IFTHEN RULES AND FUZZY INFERENCE


 George Dalton
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1 Iferece IFTHEN RULES ND FUZZY INFERENE!! "! ##$" % "&% " Represetato of koledge % "" Represetato of koledge as rles s the most poplar form. f x s the y s here ad are lgstc ales defed by fzzy sets o erses of dscorse X ad Y. rle s also called a fzzy mplcato x s s called the atecedet or premse y s s called the coseqece or coclso Represetato of koledge Examples If pressre s hgh the olme s small. If the road s slppery the drg s dageros. If a apple s red the t s rpe. If the speed s hgh the apply the brake a lttle. http//f.kast.ac.kr/lectre/cs67/textbook/ http//f.kast.ac.kr/lectre/cs67/textbook/ Koledge as Rles Ho do yo reaso? Yo at to play golf o Satrday or Sday ad yo do t at to get et he yo play. Use rles! If t ras yo get et! If yo get et yo ca t play golf If t ras o Satrday ad o t ra o Sday Yo play golf o Sday' *Fzzy ThkgThe e Scece of Fzzy Logc art Kosko Koledge as Rles Koledge s rles Rles are blackadhte lagage alet rles I has so far after oer 3 years of research ot prodced smart maches! ecase they ca t yet pt eogh rles the compter se  rles eed >k} Throg more rles at the problem *Fzzy ThkgThe e Scece of Fzzy Logc art Kosko
2 Forms of reasog Geeralzed Mods Poes Premse x s Implcato f x s the y s oseqece y s Where are fzzy sets ad x ad y are symbolc ames for objects. Forms of reasog Geeralzed Mods Toles Premse y s Implcato f x s the y s oseqece x s Where are fzzy sets ad x ad y are symbolc ames for objects. http//f.kast.ac.kr/lectre/cs67/textbook/ http//f.kast.ac.kr/lectre/cs67/textbook/ Fzzy rle as a relato f x s the y s x s y s fzzy predcates x y f x the y ca be represeted as a relato Rxy x y here Rxy ca be cosdered a fzzy set th 2dmetoal membershp fcto R xyf x y here f s fzzy mplcato fcto http//f.kast.ac.kr/lectre/cs67/textbook/ MIN fzzy mplcato Iterprets the fzzy mplcato as the mmm operato Mamda]. http//f.kast.ac.kr/lectre/cs67/textbook/ PRODUT fzzy mplcato Iterprets the fzzy mplcato as the prodct operato Larse]. EXMPLE OF FUZZY IMPLITION Fzzy rle If temperatre s hgh the hmdty s farly hgh Lets defe T erse of dscorse for temperatre H erse of dscorse for hmdty t T h H arables for temperatre ad hmdty Deote hgh as T Deote farly hgh as H The the rle becomes Rth f t s the h s or Rth Rt Rh http//f.kast.ac.kr/lectre/cs67/textbook/ http//f.kast.ac.kr/lectre/cs67/textbook/ 2
3 EXMPLE OF FUZZY IMPLITION f e ko ad e ca fd Rth t h R t h t h / t h Mamda m mplcato EXMPLE OF FUZZY IMPLITION e ko R t h for fzzy rle If temperatre s hgh the hmdty s farly hgh ccordg to ths rle hat s the hmdty he temperatre s farly hgh or t s T? t http//f.kast.ac.kr/lectre/cs67/textbook/ http//f.kast.ac.kr/lectre/cs67/textbook/ EXMPLE OF FUZZY IMPLITION We ca se composto of fzzy relatos to fd Rh! t Rt h R t h Rh Rt ο R t h OMPOSITIONL RULE OF INFERENE I order to dra coclsos from a set of rles rle base oe eeds a mechasm that ca prodce a otpt from a collecto of rles. Ths s doe sg the compostoal rle of ferece. osder a sgle fzzy rle ad ts ferece Rle f s the s Ipt s Reslt U W U ad. The fzzy rle s terpreted as a mplcato R or R Whe pt s ge to the ferece system the otpt ο R http//f.kast.ac.kr/lectre/cs67/textbook/ http//f.kast.ac.kr/lectre/cs67/textbook/ OMPOSITIONL RULE OF INFERENE ο R ο s the composto operator. The ferece procedre s called compostoal rle of ferece. The ferece mechasm s determed by to factors. Implcato operators Mamda m Larse algebrac prodct 2. omposto operators Mamda maxm Larse maxprodct OMPOSITIONL RULE OF INFERENE ompostoal rle of ferece ca be represeted graphcally as a combato of cyldrcal exteso tersecto ad projecto of fzzy sets. ld a cyldrcal exteso of xy 2. Determe tersecto of Rxy ad xy 3. ld projecto of Rxyxy http//f.kast.ac.kr/lectre/cs67/textbook/ 3
