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 rF F]G( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@`` ՜.+,0    fOn-screen ShowAdastra s.r.o."O ArialTahoma WingdingsBlendsFinding Optimal Decision Trees ContentsDecision Tree InductionOptimal decision treesThe Framework AssumptionsAssumptions (2)Main theorem Proven using several lemmasProof of Lemma 13 1/3Proof of Lemma 13 2/3Proof of Lemma 13 3/3 AlgorithmAlgorithm (2)A!_" Petr MasaPetr Masalgorithm (3)Post-pruning detailPost pruning detail (2)Post pruning detail (3)Algorithm exampleAlgorithm example (2)Algorithm example (3)Algorithm propertiesAlgorithm results  Fonts Usedpp@ <4!d!d 0<4dddd 0? %O = >Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsoUnder certain assumptions (called FF) Some restriction(s) needed due to NP-completeness of the original problem&&J&JEAssumptions (2) F  Main theorem  H Proven using several lemmas$ G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ; 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AA\ йA΅   AA\ BBA΅޽k (zɻwtSuǏE]__߾}[~ЎKIN~֭[7oޜ|2Kו*X[.;U-Ka>|H74K>@7x+gBzJ'"ŋL)pl}!J:k0L|. t.TLe:v}}-<(j $.g ܧO~Z~O?>|`w\|EFYqĚd]-2ߋ "EEegweT!ьbE;}CLe-KʻjJBvwآLc*( P敳4U&, r׮D^/1w=ݳ T-β v1]o/V6%Uk(-5ͷTyng6tpY*!VR4t³]ҹKѼ?{ηEoQK"fbөtn)ǔͿaMV`#{@Ăȷ8>!gblv۪J cP4{$YE&xLz*jjtژ.b%ѹF-cQIjK&o1qy+@[4$kLI򕻦Ũ}#csd[ꔪ$U(VTT]t:,&TA˔xL\5KhdA:εn>\.?]DE L]QaȐUqmuQfEIQhؒIA[_"[,sSEJM7:L\j(xQOQIƮTwpbi*j6^D9M< ZɺAT]J/$[J X<):NicMV l_F#bp7yfň}Z $7^yiED:4CֹEE0IU&i^(m-RrN5MHy;y%iJ,OfYj0S?zQ%Vi"}ڻPNMiq{NYDUi}^Lml;E%ۭNIf f7si :~u۷*X_>繦,J!KpGC~.݁2[ά$jZc4٭?$[ϝkҸ?ai(9f5-:%nS-tS*TȴƦ nlW#6[VY[BM+Ւ J>ؓ P8c:ϕed:$tbSrH?K174ZZUԑj蓇:^qy7j+4vdӼ2R+4ޒc6_+qufȻ0.yU:#Mi-^Uy^u&GeZN-WdKqÂfA{׹7d=Ͽ-'r*r>fiہŧ:nRA?yruN1& b_Yj!Fƈ%xn >ウ1r2vvHGdBB7?~,]yÇEJw?@\ vgA2{S]t9:O?Lw|~A.>z`jbߐ,f|ٟϞ=?g%RaIzPlLR9VH!IyֻIA2y&3amU/KǛꄔbX"<6{Z[y"FFɏv}KQc{6iȓVd[ 'PjQ;Zol)(2PZPr8%,T:Ն:ur]~Ld$f[ nek t:Wܗ/_VVuqKH"߆8lˏ2eo-66m&D&?[ץqr1;0R*'']ɺA3U+EٓI8+V%LmڒǍd NS-iG訑i]iRΒ{ESETcQ6 5L9<7UXa:,.M }>NaD 2eٟ6~Mu\-6dN۸(lПocI(s5iD gMٚ'LT\g`li&>Ͼbz4=iimRIl!6Q;:VSVKEn\0L~X_V _:[1vo~wfLRbйlawDDBiQ?~)S4VJ\MQ~8 N-VM}[SW-I6)ǎg`WNW,jvdxW?CZfIKj1u=94cf-b}:*|̨0̱r^xέ-RkJ$a4mnHo:жS7ELオ$.PaZ6y}=r(MVmcD`|.b`a&Y|ҌzD~Gyڴ/;d Pzd:ڜJ2Ӟ3ng4gV"|m!jC `7TZB tnUͻK"vRqyT1EZGŷJ8M{"j$#QȔ붖Z$C %Ni*fLg*tQ̚0RBjx;_20J+ i>)~i0D*߫(9릶$ 4N+%%)S~}Ij)0҂b1fav u=-ZHYx:%.3:;$[C4OFD&LH8.fee-oyqp-r2ay۸]VJe<2*2i$n2ZaZmگx!e]MtĐM#ҶN.I̙^Sc&fKR6(զc*TM)s퉃ҹ{Yҵ1[3&duI/rearXD*YiWt϶\%ySm'q&[˻/9lh"Qś: ,s+Rm)>kŬlF^Mҟ1jAW[9ƜR-p(FmŤ}0_ jM9)XՂo\kq.P>}t%Ye,Ȕ%Mk#tC!*uQ#6U!tn/_޹sw_4hse<ߌkēE 74M9Q`%[ҍ}i2<:yѼ+muIw¯Ŏ?5}22Ud =+ۑŁw*pֻ7:7҇s?N7Cf*&L,y;X4=L&…>@)X!tѨ|QL̥v< )'Ÿߪ!F! *_8#/;۬k" Bҡy"8ZڅK A:  t.As! BBA: йAs! uO6;qnj^])i*PNG  IHDR,&{rsRGB pHYs8t@SIDATx^m   pm`FL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsoUnder certain assumptions (called FF) Some restriction(s) needed due to NP-completeness of the original problem&&J&JEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     ((  r  S 0l U   v  NA  ??P  H  0޽h ? 3333   ((  r  S H9 U   v  NA  ??x H  0޽h ? 3333f   (  r  S йt U   v  NA  ??`v  NA ??0 H  0޽h ? 3333    $(  r  S t U   r  S 0    H  0޽h ? 3333rI@7#L ( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   pm`FL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T:ppp@ <4!d!d 0<4dddd 0? %O =T>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D Assumptions7Under certain assumptions (called FF) Binary attributes&&&,& EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm Based on greedy algorithm0G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;    \T :(  ~  s * @'   ~  s *0    v  NA ?? v  NA ??  v  NA ?? 8 |  T??  H  0޽h ? !13    $(  r  S t U   r  S 0    H  0޽h ? 3333rkDLGL( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   pm`FL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T:ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsoUnder certain assumptions (called FF) Binary attributes restriction (other attribute typess are in future work)&&J&Jx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm Based on greedy algorithm0G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;    \T :(  ~  s * @'   ~  s *0    v  NA ?? v  NA ??  v  NA ?? 8 |  T??  H  0޽h ? !13rDc }L( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   pm`FL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T:ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while predi      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}ction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsnUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in future work)&&I&Ix&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm Based on greedy algorithm0G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;    \T :(  ~  s * @'   ~  s *0    v  NA ?? v  NA ??  v  NA ?? 8 |  T??  H  0޽h ? !13rDQ,k0L( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   pm`FL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T:ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsrUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work)&&M&Mx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm Based on greedy algorithm0G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;    \T :(  ~  s * @'   ~  s *0    v  NA ?? v  NA ??  