基于条件事件代数的概率逻辑推理和概率逻辑衍推推理
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摘要
概率逻辑是以概率论和现代演绎逻辑为工具构造归纳推理逻辑的形式演绎系统。随着计算机技术的发展,概率逻辑正逐步成为面向人工智能、决策分析、知识发现和推理应用的重要发展方向。在人工智能专家系统中,通常需要利用已有或已观察到的数据进行推理应用,为实际中的分析、预测和决策等问题提供科学依据。而数据中蕴含的概率因果关系(即概率逻辑关系)可以描述各属性变量间的相互依赖、以及依赖关系的不确定性。利用这些关系进行逻辑推理就可以得到新的、更为复杂的因果关系。数据中概率因果关系的表示、挖掘、推理及应用也越来越受到许多研究者的广泛关注,成为智能数据分析、知识发现和不确定人工智能等领域中的重要研究方向。
作为描述变量间概率因果关系的有效工具,贝叶斯网被广泛应用于知识发现、决策支持、预测分析等领域中。然而,以金融趋势分析、医疗诊断预测、实时交通控制等为代表的实际应用,经典的贝叶斯网由于其精确的表示,其推理机制不能较好地满足这些应用的需求。同时,在解决实际问题的诸多应用中,用户往往并不关心过于精确的量化的知识,而仅仅需要对知识的定性描述。在概率论中常用的表示方式中,已观察到的数据的表示和相应推理结果的表示通常是用点概率或是用区间概率来描述的;而推理的过程却是一个逻辑归纳演绎的过程。对于复杂的推理直接用贝叶斯网计算通常需要经过大量的全概率和条件概率的变换。这些计算过程中能否保证知识的概率表示和逻辑表示在推理过程中保持一致?复杂的推理问题如何能等价于相对简单的推理问题?等等这些成为在实际应用中利用贝叶斯网进行推理时面临的困难。
本文提出了基于条件事件代数的概率逻辑推理和门限可信概率逻辑衍推方法。首先利用条件事件代数,通过扩展普通可测空间将经典概率理论的条件概率扩展为普通条件事件,用以表示推理过程中数据信息,实现了概率逻辑在推理过程中的相容表示;再利用条件事件代数的性质,将条件事件转换为普通事件和相应的连接事件,从而将高阶复杂推理问题转换为低阶推理问题,并结合贝叶斯网实现了高阶复杂问题的概率逻辑推理。本文针对前述问题的实际需求,研究了基于条件事件代数和贝叶斯网的概率逻辑推理关键技术,我们通过实验验证了本文所提出方法的有效性,并将这一问题应用到企业文化问卷调查分析中,初步开发了相应的应用系统。另外,本文基于一般概率逻辑的表示及推理方法,研究了将已知的前提划分为不同的可信度子集,当这些子集满足一定的门限值(或称为阀值)时,衍推出可信结论的方法。
本文的主要工作和创新之处总结如下:
●研究了利用事件的条件性使推理过程中概率和逻辑相一致以及复杂问题的表示方法。以条件事件代数为基础,在普通可测空间中引入了事件的条件性,解决了概率逻辑在推理过程中的相容性问题。并利用贝叶斯网实现了高阶条件事件推理的方法。研究的核心是通过扩展普通可测空间,利用事件的条件性,扩展条件概率的表示能力,使得高阶条件可以转换成较低阶的条件,从而提出将高阶条件推理问题转换为普通事件和相应的连接事件的推理问题,为进行复杂推理提供了一种有效的支撑技术;本文的方法不仅从一定程度上弥补了已有推理方法在概率因果关系推理方面的不足,也较好地弥补了贝叶斯网在复杂问题推理时的不足。
●研究了利用条件事件代数进行高阶正反推理的方法。在条件事件代数的基础上,结合影响图、模糊集理论,提出了利用条件的事件性表示决策过程中的因果关系的方法,并利用模糊集理论,将点概率的推理方法推广到区间概率中,结合贝叶斯网实现了正反向推理方法。该研究以决策问题的实际应用为出发点,实现了复杂决策问题的正向即从因到果,及从果到因的反向推理方法,增强了贝叶斯网在复杂推理问题中的实用性。而基于模糊集理论的处理方法使得对决策问题的描述更加符合实际应用情况,有效地避免了决策过程中专家知识的主观性。同时,该研究也是条件事件代数和模糊集理论在概率因果关系表示及推理方面的新应用。
●研究了概率逻辑门限可信衍推方法。本文基于一般概率逻辑的表示及推理方法,提出了将已知的前提划分为不同的可信度子集,当这些子集满足一定的门限值(或称为阀值),相应的结论也满足一定门限值时则称为这一结论也是可信的,即衍推。该研究提供了描述这种衍推关系的基本概念和相应的性质,给出了这种衍推关系的合理性说明,并结合模式识别和模糊集相关子集的划分方法,根据衍推关系的最大信息熵约束方法,推导出结论的可信度阀值,从而完成了概率逻辑的衍推。
本文首先解决了概率逻辑推理过程中概率和逻辑在表示规则(知识)时的相容性问题,以此为基础提出了结合贝叶斯网利用条件事件代数进行复杂决策问题的正向和反向精确推理及模糊推理的方法;并利用模式识别相关方法和比较概率子集划分的方法进行门限可信衍推推理,这些方法在其它相关工作中尚未见报道。
Based on probability theory and modern deductive logic, probability logic is used as a tool to construct the logic of deductive system of the formal inductive reasoning. With the development of computer technology, it is becoming an important development direction of artificial intelligence, decision analysis, knowledge discovery and reasoning applications.
In expert systems of artificial intelligence, existing data or observations are used in reasoning application and provided as a scientific basis for the practical analysis, forecasting and decision-making. The probabilistic causal relationship between variables of the data can be used to describe the interdependence of various properties of data, and the uncertainty of the interdependence. Using these relations to process logic reasoning, we can get a new, more complex causal relationship. The presentation, data mining, reasoning and application of probabilistic causal relationship in data are being attention by many researchers, and become the important field of the intelligent data analysis, knowledge discovery and research of uncertainty artificial intelligence.
