复不确定变量
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摘要
在电气工程、信息科学、流体力学等学科中，人们常常使用复数来表示一些物理量，比如交流电的电流、信息科学中的周期信号、流体力学里的二维势流等等．由于缺少实验条件、测量手段或者其他原因，人们并不能精确确定这些物理量，然而实际中仍然需要使用这些量进行研究，于是只能凭借人类的主观经验来估计和描述这些物理量．在前人的研究中，人们采用过复随机变量、模糊复数等概念来描述这些物理量．然而，通过众多学者的调研与实践，人们逐渐认识到概率论和模糊理论并不能解决所有的问题，用复随机变量或模糊复数进行建模有时不符合实际情况．2007年，Liu提出了不确定理论，这是研究人类主观不确定性的新的数学工具．在不确定理论的框架下，本文提出了复不确定变量的概念，定义了复不确定分布以及复不确定变量的期望值和方差，研究了复不确定变量的独立性，讨论了复不确定变量与不确定变量的关系，并给出了两类特殊的复不确定变量的定义．
本文的结构如下：首先，介绍了不确定理论的基本概念．其次，给出了不确定测度的一个判定定理，降低了判定不确定测度的难度，并利用这个判定定理，给出了乘积不确定测度的一个性质，探讨了乘积不确定空间上零测集的性质，随后引入复不确定变量的概念，给出了期望值、方差等基本概念，并研究了它们的数学性质，证明了复不确定变量期望值的线性性质，定义了复不确定变量的独立性，并探讨了复不确定变量独立和不确定变量独立的相互关系．最后，我们给出了不确定分布的一个充要条件，并将此充要条件推广到了复不确定分布和联合不确定分布情形．
综上，本文的创新点主要有：
定义了复不确定变量，用以描述实际使用的取值为复数的不确定量.
定义了复不确定变量的分布，给出了其分布的充要条件.
定义了复不确定变量的独立性，研究了复不确定变量独立的性质.
定义了复不确定变量的期望值，证明了其期望值的线性性.
In the field of electrical engineering, information science, fluid mechanics and soon, some quantities are usually modeled by complex numbers, such as the alternatingcurrents, periodic signals and two-dimensional potential flow. Because of lack of exper-imental conditions, measuring methods or something else, these quantities are usuallyuncertain, however, they are still needed to use. Thus we only can determine them bysubjective methods. Complex random variable and fuzzy complex number have beenemployed to model these quantities. However, a lot of survey show that probability the-ory and fuzzy theory are not enough for all the problems. In2007, Liu proposed anuncertainty theory, which is a new tool for studying uncertain phenomena. Under theframework of uncertainty theory, this dissertation proposes the concept of complex un-certain variable and studies some properties, such as independence, distribution, expectedvalue and variance.
Firstly, we introduce some basic concepts in uncertainty theory. Then we prove atheorem on uncertain measure, verify that a product uncertain measure is an uncertainmeasure and give a theorem on null set in a product uncertainty space. Next, we proposethe concept of complex uncertain variable, its expected value and variance, study theirproperties, discuss the independence of complex uncertain variables and study the rela-tionship between complex uncertain variable and uncertain variable. At last, we give asufcient and necessary condition of complex uncertainty distribution. This dissertationcontributes to the research field of uncertainty theory in the following aspects:(a) a def-inition of complex uncertain variable is proposed to describe complex-valued quantities.(b) a distribution of complex uncertain variable is given, and a sufcient and necessarycondition of complex uncertainty distribution is given.(c) the independence of complexuncertainty variables is given and some properties are studied.(d) the expected value ofcomplex uncertain variable is given and some properties of expected value are studied.