v  NA ?? 8 |  T??  H  0޽h ? !13r0DCL0]PL( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   pm`FL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsrUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work)&&M&Mx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm Based on the greedy algorithm0G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S t U   r  S 0    H  0޽h ? 3333rPL9luP%nL( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   pm`FL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =">Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsrUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work)&&M&Mx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm 8Based on the greedy algorithm Post-pruning phase changedXG  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S t U   r  S 0    H  0޽h ? 3333ranLD=n0Lc( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   pm`FL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsrUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work)&&M&Mx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we do not lose track)!$ G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S t U   r  S 0    H  0޽h ? 3333rlLשHëL( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   xncFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsrUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work)&&M&Mx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ M  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S t U   r  S 0    H  0޽h ? 3333   0$(  r  S Ș U   r  S d.    H  0޽h ? 3333rL ۫Mx( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   ofFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1M  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;    \T :(  ~  s * @'   ~  s *0    v  NA ??P % v  NA ??p PU v  NA ??  |  T??P  H  0޽h ? !13   @$(  r  S ܗ U   r  S  B   B H  0޽h ? 3333rDNNhTN( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@`` (  piFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high level M  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     P$(  r  S ` U   r  S p    H  0޽h ? 3333rOtnO ( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@`` (  piFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelWhich can be neighbours 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical :ZZ uM  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     P$(  r  S ` U   r  S p    H  0޽h ? 3333rOz6f8O ( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@`` (  piFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O = >Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelWhich can be neighbours 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical NYY uM  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     P$(  r  S ` U   r  S p    H  0޽h ? 3333r8OY~8r[O ( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@`` (  piFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelWhich can be neighbours 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. NYY xM  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     P$(  r  S ` U   r  S p    H  0޽h ? 3333r[O|[~O!( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@`` (  piFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelQWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with any variable of L 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, . M  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     P$(  r  S ` U   r  S p    H  0޽h ? 3333r~O~lO!( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@`` (  piFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C M  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     P$(  r  S ` U   r  S p    H  0޽h ? 3333rOlO9"( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@`` 0 ( qlFL FG     hb$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMES 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =`>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RF?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail M  G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     `$(  r  S P) U   r  S .    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Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. 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Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     ((  r  S 荇 U   v  NA ??pH  0޽h ? 3333rR7 p9R#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T:ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20066 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary \ <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RJ?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Z&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3$   J Proof of Lemma 13 2/3$   KProof of Lemma 13 3/3$   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;  r!: 9]R#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T:ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20066 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary \ <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RJ?