第三章系統(tǒng)的描述與模型建立重點(diǎn)

1、第三章系統(tǒng)的描述與模型建立Chapter 3. Descripti on of Systems and Modeli ng3.6不確定性描述（二）-模糊性3.6 Descripti on of Un certa inty (2)-Ra ndomn ess精確與模糊1. Precisi on and fuzz in ess統(tǒng)計(jì)分析不確定性描述隨機(jī)性描述丿主元分析 | 因子分析 i 模糊性描述Descripti on of un certa inty”statistic analysisDescription of randomnessPCA,Principle ComponentAnalysis

2、factor an alysisDescripti on of fuzz in ess- 精確思維及方法在科學(xué)技術(shù)發(fā)展中日益取得成功.通?！本媲缶北徽J(rèn)為是科學(xué)工作者的美德.- Thinking and perform ing with precisi on have bee n successful in developme nt of scie nee and tech no logy, and to be accurate as possible has bee n con sidered as a virtue of a scie ntific researcher.- 精確方法研究

3、的對(duì)象是無生命的機(jī)械系統(tǒng)，界限分明的機(jī)械事務(wù).- The objects being in vestigated with accuracy are n ormally non-life mecha ni cal systems, which have clear-cut boun daries.- 有關(guān)生命現(xiàn)象，社會(huì)現(xiàn)象，心理因素的科學(xué)，由于所研究的對(duì)象大多是沒有明確界限的模糊事物，很難進(jìn)行精確的測(cè)量，所以很難使用精確的定量方法.- Phenomena regarding lives， social affairs，psychology，etc，due to the objects bein

4、g in vestigated are mostly fuzzy problems without clear boun daries，it is difficult to con duct the accurate measureme nt，so hard to qua ntify with high accuracy.- 即使對(duì)于無生命的系統(tǒng)，一旦系統(tǒng)龐大，若進(jìn)行精確地測(cè)量，仍很困難.- Even to those non-life systems，once it grows enormous，it is still difficult to measure with precisi on

5、.二.模糊系統(tǒng)理論2. Fuzzy system theory- 語言變量定義及有關(guān)概念：Lan guage variable and some related con cepts:用自然語言中的詞句表示.每一個(gè)語言變量值均對(duì)應(yīng)著一個(gè)論域(Discourse uni verse).Expressed with a word or sentence of natural Ianguage，and each and every Ianguagevariable corresp onds to a Discourse Uni verse.例：人的年齡的語言值可為：非常年輕，年輕，較年輕，不年輕，較老

6、，老，很老” 等等.Ex. The Ian guage value of age of huma n may be: very young, young, somewhat young, not young, somewhat old, old, every old ” etc.模糊子集現(xiàn)設(shè)論域U=0,150,則以上每一個(gè)值都對(duì)應(yīng)著該論域的一個(gè)模糊子集可用隸屬函數(shù)（或稱類屬函數(shù)，資格函數(shù)，成員函數(shù)等）來表示.Suppose the discourse uni verse U=0,150, i n which every value should corresp ond to a fuzzy s

7、et. The fuzzy set may be described using membership fun cti on.例：給定一個(gè)一般集合A,某一元素屬a于A,則隸屬度為Ex. Given a normal set A, and a certain element a belonging to A, then the degree of bel onging isa（a）二若不屬于A,則In case it does not bel ong to A， the nA（a）二對(duì)于一模糊子集A ,某- 兀素 x除了怯（X）10外，有可能在一定程度上隸屬1,it may to a0于 A ,

8、 0 *（x）1.For a fuzzy subset A , for a certain element x other than A（X）= *certain extent belong to A , 0 A （x） certanly belong to A45 likely belong to A卜not likely belong to A89 certainly not belong to A用模糊數(shù)學(xué)表示為Expressed in fuzzy mathematics as follows:1.0 10 10 08 .05.02. 0.0 0.01234567891.5LJq Mem

9、bership fun ctio n value* 基本變量Language variableAgeGrammatical rulesLanguage valueImplication var.Basic variableUniv of discourseVery youngyoungoldiL 11020306070- 模糊數(shù)：對(duì)應(yīng)于實(shí)數(shù)軸上某一個(gè)凸?fàn)畹哪：蛹?通常是三角形，鐘形，矩形等)論域- Fuzzy number: for a certain convex-shaped fuzzy subset (normally triangle, bell, recta ngle, etc.)