本文的结构如下：首先，介绍了不确定理论的基本概念．其次，给出了不确定测度的一个判定定理，降低了判定不确定测度的难度，并利用这个判定定理，给出了乘积不确定测度的一个性质，探讨了乘积不确定空间上零测集的性质，随后引入复不确定变量的概念，给出了期望值、方差等基本概念，并研究了它们的数学性质，证明了复不确定变量期望值的线性性质，定义了复不确定变量的独立性，并探讨了复不确定变量独立和不确定变量独立的相互关系．最后，我们给出了不确定分布的一个充要条件，并将此充要条件推广到了复不确定分布和联合不确定分布情形．
综上，本文的创新点主要有：
定义了复不确定变量，用以描述实际使用的取值为复数的不确定量.
定义了复不确定变量的分布，给出了其分布的充要条件.
定义了复不确定变量的独立性，研究了复不确定变量独立的性质.
定义了复不确定变量的期望值，证明了其期望值的线性性.
In the field of electrical engineering, information science, fluid mechanics and soon, some quantities are usually modeled by complex numbers, such as the alternatingcurrents, periodic signals and two-dimensional potential flow. Because of lack of exper-imental conditions, measuring methods or something else, these quantities are usuallyuncertain, however, they are still needed to use. Thus we only can determine them bysubjective methods. Complex random variable and fuzzy complex number have beenemployed to model these quantities. However, a lot of survey show that probability the-ory and fuzzy theory are not enough for all the problems. In2007, Liu proposed anuncertainty theory, which is a new tool for studying uncertain phenomena. Under theframework of uncertainty theory, this dissertation proposes the concept of complex un-certain variable and studies some properties, such as independence, distribution, expectedvalue and variance.
Firstly, we introduce some basic concepts in uncertainty theory. Then we prove atheorem on uncertain measure, verify that a product uncertain measure is an uncertainmeasure and give a theorem on null set in a product uncertainty space. Next, we proposethe concept of complex uncertain variable, its expected value and variance, study theirproperties, discuss the independence of complex uncertain variables and study the rela-tionship between complex uncertain variable and uncertain variable. At last, we give asufcient and necessary condition of complex uncertainty distribution. This dissertationcontributes to the research field of uncertainty theory in the following aspects:(a) a def-inition of complex uncertain variable is proposed to describe complex-valued quantities.(b) a distribution of complex uncertain variable is given, and a sufcient and necessarycondition of complex uncertainty distribution is given.(c) the independence of complexuncertainty variables is given and some properties are studied.(d) the expected value ofcomplex uncertain variable is given and some properties of expected value are studied.
引文
[1] Khintchine A. Korrelationstheorie der Stationaren Stochastischen Prozesse. MathematischeAnnalen1934,109:604-615.
[2] Doob J L. Stochastic Processes. New York: Wiley,1953.
[3] Ito K. Stochastic Integral. Proceedings of the Japan Academy, Tokyo, Japan,1944.519-524.
[4] Fisher R A. Statistical Methods and Scientific Inference,3rd ed. New York: Hafner,1971.
[5] Kall P. Stochastic linear programming. Berlin: Springer-Verlag,1976.
[6] Kendall D G. Stochastic Processes Occurring in the Theory of Queues and their Analysis bythe Method of the Imbedded Markov Chain. The Annals of Mathematical Statistics,1953,24(3):338-354.
[7] Brillinger D R. Time series: Data analysis and theory. New York: Holt-Rinehart,1975.
[8] Zadeh L A. Fuzzy Sets. Information and Control,1965,8:338-353.
[9] Dubois D, Prade H. Fuzzy Sets and Systems: Theory and Applications. New York: AcademicPress,1980.
[10] Zadeh L A. Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems,1978,1:3-28.
[11] Dubois D, Prade H. Possibility Theory: An Approach to Computerized Processing of Uncer-tainty. New York: Plenum,1988.
[12] Dempster A P. Upper and Lower Probabilities Induced by a Multivalued Mapping. Annals ofMathematical Statistics,1967,38(2):325-339.
[13] Shafer G. A Mathematical Theory of Evidence. New Jersey: Princeton University Press,1976.
[14] Liu B, Liu Y. Expected Value of Fuzzy Variable and Fuzzy Expected Value Models. IEEETransactions on Fuzzy Systems,2002,10(4):445-450.