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Z&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3$   J Proof of Lemma 13 2/3$   KProof of Lemma 13 3/3$   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;  r] ]gR#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20066 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary \ <@Decision Tree InductiondDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on the prediction accuracy Advanced approaches combines their focus On the prediction accuracy vs. the tree complexity We will focus on the tree complexity while prediction accuracy remains unchanged `Z#Z)Z3ZRZZZ`#)3RJ?BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Z&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3$   J Proof of Lemma 13 2/3$   KProof of Lemma 13 3/3$   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;  rw$Rg#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0TLppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionsUnder certain assumptions (called FF) Binary attributes restriction (other attribute types are in the future work) Distribution&&Z&Zx&EAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     `<(  ~  s *XR U   ~  s *R    H  0޽h ? 3333rX@4R#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T:ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm BBased on the greedy algorithm Post-pruning phase changed We know  corollary of the crucial theorem  we can prune anytime it is possible (we cannot lose track)!$ N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;    lT :(  ~  s * @'   ~  s *@    v  NA ??P % v  NA ??p PU v  NA ??  |  T??P  H  0޽h ? !13rD&@R#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N AlgorithmRepeat Find two leaves with identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     $(  r  S V U   r  S V    H  0޽h ? 3333r|LXuR#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Tradition      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxy{|}~al algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N AlgorithmRepeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelfWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V Y Y, C PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     $(  r  S  U   r  S 諩    H  0޽h ? 3333rN:<R #( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N AlgorithmRepeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelmWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V ` `, J PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S   U   r  S H    H  0޽h ? 3333r<O_<aR #( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N AlgorithmRepeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1OAlgorithm high levelmWhich can be neighbours (v1,v2) 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V ` `, J PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;  r"ba6R#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)sTwo leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V&`&`, D PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;     $(  r  S  U   r  S 諩    H  0޽h ? 3333    $(  r  S   U   r  S H    H  0޽h ? 3333rjN mFYR#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)sTwo leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables to be shifted down 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)V&`&`, D PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;  ruR/#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)^&ZZ`ZZ&`, \ PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S   U   r  S H    H  0޽h ? 3333rOR/#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. PCE CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)^&ZZ`ZZ&`, \ PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;  rCzRA#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange CR with some variable of L (not all will pass) 5. Until L is empty (L is not empty = impossible to make them neighbours)^&ZZ`ZZ&`, n PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S   U   r  S H    H  0޽h ? 3333rO;=Rh#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them ) 5. Until L is empty (L is not empty = impossible to make them neighbours)^&ZZ`Z$Z&`$,  PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S   U   r  S H    H  0޽h ? 3333r>Oa={cR#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =o>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G  Algorithm Proven correctness for a distribution The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree from the growing phase v&? ;&? ;      $(  r  S   U   r  S H    H  0޽h ? 3333rcOcR#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T:ppp@ <4!d!d 0<4dddd 0? %O =S>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? I    lT  >(  ~  s *\ @0W   ~  s *    v  NA ??0 %7 |  T??0 PP H  0޽h ? !13rÉG R#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =S>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? I  rR#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =S>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? I  rR#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =S>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? I  rR#( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H     h !