10、- 模糊關(guān)系：不清晰不完全確定的關(guān)系.可由隸屬函數(shù)描述的模糊子集，隸屬函數(shù)值代表的密切程度.Fuzzy relation: unclear and uncertain relation. It may be represented by association degree to a fuzzy subset described by a membership function or function value.三模糊性與隨機(jī)性的區(qū)別（續(xù)見下列筆記）3. The differe nee betwee n fuzz in ess and randomn ess（con ti nue with

11、the no tes below）INTRODUCTION TO FUZZY SETS模糊集的介紹Two valued logic: - black and white-odd and eve n兩個(gè)邏輯值：黑和白奇數(shù)和偶數(shù)Situati ons that two valued logic may not be suitable:Tall man.Small errorSign ifica nt error, etc兩個(gè)邏輯值可能不匹配的情況：高大的男人 小的錯(cuò)誤 有意義的錯(cuò)誤等等。Pile of seed example:One seed is not a pile,Two seeds do

12、c on stitute a pileCan we say that 121078 seeds do not constitute a pile but 121079 seeds con stitute do?6100x10 seeds con stitute a pile.Conclusion: definition of a pile is somewhat fuzzy.打個(gè)比方：多少數(shù)量的種子才算是一堆種子一粒種子不是一堆種子。兩粒種子也不能算是一堆種子。那我們能說：121078粒種子不能算是一堆種子，但121079粒種子 是一堆種子嗎？ 結(jié)論：“一堆”的定義有些模糊。Set by be

13、longing to a set （設(shè)置屬于某個(gè)集合）In two valued logic, an object either bel ongs to set or does not bel ong to set. 在兩個(gè)邏輯值中，一個(gè)對(duì)象可能屬于這個(gè)集合也可能不屬于這個(gè)集合。例如,， Object1=O 1=3, Object2=O 2=2Oi 亡 AO2 AIn fuzzy sets, grades of bel onging cha nges betwee n(a) completely belongs (1)(b) completely excluded (2)在模糊集中，隸書度介于

14、(a)完全屬于例如:(b )部分屬于例如: Tow valued logic of high temperature （高溫的兩個(gè)邏輯值）Degree of bel onging to con cept high temperature ” Temperature*TFuzzy definition of high temperature （高溫的模糊定義）1old（X）= 1一叮-150=x=100（x表示old）0.53 old （x）例如: Degree of membership to concept young”（屬于“年輕”概念的隸屬度）卩=function to describe

15、 the membership old （描述老的程度）FORMAL DEFINITION OF FUZZY SET模糊集正式的定義A is concept such as young, medium error, large temperature, etc.A表示如年輕，平均誤差，高溫等概念。X is a uni verse of discourse such as temperature, nu mber of seeds, years, error, etc.X定義如溫度，種子的數(shù)目，年數(shù)，誤差等A(X) is an expression expressing the extend t

16、o which X fulfils the category specified by A.Sometimes we will use 卩 a(x) to indicate the membership of x to A. If there is no confusion we will simply use A(x) or sometimes 卩 a(x).有時(shí)，我們用卩a(x)顯示x與A的關(guān)系。假如那不存在模糊，我們將簡(jiǎn)單的用A(x)或卩a(x).來表示。Normal fuzzy set.一般模糊集Supx 卩 a(x).=1Sup: supremum (或運(yùn)算？)例如：A: a con

17、cept of young表示 youngX: a uni verse of discourse, years 表示一個(gè)域，yearsx: a value in the uni verse of discourse 表示域上的值A(chǔ)(x)=卩 a(x)卩 A（X）模糊集的特性SOME PROPERTIES OF FUZZY SETS并集:(A U B)(x)=max( A(x),B(x)=max(卩 a(x),卩 b(x) ) , x XUn io n: (A U B)(x)=max( A(X),B(X)=max(卩 a(x),卩 b(x) ) , x X(A U B)(x)A: large e

18、rrorB: medium error(A U B)(x): medium or large error交集:(A n B)(x)=min( A(x),B(x)=min(卩 a(x),卩 b(x),Intersection: (A n B)(x)=min( A(X),B(X)=min(卩 a(x),卩 b(x),補(bǔ)集：A(x) _A(x) , x XA( x) : not large error.A(x)(x)1INegation: A(x)=1-A(x) , x XA( x) : not large error.SOME EXAMPLES ON THE PROPERTIES OF FUZZY