4 * OMPOSITIONL RULE OF INFERENE INFERENE METHODS There are may methods to perform fzzy ferece. osder a fzzy rle R f s ad s the s Ipts ad ca be crsp pts. rsp pts ca be treated as fzzy sgletos fzzy sets ad http//f.kast.ac.kr/lectre/cs67/textbook/ MMDNI METHOD Ths method ses the mmm operato R as a fzzy mplcato ad the maxm operator for the composto. Sppose a rle base s ge the follog form R f s ad s the s 2 for U ad W. The R ad s defed by R ad MMDNI METHOD ase Ipts are crsp ad treated as fzzy sgletos. ad ] Iferece f the Reslt Example f temperatre s hgh ad hmdty s hgh the fa speed s hgh Ho to determe the fa speed for temperatre 85ºF ad hmdty 93%? http//f.kast.ac.kr/lectre/cs67/textbook/ http//f.kast.ac.kr/lectre/cs67/textbook/ MMDNI METHOD Mamda method ses m operator as fzzy mplcato fcto here s called frg stregth matchg degree satsfacto degree http//f.kast.ac.kr/lectre/cs67/textbook/ "! # $% & '! + # *! $ $% "! # # . $! $! MMDNI METHOD For mltple rles for example to rles R ad R 2 2 http//f.kast.ac.kr/lectre/cs67/textbook/ ] ] / 2 2 4
5 2 MMDNI METHOD MMDNI METHOD I geeral ] max m 2 / 2 2 ase 2 Ipts are fzzy sets here mmax max ] & & http//f.kast.ac.kr/lectre/cs67/textbook/ http//f.kast.ac.kr/lectre/cs67/textbook/ For mltple rles MMDNI METHOD & & http//f.kast.ac.kr/lectre/cs67/textbook/ ] U & & / EXMPLE OF MMDNI METHOD Let the fzzy rle base cosst of oe rle * + *./+/2 "33 Qesto What s the otpt f the pt s a crsp ale. 4 Qesto 2 What s the otpt f the pt s a fzzy set .4 http//f.kast.ac.kr/lectre/cs67/textbook/ EXMPLE OF MMDNI METHOD Fzzy ferece th pt 3 4;6<9 3> ?9 Fzzy ferece th pt 2. LRSEN METHOD Ths method ses the prodct operato R P as a fzzy mplcato ad the maxprodct operator for the composto. Sppose a rle base s ge the follog form R f s ad s the s 2 for U ad W. The R ad s defed by R ad http//f.kast.ac.kr/lectre/cs67/textbook/ http//f.kast.ac.kr/lectre/cs67/textbook/ 5
6 6 LRSEN METHOD ase Ipts are crsp ad treated as fzzy sgletos. ` Iferece f the reslt For mltple rles ] ] here ad ] ] here ad http//f.kast.ac.kr/lectre/cs67/textbook/ U ] U ] LRSEN METHOD http//f.kast.ac.kr/lectre/cs67/textbook/ / 2 Graphcal represetato of Larse method th sgleto pt LRSEN METHOD ase 2 Ipts are fzzy sets For mltple rles max mmax here max mmax here http//f.kast.ac.kr/lectre/cs67/textbook/ U ] U ] LRSEN METHOD http//f.kast.ac.kr/lectre/cs67/textbook/ Graphcal represetato of Larse method th fzzy set pts / 2 EXMPLE OF LRSEN METHOD Let the fzzy rle base cosst of oe rle * + 5 *./+/25/2 "33 Qesto What s the otpt f the pts are crsp ales. 6. /4 Qesto 2 What s the otpt f the pts are fzzy sets .+7/4 http//f.kast.ac.kr/lectre/cs67/textbook/ EXMPLE OF LRSEN METHOD http//f.kast.ac.kr/lectre/cs67/textbook/ Larse method th pt Larse method th pt
7 DEFUZZIFITION $8 33' "" 6" 633 6" "33 SUMMRY # " " 6% " " 33"9 6 % " *33" 33 " 33" 33 " "" SUMMRY $8 6 "33 " "' 33 " 33 7
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