Forml Conpt Anlysis Conpt intnts s losd sts Closur Systms nd Implitions 4 Closur Systms 0.06.005 Nxt-Closur ws dvlopd y B. Gntr (984). Lt M = {,..., n}. A M is ltilly smllr thn B M, if B A if th smllst lmnt whr A nd B diffr longs to B : A < B : i B\A: A {,,..., i-} = B {,,..., i-} Itnusd 45 to dtrmin th onpt ltti or to dtrmin th onpt ltti togthr with th stm sis or for intrtiv knowldg quisition. It dtrmins th onpt intnts in ltil ordr. 45 5 4 45 5 4 4 5 45 4 5 4 45 5 45 45 45 5 4 5 5 4 4 0.06.005 0.06.005 4
W nd th following: Algorithm Nxt-Closur for dtrmining ll onpt intnts: A < i B : i B \A A {,,..., i-} = B {,,..., i-} ) Th ltilly smllst onpt intnt is. A i := ( A {,,..., i-} ) {i} Thorm: Th smllst onpt intnt, whih ording to th ltil ordr is lrgr s givn st A M, is A i := (A i), ) Is A onpt intnt, thn w find th ltilly nxt intnt, y hking ll ttriuts i M \ A, strting with th lrgst, und thn in drsing ordr, until A < i (A i ) holds. Thn A i is th ltilly nxt onpt intnt. ) If A i = M, thn stop, ls A A i nd goto ). whr i is th lrgst lmnt of M with A < i A i. 0.06.005 5 0.06.005 6 Exmpl: on lkord Sinus 44 Noki 60 T-Fx 0 T-Fx 60 PC Hndy () Tlfon () Fx () Fx w. n. ppr (4) A i A i A i :=(A i ) A < i A i? nw onpt intnt Irg Conpt Lttis minsupp = 85% For minsupp = 85% th svn most gnrl of th.086 onpts of th Mushrooms dts http:\\kdd.is.ui.du r shown. 0.06.005 7 0.06.005 8
Irg Conpt Lttis minsupp = 85% With drsing minimum support th informtion gts rihr. minsupp = 70% minsupp = 55% 0.06.005 9 0.06.005 0 Th support of st M of ttriuts is givn y supp() G Df.: Th irg onpt ltti of forml ontxt (G,M,I) for givn miniml support minsupp is th st { (A,B) B(G,M,I) supp(b) minsupp } Th visuliztion s nstd lin digrm indits implitions. It n omputd with. [Stumm t l 00] 0.06.005 0.06.005
omputs th losur systm of ll (frqunt) onpt intnts using th support funtion: mks us of som simpl fts out th support funtion: Df.: Th support of n ttriut st (itmst) M is givn y supp() G Only onpts with support ov thrshold minsupp [0,]. 0.06.005 0.06.005 4 tris to optimiz th following thr qustions:. How n th losur of n itmst dtrmind sd on supports only?. How n th losur systm omputd with dtrmining s fw losurs s possil?. How n s mny supports s possil drivd from lrdy known supports? 0.06.005 5 0.06.005 6
. How n th losur of n itmst dtrmind sd on supports only? = { x M \ supp() = supp( { x }) }. How n th losur systm omputd with dtrmining s fw losurs s possil? W dtrmin only th losurs of th miniml gnrtors. Exmpl: {, } = {,, }, sin supp( {, } ) = / nd supp( {,, } ) = 0/ supp( {,, } ) = /, 0.06.005 7 0.06.005 8. How n th losur systm omputd with dtrmining s fw losurs s possil? W dtrmin only th losurs of th miniml gnrtors. A st is miniml gnrtor iff its support is diffrnt of th supports of ll its lowr ovrs.. How n th losur of n itmst dtrmind sd on supports only? = x M \ supp() = supp( x ). How n th losur systm omputd with dtrmining s fw losurs s possil? Approh à l Apriori Th miniml gnrtors r n ordr idl (i.., if st is not miniml gnrtor, thn non of its suprsts is ithr.) Apriori lik pproh In th xmpl, nds two runs (nd Apriori four).. How n s mny supports s possil drivd from lrdy known supports? 0.06.005 9 0.06.005 0
. How n s mny supports s possil drivd from lrdy known supports? Thorm: If is no miniml gnrtor, thn supp() = min { supp(k) K is miniml gnrtor, K }. Exmpl: supp( {,, } ) = min { 0/, /, /, /, / } = 0, sin th st is no miniml gnrtor, nd sin supp( {, } ) = 0/, supp( {, } ) = / supp( { } ) = /, supp( { } ) = / supp( { } ) = / Rmrk: It is suffiint to hk th lrgst gnrtors K with K, i.. hr {, } nd {, }.. How n th losur of n itmst dtrmind sd on supports only? = { x M \ supp() = supp( x ) }. How n th losur systm omputd with dtrmining s fw losurs s possil? Approh à l Apriori. How n s mny supports s possil drivd from lrdy known supports? If is no miniml gnrtor, thn supp() = min { supp(k) K is miniml gnrtor, K }. 0.06.005 0.06.005 i i singltons Dtrmin support for ll C Dtrmin losurs for ll C i - Prun non-miniml gnrtors from i i A l Apriori For pot. min. gnrtors: ount in dts. Els supp() = min { supp(k) K, K m.g.}. = { x M \ supp() = supp( {x}) } iff supp() supp( \ {x}) f.. x omprd with Apriori i i singltons Dtrmin support for ll C i - Dtrmin losurs for ll C i - Prun non-miniml gnrtors from i If th support is too low or qul to th support of lowr ovr, th ndidt is prund. i i + i Gnrt_Cndidts( i - ) A l Apriori i i + i Gnrt_Cndidts( i - ) no i mpty? End ys Apriori lik pproh no i mpty? End ys W only gnrt ndidts for miniml gnrtors. 0.06.005 0.06.005 4
0.06.005 5 0.06.005 6 Exmpl of 0.06.005 7 0.06.005 8
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vs. Nxt-Closur Nxt-Closur nds lmost no mmory. Nxt-Closur n xploit known symmtris twn ttriuts. Nxt-Closur n usd for knowldg quisition. hs fr ttr prformn, spilly on lrg dt sts. 0.06.005