[15] Liu B. A Survey of Credibility Theory. Fuzzy Optimization and Decision Making,2006,5(4):387-408.
[16] Liu B. Uncertainty Theory,2nd ed. Berlin: Springer-Verlag, 2007.
[17] Liu B. Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty.Berlin: Springer-Verlag,2010.
[18] Wang X, Gao Z, Guo H. Delphi Method for Estimating Uncetainty Distributions. Information:An International Interdisciplinary Journal, to be pubilished.
[19] Wang X, Gao Z, Guo H. Uncertain Hypothesis Testing for Two Experts’ Empirical Data. Math-ematical and Computer Modelling,2012,55:1478-1482.
[20] Liu B. Theory and Practice of Uncertain Programming,2nd ed. Berlin: Springer-Verlag,2009.
[21] Yan L. Optimal Portfolio Selections with Uncertain Returns. Modern Applied Science,2009,3(8):76-81.
[22] Gao Y. Uncertain Models for Single Facility Location Problems on Networks. Applied Math-ematical Modelling,2012,36(6):2592-2599.
[23] Zhang B, Peng J. Uncertain Programming Model for Chinese Postman Problem with UncertainWeights. Industrial Engineering and Management Systems,2012,11(1):18-25.
[24] Sheng Y, Yao K. A Transportation Model with Uncertain Costs and Demands. Information: AnInternational Interdisciplinary Journal, to be published.
[25] Rong L. Two New Uncertainty Programming Models of Inventory with Uncertain Costs. Jour-nal of Information and Computational Science,2011,8(2):280-288.
[26] Liu B, Yao K. Uncertain Multilevel Programming: Algorithm and Applications,http://orsc.edu.cn/online/120114.pdf
[27] Liu B, Chen X. Uncertain Multiobjective Programming and Uncertain Goal Programming,Technical Report,2012.
[28] Chen X, Ralescu D A. Expert Systems with Applications,2011,38:15582-15586.
[29] Li X, Liu B. Hybrid Logic and Uncertain Logic. Journal of Uncertain Systems,2009,3(2):83-94.
[30] Liu B. Uncertain Entailment and Modus Ponens in the Framework of Uncertain Logic. Journalof Uncertain Systems,2009,3(4):243-251.
[31] Liu B. Uncertain Logic for Modeling Human Language, Journal of Uncertain Systems,2011,5(1):3-20.
[32] Liu B. Uncertain Set Theory and Uncertain Inference Rule with Application to Uncertain Con-trol. Journal of Uncertain Systems,2010,4(2):83-98.
[33] Gao X, Gao Y, Ralescu D A. On Liu’s Inference Rule for Uncertain Systems. InternationalJournal of Uncertainty, Fuzziness and Knowledge-Based Systems,2010,18(1):1-11.
[34] Gao Y. Uncertain Inference Control for Balancing an Inverted Pendulum. Fuzzy Optimizationand Decision Making, to be published.
[35] Peng Z, Chen X. Uncertain Systems are Universal Approximators. Technical Report,2010.http://orsc.edu.cn/online/100110.pdf.
[36] Liu B. Fuzzy Process, Hybrid Process and Uncertain Process. Journal of Uncertain Systems,2008,2(1):3-16.
[37] Liu B. Extreme Value Theorems of Uncertain Process with Application to Insurance Risk Mod-el. Soft Computing, to be published.
[38] Chen X. Variation Analysis of Stationary Independent Increment Process. http-s://orsc.edu.cn/online/110422.pdf.
[39] Gao Y. Variation Analysis of Semi-canonical Process. Mathematical and Computer Modelling,2011,53(9-10):1983-1989.
[40] Liu B. Some Research Problems in Uncertainty Theory. Journal of Uncertain Systems,2009,3(1):3-10.
[41] Yao K. Uncertain Calculus with Renewal Process. Fuzzy Optimization and Decision Making,to be published.
[42] Chen X. Uncertain Calculus with Finite Variation Processes. http://orsc.edu.cn/on-line/110613.pdf.