As an effective tool for describing the probabilistic causal relationship between variables, Bayesian networks are widely used in knowledge discovery, decision support, predictive analysis and other areas. However, in the applications of the financial trend analysis, medical diagnostics forecasts, real-time traffic control, etc., classic Bayesian networks can not meet better the needs of these applications for its accurate representation and reasoning mechanism. At the same time, in many cases of applications, users often do not care too precise quantitative knowledge, but only a qualitative description of knowledge required. Normally, the representation of the observed data and reasoning results is described by point probability or interval probability, but the reasoning is a logical process. For complex reasoning, the calculation of directly using Bayesian networks usually takes a lot of transformation of the whole probability and conditional probability. Whether these calculations can ensure that the probability and logical consistency in the reasoning process, complex reasoning, how the complex reasoning problem can be transformed equivalent to the reasoning of the relatively simple issues, these are difficulties of using Bayesian networks in practical application.
Based on the methods of probabilistic logical reasoning on Bayesian networks, the general conditions probability can be used to describe the normal conditional event by extending the classical probability space with the conditional event algebra. By extending normal measurable space with conditional event, we can bring logic consistent with probability in denoting conditional probability information, and then we transform a higher-order conditional event to normal events and corresponding logical combination events via Conditional Event Algebra. By this way, we can implement the higher-order complex logic reasoning.
In this paper, we research the key technologies of reasoning based on conditional event algebra and Bayesian Network for the needs of actual application, and use this method to the corporate culture survey analysis. We experimentally verify the effectiveness of the proposed method in this paper, and design the corresponding prototype system.
The main work and innovations in this paper are summarized as following:
●We research the method of probabilistic logic reasoning by using conditional event.
Based on the conditional event algebra, we extend the normal measurable space with conditional event, and bring logic consistent with probability in denoting conditional probability information, and implement the probabilistic logic reasoning method of high-order conditional events on Bayesian Networks. By extending common space with condition event, using properties of the condition event, we extend the scalability of the conditional probability for converting high-order conditions into a lower-order terms, and propose a method for the high-order inference problem transformed into ordinary events and the corresponding connection events for reasoning problems. To carry out complex reasoning, we provide an effective support technology: the present method not only has been offset to some extent of the present probabilistic logical reasoning method, but also make up for the problem that existed in complex reasoning on Bayesian networks.
●We research the forward and backward probability logic method of high-order conditionals by using condition event algebra.
Based on the theories of influence diagram and fuzzy set, we utility conditional event to denote the casual relationship of decision-making process, and we use the theories of fuzzy set to extend point probability to interval probability. To resolve the real application of decision-making, we implement the forward and backward probability logic reasoning of complex decision-making problems.
●We research the method of entailment in probabilistic logic reasoning. Based on a general probabilistic logic representation and reasoning methods, we propose the idea of dividing the set of known premise divided into different credibility sub-sets. When these sub-sets satisfy a certain threshold value, the corresponding conclusions also satisfy a certain threshold value, then we call this conclusion is credible.
In our research, we describe the basic concepts and the corresponding attributes of such entailment relations, and give the illustrative of this entailment relationship. And we combine the methods of pattern recognition with fuzzy sets to divide a set known premise into different sub-sets. According to the constraint method of entailment relationship, we obtain the credibility threshold values of the conclusions, and complete the process of logic entailment.
In this paper, we bring the logic consistent with the probability in denoting rules by extending normal measurable space with conditional event, and propose the reasoning methods form condition to result and result to condition in decision problem, and the corresponding fuzzy reasoning methods. The methods of pattern recognition and the partition of premises based on entailment reasoning used in this paper and other related works have not been reported.The methods of pattern recognition and the partition of premises based on entailment reasoning used in this paper and other related works have not been reported.
作为描述变量间概率因果关系的有效工具,贝叶斯网被广泛应用于知识发现、决策支持、预测分析等领域中。然而,以金融趋势分析、医疗诊断预测、实时交通控制等为代表的实际应用,经典的贝叶斯网由于其精确的表示,其推理机制不能较好地满足这些应用的需求。同时,在解决实际问题的诸多应用中,用户往往并不关心过于精确的量化的知识,而仅仅需要对知识的定性描述。在概率论中常用的表示方式中,已观察到的数据的表示和相应推理结果的表示通常是用点概率或是用区间概率来描述的;而推理的过程却是一个逻辑归纳演绎的过程。对于复杂的推理直接用贝叶斯网计算通常需要经过大量的全概率和条件概率的变换。这些计算过程中能否保证知识的概率表示和逻辑表示在推理过程中保持一致?复杂的推理问题如何能等价于相对简单的推理问题?等等这些成为在实际应用中利用贝叶斯网进行推理时面临的困难。
本文提出了基于条件事件代数的概率逻辑推理和门限可信概率逻辑衍推方法。首先利用条件事件代数,通过扩展普通可测空间将经典概率理论的条件概率扩展为普通条件事件,用以表示推理过程中数据信息,实现了概率逻辑在推理过程中的相容表示;再利用条件事件代数的性质,将条件事件转换为普通事件和相应的连接事件,从而将高阶复杂推理问题转换为低阶推理问题,并结合贝叶斯网实现了高阶复杂问题的概率逻辑推理。本文针对前述问题的实际需求,研究了基于条件事件代数和贝叶斯网的概率逻辑推理关键技术,我们通过实验验证了本文所提出方法的有效性,并将这一问题应用到企业文化问卷调查分析中,初步开发了相应的应用系统。另外,本文基于一般概率逻辑的表示及推理方法,研究了将已知的前提划分为不同的可信度子集,当这些子集满足一定的门限值(或称为阀值)时,衍推出可信结论的方法。
本文的主要工作和创新之处总结如下:
●研究了利用事件的条件性使推理过程中概率和逻辑相一致以及复杂问题的表示方法。以条件事件代数为基础,在普通可测空间中引入了事件的条件性,解决了概率逻辑在推理过程中的相容性问题。并利用贝叶斯网实现了高阶条件事件推理的方法。研究的核心是通过扩展普通可测空间,利用事件的条件性,扩展条件概率的表示能力,使得高阶条件可以转换成较低阶的条件,从而提出将高阶条件推理问题转换为普通事件和相应的连接事件的推理问题,为进行复杂推理提供了一种有效的支撑技术;本文的方法不仅从一定程度上弥补了已有推理方法在概率因果关系推理方面的不足,也较好地弥补了贝叶斯网在复杂问题推理时的不足。
●研究了利用条件事件代数进行高阶正反推理的方法。在条件事件代数的基础上,结合影响图、模糊集理论,提出了利用条件的事件性表示决策过程中的因果关系的方法,并利用模糊集理论,将点概率的推理方法推广到区间概率中,结合贝叶斯网实现了正反向推理方法。该研究以决策问题的实际应用为出发点,实现了复杂决策问题的正向即从因到果,及从果到因的反向推理方法,增强了贝叶斯网在复杂推理问题中的实用性。而基于模糊集理论的处理方法使得对决策问题的描述更加符合实际应用情况,有效地避免了决策过程中专家知识的主观性。同时,该研究也是条件事件代数和模糊集理论在概率因果关系表示及推理方面的新应用。
●研究了概率逻辑门限可信衍推方法。本文基于一般概率逻辑的表示及推理方法,提出了将已知的前提划分为不同的可信度子集,当这些子集满足一定的门限值(或称为阀值),相应的结论也满足一定门限值时则称为这一结论也是可信的,即衍推。该研究提供了描述这种衍推关系的基本概念和相应的性质,给出了这种衍推关系的合理性说明,并结合模式识别和模糊集相关子集的划分方法,根据衍推关系的最大信息熵约束方法,推导出结论的可信度阀值,从而完成了概率逻辑的衍推。
本文首先解决了概率逻辑推理过程中概率和逻辑在表示规则(知识)时的相容性问题,以此为基础提出了结合贝叶斯网利用条件事件代数进行复杂决策问题的正向和反向精确推理及模糊推理的方法;并利用模式识别相关方法和比较概率子集划分的方法进行门限可信衍推推理,这些方法在其它相关工作中尚未见报道。
Based on probability theory and modern deductive logic, probability logic is used as a tool to construct the logic of deductive system of the formal inductive reasoning. With the development of computer technology, it is becoming an important development direction of artificial intelligence, decision analysis, knowledge discovery and reasoning applications.
