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ju2$/s_ڵWe.2$Sk05wbs^r]!2$'d)KVb$Jǵcto1mb$K\sP^MwѯM6.2$'3uɧvQe{2$ n'gD~T2$r??Yl!2$N46a# <AA f3f "f&@8' ʚ;wǀ3ʚ;g4BdBd< 0vppp@ <4!d!d k 0T+<4dddd k 0T+<4BdBd k 0T+0___PPT10 ppd___PPT9F>&?`+ LChap7 Fuzzy Set and Logic Zhongzhi ShiO =WAdvanced Computing Seminar Data Mining and Its Industrial Applications  Chapter 7  Fuzzy Sets fG$ "+G$ "F2 "F "F "B G6&F90 >    Zhongzhi Shi, Markus Stumptner, Yalei Hao, Gerald Quirchmayr Knowledge and Software Engineering Lab Advanced Computing Research Centre School of Computer and Information Science University of South Australia H>n<>BBBX          0>OutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering SummaryY LH       h Introduction  6 "    Idea of Fuzzy  i Introduction  G6 "   What does it offer? generate precise solutions from certain or approximate information While other approaches require accurate equations to model real-world behaviors, fuzzy design can accommodate the ambiguities of real-world human language and logic. It provides both an intuitive method for describing systems in human terms and automates the conversion of those system specifications into effective models. 4ZZ  j Introduction  G6 "   Aristotle "Law of the Excluded Middle," every proposition must either be True or False, A or not-A Ex., rose is either red or not red It cannot be red and not red Plato there was a third region (beyond True and False) where these opposites "tumbled about. ;@W ;     W$    k Introduction  G6 "   WIn the early 1900s, Lukasiewicz described a three-valued logic, which can best be translated as the term `possible', and assigned it a numeric value between True and False. Knuth, a former student of Lukasiewicz proposed a three-valued logic apparently missed by Lukasiewicz, whiched used an integral range [-1, 0 +1] rather than [0, 1, 2]. L!(!(      5     :  l Introduction  G6 "   1960 s Lotfi A. Zadeh, a professor of UC Berkeley Observed that conventional computer logic was incapable of manipulating data representing subjective or vague human ideas such as "an atractive person" or "pretty hot". Fuzzy logic, hence was designed to allow computers to determine the distinctions among data with shades of gray, similar to the process of human reasoning.@Z,ZFZ,Fh         m Introduction  G6 "   In 1965, Zadeh published his seminal work "Fuzzy Sets Described the mathematics of fuzzy set theory, and by extension fuzzy logic. This theory proposed making the membership function (or the values False and True) operate over the range of real numbers [0.0, 1.0]. : . .2     n Introduction  G6 "   _US and certain parts of Europe ignored it, fuzzy logic was excepted with open arms in Japan, China and most Oriental countries. The world's largest number of fuzzy researchers are in China with over 10,000 scientists. The popularity of fuzzy logic in the Orient reflects the fact that Oriental thinking more easily accepts the concept of "fuzziness". _   pOutlineG6 "  YIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary *Z  CH    !  @ Fuzzy Sets  G0 "  8Universal Set X  always a crisp set. Crisp set assigns value {0,1} to members in X Fuzzy set assigns value [0,1] to members in X These values are called the membership functions m. Membership function of a fuzzy set A is denoted by : A: X [0,1] A: [x1/m1, x2/m2, & , xn/mn}(0,,C % C C C C &   s  Fuzzy Sets $  E "   [Fuzzy sets How fuzzy sets quantifying the same information can describe this natural drift.& Q Q \ t Fuzzy Sets $  E "   4Let's talk about people and "youthness Set S (the universe of discourse) is the set of people Fuzzy subset YOUNG "to what degree is person x young?" To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset YOUNG. A membership function based on the person's age young(x) = { 1, if age(x) <= 20, (30-age(x))/10, if 20 < age(x) <= 30, 0, if age(x) > 30 } (91^(91  ^J   O  $  u Fuzzy Sets$  E "   gGraph of membership function Example: the degree of truth of the statement "Parthiban is YOUNG" is 0.50&KK&L    X Fuzzy Sets  C0  Sets with fuzzy boundariesZ  YMembership Functions (MFs)A&   ECharacteristics of MFs: Subjective measures Not probability functions&..&  0 Z Fuzzy Sets  C0  OFormal definition: A fuzzy set A in X is expressed as a set of ordered pairs: xZ<ZZ  " "*O  ["Fuzzy Sets with Discrete Universes##A # >Fuzzy set C =  desirable city to live in X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A =  sensible number of children X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}`*Y,q*Y ",q "&M    \Fuzzy Sets with Cont. Universes C$  Fuzzy set B =  about 50 years old X = Set of positive real numbers (continuous) B = {(x, mB(x)) | x in X}d#H#7 "G  " " "&Z    ]Alternative NotationA  6A fuzzy set A can be alternatively denoted as follows:7Z7 7 ^Fuzzy PartitionA  Fuzzy partitions formed by the linguistic values  young ,  middle aged , and  old : T _More DefinitionsA  KSupport Core Normality Crossover points Fuzzy singleton a-cut, strong a-cutDL8C  C  L  IConvexity Fuzzy numbers Bandwidth Symmetricity Open left or right, closedJJ&"    `MF TerminologyA   aConvexity of Fuzzy Sets  1A fuzzy set A is convex if for any l in [0, 1], H2Z  "C  1  bSet-Theoretic OperationsC0  +Subset: Complement: Union: Intersection: , cSet-Theoretic OperationsA   dMF FormulationA  Triangular MF:Z  eMF FormulationA   fMF Formulation   Sigmoidal MF:Z   gMF FormulationA  L-R MF:  hCylindrical ExtensionA   i2D MF Projection   jFuzzy ComplementA  rGeneral requirements: Boundary: N(0)=1 and N(1) = 0 Monotonicity: N(a) > N(b) if a < b Involution: N(N(a) = a Two types of fuzzy complements: Sugeno s complement: Yager s complement: ZXZ ZZZZZX f4   N       kFuzzy ComplementA   lFuzzy Intersection: T-normA  <Basic requirements: Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a Monotonicity: T(a, b) < T(c, d) if a < c and b < d Commutativity: T(a, b) = T(b, a) Associativity: T(a, T(b, c)) = T(T(a, b), c) Four examples (page 37): Minimum: Tm(a, b) Algebraic product: Ta(a, b) Bounded product: Tb(a, b) Drastic product: Td(a, b)ZZZbZ ZA   '       mT-norm Operator   nFuzzy Union: T-conorm or S-norm&    0Basic requirements: Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a Monotonicity: S(a, b) < S(c, d) if a < c and b < d Commutativity: S(a, b) = S(b, a) Associativity: S(a, S(b, c)) = S(S(a, b), c) Four examples (page 38): Maximum: Sm(a, b) Algebraic sum: Sa(a, b) Bounded sum: Sb(a, b) Drastic sum: Sd(a, b)ZZZVZ A   '      B  *     oT-conorm or S-norm&     CFuzzy VariablesC0  Variables whose states are defined by linguistic concepts like low, medium, high. These linguistic concepts are fuzzy sets themselves.   qOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary2Y 5<    ! < Fuzzy Logic     Fuzzy logic is a superset of conventional(Boolean) logic It extended to handle the concept of partial truth- truth values between "completely true" and "completely false". It is the logic underlying modes of reasoning which are approximate rather than exact. The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature.,  = Fuzzy Logic   Essential characteristics of fuzzy logic: In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning. In fuzzy logic everything is a matter of degree. Any logical system can be fuzzified (thus, define Boolean logic as a subset of Fuzzy logic ) In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables Inference is viewed as a process of propagation of elastic constraints.