[43]陈孝伟．有界变差过程不确定分析[博士学位论文]．北京：清华大学，2011．
[44] Chen X, Liu B. Existence and Uniqueness Theorem for Uncertain Diferential Equations.Fuzzy Optimization and Decision Making,2010,9(1):69-81.
[45] Gao Y. Existence and Uniqueness Theorem on Uncertain Diferential Equations with LocalLipschitz Condition. Journal of Uncertain Systems, to be published.
[46] Yao K. A Type of Nonlinear Uncertain Diferential Equations with Analytic Solution.http://orsc.edu.cn/online/110928.pdf.
[47] Chen X. American Option Pricing Formula for Uncertain Financial Market. International Jour-nal of Operations Research,2011,8(2):32-37.
[48] Peng J, Yao K. A New Option Pricing Model for Stocks in Uncertainty Markets. InternationalJournal of Operations Research,2011,8(2):18-26.
[49] Yao K. A No-arbitrage Determinant Theorem for Uncertain Stock Model.http://orsc.edu.cn/online/100903.pdf.
[50] Zhu Y. Uncertain Optimal Control for Multistage Fuzzy Systems. IEEE Transactions on Sys-tems Man and Cybernetics Part B,2011,41(4):964-975.
[51] Xu X, Zhu Y. Uncertain Bang-bang Control for Continuous Time Model. http://orsc.edu.cn/online/110525.pdf.
[52] Liu B. Uncertain Risk Analysis and Uncertain Reliability Analysis. Journal of Uncertain Sys-tems,2010,4(3):163-170.
[53] Peng J. Value at Risk and Tail Value at Risk in Uncertain Environment. Proceedings of theEighth International Conference on Information and Management Sciences, Kunming, China,2009.787-793.
[54] Wooding R A. The Multivariate Distribution of Complex Normal Variables. Biometrica,1956,43:212-215
[55] Turin G L. The Characteristic Function of Herimitian quadratic forms in complex normal vari-able. Biometrika,1960,47:199-201.
[56] Goodman N R. Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distri-bution (An Introduction). The Annals of Mathematical Statistics,1963,34(1):152-177.
[57] Pearlman W. Polar Quantization of a Complex Gaussian Random Variable. IEEE Transactionson Communications,1979,27(6):892-899.
[58] Miller K S. Complex Stochastic Processes: An Introduction to Theory and Application. Boston:Addison-Wesley,1974.
[59] Neeser F D, Massey J L. Proper Complex Random Processes with Applications to InformationTheory. IEEE Information Theory Society,1993,39(4):1293-1302.
[60] Matheron, G. Random Sets and Integral Geometry. New York: Wiley,1975.
[61] Hofmann J. Probability in Banach Space. In Lecture Notes in Mathematics,1977,598:1-186.
[62] Buckley J J. Fuzzy Complex Numbers. Fuzzy Sets and Systems,1989,33(3):333-345.
[63] Buckley J J, Qu Y. Fuzzy Complex Analysis I: Diferentiation. Fuzzy Sets and Systems,1991,41(3):269-284.
[64] Buckley J J. Fuzzy Complex Analysis II: Integration. Fuzzy Sets and Systems,1992,49(2):171-179.
[65] Zhang G. Fuzzy Limit Theory of Fuzzy Complex Numbers. Fuzzy Sets and Systems,1992,46(2):227-235.
[66] Wu C, Qiu J. Some Remarks for Fuzzy Complex Analysis, Fuzzy Sets and Systems,1999,106(2):231-238.
[67] Ramot D, Milo R, Friedman M, Kandel A. Complex Fuzzy Sets. IEEE Transactions on FuzzySystems,2002,10(2):171-186.
[68] Ramot D, Friedman M, Langholz G, Kandel A. Complex Fuzzy Logic. IEEE Transactions onFuzzy Systems,2003,11(4):450-461.
[69]马生全．模糊复分析理论基础．北京：科学出版社，2010.