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefgijklmnopqrstuvwxyz{|}~ )0`Y 0 a.  @n?" dd@  @@``   soFL FG     b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =S>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)! !N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? 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N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . 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N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) SAlgorithm example UAlgorithm example (2) TAlgorithm example (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? I     `T(  r  S ٠ U   v  NA ??   `P ??0P p>Leaves v1,v2 have the same distribution of the target variable??  Z??@4& 51  Z ??  52B   fDԔ??P 0 P H  0޽h ? 3333   :2(  r  S H U   v  NA ??   v  NA ?? B   fDԔ??P 0 P   ZA??@4& 52  Z?? & 53H  0޽h ? 3333    pL(  r  S hi U   v  NA ??2  Z)??@4& 53  ZH#?? & 54B   fDԔ??P ` P H  0޽h ? 3333r nS0nܘmU%( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``    vFL FG        _b$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS2$&/C_I S2$ DBg+ P S2$4OO-$1 3: S2$Ou"_YE+P2n S 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =x>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)!  N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) SAlgorithm example UAlgorithm example (2) TAlgorithm example (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? I     `T(  r  S ٠ U   v  NA ??   `P ??0P p>Leaves v1,v2 have the same distribution of the target variable??  Z??@4& 51  Z ??  52B   fDԔ?? 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" # $ % & ' `H )0`Y 0 a.  @n?" dd@  @@``   wFL FG        ob$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS2$&/C_I S2$ DBg+ P S2$4OO-$1 3: S2$Ou"_YE+P2n S2$!&<[?GDa2 S 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)!  N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) SAlgorithm example UAlgorithm example (2) TAlgorithm example (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? I  VAlgorithm resultsTested on real datab    (  r  S ` U   r  S _    v  NA ?? H  0޽h ? 3333rV<VI&( / 0DArialngs4`H )0`Y 0"DTahomags4`H )0`Y 0" DWingdings4`H )0`Y 0 a.  @n?" dd@  @@``   wFL FG        ob$pw5wD@l6]^ 93S2$1:6T_LVK,3S2$b`<~_/h`Sb$  W5pNyn)G[S2$SKUEjDЭDIV3/bfS2$Eݧzc=S2$jn2[(Xz\PS2$WQ{t>f*zjS2$*"O$]Ą#Sb$^Qi ?>v$SiSb$tR|c @c{ S2$2PWGHWt&S2$0Vs;|P,HS2$tos5/QMESb$j^])i*Sb$}loD0Hћ]S2$ cj;oLzf"gS2$&/C_I S2$ DBg+ P S2$4OO-$1 3: S2$Ou"_YE+P2n S2$!&<[?GDa2 S 0e0e A@A5%8c8c     ?1d0u0@Ty2 NP'p<'p@A)BCD|E?@89 : 7 uʚ;2Nʚ;g4CdCd )0T ppp@ <4!d!d 0<4dddd 0? %O =>Finding Optimal Decision Trees(.Petr Maa April 4, 20064 ?Contentsf Introduction - problem definition Basic definitions and operations Crucial theorem Algorithm Summary V <@Decision Tree InductionrDecision Tree Induction is a widely used method Traditional algorithms are based on  heuristic Focused on representing a distribution Advanced approaches combines their focus Representing a distribution vs. the tree complexity We will focus on the tree complexity of trees which represent a given distribution `Z'Z)Z4ZTZZZ`')4TF2BOptimal decision treesOOptimal decision tree Represents data Is the smallest tree that represents data&::OC The FrameworkIt is showed that sufficient is Growing phase not changed Post-pruning phase with 2 operations (and suitable use of them) - Prune operation and one more operation called PCE &  D AssumptionseBinary attributes restriction (other attribute types are in the future work) Distribution restrictionpEAssumptions (2) F  Main theorem  H Proven using several lemmas I Proof of Lemma 13 1/3"   J Proof of Lemma 13 2/3"   KProof of Lemma 13 3/3"   L Algorithm .Based on the greedy algorithm Post-pruning phase changed We can prune anytime it is possible (we cannot lose track  corollary of the crucial theorem)!  N Algorithm (2)Repeat Find two leaves With identical probability distribution of the target variable Which can be neighbours after sequence of PCE Make them neighbours and prune them Until no two leaves can be mergedbm$"m$" ,c  1O Algorithm (3)Two leaves v1,v2 - can be neighbours? 1.continue/fail condition Definition of leaves v1, v2 have all variables identical and all but one their values identical 2. Find the common root of v1,v2 3. Define L= list of variables under the common root (to be shifted down) 4. Parent-child exchange the common root with some variable of L (not all will pass, we will try all of them until the situation that no one pass) 5. Final check If L is empty = v1 and v2 are neighbours; If L is not empty = impossible to make them neighboursv&ZZ`Z ZaZ&` a  >  . PPost-pruning detail QPost pruning detail (2) RPost pruning detail (3) SAlgorithm example UAlgorithm example (2) TAlgorithm example (3) G Algorithm propertiesProven correctness for distributions The resulting tree both represents data and Is the smallest tree that represents data Polynomial In the number of leaves of the tree T1 (the tree from the growing phase) v%? I%? I  VAlgorithm resultsTested on real datab    (  r  S ` U   r  S _    v  NA ??`@H  0޽h ? 3333rV3 "V