**ZZ*2     > Fuzzy Logic   8Bivalent sets To characterize the temperature of a room***8  ? Fuzzy logic   Limitations of bivalent sets mutually exclusive it is not possible to have membership of more than one set ( opinion would widely vary as to whether 50 degrees Fahrenheit is 'cold' or 'cool' hence the expert knowledge we need to define our system is mathematically at odds with the humanistic world). Clearly, it is not accurate to define a transition from a quantity such as 'warm' to 'hot' by the application of one degree Fahrenheit of heat. In the real world a smooth drift from warm to hot would occur. hZZZ@Z@  x Fuzzy Logic   Fuzzy Set Operations Union The membership function of the Union of two fuzzy sets A and B with membership functions mA and mB respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebranA]@AY    f =S  y Fuzzy logic   .Union maximum criterion$   z Fuzzy Logic   Fuzzy Set Operations Union The membership function of the Union of two fuzzy sets A and B with membership functions mA and mB respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra9Y    fHS  { Fuzzy Logic   .Union maximum criterion$   | Fuzzy Loic&   Fuzzy Set Operations Intersection The membership function of the Intersection of two fuzzy sets A and B with membership functions mA and mB is defined as the minimum of the two individual membership functions. This is called the minimum criterion. The Intersection operation in Fuzzy set theory is the equivalent of the AND operation in Boolean algebra."j`F"`    Y H c } Fuzzy logic   <Intersection minimum criterion$    ~ Fuzzy Logic   Fuzzy Set Operations Complement The membership function of the Complement of a Fuzzy set A with membership function mA is defined as the negation of the specified membership function. This is called the negation criterion. The Complement operation in Fuzzy set theory is the equivalent of the NOT operation in Boolean algebra.  '@A T  U Q H  Fuzzy logic   :Complement negation criterion$     Fuzzy Logic   XCommon rules in classical set theory and Fuzzy set theory De Morgans law Associativity &::L=        Fuzzy Logic   WCommon rules in classical set theory and Fuzzy set theory Commutativity Distributivity&::@:       rOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary2Y$!<    ! DClusterC6  A number of similar individuals that occur together as a: two or more consecutive consonants or vowels in a segment of speech b: a group of houses (...) c: an aggregation of stars or galaxies that appear close together in the sky and are gravitationally associated.R 7Fo@  ECluster Analysis .C6@ A statistical classification technique for discovering whether the individuals of a population fall into different groups by making quantitative comparisons of multiple characteristics.@ FVehicle Example&@@&  GVehicle Clusters@   H Terminology @   IMembership matrix M2@@$  J c-partition @  KObjective function@  MNOPQRSLTFuzzy membership matrix M:@@L    UFuzzy membership matrix M:@@@     sOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary*Y8<    ! tFuzzy C-Means Clustering  KInput: Unlabeled data set Main Output Common Additional Output* 'L L 0uFuzzy C-Means Clustering   1vFuzzy C-Means Clustering  |Optimization of an  objective function or  performance index ?? ? 2wFuzzy C-Means Clustering  ^Zeroing the gradient of with respect to Zeroing the gradient of with respect to __ _ QxFuzzy C-Means Clustering  Initial Choices Number of clusters Maximum number of iterations (Typ.: 100) Weighting exponent (Fuzziness degree) m=1: crisp m=2: Typical Termination measure 1-norm Termination threshold (Typ. 0.01)dbXb-#@A     RyFuzzy C-Means Clustering  Guess Initial Cluster Centers Alternating Optimization (AO) REPEAT UNTIL ( or ) F<H<)  SzFuzzy C-Means Clustering   T{Fuzzy C-Means Clustering  Process could be shifted one half cycle Initialization is done on Iterates become Termination criterion The convergence theory is the same in either case Initializing and terminating on V is advantageous Convenience Speed Storagef(Z@ZdZZ(@d  V|Fuzzy C-Means Clustering  Advantages Unsupervised Always converges Disadvantages Long computational time Sensitivity to the initial guess (speed, local minima) Sensitivity to noise One expects low (or even no) membership degree for outliers (noisy points)  dK dJ  ;$Summary Advantages of Fuzzy Systems$    Conceptually straightforward to understand. The mathematical concepts behind fuzzy reasoning are simple. Flexible:You can modify and add on fuzzy rules without starting from scratch. Tolerant of imprecise data. Everything is imprecise if you look closely enough, but more than that, many things are imprecise even at first glance Can model nonlinear functions of arbitrary complexity. Can create a fuzzy system to match any set of input/output data. Can be built on top of the experience of experts. Is close to natural language.ZB  W}'Summary Disadvantages of Fuzzy Systems$    tCreating the fuzzy rules base can be troublesome It is difficult to create the fuzzy rules base from input/output data if no fuzzy rule extraction technique is used Accuracy of the inference depends directly on the number of fuzzy rules used in complex problem Increasing input variables and fuzzy membership fns used will increase the number of fuzzy rules exponentially.uuB u X~ References C,/ [1] Lofti A. Zadeh. Fuzzy sets, Inf. Control 8, 338-353, 1965. [2] Jan Jantzen. Tutorial On Fuzzy Clustering. http://fuzzy.iau.dtu.dk/tutor/fcm/cluster.ppt [3] Nikhil R. Pal, James C. Bezdek, and Richard J. Hathaway. Sequential Competitive Learning and the Fuzzy c-Means Clustering Algorithms. Neural Networks, Vol. 9, No. 5, pp. 787-796, 1996 [4] Lofti A. Zadeh. Fuzzy Logic, Computer, v.21 n.4, p.83-93, April 1988 [5] Novak, V., Perfilieva, I., Mockor, J. (2000):, Mathematical principles of fuzzy logic. 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[1.0&8 AG xB   H "p?0` 0 xB  B H "p?0` 0 ` xB B H "p? 0   Tgֳgֳ "? a Crisp set A& 8 _ G  pB  H "8c?  pB  H "8c?   T@gֳgֳ?e Jl  }Membership function:8 A G G    Tgֳgֳ?5 6  ]Heights&8 AG  vB  N1?  vB  N1?  vB  N1?  vB  N1?    Tgֳgֳ?   d 5 10  &8 AG   Txgֳgֳ?   b 6 2  &8 AG   T\gֳgֳ? 3  Z.5&8 AG   TLgֳgֳ? C  Z.9&8 AG vB  N1?` p`    Tdgֳgֳ?   Fuzzy set AN 8 _GGG    TPgֳgֳ? X ~  [1.0&8 AG 2  S ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`T q  2   C ENGHZJQp? `T`T`T`T`T`T`T`T`T`T  pB ! Hp? 2 " S ENGHZJQp? `T`T`T`T`T`T`T`T`T`TW  pB #@ Hp?c c H  0޽h ??0 " a(B:___PPT10.+D' = @B D=' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+p+0+0 ++0+0 +  0 L0 + zn(      ¾gֳgֳ 8c8c?T'     C xȾgֳgֳ ?T9  * @`pB  H "8c? @ pB  H "8c?p   T|̾gֳgֳ?E<  gMFs&8 AG "    T@Ѿgֳgֳ?85~ ]Heights&8 AG "  vB  N31?  vB   N31?  vB   N31?    NԾgֳgֳ?  d 5 10  &8 AG "    Nپgֳgֳ? ~wW  Z.5&8 AG "    Tܾgֳgֳ? ~wL  Z.8&8 AG " vB  N31?    TXgֳgֳ?5 ~w  Z.1&8 AG " pB  H8c?   2  C ENGHZJQ8c? `T`T`T`T`T`T`T`T`T`T  2  C E$GH I`TQ8c? `TDT`TDT`T`T`TDT`T`T!  2  C E$GH, I`TQ31? `T@T`T@T`T`T`T@T`T`T   T\gֳgֳ?1F   V tall in Asia4 GG  pB  H8c?  2  S ENGZHI`TJQ "1? `T`T`T`T`T`T`T`T`T`T q   2  C ENGHZJQ "8c? `T`T`T`T`T`T`T`T`T`T  2  C E$GH I`TQ "8c? `TDT`TDT`T`T`TDT`T`T   Tgֳgֳ?a H  "V tall in the US" G "  2  S ENGHZJQ8c? `T`T`T`T`T`T`T`T`T`T  2  S E$GH I`TQ8c? `TDT`TDT`T`T`TDT`T`T!  2  c ENGZHI`TJQf1? `T`T`T`T`T`T`T`T`T`T    Zgֳgֳ?   V tall in NBA4 GG   vB  N8c?  H  0޽h ?  a(  0 + IAq(     @gֳgֳ 8c8c?T'g     C xgֳgֳ ?T9    HA5 W?1?4X W$ 0 0P2   E$GHq I`TQ31? `TT`TT`T`T`TT`T`T   H' " 2   fG31? PQ   T gֳgֳ? %j  s!Universe or universe of discourse"" "G " "   Tgֳgֳ?A (  [ Fuzzy set"   G "     Tgֳgֳ?  "  jMembership function (MF)" G "  z  < 1    < 1 ,$ 02   C BUENG ITQ38c? UU`TUU`TT`TU`TT`T {   2   c CUENG/gHZJ2Q38c? Ҭ`TU^TUҬ`TU^TU2^TU2 y P &2   s BUCUENGV.gHZIQJ5Q38c? լUUUUլUUUUQ5UUQ5 < 1 xB  H38c?P ~ ~ xB  H38c?; ~ ~ 2  S BUENGs# IEQ38c? UU`TUU`TE`TU`TE`T < < "   gֳgֳG ^&p?^ff3 `@,$D  0 CA fuzzy set is totally characterized by a membership function (MF). D DE  D H  0޽h ?o`    a(___PPT10+OD' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +B  0 + n(     `)gֳgֳ 8c8c?T'g     3 r8*gֳgֳ ?TX0   p  HA61?T \H  0޽h ? a(  0 + :(     7gֳgֳ 8c8c?T     3 r8gֳgֳ ?T(  p  HA71? 