[70] Yang L. Uncertain Variables Taking Values in a Normed Linear Space. http://orsc.edu.cn/on-line/090806.pdf.
[71] Liu Y. Uncertain Random Variables: A Mixture of Uncertainty and Randomness. Soft Com-puting, to be published.
[72] Gao X. Some Properties of Continuous Uncertain Measure. International Journal of Uncertain-ty, Fuzziness and Knowledge-Based Systems,2009,17(3):419-426.
[73]高欣．不确定测度及其应用[博士学位论文]．北京：清华大学，2009．
[74] You C. Some Convergence Theorems of Uncertain Sequences. Mathematical and ComputerModelling,2009,49(3-4):482-487.
[75] Zhang Z. Some Discussions on Uncertain Measure. Fuzzy Optimization and Decision Making,2011,10(1):31-43.
[76] Liu Y, Ha M. Expected Value of Function of Uncertain Variables. Journal of Uncertain Sys-tems,2010,4(3):181-186.
[77] Liu W, Xu J. Some Properties on Expected Value Operator for Uncertain Variables. Informa-tion: An International Interdisciplinary Journal,2010,13(5):1693-1699.
[78] Yang X. Moments and Tails Inequality within the Framework of Uncertainty Theory. Informa-tion: An International Interdisciplinary Journal,2011,14(8):2599-2604.
[79] Chen X, Dai W. Maximum Entropy Principle for Uncertain Variables. International Journal ofFuzzy Systems,2011,13(3):232-236.
[80] Dai W, Chen X. Entropy of Function of Uncertain Variables. Mathematical and ComputerModelling,2012,55(3-4):754-760.
[81]戴韡．不确定理论中的极大熵原理[博士学位论文]．北京：清华大学，2010．
[82] Dai W. Quadratic Entropy of Uncertain Variables. Information: An International Interdisci-plinary Journal, to be published.
[83] Chen X, Samarjit K, Ralescu D A. Cross-Entropy Measure of Uncertain Variables. InformationSciences, to be published.
[84] Peng Z, Iwamura K. A Sufcient and Necessary Condition of Uncertain Measure. Information:An International Interdisciplinary Journal, to be published.
[85] Peng Z. A Sufcient and Necessary Condition of Product Uncertain Null Set. Proceedingsof the Eighth International Conference on Information and Management Sciences, Kunming,China,2009.798-801.
[86] Peng Z, Iwamura K. A Sufcient and Necessary Condition of Uncertainty Distribution. Journalof Interdisciplinary Mathematics,2010,13(3):277-285.
[2] Doob J L. Stochastic Processes. New York: Wiley,1953.
[3] Ito K. Stochastic Integral. Proceedings of the Japan Academy, Tokyo, Japan,1944.519-524.
[4] Fisher R A. Statistical Methods and Scientific Inference,3rd ed. New York: Hafner,1971.
[5] Kall P. Stochastic linear programming. Berlin: Springer-Verlag,1976.
[6] Kendall D G. Stochastic Processes Occurring in the Theory of Queues and their Analysis bythe Method of the Imbedded Markov Chain. The Annals of Mathematical Statistics,1953,24(3):338-354.
[7] Brillinger D R. Time series: Data analysis and theory. New York: Holt-Rinehart,1975.
[8] Zadeh L A. Fuzzy Sets. Information and Control,1965,8:338-353.
[9] Dubois D, Prade H. Fuzzy Sets and Systems: Theory and Applications. New York: AcademicPress,1980.
[10] Zadeh L A. Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems,1978,1:3-28.
[11] Dubois D, Prade H. Possibility Theory: An Approach to Computerized Processing of Uncer-tainty. New York: Plenum,1988.
[12] Dempster A P. Upper and Lower Probabilities Induced by a Multivalued Mapping. Annals ofMathematical Statistics,1967,38(2):325-339.
[13] Shafer G. A Mathematical Theory of Evidence. New Jersey: Princeton University Press,1976.
