangular norms[J]. Nonlinear Anal.,2008,68 (11).
[24]Ouyang Y. On the construction of boundary weak triangular norms through additive generators [J]. Nonlinear Anal.,2007,66:125-130.
[25]Schweizer B, Sklar A, Probabilistic Metric Spaces[M]. New York:Elsevier Science,1983.
[26]Cuculescu I, Theodorescu R. Copulas:diagonals and tracks[J].Rev. Roumaine Math. Pures Appl.2001,46:731-742.
[27]Durante F, Mesiar R, Papini P L. Sempi C,2-Increasing binary aggregation operators[J]. Inform. Sci.,2007,177:111-129.
[28]Darsow W F, Nguyen B, and Olsen E T. Copulas and Markov processes[J]. Illinois J. Math.,1992, vol.36:600-642.
[29]Durante F.Solution of an open problem for associative copulas[J]. Fuzzy Sets and Systems,2005,152:411-415.
[30]Bassan B, and Spizzichino F.Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes[J]. Journal of Multivariate Analysis, 2005,93:313-339.
[31]Klement E P.Mesiar R, Pap E.Transformations of copulas[J].Kybernetika,2005, 41:425-434.
[32]Morillas P M.A method to obtain new copulas from a given one[J]. Metrika, 2005,61:169-184.
[33]Mayor G, J. Torrens.On a family of t-norms[J]. Fuzzy Sets and Systems,1991, 41:161-166.
[34]Durante F, Sempi C. Semi-copula[J]. Kybernetika,2005,41 (3):315-328.
[35]沈银芳.多元拟连接函数的刻画[J].浙江大学学报,2007,34(2)：144-147.
[36]曾霞,何平.基于g函数的多元copula的构造[J].数理统计与管理,2008,27(5)：844-849.
[37]王传玉.离散数学与算法[M].合肥：安徽大学出版社,2001：19,140-142.
[2]Nelsen R B. An Introduction to Copulas.[M]. New York:Springer,1999.
[3]Alsina C. Nelsen R B, Schweiser B. On the characterization of a class of binary operations on distribution functions[J]. Statist. Probab. Lett,1993,17:85-89.
[4]Genest C, Quesada Molina J, Rodriguez Lallena J A, Sempi C. A characterization of quasi-copulas[J]. Journal of Multivariate Analysis,1999,69:193-205.
[5]Cuculescu I, Theodorescu R. Copulas:diagonals, tracks [J]. Rev. Roumaine Math. Pure Appl,46:731-742.
[6]Nelsen R B, Manuel Ubeda Flores. The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas[J]. C R Acad Sci Paris Ser.,2005,1341: 583-586.
[7]Kolesarova A, Mesiar R, MordelovaJ, Sempi C. Discrete copulas[J]. IEEE Transactions on Fuzzy System,2006,14 (5):698-705.
[8]Quesada Molina J J, Sempi C. Discrete quasi-copulas[J]. Insurance:Mathematics & Economics,2005,37 (1):27-41.
[9]Aguilo I, Suner J, and Torrens J. Matrix representation of discrete quasi-copulas[J]. Fuzzy Sets and Systems,2008,159 (13):1658-1672.
[10]Aguilo I, Suner J, and Torrens J. Matrix representation of copulas and quasi-copulas defined on non-square grids of the unit square[J]. Fuzzy Sets and Systems,2010,161 (2):254-268.
[11]Aguilo I, Suner J, and Torrens J. Representation of discrete quasi-copulas through non-square matrices[J]. in:T. Alsinet, J. Puyol-Gruart, C. Torras (Eds.), Artificial Intelligence Research and Development, Vol.184, IOS Press, Amsterdam,2008, pp.273-282.
[12]Kolesarova A, Mordelova J. Quasi-copulas and copulas on a discrete scale[J]. Soft Computing,2006,10:495-501.
[13]Mayor G, Suner J, and Torrens. Copula-like operations on finite settings[J]. IEEE Transactions on Fuzzy Systems,2005,13 (4):468-477.
[14]Mayor G, Suner J, and Torrens J. Sklar's theorem in finite settings[J]. IEEE Transactions on Fuzzy Systems,2007,15 (3):410-416.
[15]Carley H, Maximum and minimum extensions of finite sub-copulas [J]. Commun. Stat. Theory Meth,2002,31:2151-2166.
[16]Klement E P, Kolesarova A. Extension to copulas and quasi-copulas as special 1-Lipschitz aggregation operators[J]. Kybernetika,2005,41 (3):329-348.
[17]Kolesarova A.1-Lipschitz aggregation operators and quasi-copulas [J]. Kybernetika,2003,39:615-629.
[18]Mordelova J, Kolesarova A. Some results on discrete copulas[J]. in:Proc. EUSFLAT-2007, pp.145-150.
[19]Robbins D P, Rumsey H. Determinants and alternating-sign matrices [J]. Advances in Mathematics,1986,62:169-184.
[20]Nelsen R B, Quesada Molina J J, Rodriguez Lallena J A, Ubeda Flores M. Best-possible bounds on sets of bivariate distribution functions[J]. Journal of Multivariate Analysis,2004,90:348-358.
[21]Durante F, Mesiar R, Sempi C. On a family of copulas constructed from the diagonal section [J]. Soft Computing,2006,10(6):490-494.
[22]. Davey B A, Priestley H A. Introduction to Lattices and Order[M].seconded., Cambridge University Press, Cambridge,2002.
[23]Ouyang Y, Fang J. Some observations about the convex combination of continuous triangular norms[J]. Nonlinear Anal.,2008,68 (11).
[24]Ouyang Y. On the construction of boundary weak triangular norms through additive generators [J]. Nonlinear Anal.,2007,66:125-130.
[25]Schweizer B, Sklar A, Probabilistic Metric Spaces[M]. New York:Elsevier Science,1983.
[26]Cuculescu I, Theodorescu R. Copulas:diagonals and tracks[J].Rev. Roumaine Math. Pures Appl.2001,46:731-742.
[27]Durante F, Mesiar R, Papini P L. Sempi C,2-Increasing binary aggregation operators[J]. Inform. Sci.,2007,177:111-129.
[28]Darsow W F, Nguyen B, and Olsen E T. Copulas and Markov processes[J]. Illinois J. Math.,1992, vol.36:600-642.
[29]Durante F.Solution of an open problem for associative copulas[J]. Fuzzy Sets and Systems,2005,152:411-415.
[30]Bassan B, and Spizzichino F.Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes[J]. Journal of Multivariate Analysis, 2005,93:313-339.
[31]Klement E P.Mesiar R, Pap E.Transformations of copulas[J].Kybernetika,2005, 41:425-434.
[32]Morillas P M.A method to obtain new copulas from a given one[J]. Metrika, 2005,61:169-184.
[33]Mayor G, J. Torrens.On a family of t-norms[J]. Fuzzy Sets and Systems,1991, 41:161-166.
[34]Durante F, Sempi C. Semi-copula[J]. Kybernetika,2005,41 (3):315-328.
[35]沈银芳.多元拟连接函数的刻画[J].浙江大学学报,2007,34(2)：144-147.
[36]曾霞,何平.基于g函数的多元copula的构造[J].数理统计与管理,2008,27(5)：844-849.
[37]王传玉.离散数学与算法[M].合肥：安徽大学出版社,2001：19,140-142.