In expert systems of artificial intelligence, existing data or observations are used in reasoning application and provided as a scientific basis for the practical analysis, forecasting and decision-making. The probabilistic causal relationship between variables of the data can be used to describe the interdependence of various properties of data, and the uncertainty of the interdependence. Using these relations to process logic reasoning, we can get a new, more complex causal relationship. The presentation, data mining, reasoning and application of probabilistic causal relationship in data are being attention by many researchers, and become the important field of the intelligent data analysis, knowledge discovery and research of uncertainty artificial intelligence.
As an effective tool for describing the probabilistic causal relationship between variables, Bayesian networks are widely used in knowledge discovery, decision support, predictive analysis and other areas. However, in the applications of the financial trend analysis, medical diagnostics forecasts, real-time traffic control, etc., classic Bayesian networks can not meet better the needs of these applications for its accurate representation and reasoning mechanism. At the same time, in many cases of applications, users often do not care too precise quantitative knowledge, but only a qualitative description of knowledge required. Normally, the representation of the observed data and reasoning results is described by point probability or interval probability, but the reasoning is a logical process. For complex reasoning, the calculation of directly using Bayesian networks usually takes a lot of transformation of the whole probability and conditional probability. Whether these calculations can ensure that the probability and logical consistency in the reasoning process, complex reasoning, how the complex reasoning problem can be transformed equivalent to the reasoning of the relatively simple issues, these are difficulties of using Bayesian networks in practical application.
Based on the methods of probabilistic logical reasoning on Bayesian networks, the general conditions probability can be used to describe the normal conditional event by extending the classical probability space with the conditional event algebra. By extending normal measurable space with conditional event, we can bring logic consistent with probability in denoting conditional probability information, and then we transform a higher-order conditional event to normal events and corresponding logical combination events via Conditional Event Algebra. By this way, we can implement the higher-order complex logic reasoning.
In this paper, we research the key technologies of reasoning based on conditional event algebra and Bayesian Network for the needs of actual application, and use this method to the corporate culture survey analysis. We experimentally verify the effectiveness of the proposed method in this paper, and design the corresponding prototype system.
The main work and innovations in this paper are summarized as following:
●We research the method of probabilistic logic reasoning by using conditional event.
Based on the conditional event algebra, we extend the normal measurable space with conditional event, and bring logic consistent with probability in denoting conditional probability information, and implement the probabilistic logic reasoning method of high-order conditional events on Bayesian Networks. By extending common space with condition event, using properties of the condition event, we extend the scalability of the conditional probability for converting high-order conditions into a lower-order terms, and propose a method for the high-order inference problem transformed into ordinary events and the corresponding connection events for reasoning problems. To carry out complex reasoning, we provide an effective support technology: the present method not only has been offset to some extent of the present probabilistic logical reasoning method, but also make up for the problem that existed in complex reasoning on Bayesian networks.
●We research the forward and backward probability logic method of high-order conditionals by using condition event algebra.
Based on the theories of influence diagram and fuzzy set, we utility conditional event to denote the casual relationship of decision-making process, and we use the theories of fuzzy set to extend point probability to interval probability. To resolve the real application of decision-making, we implement the forward and backward probability logic reasoning of complex decision-making problems.
●We research the method of entailment in probabilistic logic reasoning. Based on a general probabilistic logic representation and reasoning methods, we propose the idea of dividing the set of known premise divided into different credibility sub-sets. When these sub-sets satisfy a certain threshold value, the corresponding conclusions also satisfy a certain threshold value, then we call this conclusion is credible.
In our research, we describe the basic concepts and the corresponding attributes of such entailment relations, and give the illustrative of this entailment relationship. And we combine the methods of pattern recognition with fuzzy sets to divide a set known premise into different sub-sets. According to the constraint method of entailment relationship, we obtain the credibility threshold values of the conclusions, and complete the process of logic entailment.
In this paper, we bring the logic consistent with the probability in denoting rules by extending normal measurable space with conditional event, and propose the reasoning methods form condition to result and result to condition in decision problem, and the corresponding fuzzy reasoning methods. The methods of pattern recognition and the partition of premises based on entailment reasoning used in this paper and other related works have not been reported.The methods of pattern recognition and the partition of premises based on entailment reasoning used in this paper and other related works have not been reported.
引文
[1] D.Bamber, I.R.Goodman. Reasoning with assertions of high conditional probability:Entailment with universal near surety[Z]. Ithaca,New York: The 2nd International symposium on Imprecise Probabilities and Their Applications, 2001.