[14] Liu B, Liu Y. Expected Value of Fuzzy Variable and Fuzzy Expected Value Models. IEEETransactions on Fuzzy Systems,2002,10(4):445-450.
[15] Liu B. A Survey of Credibility Theory. Fuzzy Optimization and Decision Making,2006,5(4):387-408.
[16] Liu B. Uncertainty Theory,2nd ed. Berlin: Springer-Verlag, 2007.
[17] Liu B. Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty.Berlin: Springer-Verlag,2010.
[18] Wang X, Gao Z, Guo H. Delphi Method for Estimating Uncetainty Distributions. Information:An International Interdisciplinary Journal, to be pubilished.
[19] Wang X, Gao Z, Guo H. Uncertain Hypothesis Testing for Two Experts’ Empirical Data. Math-ematical and Computer Modelling,2012,55:1478-1482.
[20] Liu B. Theory and Practice of Uncertain Programming,2nd ed. Berlin: Springer-Verlag,2009.
[21] Yan L. Optimal Portfolio Selections with Uncertain Returns. Modern Applied Science,2009,3(8):76-81.
[22] Gao Y. Uncertain Models for Single Facility Location Problems on Networks. Applied Math-ematical Modelling,2012,36(6):2592-2599.
[23] Zhang B, Peng J. Uncertain Programming Model for Chinese Postman Problem with UncertainWeights. Industrial Engineering and Management Systems,2012,11(1):18-25.
[24] Sheng Y, Yao K. A Transportation Model with Uncertain Costs and Demands. Information: AnInternational Interdisciplinary Journal, to be published.
[25] Rong L. Two New Uncertainty Programming Models of Inventory with Uncertain Costs. Jour-nal of Information and Computational Science,2011,8(2):280-288.
[26] Liu B, Yao K. Uncertain Multilevel Programming: Algorithm and Applications,http://orsc.edu.cn/online/120114.pdf
[27] Liu B, Chen X. Uncertain Multiobjective Programming and Uncertain Goal Programming,Technical Report,2012.
[28] Chen X, Ralescu D A. Expert Systems with Applications,2011,38:15582-15586.
[29] Li X, Liu B. Hybrid Logic and Uncertain Logic. Journal of Uncertain Systems,2009,3(2):83-94.
[30] Liu B. Uncertain Entailment and Modus Ponens in the Framework of Uncertain Logic. Journalof Uncertain Systems,2009,3(4):243-251.
[31] Liu B. Uncertain Logic for Modeling Human Language, Journal of Uncertain Systems,2011,5(1):3-20.
[32] Liu B. Uncertain Set Theory and Uncertain Inference Rule with Application to Uncertain Con-trol. Journal of Uncertain Systems,2010,4(2):83-98.
[33] Gao X, Gao Y, Ralescu D A. On Liu’s Inference Rule for Uncertain Systems. InternationalJournal of Uncertainty, Fuzziness and Knowledge-Based Systems,2010,18(1):1-11.
[34] Gao Y. Uncertain Inference Control for Balancing an Inverted Pendulum. Fuzzy Optimizationand Decision Making, to be published.
[35] Peng Z, Chen X. Uncertain Systems are Universal Approximators. Technical Report,2010.http://orsc.edu.cn/online/100110.pdf.
[36] Liu B. Fuzzy Process, Hybrid Process and Uncertain Process. Journal of Uncertain Systems,2008,2(1):3-16.
[37] Liu B. Extreme Value Theorems of Uncertain Process with Application to Insurance Risk Mod-el. Soft Computing, to be published.
[38] Chen X. Variation Analysis of Stationary Independent Increment Process. http-s://orsc.edu.cn/online/110422.pdf.
[39] Gao Y. Variation Analysis of Semi-canonical Process. Mathematical and Computer Modelling,2011,53(9-10):1983-1989.
[40] Liu B. Some Research Problems in Uncertainty Theory. Journal of Uncertain Systems,2009,3(1):3-10.
[41] Yao K. Uncertain Calculus with Renewal Process. Fuzzy Optimization and Decision Making,to be published.
