第三章模糊集理论.ppt

Chapter 3 Fuzzy Set Theory ,Fuzzy Reasoning System  Fuzzy Modeling Outline Fuzzy Set Theory Fuzzy Reasoning System Fuzzy Modeling Fuzzy Clustering Fuzzy Comprehensive Evaluation Reference 工程模糊数学及应用 李士勇 哈尔滨工业大学出版社 模糊数学 原理及应用 杨轮标等 华南理工大学出版社 数学建模方法及应用 韩中庚 高等教育出版社 第19章 Fuzzy Sets Theory: Outline Introduction Basic definitions and terminology Set-theoretic operations MF formulation and parameterization MFs of one and two dimensions Derivatives of parameterized MFs More on fuzzy union, intersection, and complement Fuzzy complement Fuzzy intersection and union Parameterized T-norm and T-conorm Usefulness of Fuzzy Set Fuzzy if-then rules, which are commonly used in our daily expressions, is like our daily language. Such as ‘large, small median’ We can use a collection of fuzzy rules to describe a system behavior; this forms the fuzzy inference system, or fuzzy controller if used in control systems. Such as controller of air-condition: if the temperature is high then stop; if the temperature is low then run, ect. In particular, we can apply neural networks, learning method in a fuzzy inference system. A fuzzy inference system with learning capability is called ANFIS, stands for adaptive neuro-fuzzy inference system. Fuzzy Sets Sets with fuzzy boundaries Membership Functions (MFs) Characteristics of MFs: Subjective measures Not probability functions Example: Negative, Zero and Positive illustrates the features of the triangular membership function which is used in this example because of its mathematical simplicity. Other shapes can be used but the triangular shape lends itself to this illustration Fuzzy Sets Formal definition: A fuzzy set A in X is expressed as a set of ordered pairs: Fuzzy Sets with Discrete Universes Fuzzy set C = “desirable city to live in” X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (Los Angeles, 0.6)} Fuzzy set A = “sensible number of children” X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0, .1), (1,