[2] E.Waltz, L.James.赵宗贵译.多传感器数据融合[M].北京:电子部28所,1990.
[3]美国国防部关键技术计划编写组.美国国防部关键技术(1992财年)[M].北京:国防科技信息中心,1991.
[4] E.W.Adams. Probability and the Logic of Conditionals. In: Hintikka j., suppes P. eds. Aspects of inductive logic[M]. North-Holland, Amsterdam: Elsevier Science, 1966, 265-316.
[5] E.W.Adams. The logic of conditionals[M]. Dordrecht,Holland:D.Reidel Co.,1975.
[6] P.G.Calabrese. An algebraic synthesis of the foundations of logic and probability[J]. Information Science, 1987, 42, (2):187-316.
[7] P.G.Calabrese. A theory of conditional information with applications[J]. IEEE Transactions on system, Man and Cybernetics, 1994, 24, (12):1676-1664.
[8] I.R.Goodman, H.T.Nguyen,E.A.Walker. Conditional inference and logic for intelligent systems: A theory of measure-free conditioning[J]. Amsterdam, North-Holland: Elsevier Science, 1991.
[9] E.W.Adams. Updating on conditional information[J]. IEEE Transactions on system,Man and Cybernetics, 1994, 24, (12):1708-1713.
[10] P.G.Calabrese. Deduction with uncertain conditionals[J]. Information Science, 2002, 147:143-191.
[11] I.R.Goodman, G.F.Kramer. Recent use of relational event algebra, including comparisons, estimation and deductions for probability-functional models[Z]. Gaithersbur: Proceedings of the 1998 IEEE ISIC|CIRA|ISAS joint Conference, 1998.
[12] C.T.William, D.Bamber, I.R.Goodman, H.T.Nguyen. A new method for representing linguistic quantifications by random sets with applications to tracking and data fusion[Z]. New York: ISIF, 2002.
[13]邓勇,刘琪,施文康.条件事件代数研究综述[J].计算机学报,2003,26,(6):650-661.
[14]邓勇,施文康.条件事件代数在数据融合中的应用[J].红外与激光工程,2001,30,(6):451-456.
[15]李兵,罗雪山,罗爱民.条件随机变量与条件数据融合[J].国防科技大学学报, 1999,21, (1):109-112.
[16]李兵,郁文东,胡卫东.基于条件事件代数系统的条件证据与条件信任组合[J].模糊系统与数学, 2001, 15, (1):103-107.
[17]王万森,何华灿.基于泛逻辑学的逻辑关系柔性化研究[J].软件学报, 2005,16, (5):754-750.
[18] H.T.Nguyen,V.Kreinovich,I.R.Goodman. Why unary and binary operations in logic: General result motivated by interval value logics[J]. IEEE Transactions on system, Man and Cybernetics, 2001:1991-1996.
[19] D.Bamber, I.R.Goodman. Deduction from conditional knowledge[J]. Soft Computing, 2004, (8):247-255.
[20] J.D.Marek, A.S.Herbert. Causality in Bayesian Belief Networks[Z]. Washington D.C.: Proceedings of the ninth annual conference on uncertainty in artificial intelligence (UAI-93), 1993.
[21] E.W.Adams, A.S.Herbert. Remarks on wishes and counterfactuals[J]. Pacific Philosophical Quarterly, 1998, 79:191-195.
[22] D.T.Heinze. A cognitive grammar motivated ontology for processing with term cycles and non-classical Negation[J]. Journal of Applied Logic, 2002, 31,(1):65–71.
[23] E.W.Adams. On the logic of high probability[J]. Journal of Philosophical logic, 1986, 15:235-279.
[24] E.W.Adams. Four probability-preserving properties of inferences[J]. Journal of Philosophical logic, 1996, 25:1-24.
[25] I.R.Goodman, R.P.S.Mahler, H.T.Nguyen. What is conditional event algebra and why should you care?[Z]. San Diego: Proceedings of SPIE Conference on Signal Processing, Sensor Fusion and Target Recognition, 1999: 2-13.
[26] I.R.Goodman. Toward a comprehensive theory of linguistic and probabilistic evidence:Two new approaches to conditional event algebra[J]. IEEE Transactions on System, Man, and Cybernetics, 1994, 24, (12):1685-1698.
[27] I.R.Goodman, H.T.Nguyen. Mathematical foundations of conditionals and their probabilistic assignments[J]. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 1995, 3, (3):247-339.
[28] I.R.Goodman, G.F.Kramer. Extension of relational event algebra to random sets with applications to data fusion[Z]. Berlin:Proceedings of Workshop on Applications and Theory of Random Sets, Institute for Mathematics and Its Applications(IMA), 1996: 209-242.
[29] T.Hailperin. Best possible in equalities for the probability of a logical function of events[J]. American Mathematical Monthly,1965,72: 343-359.
[30] T.Hailperin.Probability logic[J]. Notre Dame Journal of Formal Logic, 1984, 25, (3): 198-212.
[31] R.A.Howard, J.E.Matheson. Influence Diagrams. In Howard R.A., Matheson J. Eds. Reading on the principles and application of decision analysis[C]. Menlo Park Calif., Strategic Decision Group, 1984: 721-762.
[32] D.Bamber, I.R.Goodman, H.T.Nguyen. Robust reasoning with rules that have exception: From second-order probability to argumentation via upper envelops of probability and possibility plus directed graphs[J]. Annals of Mathematics and Artificial Intelligence, 2005, 45:83-171.
[33] R.Shachter. Evaluating influence diagram[J]. Operations Research, 1986, 34:871-882.
[34] I.R.Goodman. An even more realistic (non-association) logic and its relation to psychology of human reasoning[J]. Transactions on System, Man, and Cybernetics, 2001, 20, (11):1586-1591.
[35] H.T.Nguyen, V.Kreinovich, and Q.Zuo. Interval-valued degrees of belief:Application of interval computations to expert systems and intelligent control[J]. International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems(LJUFKS), 1997, 5, (3): 317-358.
[36] D.Bamber.Probabilistic entailment of conditionals by conditionals[J]. IEEE Transaction on systems, man, and cybernetics, 1994, 24, (12): 168-174.
[37] D.Bamber. Entailment with surety of scaled assertions of high conditional probability[J]. Journal of Philosophical Logic. 2000, 29:1-74.