19、 SETS幾個(gè)關(guān)于模糊集特性的例子We have a discrete uni verse of discourse her(:這里有個(gè)離散域的推理過程)A=(1.00.8 0.4 0.5)11 11234B=(0.9 0.4 0.0 0.7)A U B=(1.00.8 0.4 0.7)A A B=(0.90.4 0.0 0.5)Ac=(0.0 0.2 0.6 0.5)cA A A =(0.0 0.2 0.4 0.5)1234A U Ac =(1.0 0.8 0.6 0.5)注意:：假如是二值邏輯集Note: if it is the two-value logic.A=(1111)cA =(

20、0 0 0 0)A A Ac =(0.0 0.0 0.00.0)A U Ac =(1.01.01.01.0)我們注意到AA Ac不是零值.,A U Ac不是唯一的。這對(duì)嚴(yán)格定義的模糊集都是 適用的。對(duì)比這個(gè)例子，可知二值邏輯不是模糊邏輯。cNotice that the overlap membership A A A is not zero. Also note that the underlapcmembership AU A is not unity. This holds for all proper fuzzy sets. Compare this example with the

21、one prese nted in the subset above where we have 2-valued logic, i.e., no fuzzy logic.FUZZINESS AND RANDOMNESS: HANDLING UNCERTAINTY隨意性和模糊性：操作的不確定X: a finite uni verse of discourse,x X (一個(gè)有限的域)A(x)=a: a value of membership function jof A for certain element of X is equal to a”.P( x A)=a probability

22、of x belonging to A is equal to a”.We now perform an experime nt in which we pick up x and observe the outcome:Before experime ntAfter experime ntFuzz in ess:A(x)=aA(x)=aRandomnessP( x A)=a1 if x A0 if x 一 AConsider the following situation: x is the outcome of a die throwing experiment, A isan eve n

23、 nu mber.Upon observation of x, the a priori probability:P( x A)=a becomes a posterior, i.e., 1 if X A, 0 otherwise.At the same time A(x), being a measure of the extent to which x belongs to A, remain the same.RANDOMNESS: statistical in exact ness due to ran dom eve nts.FUZZINESS: in exact ness due

24、to percepti on process of huma n being.Both concepts describe uncertainty with numbers in the interval 0, 1.模糊關(guān)系FUZZY RELATIONSR: X Y f 0,1x X, y Y對(duì)于每對(duì)x,y對(duì)應(yīng)于一個(gè)屬于（0,1）的值To every pair x X and y Y and member from a range 0,1 is assigned.R表示序偶x,y的隸屬程度，也描述了 u,v間具有模糊關(guān)系R的量級(jí)R expresses the stre ngth of ties

25、 betwee n them.模糊關(guān)系EXAMING ON FUZZY RELATIONS例1R(x,y): x相似y.的程度函數(shù)EX1 R(x,y): x is similar to y.1R(X，y)= 1 (x-y)4y例2例1的離散域的論域.EX2Discrete uni verse of discoursey1.00.70.0R=1(0.10.30.5xt-norm is used to derive the relati on ship betwee n fuzzy sets.我們?cè)趺磥泶_定模糊關(guān)系HOW DO WE DETERMINE FUZZY RELATIONS定義一個(gè)標(biāo)準(zhǔn)模

26、糊關(guān)系t:Defin iti on of t-norm:t： 0,1x0,1 - 0,1它滿足以下特性：It has the follow ing properties(i) 非遞減性：non-decreasing:x t w - y t z for x _y and w_z (注意：t代表標(biāo)準(zhǔn)t域，不是一個(gè)變 量)例如：xy是一個(gè)標(biāo)準(zhǔn)t域，也就是.,簡(jiǎn)單相乘是一個(gè)標(biāo)準(zhǔn)t域，因此，xwx and zw(ii) 滿足交換律：commutative:x t y =y t x EX. xy=yx(iii) 滿足結(jié)合律：associative:(x t y) t z = x t ( y t z) EX. (x y) z = x ( y z)(iv) 滿足限制條件： satisfy the bounding conditionx t 0=0 EX. x0=0x t 仁xEX.x仁x一些關(guān)于標(biāo)準(zhǔn)t域的其他例子：SOME OTHER EXAMPLES OF t-