[42] Chen X. Uncertain Calculus with Finite Variation Processes. http://orsc.edu.cn/on-line/110613.pdf.
[43]陈孝伟．有界变差过程不确定分析[博士学位论文]．北京：清华大学，2011．
[44] Chen X, Liu B. Existence and Uniqueness Theorem for Uncertain Diferential Equations.Fuzzy Optimization and Decision Making,2010,9(1):69-81.
[45] Gao Y. Existence and Uniqueness Theorem on Uncertain Diferential Equations with LocalLipschitz Condition. Journal of Uncertain Systems, to be published.
[46] Yao K. A Type of Nonlinear Uncertain Diferential Equations with Analytic Solution.http://orsc.edu.cn/online/110928.pdf.
[47] Chen X. American Option Pricing Formula for Uncertain Financial Market. International Jour-nal of Operations Research,2011,8(2):32-37.
[48] Peng J, Yao K. A New Option Pricing Model for Stocks in Uncertainty Markets. InternationalJournal of Operations Research,2011,8(2):18-26.
[49] Yao K. A No-arbitrage Determinant Theorem for Uncertain Stock Model.http://orsc.edu.cn/online/100903.pdf.
[50] Zhu Y. Uncertain Optimal Control for Multistage Fuzzy Systems. IEEE Transactions on Sys-tems Man and Cybernetics Part B,2011,41(4):964-975.
[51] Xu X, Zhu Y. Uncertain Bang-bang Control for Continuous Time Model. http://orsc.edu.cn/online/110525.pdf.
[52] Liu B. Uncertain Risk Analysis and Uncertain Reliability Analysis. Journal of Uncertain Sys-tems,2010,4(3):163-170.
[53] Peng J. Value at Risk and Tail Value at Risk in Uncertain Environment. Proceedings of theEighth International Conference on Information and Management Sciences, Kunming, China,2009.787-793.
[54] Wooding R A. The Multivariate Distribution of Complex Normal Variables. Biometrica,1956,43:212-215
[55] Turin G L. The Characteristic Function of Herimitian quadratic forms in complex normal vari-able. Biometrika,1960,47:199-201.
[56] Goodman N R. Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distri-bution (An Introduction). The Annals of Mathematical Statistics,1963,34(1):152-177.
[57] Pearlman W. Polar Quantization of a Complex Gaussian Random Variable. IEEE Transactionson Communications,1979,27(6):892-899.
[58] Miller K S. Complex Stochastic Processes: An Introduction to Theory and Application. Boston:Addison-Wesley,1974.
[59] Neeser F D, Massey J L. Proper Complex Random Processes with Applications to InformationTheory. IEEE Information Theory Society,1993,39(4):1293-1302.
[60] Matheron, G. Random Sets and Integral Geometry. New York: Wiley,1975.
[61] Hofmann J. Probability in Banach Space. In Lecture Notes in Mathematics,1977,598:1-186.
[62] Buckley J J. Fuzzy Complex Numbers. Fuzzy Sets and Systems,1989,33(3):333-345.
[63] Buckley J J, Qu Y. Fuzzy Complex Analysis I: Diferentiation. Fuzzy Sets and Systems,1991,41(3):269-284.
[64] Buckley J J. Fuzzy Complex Analysis II: Integration. Fuzzy Sets and Systems,1992,49(2):171-179.
[65] Zhang G. Fuzzy Limit Theory of Fuzzy Complex Numbers. Fuzzy Sets and Systems,1992,46(2):227-235.
[66] Wu C, Qiu J. Some Remarks for Fuzzy Complex Analysis, Fuzzy Sets and Systems,1999,106(2):231-238.
[67] Ramot D, Milo R, Friedman M, Kandel A. Complex Fuzzy Sets. IEEE Transactions on FuzzySystems,2002,10(2):171-186.
[68] Ramot D, Friedman M, Langholz G, Kandel A. Complex Fuzzy Logic. IEEE Transactions onFuzzy Systems,2003,11(4):450-461.
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