[38] W.Doyle. Operation useful for similarity-invariant pattern recognition[J]. Journal of Associated Computer, 1962, 9: 259-267.
[39] H.D.Cheng, Jim-Rong Chen and Jiguang Li. Threshold selection based on fuzzy c-partition entropy approach.Pattern Recognition[J]. 1998, 31, (7): 857-870.
[40] J.S.Weszka and A.Rpsenfiled. Threshold evaluation technique[J]. IEEE Transaction on Systems, Man and Cybernet. 1978, 8, (8): 622-629.
[41] I.R.Goodman and H.T.Nguyen. Adams′high probability deduction and combination of information in the context of product probability conditional event algebra[C]. Las Vegas, NV: Proceedings of the international conference on multisource-multisensor information fusion (Fusion-98). 1998: 1-8.
[42] P.Roeper, H.Leblanc. Probability theory and probability logic[M]. University of Toronto Press Incorporated, 1999.
[43] D.Lewis. Probabilities of conditionals and conditional[J]. Philosophy Review, 1976, 85, (3):297-315.
[44] I.R.Goodman, R.P.S.Mahler and H.T.Nguyen. Mathematics of Data Fusion[M]. North-Holland, Amsterdam:Kluwer Academic Publisher, 1997.
[45] Eilas Gyftodimos, Peter A.Flach. Combining Bayesian Networks with higher-order data representations[C]. Springer-Verlag, A.F.Famili et al (EDs):IDA 2005, LNCS 3646, 2005: 145-156.
[46] J.Pearl. Probabilistic reasoning in intelligent systems: Networks of plausible inference[M]. San Mateo CA: Morgan Kaufman Publishers, 1988.
[47] I.R.Goodman. Toward a comprehension theory of linguistic and probabilistic evidence: Two new approaches to conditional event algebra[J]. IEEE Transaction on Systems,Man and Cybernetics, 1994, 24, (12): 1685-1697.
[48] G.O.Roberts, S.K.Sahu. Updating schemes, correlation structure, blocking and parameterisation for the Gibbs sampler[J]. Journal of the Royal Statistical Society, 1997, 59: 291-317.
[49] M.Tanner. Tools for statistical inference, method for exploration of posterior distribution and likelihood function[M]. Springer-Veriag, 3rd Edition, 1996.
[50] D.Bamber, I.R.Goodman. New uses of second order probability techniques in estimating critical probabilities in command & control decision-making[C]. New York: Proceedings of the 2000 Command & Control Research & Technology Symposium, 2000: 26-28.
[51] I.R.Goodman, H.T.Nguyen. Probability updating using second order probabilities and conditional event algebra[J]. Information Sciences, 1999, 131, (3):295-347.
[52] P.M.S.David. Actuarial modeling with MCMC and BUGS[J]. North American Actuarial Journal, 2001, 5, (2): 96-125.
[53] L.Bauwens, M.Lubrano. Bayesian inference on GARCH models using the Gibbs sampler[J]. Econometrics Journal, 1998, 1: 23-46.
[54] D.Bamber, I.R.Goodman, W.C.Torrez, and H.T.Nguyen. Complexity reducing algorithm for Near Optimal Fusion (CRANOF) with application to tracking and information fusion[C]. SPIE: Signal Processing, Sensor Fusion, and Target Recognition X, 2001: 269-280.
[55] R.A.Howard, J.E.Matheson. The principles and applications of decision analysis[Z]. Amsterdam: Strategic Decision Group, 1981: 721-762.
[56] L.J.Rodríguez-mu?iz, M.López-Díaz, M.A.Gil. Solving influence diagrams with fuzzy chance and value nodes[J]. European Journal of Operational Research, 2005, 167: 444-460.
[57] Y.Iwasaki, H.A.Simon. Causality in device behavior.Artificial Intelligence[J]. 1986, 29, (1): 3-12.
[58] K.Weichselberger. The theory of interval probability as unifying concept for uncertainty[J]. Journal of Approximate Reasoning, 2000, 24:149–170.
[59] J.Pearl. Evidential reasoning using stochastic simulation of casual models[J]. Journal of Artificial Intelligence, 1987, 32:245–257.
[60] S.K.M.Wong and C.J.Buta. Contextual weak independence in Bayesian networks[C]. New York: Proceeding Of the 15th Conference UAI, 1999: 670–690.
[61] Z.Pawlak. Rough sets[J]. Journal of Computer and Information Science, 1982, 11: 341–356.
[62] F.Cozman et al.. Combing probability and logic[J]. Journal of Applied Logic, 2008, 1: 231-254.
[63] D.Batens, C.Mortensen, G.Priest. From: Fronties of paraconsistent logic[C]. Balddock: Van Bendegem, J.P.(ed.) Studies in Logic and Computation, Research Studies Press, 2008, 8.
[64] M.Goldszmidt, P.Morris, J.Pearl. A maximum entropy approach to nonmonotonic reasoning[j]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1993, 15: 220–232.
[65] R.Fagin, J.Y.Halpern, N.Maggido. A logic for reasoning about probabilities[J]. Information and Computation, 1990, 87: 78-128.
[66] J.B.Paris. The uncertain reasoner′s companion[M]. Cambridge: Cambridge University Press, 1994.
[67] J.B.Paris. Commonsense and maximum entropy[J]. Synthesis, 1998, 117, (11): 119-132.
[68] E.W.Adams. Probabilistic enthymemes[J]. Journal of Pragmatics, 1983, 7: 283-295.
[69] K.M.Knight. Measuring inconsistency[J]. Journal of Philosophical Logic, 2002, 31, (1): 77–98.
[70] L.A.Zadeh. Probability measures of fuzzy events[J]. Journal of Mathmatics and Application, 1968, 23: 421-427.
[71] L.A.Zadeh. Outline of a new approach to the analysis of complex systems and decision processes[J]. IEEE Transactions on system, Man and Cybernetics, 1973, 3, (1): 28-44.
[72] J.C.Bezdek. Pattern recognition with fuzzy objective function, algorithms[M]. New York: Plenum press, 1981.
[73] S.Benferhat, D.Dubois and H.Prade. Nonmonotonic reasoning, conditional objects and possibility theory[J]. Artificial Intelligence, 1997, 92: 259-276.
