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利用模糊推理建立了一种基于输入-输出数据构造联合概率密度函数的方法。首先,将一组单输入-单输出数据转换成模糊推理规则,通过选择适当的模糊蕴涵算子生成模糊关系,再利用这种模糊关系求出二维随机变量的联合概率密度函数。当将模糊蕴涵分别取为Larsen蕴涵和Mamdani蕴涵时,分别得到了两种具体的概率密度函数(称之为Larsen分布和Mamdani分布)。其次,分别研究这两种概率分布的边缘分布和数字特征,指出这两种概率分布有相同的数学期望,有几乎相同的方差和协方差。从而进一步揭示模糊系统的概率论意义。
Using fuzzy reasoning, a method of constructing joint probability density function based on input - output data was established. First, a set of single-input-single-output data is converted into fuzzy reasoning rules, fuzzy relations are generated by selecting appropriate fuzzy implication operators, and then the joint probability density function of two-dimensional random variables is obtained by using the fuzzy relationship. When the fuzzy implication is taken as Larsen implication and Mamdani implication respectively, two specific probability density functions (called Larsen distribution and Mamdani distribution) are respectively obtained. Secondly, we study the edge distribution and the digital characteristics of these two probability distributions separately, and point out that the two probability distributions have the same mathematical expectation, almost the same variance and covariance. Which further reveals the significance of probability theory of fuzzy systems.