2  BA8 X??p X X$ 0 0H  0޽h ? a(t   0 +    (     Hgֳgֳ 8c8c?T'g     C xxIgֳgֳ ?T9    HA9 Y?1? |qX Y$ 0 0  HA: Z?1? \ X Z$ 0 0  T4Lgֳgֳ?)L 6 ] X is discrete  G "    ThQgֳgֳ?i q v  _X is continuous  G "    ZG/*d @1?d L    ZG/*d @1?$    NpUgֳgֳ?  Note that S and integral signs stand for the union of membership grades;  / stands for a marker and does not imply division.\~  Gf&G f&qGf&G ~ H  0޽h ? a(:  0 + f(     (fgֳgֳ 8c8c?T'g     C xggֳgֳ ?T9  p  HA;1?2  Thgֳgֳ?y  flingmf.m  G  H  0޽h ? a(  0 + :2(     Xrgֳgֳ 8c8c?T'g     C x|lgֳgֳ ?T     3 rugֳgֳ ?T($    H  0޽h ? a(J  0 + &'(     {gֳgֳ 8c8c?T'g   vB  N "8c?P PP vB  N "8c?PPA   ZPgֳgֳ "? XMF&8 AG    Zgֳgֳ "? %F  WX&8 AG  |B  T "1?pP0p|B  T "1?PP |B   T "1?p00P    Zgֳgֳ "?%NG Z.5&8 AG    Zgֳgֳ "?ud1, Y1&8 AG    Z؋gֳgֳ "?5 d1  Y0&8 AG 2   S ENGHZJQ "8c? `T`T`T`T`T`T`T`T`T`TpP 2  S E$GH I`TQ "8c? `T/T`T/T`T`T`T/T`T`TQ q2  S ENGHZJQ "1? `T`T`T`T`T`T`T`T`T`TP p0   Zgֳgֳ "?q  X  TCore  G  vB  N "8c?P 2  S BTENG I1Q "8c? ѫTT`TѫTT`T1`TT`T1`Tx 1q2  c ENGZHI`TJQ "8c? `T`T`T`T`T`T`T`T`T`Tp55P |B  T "1?P P  |B  T "1?  |B  T "1?pP |B  T "1?P 2  c ENGZHI`TJQ "1? `T`T`T`T`T`T`T`T`T`TP a |B  T "1?P 00vB @ N "1?@ 0@vB @ N "1?@ @  Z gֳgֳ "? V jx  `Crossover points  G    ZHgֳgֳ "? r  WSupport  G  vB @ N "1? vB  N "1? P  |B   T "1? P  vB !@ N "1?  vB "@ N "1?   # Zgֳgֳ "? 6 Y  ka - cut4 G G  |B $ T "1?  |B % T "1?    & Z,gֳgֳ "?u ^7,  Wa&8 AG   H  0޽h ?o`  a(  0 +  (     @gֳgֳ 8c8c?T'g     C xgֳgֳ ?T9    BA< [??@0PoX [$ 0 0p  HA=1?4 eQn  Ngֳgֳ? 8Alternatively, A is convex is all its a-cuts are convex.t90 _G "G "G "G  "G " 9    Tعgֳgֳ?{  h convexmf.m  G   H  0޽h ? a(  0 +  D(     gֳgֳ 8c8c?T'     C x4gֳgֳ ?T9    BA> \??@ap oX \$ 0 0  BA? ]?? 0I X ]$ 0 0  BA@ ^?? @ X ^$ 0 0  BAA _??o X _$ 0 0H  0޽h ? a(  0 + `X0(     gֳgֳ 8c8c?T'g   p  HAB1?p  HAC1?2<   Tgֳgֳ?i v  Xsubset.m  G    Tgֳgֳ?ik v h fuzsetop.m  G   H  0޽h ? a(   0 + @ 7(       Pgֳgֳ 8c8c?T'g      3 r(gֳgֳ ?TXp     BAD `??iX `$ 0 0   Ngֳgֳ? wTrapezoidal MF:$0 _G "      BAE a??aX a$ 0 0   N\gֳgֳ?8  P  |Generalized bell MF:$0 _G "      BAF b?? s zZX b$ 0 0   N gֳgֳ?h   t Gaussian MF:$ 0 _ G "      BAG c?? X c$ 0 0H   0޽h ? a(g  0 + P(     0 gֳgֳ 8c8c?T'g   p  HAH1?J  T gֳgֳ?   g disp_mf.m  G   H  0޽h ? a(+   0 + ` W(      gֳgֳ 8c8c?T'g     C x gֳgֳ ?T     BAI d??0X d$ 0 0  N gֳgֳ?8P s Extensions:$ 0 _ G "   p  HAJ1?$/   N/ gֳgֳ?  Abs. difference of two sig. MF$0 ZG "     Np; gֳgֳ?   ~Product of two sig. MF$0 ZG "      T(3 gֳgֳ? h disp_sig.m  G      ZG/*d @1?d  L    ZG/*d @1? H  0޽h ? a(   0 + 4 , p (     J gֳgֳ 8c8c?T'g     3 rdK gֳgֳ ?T@     BAK e??IX e$ 0 0  NT gֳgֳ?   pExample:$ 0 _ G "     BAL f?? 0 X f$ 0 0  BAM g??`0 X g$ 0 0p  HAN1?    THV gֳgֳ?9 F fdifflr.m  G     N@\ gֳgֳ?  vc=65 a=60 b=10$0 _G "      Ne gֳgֳ? `p vc=25 a=10 b=40$0 _G "   H  0޽h ? a(b  0 + (     Xx gֳgֳ 8c8c?T'g   p  HAO1?vL   Nz gֳgֳ?x ` r Base set A$ 0 _ G "     N\ gֳgֳ?x0 P` }Cylindrical Ext. of A$0 _G "     T0 gֳgֳ?; & g cyl_ext.m  G "   H  0޽h ? a(  0 + og (        gֳgֳ 8c8c?T'g   p   HAP1?4    N gֳgֳ?(@p |Two-dimensional MF&8 _G "      Np gֳgֳ?( @ {Projection onto X&8 _G "      N gֳgֳ?(  {Projection onto Y&8 _G "      BAQ h??{ X h$ 0 0   BAR i??c u%X i$ 0 0   BAS j??l vX j$ 0 0   T gֳgֳ?7 Y project.m  G "  H   0޽h ? a(t  0 + $(  $ $  Ժ gֳgֳ 8c8c?T'g    $ C x gֳgֳ ?T9     $ BAT k?? v X k$ 0 0 $ BAU l??  4X l$ 0 0H $ 0޽h ? a(p  0 +  ((  ( (   gֳgֳ 8c8c?T'g   p ( HAV1?$I ( T gֳgֳ?ik v Z negation.m  G   ( BAT m??M X m$ 0 0 ( BAU n??G [X n$ 0 02 ( N gֳgֳ?@  (Sugeno s complement::8 _G "G1    0 ( N gֳgֳ?  &Yager s complement::8 _G "G1    H ( 0޽h ? a(  0 + ld,(  , ,  h gֳgֳ 8c8c?T'g    , C x gֳgֳ ?T9   H , 0޽h ? a(S  0 + P 0(  0 0  p gֳgֳ 8c8c?T'g   p 0 HAW1?4< p 0 HAX1?p { 0 Th gֳgֳ?Qo Minimum: Tm(a, b)J  G "G "G "   0 T gֳgֳ?f  Algebraic product: Ta(a, b)J G "G "G "   0 TH gֳgֳ?  Bounded product: Tb(a, b)J G "G "G "   0 T gֳgֳ?L Drastic product: Td(a, b)J G "G "G "    0 T gֳgֳ?9 F etnorm.m  G  H 0 0޽h ? a(  0 + ld`4(  4 4   gֳgֳ 8c8c?T'g    4 C x\ gֳgֳ ?T9   H 4 0޽h ? a(  0 + 3+p 8(  8 8  ) gֳgֳ 8c8c?T'g    8 T* gֳgֳ?9K L F g tconorm.m  G   p 8 HAY1?d< p 8 HAZ1? {) 8 T0 gֳgֳ?>o Maximum: Sm(a, b)J  G "G "G "&     8 TD7 gֳgֳ?f  Algebraic sum: Sa(a, b)J G "G "G "  - 8 T= gֳgֳ?  Bounded sum: Sb(a, b)J G "G "G "&    -  8 TC gֳgֳ?D Drastic sum: Sd(a, b)J G "G "G "&    H 8 0޽h ? a(n  0 |(  |x | c $L T`   x | c $M T`    LB | c $D ttLB | c $Dt4RB | s *DԔl tl RB | s *DԔl tzL @  |# l t ZB  | s *DԔ@ ZB  | s *DԔ@ P@ ZB  | s *DԔP@ zL @   |# l  ZB  | s *DԔ@ ZB | s *DԔ@ P@ ZB | s *DԔP@ RB | s *DԔl d RB | s *DԔl 4 l RB | s *DԔl 4 RB | s *DԔl d RB | s *DԔl 4l RB | s *DԔl 44 | <P :.  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H | 0޽h ? ̙33  0 @<(  @~ @ s *z T'g   ~ @ s *d{ TPP    H @ 0޽h ? f̙f  0 `<(  `~ ` s * T'g   ~ ` s *\ T9   H ` 0޽h ? f̙f  0 d<(  d~ d s * T'g   ~ d s *x T9   H d 0޽h ? f̙fb  0  h(  h~ h s * T'g   ~ h s *n T   ^ h 6A?pp~H h 0޽h ? f̙f  0  l<(  l~ l s * T'g   ~ l s *Ы T9   H l 0޽h ? f̙f  0 00(  x  c $( T'g   x  c $ T9   H  0޽h ? f̙f  0 ld@(  x  c $ T'g   x  c $H T9   ^  6A?@H^  6A?PH  0޽h ? f̙f  0 P<(  ~  s * T'g   ~  s *` T9   H  0޽h ? f̙f  0 xp`(  ~  s *0 T'g   ~  s * T9   ^  6A?@H^  6A?PH  0޽h ? f̙f  0 p<(  ~  s *T T'g   ~  s *, T9   H  0޽h ? f̙fZ  0   (  ~  s * T'g   ~  s * T9   ^  6A ?  6 M ] *      6    [ (   \   c 4A report`H  0޽h ? f̙f  0 <(  ~  s * T'g   ~  s * T9   H  0޽h ? f̙fX  0  (  ~  s * T'g   ~  s * T9     6 M ] *      6    [ (   \  c 4A report`@\   c 4A reportppCH  0޽h ? f̙f  0  (  x  c $ T'g   x  c $X T9      6    [ (   \   c 4A report  \   c 4Areport   6@ M ] *   \  c 4AreportP P >   6 h ] *   \  c 4Areport PS H  0޽h ? f̙f   0  }  (  ~  s *( T'g   ~  s *\) T9     6    [ (     6/ M ] *     6H5    [ (      6; h ] *      6@ x   [ (      6|F M ] *     6K    [ (   \  c 4Areport   6@Q M ] *     6W    [ (   \  c 4AreportP >>   6x\ _ ] *     6Hb    [ (   \  c 4Areport J H  0޽h ? f̙f  0 D<(  D~ D s * < 1?0hl2 ? < 1?)3`j @ Z xaxa1?%   dTop speed [km/h]$ g    A  `x xaxa1?H y  _ Weight [kg]$   g   l2 B <1? T u  C Z xaxa1?s J   [ Sports cars   c   l2 D <1?T4  E Z xaxa1?C* !  bMedium market cars  c   l2 F <1?4<,  G Z xaxa1?z1 - WLorries  c   H  0޽h ? GGGggg-  0L0 ,-$-MMT,(    S ~ xaxa1 ?T'g   $L  # ZB  s *1?FFZB  s *1?SSZB  s *1?OJZB  s *1?OJZB  s *1?SSZB   s *1?FFZB   s *1?FFZB   s *1?OJZB   s *1?FFZB   s *1?'JZB  s *1?Wj  Z xaxa1? i& Y100$ g  ZB  s *1? ' JZB  s *1? W j  Z xaxa1?i  Y150$ g  ZB  s *1?g 'g JZB  s *1?g Wg j  Z xaxa1? i  Y200$ g  ZB  s *1?='=JZB  s *1?=W=j  Z xaxa1?i Y250$ g  ZB  s *1?'JZB  s *1?Wj  Z4 xaxa1?ji Y300$ g  ZB  s *1?FFZB B s *1?FF  Z xaxa1?y  Y500$ g  ZB  s *1?  ZB  B s *1?   ! Z xaxa1?0x ~+  Z1000$ g  ZB " s *1?H H ZB #B s *1?H H  $ Z xaxa1?0 ~  Z1500$ g  ZB % s *1?  ZB &B s *1?   ' Z xaxa1?0 ~4  Z2000$ g  ZB ( s *1?QQZB )B s *1?QQ * Z xaxa1?0~ Z2500$ g  ZB + s *1?ZB ,B s *1? - Z xaxa1?0~6 Z3000$ g  ZB . s *1?SSZB /B s *1?SS 0 Z xaxa1?0~ Z3500$ g  ZB 1 s *1?FFZB 2 s *1?SSZB 3 s *1?OJZB 4 s *1?OJZB 5 s *1?FFZB 6 s *1?SSl2 7 < 1?h l2 8 < 1? x , l2 9 < 1?' ^ l2 : < 1?A  x l2 ; < 1?  l2 < < 1?J l2 = < 1?l2 > < 1?0hl2 ? < 1?)3`j @ Z xaxa1?%   dTop speed [km/h]$ g    A  ` xaxa1?H y  _ Weight [kg]$   g   l2 B <1? T u  C Z`! xaxa1?s J   [ Sports cars   c   l2 D <1?T4  E Z& xaxa1?C* !  bMedium market cars  c   l2 F <1?4<,  G Z* xaxa1?z1 - WLorries  c    H Zt. GH1?  Object or data point2 C c   I Z2 GpHd1? $  wfeature2 C c   J Z$7 GH{1?%G } feature space2  C c    K Z< GH1?m$ :  wcluster2 C c   L Z|@ GtH1?` wfeature2 C c   M ZLE GHv1?n ; ulabel2 C c  H  0޽h ?o`HIJKLM GGGggg  0 B: (  ~  s *L T'g   `  c $A%  ?? f     ZXN GFH1?:   data point kB   C C c     ZS GBH,1?  