[74] D.Dubois, H.Prade. Qualitative and quantitative possibility theory and its applications to reasoning and decision under uncertainty[M], In Handbook of Defeasible Reasoning and Uncertainty Management Systems, P. Smets ed., Kluwer Academic Publishers, Dordrecht, 1998, 1: 169-226.
[75] W.Spohn,Ordinal conditional functions: A dynamic theory of epistemic states[M]. Incausation in decision,belief change and statistics, W.L.Harper and B.Skyrms, eds., Kluwer Academic Publishers,Dordrecht, 1988, 2: 105-134.
[76] S.Kraus, D.Lehmann, and M.Magidor. Nonmonotonic reasoning, preferential models and cumulative logics[J]. Artificial Intelligence, 1990, 44: 167-207.
[77] D.Lehmann and M.Magidor. What does a conditional knowledge base entail?[J]. Artificial Intelligence, 1992, 55:1-60.
[78] D.Dubois, H.Prade. Possibilistic logic, preferential models, nonmonotonicity and related issues[C]. Armsterdam: Proceedings of 12th International Joint Conference on Artificial Intelligence (IJCA′91), 1991: 419-424.
[79] S.Benferhat,D.Dubois and H.Prade. Representing default rules in possibilistic logic[C]. Cambridge: Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning (KR′92), Cambridge, MA,1992: 673-684.
[80] D.Poole. Probabilistic Horn abduction and Bayesian network[J]. Artificial Intelligence, 1993, 64: 81–129.
[81] G.Kern-Isberner, T.Lukasiewicz. Combining probabilistic logic programming with the power of maximum entropy[J]. Artificial Intelligence, 2004, 157:139-202.
[82] W.S.Jevous. The Principles of Science[M]. London: Dover Press,1877: 197.
[83] Hintikka, J.(eds.). Rudolf Carnap, logical empiricist[M]. D.Reidel Publication Co., 1995.
[84] Schilpp, P.A.(eds.). The philosophy of Rudolf Carnap[M]. Open Court, 1978: 72.
[85] S. J. Russel, P. Norvig. Artificial intelligence - a modern approach [M]. Boston: Pearson Education, Publishing as Prentice-Hall, 2002.
[86] D. Heckerman, M. P. Wellman. Bayesian Networks [J]. Communications of ACM, 1995, 38 (3): 27-30.
[87] W. L. Buntine. A guide to the literature on learning probabilistic networks from data [J]. IEEE Transactions on Knowledge and Data Engineering, 1996, 8 (2):195-210.
[88] J. Cheng, R. Greiner, J. Kelly, D. Bell, W. Liu. Learning Bayesian network from data: An information-theory based approach [J]. Artificial Intelligence, 2002, 137(2): 43-90.
[89] J. Cheng. Power Constructor system [EB/OL].http://www.cs.ualberta.ca/jcheng/bnpc.htm, 1998.
[90] J. Pearl. Propagation and Structuring in Belief Networks [J]. Artificial Intelligence, 1986, 29(3): 241-288.
[91] J.Pearl. Evidential reasoning using stochastic simulation of causal models [J].Artificial Intelligence, 1987, 32: 245-257.
[92]刘惟一,李维华,岳昆.《智能数据分析》[M].北京:科学出版社, 2007.
[2] E.Waltz, L.James.赵宗贵译.多传感器数据融合[M].北京:电子部28所,1990.
[3]美国国防部关键技术计划编写组.美国国防部关键技术(1992财年)[M].北京:国防科技信息中心,1991.
[4] E.W.Adams. Probability and the Logic of Conditionals. In: Hintikka j., suppes P. eds. Aspects of inductive logic[M]. North-Holland, Amsterdam: Elsevier Science, 1966, 265-316.
[5] E.W.Adams. The logic of conditionals[M]. Dordrecht,Holland:D.Reidel Co.,1975.
[6] P.G.Calabrese. An algebraic synthesis of the foundations of logic and probability[J]. Information Science, 1987, 42, (2):187-316.
[7] P.G.Calabrese. A theory of conditional information with applications[J]. IEEE Transactions on system, Man and Cybernetics, 1994, 24, (12):1676-1664.
[8] I.R.Goodman, H.T.Nguyen,E.A.Walker. Conditional inference and logic for intelligent systems: A theory of measure-free conditioning[J]. Amsterdam, North-Holland: Elsevier Science, 1991.
[9] E.W.Adams. Updating on conditional information[J]. IEEE Transactions on system,Man and Cybernetics, 1994, 24, (12):1708-1713.
[10] P.G.Calabrese. Deduction with uncertain conditionals[J]. Information Science, 2002, 147:143-191.
[11] I.R.Goodman, G.F.Kramer. Recent use of relational event algebra, including comparisons, estimation and deductions for probability-functional models[Z]. Gaithersbur: Proceedings of the 1998 IEEE ISIC|CIRA|ISAS joint Conference, 1998.
[12] C.T.William, D.Bamber, I.R.Goodman, H.T.Nguyen. A new method for representing linguistic quantifications by random sets with applications to tracking and data fusion[Z]. New York: ISIF, 2002.
[13]邓勇,刘琪,施文康.条件事件代数研究综述[J].计算机学报,2003,26,(6):650-661.
[14]邓勇,施文康.条件事件代数在数据融合中的应用[J].红外与激光工程,2001,30,(6):451-456.
[15]李兵,罗雪山,罗爱民.条件随机变量与条件数据融合[J].国防科技大学学报, 1999,21, (1):109-112.
[16]李兵,郁文东,胡卫东.基于条件事件代数系统的条件证据与条件信任组合[J].模糊系统与数学, 2001, 15, (1):103-107.
[17]王万森,何华灿.基于泛逻辑学的逻辑关系柔性化研究[J].软件学报, 2005,16, (5):754-750.
[18] H.T.Nguyen,V.Kreinovich,I.R.Goodman. Why unary and binary operations in logic: General result motivated by interval value logics[J]. IEEE Transactions on system, Man and Cybernetics, 2001:1991-1996.
[19] D.Bamber, I.R.Goodman. Deduction from conditional knowledge[J]. Soft Computing, 2004, (8):247-255.
[20] J.D.Marek, A.S.Herbert. Causality in Bayesian Belief Networks[Z]. Washington D.C.: Proceedings of the ninth annual conference on uncertainty in artificial intelligence (UAI-93), 1993.
[21] E.W.Adams, A.S.Herbert. Remarks on wishes and counterfactuals[J]. Pacific Philosophical Quarterly, 1998, 79:191-195.
[22] D.T.Heinze. A cognitive grammar motivated ontology for processing with term cycles and non-classical Negation[J]. Journal of Applied Logic, 2002, 31,(1):65–71.
[23] E.W.Adams. On the logic of high probability[J]. Journal of Philosophical logic, 1986, 15:235-279.
[24] E.W.Adams. Four probability-preserving properties of inferences[J]. Journal of Philosophical logic, 1996, 25:1-24.
[25] I.R.Goodman, R.P.S.Mahler, H.T.Nguyen. What is conditional event algebra and why should you care?[Z]. San Diego: Proceedings of SPIE Conference on Signal Processing, Sensor Fusion and Target Recognition, 1999: 2-13.
[26] I.R.Goodman. Toward a comprehensive theory of linguistic and probabilistic evidence:Two new approaches to conditional event algebra[J]. IEEE Transactions on System, Man, and Cybernetics, 1994, 24, (12):1685-1698.
[27] I.R.Goodman, H.T.Nguyen. Mathematical foundations of conditionals and their probabilistic assignments[J]. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 1995, 3, (3):247-339.
[28] I.R.Goodman, G.F.Kramer. Extension of relational event algebra to random sets with applications to data fusion[Z]. Berlin:Proceedings of Workshop on Applications and Theory of Random Sets, Institute for Mathematics and Its Applications(IMA), 1996: 209-242.
[29] T.Hailperin. Best possible in equalities for the probability of a logical function of events[J]. American Mathematical Monthly,1965,72: 343-359.
[30] T.Hailperin.Probability logic[J]. Notre Dame Journal of Formal Logic, 1984, 25, (3): 198-212.
[31] R.A.Howard, J.E.Matheson. Influence Diagrams. In Howard R.A., Matheson J. Eds. Reading on the principles and application of decision analysis[C]. Menlo Park Calif., Strategic Decision Group, 1984: 721-762.
[32] D.Bamber, I.R.Goodman, H.T.Nguyen. Robust reasoning with rules that have exception: From second-order probability to argumentation via upper envelops of probability and possibility plus directed graphs[J]. Annals of Mathematics and Artificial Intelligence, 2005, 45:83-171.
[33] R.Shachter. Evaluating influence diagram[J]. Operations Research, 1986, 34:871-882.
[34] I.R.Goodman. An even more realistic (non-association) logic and its relation to psychology of human reasoning[J]. Transactions on System, Man, and Cybernetics, 2001, 20, (11):1586-1591.
[35] H.T.Nguyen, V.Kreinovich, and Q.Zuo. Interval-valued degrees of belief:Application of interval computations to expert systems and intelligent control[J]. International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems(LJUFKS), 1997, 5, (3): 317-358.
[36] D.Bamber.Probabilistic entailment of conditionals by conditionals[J]. IEEE Transaction on systems, man, and cybernetics, 1994, 24, (12): 168-174.
[37] D.Bamber. Entailment with surety of scaled assertions of high conditional probability[J]. Journal of Philosophical Logic. 2000, 29:1-74.
[38] W.Doyle. Operation useful for similarity-invariant pattern recognition[J]. Journal of Associated Computer, 1962, 9: 259-267.
[39] H.D.Cheng, Jim-Rong Chen and Jiguang Li. Threshold selection based on fuzzy c-partition entropy approach.Pattern Recognition[J]. 1998, 31, (7): 857-870.
[40] J.S.Weszka and A.Rpsenfiled. Threshold evaluation technique[J]. IEEE Transaction on Systems, Man and Cybernet. 1978, 8, (8): 622-629.
[41] I.R.Goodman and H.T.Nguyen. Adams′high probability deduction and combination of information in the context of product probability conditional event algebra[C]. Las Vegas, NV: Proceedings of the international conference on multisource-multisensor information fusion (Fusion-98). 1998: 1-8.
[42] P.Roeper, H.Leblanc. Probability theory and probability logic[M]. University of Toronto Press Incorporated, 1999.
[43] D.Lewis. Probabilities of conditionals and conditional[J]. Philosophy Review, 1976, 85, (3):297-315.
[44] I.R.Goodman, R.P.S.Mahler and H.T.Nguyen. Mathematics of Data Fusion[M]. North-Holland, Amsterdam:Kluwer Academic Publisher, 1997.
[45] Eilas Gyftodimos, Peter A.Flach. Combining Bayesian Networks with higher-order data representations[C]. Springer-Verlag, A.F.Famili et al (EDs):IDA 2005, LNCS 3646, 2005: 145-156.
[46] J.Pearl. Probabilistic reasoning in intelligent systems: Networks of plausible inference[M]. San Mateo CA: Morgan Kaufman Publishers, 1988.
[47] I.R.Goodman. Toward a comprehension theory of linguistic and probabilistic evidence: Two new approaches to conditional event algebra[J]. IEEE Transaction on Systems,Man and Cybernetics, 1994, 24, (12): 1685-1697.
[48] G.O.Roberts, S.K.Sahu. Updating schemes, correlation structure, blocking and parameterisation for the Gibbs sampler[J]. Journal of the Royal Statistical Society, 1997, 59: 291-317.
[49] M.Tanner. Tools for statistical inference, method for exploration of posterior distribution and likelihood function[M]. Springer-Veriag, 3rd Edition, 1996.
[50] D.Bamber, I.R.Goodman. New uses of second order probability techniques in estimating critical probabilities in command & control decision-making[C]. New York: Proceedings of the 2000 Command & Control Research & Technology Symposium, 2000: 26-28.
[51] I.R.Goodman, H.T.Nguyen. Probability updating using second order probabilities and conditional event algebra[J]. Information Sciences, 1999, 131, (3):295-347.
[52] P.M.S.David. Actuarial modeling with MCMC and BUGS[J]. North American Actuarial Journal, 2001, 5, (2): 96-125.
[53] L.Bauwens, M.Lubrano. Bayesian inference on GARCH models using the Gibbs sampler[J]. Econometrics Journal, 1998, 1: 23-46.
[54] D.Bamber, I.R.Goodman, W.C.Torrez, and H.T.Nguyen. Complexity reducing algorithm for Near Optimal Fusion (CRANOF) with application to tracking and information fusion[C]. SPIE: Signal Processing, Sensor Fusion, and Target Recognition X, 2001: 269-280.