cluster centre iB C C c    ZY G?H4@1?8 > c   xdistance2  C c    Z^ GH,1?P  cluster centre jB C C c  H  0޽h ?O@ GGGggg  0 B:0(  ~  s *f T'g   `  c $A& !??3P  !\  Zhk GMH#z1?Z 2All clusters C together fills the whole universe Ub3  C C "C C c 2   Zp GH1?. T Clusters do not overlap2 C c  p  Zu G^yH 1? 3  FA cluster C is never empty and it is smaller than the whole universe UbG  C C :C C c F o  ZD} GH 1? 4 ;  eThere must be at least 2 clusters in a c-partition and at most as many as the number of data points KBf dC C c e H  0޽h ?O@ GGGggg  0 @G(  ~  s *0 T'g   `  c $A' "??2\  "!  Z܈ GTH1?q  'Minimise the total sum of all distances2( 'C c ' H  0޽h ? GGGggg 0 PN(    0XN RJB___PPT9$ \Place two cluster centres Assign a fuzzy membership to each data point depending on distance4]  2\Cc\  @`p  HA) ?D9/ B  s *޽h ? ̙33 0 `G(    N$X ?c pRJB___PPT9$ 9Compute the new centre of each class Move the crosses (x)2:  9Cc9  @`p  HA* ?D9/ B  s *޽h ? ̙33y 0 )!p(    NX ? p  Iteration 22   Cc   @`p  HA+ ?D9/ B  s *޽h ? ̙33y 0 )!(    N(` ? p  Iteration 52   Cc   @`p  HA, ?D9/ B  s *޽h ? ̙33z 0 *"(    N ` ? p  Iteration 102   Cc   @`p  HA- ?D9/ B  s *޽h ? ̙33 0 (  o  N ` ?c p kIteration 13 (then stop, because no visible change) Each data point belongs to the two clusters to a degree2l kCck  @`p  HA. ?D9/ B  s *޽h ? ̙33 0 #(  =  Nt(` ? ^VN___PPT90( YThe membership matrix M: The last five data points (rows) belong mostly to the first cluster (column) The first five data points (rows) belong mostly to the second cluster (column)P   CCc  @`  NL7` ?w8  (M = 0.0025 0.9975 0.0091 0.9909 0.0129 0.9871 0.0001 0.9999 0.0107 0.9893 0.9393 0.0607 0.9638 0.0362 0.9574 0.0426 0.9906 0.0094 0.9807 0.0193" 2c    B  s *޽h ? ̙33v 0 &(     0x`N " <Each data point belongs to two clusters to different degrees2= 2<Cc< p  HA( ?D9/ B  s *޽h ? ̙33  0 P(  ~  s * T'g   `  c $A/ #??5  #Z  S A0 $?d   $[  Z GHn1?; Q@ 1Distance from point k to current cluster centre ib2 C C C C c 1 Z  Z GH1? 7  0Distance from point k to other cluster centres jb1 C C C C c 0 n  Z GSH1? M DPoint k s membership of cluster ib# C C C C c "    Z GHN1?  L Fuzziness exponent2 C c  H  0޽h ?O@ GGGggg  0 f^`(  ~  s * T'g   `  c $A1 %??% %`  c $A2 &??Z  &P  Z GHKL1?  K  6Gravitation to cluster i relative to total gravitationR7 C C C c 6 H  0޽h ? GGGggg  0 H<(  H~ H s *$ T'g   ~ H s * TP    H H 0޽h ? f̙f  0L0 % 0(L(  L L Z` gֳgֳ ?T"    L N gֳgֳ ?"0P > \ L S A[ ?? 1 h L s ,A\ ??T ( L T@ gֳgֳ ?"6@`NNN?N4  fis the number of data point in"  h L s ,A] ??` ^  h L s ,A^ ?? h L s ,A_ ?? 2  L T gֳgֳ ?"6@`NNN?N p(is the number of features in each vector))" ) %  L T8 gֳgֳ ?"6@`NNN?N  cA c-partition of X, which is      L T gֳgֳ ?"6@`NNN?Np  Nmatrix U   h  L s ,A` ?? z    L T gֳgֳ ?"6@`NNN?N6 NV  TSet of vectors   h L s ,Aa ?? @  h L s ,Ab ??8   < L T gֳgֳ ?"6@`NNN?Nh   z4is called  cluster center     L N gֳgֳ ?"0 Q Input, Output   L c l gֳgֳ ?T3"0e'g<$  0   h L s ,A[ ??P  B L s *޽h ? f31)___PPT10 .+TD'  = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*L+8+0+L0 +`   0 % s k P (  P P N gֳgֳ ?"0P > \ P S A] ??uC \ P S Ac ?? I%  , P T gֳgֳ ?"6@`NNN?N6   j Rows of U (Membership Functions)!!" ! \ P S Ad ??BW \ P S Ae ??0 E  \ P S Af ??   P N  gֳgֳ ?"0 WSample Illustration  T  P C ,Agfcm_d1_in V  P C .Ahfcm_d1_c4r  Hd  P C <Ai$fcm_d1_mf_subplot H  P  ` gֳgֳ ?T"'g    F  P p P s ,A] ??uC p P s ,Ad ??BW p P s ,Af ?? >F I E  P  IE p P s ,Ac ??I %  p P s ,Ae ??0 E  B P s *޽h ? f3y___PPT10Y+D='  = @B +=   0 % P H T (  T T N gֳgֳ ?"0P > \ T S Aj ??`   T Nh gֳgֳ ?"0 i(FCM), Objective Function  \ T S Ak ?? G   T T gֳgֳ ?"6@`NNN?N0 >*  RDistance  "  \ T S Al ?? : = T T gֳgֳ ?"6@`NNN?NP   {Degree of Fuzzification"&     \  T S Am ??0     T T gֳgֳ ?"6@`NNN?N u  T Constraint  "  h  T s ,An ??0 V    T T  gֳgֳ ?"6@`NNN?N   PA-norm"    T  `$ gֳgֳ ?T"'g     T Z$ gֳgֳ ?T"   h T s ,Aj ??(  h T s ,Am ??  2  h T s ,Ak ??, *  h T s ,Al ?? 6 B T s *޽h ? f3y___PPT10Y+D='  = @B +   0 %   0X (  X X Z(2 gֳgֳ ?T"    X N3 gֳgֳ ?"0P >  X Nd6 gֳgֳ ?"0 aMinimizing Objective Function  h X s ,Ao ??00  \ X S Ap ?? E \ X S Aq ?? `|  \ X S Ap ?? U  \  X S Ar ??`| h  X s ,As ??l \  X S At ??   h  X s ,Au ??~ Et F \  X S Av ?? p h X s ,Aw ?? /  - X T< gֳgֳ ?"6@`NNN?N   k!Note: It is the Center of Gravity""" "  X  `@ gֳgֳ ?T"'g    h X s ,Av ??  h X s ,At ??0 0  B X s *޽h ? f3y___PPT10Y+D='  = @B +#  0 % 6.@ \(  \ \ ZT gֳgֳ ?T"   ( @` \ N[ gֳgֳ ?"0P > \ \ S Ax ??   \ \ S Ay ???~@ \ \ S Az ??ppJ  \ \ S A{ ??L b  \ \ S A| ?? vm    \ NDXgֳgֳ ?"0 HPick    \  `Xgֳgֳ ?T"'g    h  \ s ,Ax ??0 E h  \ s ,A| ??p   h  \ s ,A{ ?? 6 B \ s *޽h ? f3y___PPT10Y+D='  = @B +]   0 % p h `d (  d d Z{Wgֳgֳ ?T"   ( @` d N( Xgֳgֳ ?"0P > \ d S A} ?? )0 \ d S A~ ??P 0K \ d S A ??0 @  h d s ,A ?? 9 D  \ d S A ??@ @ ]  \  d S A ??O   h  d s ,A ?? +  h  d s ,A ?? P @   d Ntp gֳgֳ ?"0 RGuess, Iterate    d  `r gֳgֳ ?T"'g    h d s ,A} ??i h d s ,A~ ??PK h d s ,A ??0   h d s ,A ??@ @ ]  h d s ,A ??` 1  B d s *޽h ? f3y___PPT10Y+D='  = @B +  0 % #p h(  h h N gֳgֳ ?"0P >  h N\ gֳgֳ ?"0 cSample Termination Measure Plot   Z h C 2Afcm_d1_termm p^ h C 6Afcm_d1_mf_stemP d$ h TȆ gֳgֳ ?"6@`NNN?N @z bFinal Membership Degrees"  & h T gֳgֳ ?"6@`NNN?N  dTermination Measure Values"   h  `X gֳgֳ ?T"'g    \  h S A ??` h  h s ,A ??0_ B h s *޽h ? f3y___PPT10Y+D='  = @B +k  0 % ~v l(  l l N gֳgֳ ?"0P > \ l S A ??  \ l S A ??  \ l S A ?? P  l  `ܕ gֳgֳ ?T"'g     l Z@ gֳgֳ ?T"   ( @` l N  gֳgֳ ?"0 XImplementation Notes  h  l s ,A ??p $ h  l s ,A ??( ?L h  l s ,A ?? E  B l s *޽h ? f3y___PPT10Y+D='  = @B +  0 % )!p(  p p N gֳgֳ ?"0P Q Pros and Cons   p  ` gֳgֳ ?T"'g     p Z!Xgֳgֳ ?T"   ( @`B p s *޽h ? f3y___PPT10Y+D='  = @B +  0 P0(  Px P c $ T'g    x P c $l T9   H P 0޽h ? f̙f  0 t<(  t~ t s *d T'g    ~ t s *( T9   H t 0޽h ? f̙f  0  x<(  x~ x s *KXT'g  X ~ x s *LXTq  X H x 0޽h ? ff[[ffRR|___PPT10i.(Here I like to emphasize some important properties of membership functions. First of all, it subjective measure; my membership function of Vall?is likely to be different from yours. Also it context sensitive. For example, I 5?1? and I considered pretty tall in Taiwan. But in the States, I only considered medium build, so may be only tall to the degree of .5. But if I an NBA player, Il be considered pretty short, cannot even do a slam dunk! So as you can see here, we have three different MFs for Vall?in different contexts. Although they are different, they do share some common characteristics --- for one thing, they are all monotonically increasing from 0 to 1. Because the membership function represents a subjective measure, it not probability function at all.    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H>n<>BBBX          0>OutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering SummaryY LH       h Introduction  6 "    Idea of Fuzzy  i Introduction  G6 "   What does it offer? generate precise solutions from certain or approximate information While other approaches require accurate equations to model real-world behaviors, fuzzy design can accommodate the ambiguities of real-world human language and logic. It provides both an intuitive method for describing systems in human terms and automates the conversion of those system specifications into effective models. 4ZZ  j Introduction  G6 "   Aristotle "Law of the Excluded Middle," every proposition must either be True or False, A or not-A Ex., rose is either red or not red It cannot be red and not red Plato there was a third region (beyond True and False) where these opposites "tumbled about. ;@W ;     W$    k Introduction  G6 "   WIn the early 1900s, Lukasiewicz described a three-valued logic, which can best be translated as the term `possible', and assigned it a numeric value between True and False. Knuth, a former student of Lukasiewicz proposed a three-valued logic apparently missed by Lukasiewicz, whiched used an integral range [-1, 0 +1] rather than [0, 1, 2]. L!