[55] R.A.Howard, J.E.Matheson. The principles and applications of decision analysis[Z]. Amsterdam: Strategic Decision Group, 1981: 721-762.
[56] L.J.Rodríguez-mu?iz, M.López-Díaz, M.A.Gil. Solving influence diagrams with fuzzy chance and value nodes[J]. European Journal of Operational Research, 2005, 167: 444-460.
[57] Y.Iwasaki, H.A.Simon. Causality in device behavior.Artificial Intelligence[J]. 1986, 29, (1): 3-12.
[58] K.Weichselberger. The theory of interval probability as unifying concept for uncertainty[J]. Journal of Approximate Reasoning, 2000, 24:149–170.
[59] J.Pearl. Evidential reasoning using stochastic simulation of casual models[J]. Journal of Artificial Intelligence, 1987, 32:245–257.
[60] S.K.M.Wong and C.J.Buta. Contextual weak independence in Bayesian networks[C]. New York: Proceeding Of the 15th Conference UAI, 1999: 670–690.
[61] Z.Pawlak. Rough sets[J]. Journal of Computer and Information Science, 1982, 11: 341–356.
[62] F.Cozman et al.. Combing probability and logic[J]. Journal of Applied Logic, 2008, 1: 231-254.
[63] D.Batens, C.Mortensen, G.Priest. From: Fronties of paraconsistent logic[C]. Balddock: Van Bendegem, J.P.(ed.) Studies in Logic and Computation, Research Studies Press, 2008, 8.
[64] M.Goldszmidt, P.Morris, J.Pearl. A maximum entropy approach to nonmonotonic reasoning[j]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1993, 15: 220–232.
[65] R.Fagin, J.Y.Halpern, N.Maggido. A logic for reasoning about probabilities[J]. Information and Computation, 1990, 87: 78-128.
[66] J.B.Paris. The uncertain reasoner′s companion[M]. Cambridge: Cambridge University Press, 1994.
[67] J.B.Paris. Commonsense and maximum entropy[J]. Synthesis, 1998, 117, (11): 119-132.
[68] E.W.Adams. Probabilistic enthymemes[J]. Journal of Pragmatics, 1983, 7: 283-295.
[69] K.M.Knight. Measuring inconsistency[J]. Journal of Philosophical Logic, 2002, 31, (1): 77–98.
[70] L.A.Zadeh. Probability measures of fuzzy events[J]. Journal of Mathmatics and Application, 1968, 23: 421-427.
[71] L.A.Zadeh. Outline of a new approach to the analysis of complex systems and decision processes[J]. IEEE Transactions on system, Man and Cybernetics, 1973, 3, (1): 28-44.
[72] J.C.Bezdek. Pattern recognition with fuzzy objective function, algorithms[M]. New York: Plenum press, 1981.
[73] S.Benferhat, D.Dubois and H.Prade. Nonmonotonic reasoning, conditional objects and possibility theory[J]. Artificial Intelligence, 1997, 92: 259-276.
[74] D.Dubois, H.Prade. Qualitative and quantitative possibility theory and its applications to reasoning and decision under uncertainty[M], In Handbook of Defeasible Reasoning and Uncertainty Management Systems, P. Smets ed., Kluwer Academic Publishers, Dordrecht, 1998, 1: 169-226.
[75] W.Spohn,Ordinal conditional functions: A dynamic theory of epistemic states[M]. Incausation in decision,belief change and statistics, W.L.Harper and B.Skyrms, eds., Kluwer Academic Publishers,Dordrecht, 1988, 2: 105-134.
[76] S.Kraus, D.Lehmann, and M.Magidor. Nonmonotonic reasoning, preferential models and cumulative logics[J]. Artificial Intelligence, 1990, 44: 167-207.
[77] D.Lehmann and M.Magidor. What does a conditional knowledge base entail?[J]. Artificial Intelligence, 1992, 55:1-60.
[78] D.Dubois, H.Prade. Possibilistic logic, preferential models, nonmonotonicity and related issues[C]. Armsterdam: Proceedings of 12th International Joint Conference on Artificial Intelligence (IJCA′91), 1991: 419-424.
[79] S.Benferhat,D.Dubois and H.Prade. Representing default rules in possibilistic logic[C]. Cambridge: Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning (KR′92), Cambridge, MA,1992: 673-684.
[80] D.Poole. Probabilistic Horn abduction and Bayesian network[J]. Artificial Intelligence, 1993, 64: 81–129.
[81] G.Kern-Isberner, T.Lukasiewicz. Combining probabilistic logic programming with the power of maximum entropy[J]. Artificial Intelligence, 2004, 157:139-202.
[82] W.S.Jevous. The Principles of Science[M]. London: Dover Press,1877: 197.
[83] Hintikka, J.(eds.). Rudolf Carnap, logical empiricist[M]. D.Reidel Publication Co., 1995.
[84] Schilpp, P.A.(eds.). The philosophy of Rudolf Carnap[M]. Open Court, 1978: 72.
[85] S. J. Russel, P. Norvig. Artificial intelligence - a modern approach [M]. Boston: Pearson Education, Publishing as Prentice-Hall, 2002.
[86] D. Heckerman, M. P. Wellman. Bayesian Networks [J]. Communications of ACM, 1995, 38 (3): 27-30.
[87] W. L. Buntine. A guide to the literature on learning probabilistic networks from data [J]. IEEE Transactions on Knowledge and Data Engineering, 1996, 8 (2):195-210.
[88] J. Cheng, R. Greiner, J. Kelly, D. Bell, W. Liu. Learning Bayesian network from data: An information-theory based approach [J]. Artificial Intelligence, 2002, 137(2): 43-90.
[89] J. Cheng. Power Constructor system [EB/OL].http://www.cs.ualberta.ca/jcheng/bnpc.htm, 1998.
[90] J. Pearl. Propagation and Structuring in Belief Networks [J]. Artificial Intelligence, 1986, 29(3): 241-288.
[91] J.Pearl. Evidential reasoning using stochastic simulation of causal models [J].Artificial Intelligence, 1987, 32: 245-257.
[92]刘惟一,李维华,岳昆.《智能数据分析》[M].北京:科学出版社, 2007.