(!(      5     :  l Introduction  G6 "   1960 s Lotfi A. Zadeh, a professor of UC Berkeley Observed that conventional computer logic was incapable of manipulating data representing subjective or vague human ideas such as "an atractive person" or "pretty hot". Fuzzy logic, hence was designed to allow computers to determine the distinctions among data with shades of gray, similar to the process of human reasoning.@Z,ZFZ,Fh         m Introduction  G6 "   In 1965, Zadeh published his seminal work "Fuzzy Sets Described the mathematics of fuzzy set theory, and by extension fuzzy logic. This theory proposed making the membership function (or the values False and True) operate over the range of real numbers [0.0, 1.0]. : . .2     n Introduction  G6 "   _US and certain parts of Europe ignored it, fuzzy logic was excepted with open arms in Japan, China and most Oriental countries. The world's largest number of fuzzy researchers are in China with over 10,000 scientists. The popularity of fuzzy logic in the Orient reflects the fact that Oriental thinking more easily accepts the concept of "fuzziness". _   pOutlineG6 "  YIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary *Z  CH    !  @ Fuzzy Sets  G0 "  8Universal Set X  always a crisp set. Crisp set assigns value {0,1} to members in X Fuzzy set assigns value [0,1] to members in X These values are called the membership functions m. Membership function of a fuzzy set A is denoted by : A: X [0,1] A: [x1/m1, x2/m2, & , xn/mn}(0,,C % C C C C &   s  Fuzzy Sets $  E "   [Fuzzy sets How fuzzy sets quantifying the same information can describe this natural drift.& Q Q \ t Fuzzy Sets $  E "   4Let's talk about people and "youthness Set S (the universe of discourse) is the set of people Fuzzy subset YOUNG "to what degree is person x young?" To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset YOUNG. A membership function based on the person's age young(x) = { 1, if age(x) <= 20, (30-age(x))/10, if 20 < age(x) <= 30, 0, if age(x) > 30 } (91^(91  ^J   O  $  u Fuzzy Sets$  E "   gGraph of membership function Example: the degree of truth of the statement "Parthiban is YOUNG" is 0.50&KK&L    X Fuzzy Sets  C0  Sets with fuzzy boundariesZ  YMembership Functions (MFs)A&   ECharacteristics of MFs: Subjective measures Not probability functions&..&  0 Z Fuzzy Sets  C0  OFormal definition: A fuzzy set A in X is expressed as a set of ordered pairs: xZ<ZZ  " "*O  ["Fuzzy Sets with Discrete Universes##A # >Fuzzy set C =  desirable city to live in X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A =  sensible number of children X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}`*Y,q*Y ",q "&M    \Fuzzy Sets with Cont. Universes C$  Fuzzy set B =  about 50 years old X = Set of positive real numbers (continuous) B = {(x, mB(x)) | x in X}d#H#7 "G  " " "&Z    ]Alternative NotationA  6A fuzzy set A can be alternatively denoted as follows:7Z7 7 ^Fuzzy PartitionA  Fuzzy partitions formed by the linguistic values  young ,  middle aged , and  old : T _More DefinitionsA  KSupport Core Normality Crossover points Fuzzy singleton a-cut, strong a-cutDL8C  C  L  IConvexity Fuzzy numbers Bandwidth Symmetricity Open left or right, closedJJ&"    `MF TerminologyA   aConvexity of Fuzzy Sets  1A fuzzy set A is convex if for any l in [0, 1], H2Z  "C  1  bSet-Theoretic OperationsC0  +Subset: Complement: Union: Intersection: , cSet-Theoretic OperationsA   dMF FormulationA  Triangular MF:Z  eMF FormulationA   fMF Formulation   Sigmoidal MF:Z   gMF FormulationA  L-R MF:  hCylindrical ExtensionA   i2D MF Projection   jFuzzy ComplementA  rGeneral requirements: Boundary: N(0)=1 and N(1) = 0 Monotonicity: N(a) > N(b) if a < b Involution: N(N(a) = a Two types of fuzzy complements: Sugeno s complement: Yager s complement: ZXZ ZZZZZX f4   N       kFuzzy ComplementA   lFuzzy Intersection: T-normA  <Basic requirements: Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a Monotonicity: T(a, b) < T(c, d) if a < c and b < d Commutativity: T(a, b) = T(b, a) Associativity: T(a, T(b, c)) = T(T(a, b), c) Four examples (page 37): Minimum: Tm(a, b) Algebraic product: Ta(a, b) Bounded product: Tb(a, b) Drastic product: Td(a, b)ZZZbZ ZA   '       mT-norm Operator   nFuzzy Union: T-conorm or S-norm&    0Basic requirements: Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a Monotonicity: S(a, b) < S(c, d) if a < c and b < d Commutativity: S(a, b) = S(b, a) Associativity: S(a, S(b, c)) = S(S(a, b), c) Four examples (page 38): Maximum: Sm(a, b) Algebraic sum: Sa(a, b) Bounded sum: Sb(a, b) Drastic sum: Sd(a, b)ZZZVZ A   '      B  *     oT-conorm or S-norm&     CFuzzy VariablesC0  Variables whose states are defined by linguistic concepts like low, medium, high. These linguistic concepts are fuzzy sets themselves.   qOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary2Y 5<    ! < Fuzzy Logic     Fuzzy logic is a superset of conventional(Boolean) logic It extended to handle the concept of partial truth- truth values between "completely true" and "completely false". It is the logic underlying modes of reasoning which are approximate rather than exact. The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature.,  = Fuzzy Logic   Essential characteristics of fuzzy logic: In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning. In fuzzy logic everything is a matter of degree. Any logical system can be fuzzified (thus, define Boolean logic as a subset of Fuzzy logic ) In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables Inference is viewed as a process of propagation of elastic constraints.**ZZ*2     > Fuzzy Logic   8Bivalent sets To characterize the temperature of a room***8  ? Fuzzy logic   Limitations of bivalent sets mutually exclusive it is not possible to have membership of more than one set ( opinion would widely vary as to whether 50 degrees Fahrenheit is 'cold' or 'cool' hence the expert knowledge we need to define our system is mathematically at odds with the humanistic world). Clearly, it is not accurate to define a transition from a quantity such as 'warm' to 'hot' by the application of one degree Fahrenheit of heat. In the real world a smooth drift from warm to hot would occur. hZZZ@Z@  x Fuzzy Logic   Fuzzy Set Operations Union The membership function of the Union of two fuzzy sets A and B with membership functions mA and mB respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebranA]@AY    f =S  y Fuzzy logic   .Union maximum criterion$   z Fuzzy Logic   Fuzzy Set Operations Union The membership function of the Union of two fuzzy sets A and B with membership functions mA and mB respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra9Y    fHS  { Fuzzy Logic   .Union maximum criterion$   | Fuzzy Loic&   Fuzzy Set Operations Intersection The membership function of the Intersection of two fuzzy sets A and B with membership functions mA and mB is defined as the minimum of the two individual membership functions. This is called the minimum criterion. The Intersection operation in Fuzzy set theory is the equivalent of the AND operation in Boolean algebra."j`F"`    Y H c } Fuzzy logic   <Intersection minimum criterion$    ~ Fuzzy Logic   Fuzzy Set Operations Complement The membership function of the Complement of a Fuzzy set A with membership function mA is defined as the negation of the specified membership function. This is called the negation criterion. The Complement operation in Fuzzy set theory is the equivalent of the NOT operation in Boolean algebra.  '@A T  U Q H  Fuzzy logic   :Complement negation criterion$     Fuzzy Logic   XCommon rules in classical set theory and Fuzzy set theory De Morgans law Associativity &::L=        Fuzzy Logic   WCommon rules in classical set theory and Fuzzy set theory Commutativity Distributivity&::@:       rOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary2Y$!<    ! DClusterC6  A number of similar individuals that occur together as a: two or more consecutive consonants or vowels in a segment of speech b: a group of houses (...) c: an aggregation of stars or galaxies that appear close together in the sky and are gravitationally associated.R 7Fo@  ECluster Analysis .C6@ A statistical classification technique for discovering whether the individuals of a population fall into different groups by making quantitative comparisons of multiple characteristics.@ FVehicle Example&@@&  GVehicle Clusters@   H Terminology @   IMembership matrix M2@@$  J c-partition @  KObjective function@  MNOPQRSLTFuzzy membership matrix M:@@L    UFuzzy membership matrix M:@@@    w !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{     sOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary*Y8<    ! tFuzzy C-Means Clustering  KInput: Unlabeled data set Main Output Common Additional Output* 'L L 0uFuzzy C-Means Clustering   1vFuzzy C-Means Clustering  |Optimization of an  objective function or  performance index ?? ? 2wFuzzy C-Means Clustering  ^Zeroing the gradient of with respect to Zeroing the gradient of with respect to __ _ QxFuzzy C-Means Clustering  Initial Choices Number of clusters Maximum number of iterations (Typ.: 100) Weighting exponent (Fuzziness degree) m=1: crisp m=2: Typical Termination measure 1-norm Termination threshold (Typ. 0.01)dbXb-#@A     RyFuzzy C-Means Clustering  Guess Initial Cluster Centers Alternating Optimization (AO) REPEAT UNTIL ( or ) F<H<)  SzFuzzy C-Means Clustering   T{Fuzzy C-Means Clustering  Process could be shifted one half cycle Initialization is done on Iterates become Termination criterion The convergence theory is the same in either case Initializing and terminating on V is advantageous Convenience Speed Storagef(Z@ZdZZ(@d  V|Fuzzy C-Means Clustering  Advantages Unsupervised Always converges Disadvantages Long computational time Sensitivity to the initial guess (speed, local minima) Sensitivity to noise One expects low (or even no) membership degree for outliers (noisy points)  dK dJ  ;$Summary Advantages of Fuzzy Systems$    Conceptually straightforward to understand. The mathematical concepts behind fuzzy reasoning are simple. Flexible:You can modify and add on fuzzy rules without starting from scratch. Tolerant of imprecise data. Everything is imprecise if you look closely enough, but more than that, many things are imprecise even at first glance Can model nonlinear functions of arbitrary complexity. Can create a fuzzy system to match any set of input/output data. Can be built on top of the experience of experts. Is close to natural language.ZB  W}'Summary Disadvantages of Fuzzy Systems$    tCreating the fuzzy rules base can be troublesome It is difficult to create the fuzzy rules base from input/output data if no fuzzy rule extraction technique is used Accuracy of the inference depends directly on the number of fuzzy rules used in complex problem Increasing input variables and fuzzy membership fns used will increase the number of fuzzy rules exponentially.uuB u X~ References C,/ [1] Lofti A. Zadeh. Fuzzy sets, Inf. Control 8, 338-353, 1965. [2] Jan Jantzen. Tutorial On Fuzzy Clustering. http://fuzzy.iau.dtu.dk/tutor/fcm/cluster.ppt [3] Nikhil R. Pal, James C. Bezdek, and Richard J. Hathaway. Sequential Competitive Learning and the Fuzzy c-Means Clustering Algorithms. Neural Networks, Vol. 9, No. 5, pp. 787-796, 1996 [4] Lofti A. Zadeh. Fuzzy Logic, Computer, v.21 n.4, p.83-93, April 1988 [5] Novak, V., Perfilieva, I., Mockor, J. (2000):, Mathematical principles of fuzzy logic. Kluwer 2000. 0P CLCCCC'CC n    &  *&         6     6    0 0{Vwww.fzcyps.com/shizzB.   0 Thank you !!!0 F;3F;3  /pUrHo~$o]X17(  `?   !"#$%&'()*+,-՜.+,D՜.+, @     On-screen Show.L3W kTimes New Roman 新細明體 Garamond宋体Arial Wingdings Arial NarrowTahoma Arial BlackSymbol Courier NewEdgeMicrosoft Word 文件 DocumentMicrosoft Equation 3.0 位图图像Microsoft Clip Gallery Equation 方程式MathType 5.0 EquationjAdvanced Computing Seminar Data Mining and Its Industrial Applications — Chapter 7 — Fuzzy Sets Outline Introduction Introduction Introduction Introduction Introduction Introduction IntroductionOutline Fuzzy Sets Fuzzy Sets Fuzzy Sets Fuzzy Sets Fuzzy SetsMembership Functions (MFs) Fuzzy Sets#Fuzzy Sets with Discrete Universes Fuzzy Sets with Cont. UniversesAlternative NotationFuzzy PartitionMore DefinitionsMF TerminologyConvexity of Fuzzy SetsSet-Theoretic OperationsSet-Theoretic OperationsMF FormulationMF FormulationMF FormulationMF FormulationCylindrical Extension2D MF ProjectionFuzzy ComplementFuzzy ComplementFuzzy Intersection: T-normT-norm Operator Fuzzy Union: T-conorm or S-normT-conorm or S-normFuzzy VariablesOutline Fuzzy Logic Fuzzy Logic Fuzzy Logic Fuzzy logic Fuzzy Logic Fuzzy logic Fuzzy Logic Fuzzy Logic Fuzzy Loic Fuzzy logic Fuzzy Logic Fuzzy logic Fuzzy Logic Fuzzy LogicOutlineClusterCluster Analysis Vehicle ExampleVehicle Clusters TerminologyMembership matrix M c-partitionObjective function Slide 64 Slide 65 Slide 66 Slide 67 Slide 68 Slide 69 Slide 70 Slide 71Fuzzy membership matrix MFuzzy membership matrix MOutlineFuzzy C-Means ClusteringFuzzy C-Means ClusteringFuzzy C-Means ClusteringFuzzy C-Means ClusteringFuzzy C-Means ClusteringFuzzy C-Means ClusteringFuzzy C-Means ClusteringFuzzy C-Means ClusteringFuzzy C-Means Clustering%Summary Advantages of Fuzzy Systems(Summary Disadvantages of Fuzzy Systems Referenceswww.fzcyps.com/shizz  Fonts Used Design TemplateEmbedded OLE Servers Slide TitlesW 8@ _PID_HLINKSAx http://www.unisa.edu.au/~sShizz_ L0UniSAUniSASMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 EquationEquation Equation.DSMT40*MathType 5.0 Equation Equation Equation.DSMT40*MathType 5.0 Equation!Equation Equation.DSMT40*MathType 5.0 Equation"Equation Equation.DSMT40*MathType 5.0 Equation#Equation Equation.DSMT40*MathType 5.0 Equation$Equation Equation.DSMT40*MathType 5.0 Equation%Equation Equation.DSMT40*MathType 5.0 Equation&Equation Equation.DSMT40*MathType 5.0 Equation'Equation Equation.DSMT40*MathType 5.0 Equation(Equation Equation.DSMT40*MathType 5.0 Equation)Equation Equation.DSMT40*MathType 5.0 Equation*Equation Equation.DSMT40*MathType 5.0 Equation+Equation Equation.DSMT40*MathType 5.0 Equation,Equation Equation.DSMT40*MathType 5.0 Equation-Equation Equation.DSMT40*MathType 5.0 Equation.Equation Equation.DSMT40*MathType 5.0 Equation/Equation Equation.DSMT40*MathType 5.0 Equation3Equation Equation.DSMT40*MathType 5.0 Equation4Equation Equation.DSMT40*MathType 5.0 Equation5Equation Equation.DSMT40*MathType 5.0 Equation6Equation Equation.DSMT40*MathType 5.0 Equation7Equation Equation.DSMT40*MathType 5.0 Equation8Equation Equation.DSMT40*MathType 5.0 Equation9Equation Equation.DSMT40*MathType 5.0 Equation:Equation Equation.DSMT40*MathType 5.0 Equation;Equation Equation.DSMT40*MathType 5.0 Equation<Equation Equation.DSMT40*MathType 5.0 Equation=Equation Equation.DSMT40*MathType 5.0 Equation>Equation Equation.DSMT40*MathType 5.0 Equation?Equation Equation.DSMT40*MathType 5.0 Equation@Equation Equation.DSMT40*MathType 5.0 EquationAEquation Equation.DSMT40*MathType 5.0 EquationBEquation Equation.DSMT40*MathType 5.0 EquationCEquation Equation.DSMT40*MathType 5.0 EquationDEquation Equation.DSMT40*MathType 5.0 EquationEEquation Equation.DSMT40*MathType 5.0 EquationFEquation Equation.DSMT40*MathType 5.0 EquationGEquation Equation.DSMT40*MathType 5.0 EquationHEquation Equation.DSMT40*MathType 5.0 EquationIEquation Equation.DSMT40*MathType 5.0 EquationJEquation Equation.DSMT40*MathType 5.0 EquationKEquation Equation.DSMT40*MathType 5.0 EquationLEquation Equation.DSMT40*MathType 5.0 EquationMEquation Equation.DSMT40*MathType 5.0 EquationNEquation Equation.DSMT40*MathType 5.0 EquationOEquation Equation.DSMT40*MathType 5.0 EquationPEquation Equation.DSMT40*MathType 5.0 Equation/ 0DDTimes New RomanTT+ܖ 0ܖDe0}fԚ New RomanTT+ܖ 0ܖ DGaramond RomanTT+ܖ 0ܖ0D[SOamond RomanTT+ܖ 0ܖ@DArialnd RomanTT+ܖ 0ܖPDWingdingsRomanTT+ܖ 0ܖ`DArial NarrowanTT+ܖ 0ܖ"pDTahomaarrowanTT+ܖ 0ܖ"DArial BlackanTT+ܖ 0ܖ"DSymbollackanTT+ܖ 0ܖDCourier NewanTT+ܖ 0ܖ1c .  @n?" dd@  @@`` TLx)^ k?$ #$%&' )? +,  CDk;   !"-./0H1N234   5 6|$WX YZ[\] ^`(abcd ef g h jk   mnopqrstuvw x yz2$ EЃ>Hb$(#RP(*5yz9$b$\2Fk҂.Xdf)b$0RcQ3b$>LM[7zFl b$HN}//>b$bv{F56 "2$4c(aR;%)#Ĩ;b$mMkzf|{,?b$7}Kׇ)b$߂o#p+[&e/b$k&~jl|z54b$]Nz4׵"-Lb$ :'8xXRb$ܭ HXb$s!m7_^b$Mss\ޗ_^web$fmb:+<;/_]`lb$[-Vt'拻usR$X3&Cqw 2{$$$$$$$$$$$$$$$$2$8ʌ捡Z"@?2$>CwT3󰊭A2$!E4 'F2$@Da 'KYC|oT2$>gFDΌh12$y{~Vzağ292$Xc?aL8k2$IZ"A ؎<2$X vn&{G2$1E /o+,O&2$W "O[|Pu2$>]3H~_"?c Ѱ2$Q]/ȆY2$ tCN!'>+ub$rȉmr89j b$+*KFB Y2$h?^Q3[ \b$C<tU> =^b$si45p|T6 {k2$ :*6Ͱ7^x2$$e\Vg)>^z2$B!ANuW0,x|b$U>C;̎X~2$ ߒH8Q+3b$rgI 2$E]H67!UZCp2$} ]tO.12$'B.~$l^2$Jcʹm8_ m5b$z$Ŧlm=b$`BQf}ł[C9E 2$3XuՔZ 2$TV6 0 jf2$@ApV)2$-ruŧ'5b$Zzݒ]rƶ2$2Pz4jeWb$2VU,i2$BDv.Qzb2$h4)"]iL2$Z 1In b$R؂=ӭ g b$Uʤ9*L^FEb$lC#2BQ*12$i+0w<'_[2$))Sڨ;]2$&i"s%^2$m\R ժMA``2$LP#2G!/bb$ yWr.H6>!cb$0ͥo/"{P h8ބb$-s}o|p*b$uՓNQk9b$Pԃ+J> -m!2$n ybG t N2$6*n=*丞NQ2$@jm OšzI4(eS2$;yklɷ$dE\1*U2$ؑk8/6mM/[W2$ b',j;x!Y2$]]n#\z[2$Im8=u]2$qMH׷_2$Ve$c?rr Aa2$} MpB_c2$ ɅLmeb$r+ SFgb$-TТ6vcҵyb$n 2$L 14ZLxr,2$ނ"h.~3y_2$ j]bǯ7;T2$}_&n2ET2$kS8= j2$Էϒ 0LH'^2$"%\ݟ8z2$`!-&=2$ LUGMId'´2$!Ut}ȹ\ԛz2$ouLxTe?V2u2$}1 7;v2$dEq6CG, j2$*Y*y2$n?<{_C2$tw+Qy+W34Z,vC2$c,f ju2$/s_ڵWe.2$Sk05wbs^r]!2$'d)KVb$Jǵcto1mb$K\sP^MwѯM6.2$'3uɧvQe{2$ n'gD~T2$r??Yl!2$N46a# <AA f3f "f&@8' ʚ;wǀ3ʚ;g4BdBd< 0vppp@ <4!d!d k 0T+<4dddd k 0T+<4BdBd k 0T+0___PPT10 ppd___PPT9F>&?`+ LChap7 Fuzzy Set and Logic Zhongzhi ShiO =WAdvanced Computing Seminar Data Mining and Its Industrial Applications  Chapter 7  Fuzzy Sets fG$ "+G$ "F2 "F "F "B G6&F90 >    Zhongzhi Shi, Markus Stumptner, Yalei Hao, Gerald Quirchmayr Knowledge and Software Engineering Lab Advanced Computing Research Centre School of Computer and Information Science University of South Australia H>n<>BBBX          0>OutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering SummaryY LH       h Introduction  6 "    Idea of Fuzzy  i Introduction  G6 "   What does it offer? generate precise solutions from certain or approximate information While other approaches require accurate equations to model real-world behaviors, fuzzy design can accommodate the ambiguities of real-world human language and logic. It provides both an intuitive method for describing systems in human terms and automates the conversion of those system specifications into effective models. 4ZZ  j Introduction  G6 "   Aristotle "Law of the Excluded Middle," every proposition must either be True or False, A or not-A Ex., rose is either red or not red It cannot be red and not red Plato there was a third region (beyond True and False) where these opposites "tumbled about. ;@W ;     W$    k Introduction  G6 "   WIn the early 1900s, Lukasiewicz described a three-valued logic, which can best be translated as the term `possible', and assigned it a numeric value between True and False. Knuth, a former student of Lukasiewicz proposed a three-valued logic apparently missed by Lukasiewicz, whiched used an integral range [-1, 0 +1] rather than [0, 1, 2]. L!(!(      5     :  l Introduction  G6 "   1960 s Lotfi A. Zadeh, a professor of UC Berkeley Observed that conventional computer logic was incapable of manipulating data representing subjective or vague human ideas such as "an atractive person" or "pretty hot". Fuzzy logic, hence was designed to allow computers to determine the distinctions among data with shades of gray, similar to the process of human reasoning.@Z,ZFZ,Fh         m Introduction  G6 "   In 1965, Zadeh published his seminal work "Fuzzy Sets Described the mathematics of fuzzy set theory, and by extension fuzzy logic. This theory proposed making the membership function (or the values False and True) operate over the range of real numbers [0.0, 1.0]. : . .2     n Introduction  G6 "   _US and certain parts of Europe ignored it, fuzzy logic was excepted with open arms in Japan, China and most Oriental countries. The world's largest number of fuzzy researchers are in China with over 10,000 scientists. The popularity of fuzzy logic in the Orient reflects the fact that Oriental thinking more easily accepts the concept of "fuzziness". _   pOutlineG6 "  YIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary *Z  CH    !  @ Fuzzy Sets  G0 "  8Universal Set X  always a crisp set. Crisp set assigns value {0,1} to members in X Fuzzy set assigns value [0,1] to members in X These values are called the membership functions m. Membership function of a fuzzy set A is denoted by : A: X [0,1] A: [x1/m1, x2/m2, & , xn/mn}(0,,C % C C C C &   s  Fuzzy Sets $  E "   [Fuzzy sets How fuzzy sets quantifying the same information can describe this natural drift.& Q Q \ t Fuzzy Sets $  E "   4Let's talk about people and "youthness Set S (the universe of discourse) is the set of people Fuzzy subset YOUNG "to what degree is person x young?" To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset YOUNG. A membership function based on the person's age young(x) = { 1, if age(x) <= 20, (30-age(x))/10, if 20 < age(x) <= 30, 0, if age(x) > 30 } (91^(91  ^J   O  $  u Fuzzy Sets$  E "   gGraph of membership function Example: the degree of truth of the statement "Parthiban is YOUNG" is 0.50&KK&L    X Fuzzy Sets  C0  Sets with fuzzy boundariesZ  YMembership Functions (MFs)A&   ECharacteristics of MFs: Subjective measures Not probability functions&..&  0 Z Fuzzy Sets  C0  OFormal definition: A fuzzy set A in X is expressed as a set of ordered pairs: xZ<ZZ  " "*O  ["Fuzzy Sets with Discrete Universes##A # >Fuzzy set C =  desirable city to live in X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A =  sensible number of children X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}`*Y,q*Y ",q "&M    \Fuzzy Sets with Cont. Universes C$  Fuzzy set B =  about 50 years old X = Set of positive real numbers (continuous) B = {(x, mB(x)) | x in X}d#H#7 "G  " " "&Z    ]Alternative NotationA  6A fuzzy set A can be alternatively denoted as follows:7Z7 7 ^Fuzzy PartitionA  Fuzzy partitions formed by the linguistic values  young ,  middle aged , and  old : T _More DefinitionsA  KSupport Core Normality Crossover points Fuzzy singleton a-cut, strong a-cutDL8C  C  L  IConvexity Fuzzy numbers Bandwidth Symmetricity Open left or right, closedJJ&"    `MF TerminologyA   aConvexity of Fuzzy Sets  1A fuzzy set A is convex if for any l in [0, 1], H2Z  "C  1  bSet-Theoretic OperationsC0  +Subset: Complement: Union: Intersection: , cSet-Theoretic OperationsA   dMF FormulationA  Triangular MF:Z  eMF FormulationA   fMF Formulation   Sigmoidal MF:Z   gMF FormulationA  L-R MF:  hCylindrical ExtensionA   i2D MF Projection   jFuzzy ComplementA  rGeneral requirements: Boundary: N(0)=1 and N(1) = 0 Monotonicity: N(a) > N(b) if a < b Involution: N(N(a) = a Two types of fuzzy complements: Sugeno s complement: Yager s complement: ZXZ ZZZZZX f4   N       kFuzzy ComplementA   lFuzzy Intersection: T-normA  <Basic requirements: Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a Monotonicity: T(a, b) < T(c, d) if a < c and b < d Commutativity: T(a, b) = T(b, a) Associativity: T(a, T(b, c)) = T(T(a, b), c) Four examples (page 37): Minimum: Tm(a, b) Algebraic product: Ta(a, b) Bounded product: Tb(a, b) Drastic product: Td(a, b)ZZZbZ ZA   '       mT-norm Operator   nFuzzy Union: T-conorm or S-norm&    0Basic requirements: Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a Monotonicity: S(a, b) < S(c, d) if a < c and b < d Commutativity: S(a, b) = S(b, a) Associativity: S(a, S(b, c)) = S(S(a, b), c) Four examples (page 38): Maximum: Sm(a, b) Algebraic sum: Sa(a, b) Bounded sum: Sb(a, b) Drastic sum: Sd(a, b)ZZZVZ A   '      B  *     oT-conorm or S-norm&     CFuzzy VariablesC0  Variables whose states are defined by linguistic concepts like low, medium, high. These linguistic concepts are fuzzy sets themselves.   qOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary2Y 5<    ! < Fuzzy Logic     Fuzzy logic is a superset of conventional(Boolean) logic It extended to handle the concept of partial truth- truth values between "completely true" and "completely false". It is the logic underlying modes of reasoning which are approximate rather than exact. The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature.,  = Fuzzy Logic   Essential characteristics of fuzzy logic: In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning. In fuzzy logic everything is a matter of degree. Any logical system can be fuzzified (thus, define Boolean logic as a subset of Fuzzy logic ) In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables Inference is viewed as a process of propagation of elastic constraints.**ZZ*2     > Fuzzy Logic   8Bivalent sets To characterize the temperature of a room***8  ? Fuzzy logic   Limitations of bivalent sets mutually exclusive it is not possible to have membership of more than one set ( opinion would widely vary as to whether 50 degrees Fahrenheit is 'cold' or 'cool' hence the expert knowledge we need to define our system is mathematically at odds with the humanistic world). Clearly, it is not accurate to define a transition from a quantity such as 'warm' to 'hot' by the application of one degree Fahrenheit of heat. In the real world a smooth drift from warm to hot would occur. hZZZ@Z@  x Fuzzy Logic   Fuzzy Set Operations Union The membership function of the Union of two fuzzy sets A and B with membership functions mA and mB respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebranA]@AY    f =S  y Fuzzy logic   .Union maximum criterion$   z Fuzzy Logic   Fuzzy Set Operations Union The membership function of the Union of two fuzzy sets A and B with membership functions mA and mB respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra9Y    fHS  { Fuzzy Logic   .Union maximum criterion$   | Fuzzy Loic&   Fuzzy Set Operations Intersection The membership function of the Intersection of two fuzzy sets A and B with membership functions mA and mB is defined as the minimum of the two individual membership functions. This is called the minimum criterion. The Intersection operation in Fuzzy set theory is the equivalent of the AND operation in Boolean algebra."j`F"`    Y H c } Fuzzy logic   <Intersection minimum criterion$    ~ Fuzzy Logic   Fuzzy Set Operations Complement The membership function of the Complement of a Fuzzy set A with membership function mA is defined as the negation of the specified membership function. This is called the negation criterion. The Complement operation in Fuzzy set theory is the equivalent of the NOT operation in Boolean algebra.  '@A T  U Q H  Fuzzy logic   :Complement negation criterion$     Fuzzy Logic   XCommon rules in classical set theory and Fuzzy set theory De Morgans law Associativity &::L=        Fuzzy Logic   WCommon rules in classical set theory and Fuzzy set theory Commutativity Distributivity&::@:       rOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary2Y$!<    ! DClusterC6  A number of similar individuals that occur together as a: two or more consecutive consonants or vowels in a segment of speech b: a group of houses (...) c: an aggregation of stars or galaxies that appear close together in the sky and are gravitationally associated.R 7Fo@  ECluster Analysis .C6@ A statistical classification technique for discovering whether the individuals of a population fall into different groups by making quantitative comparisons of multiple characteristics.@ FVehicle Example&@@&  GVehicle Clusters@   H Terminology @   IMembership matrix M2@@$  J c-partition @  KObjective function@  MNOPQRSLTFuzzy membership matrix M:@@L    UFuzzy membership matrix M:@@@     sOutlineG6 "  XIntroduction Fuzzy Sets Fuzzy Logic Fuzzy Clustering Fuzzy C-Means Clustering Summary*Y8<    ! tFuzzy C-Means Clustering  KInput: Unlabeled data set Main Output Common Additional Output* 'L L 0uFuzzy C-Means Clustering   1vFuzzy C-Means Clustering  |Optimization of an  objective function or  performance index ?? ? 2wFuzzy C-Means Clustering  ^Zeroing the gradient of with respect to Zeroing the gradient of with respect to __ _ QxFuzzy C-Means Clustering  Initial Choices Number of clusters Maximum number of iterations (Typ.: 100) Weighting exponent (Fuzziness degree) m=1: crisp m=2: Typical Termination measure 1-norm Termination threshold (Typ. 0.01)dbXb-#@A     RyFuzzy C-Means Clustering  Guess Initial Cluster Centers Alternating Optimization (AO) REPEAT UNTIL ( or ) F<H<)  SzFuzzy C-Means Clustering   T{Fuzzy C-Means Clustering  Process could be shifted one half cycle Initialization is done on Iterates become Termination criterion The convergence theory is the same in either case Initializing and terminating on V is advantageous Convenience Speed Storagef(Z@ZdZZ(@d  V|Fuzzy C-Means Clustering  Advantages Unsupervised Always converges Disadvantages Long computational time Sensitivity to the initial guess (speed, local minima) Sensitivity to noise One expects low (or even no) membership degree for outliers (noisy points)  dK dJ  ;$Summary Advantages of Fuzzy Systems$    Conceptually straightforward to understand. The mathematical concepts behind fuzzy reasoning are simple. Flexible:You can modify and add on fuzzy rules without starting from scratch. Tolerant of imprecise data. Everything is imprecise if you look closely enough, but more than that, many things are imprecise even at first glance Can model nonlinear functions of arbitrary complexity. Can create a fuzzy system to match any set of input/output data. Can be built on top of the experience of experts. Is close to natural language.ZB  W}'Summary Disadvantages of Fuzzy Systems$    tCreating the fuzzy rules base can be troublesome It is difficult to create the fuzzy rules base from input/output data if no fuzzy rule extraction technique is used Accuracy of the inference depends directly on the number of fuzzy rules used in complex problem Increasing input variables and fuzzy membership fns used will increase the number of fuzzy rules exponentially.uuB u X~ References C,/ [1] Lofti A. Zadeh. Fuzzy sets, Inf. Control 8, 338-353, 1965. [2] Jan Jantzen. Tutorial On Fuzzy Clustering. http://fuzzy.iau.dtu.dk/tutor/fcm/cluster.ppt [3] Nikhil R. Pal, James C. Bezdek, and Richard J. Hathaway. Sequential Competitive Learning and the Fuzzy c-Means Clustering Algorithms. Neural Networks, Vol. 9, No. 5, pp. 787-796, 1996 [4] Lofti A. Zadeh. Fuzzy Logic, Computer, v.21 n.4, p.83-93, April 1988 [5] Novak, V., Perfilieva, I., Mockor, J. (2000):, Mathematical principles of fuzzy logic. Kluwer 2000. 0P CLCCCC'CC n    &  *&         6     6    0 0{Vwww.fzcyps.com/shizzB.   0 Thank you !!!0 F;3F;3  /pUr]~]KX1Root EntrydO)e@ Picturese&Current User./SummaryInformation(L8U    w !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{  !"#$%&'()*+,- Fonts Used Design TemplateEmbedded OLE Servers Slide TitlesW 8@ _PID_HLINKSAx http://www.unisa.edu.au/~sShizz_ Lshizzshizz