多條件耦合之半監督式學習於中文知識擷取之研究

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(37) Knowledge Base. Facts Promoted Instances. Candidates Corpus. Seed Instances Candidate Instances Candidate Instances. Internet. CCPL CSEAL Sub-Components. 'JHVSF 4ZTUFN "SDIJUFDUVSF PG $I/&-- $I/&-- IBT UXP TVCTZTUFN DPN QPOFOUT $PVQMFE $IJOFTF 1BUUFSO -FBSOFS $$1- BOE $PVQMFE 4FU &YQBOEFS GPS "OZ -BOHVBHF $4&"- JOUSPEVDUJPOT PG $$1- BOE $4&"- BSF JO DIBQUFS BOE 5IF "SSPXT JO ࠂHVSF SFQSFTFOU UIF ࠃPX PG EBUB. .

(38) $IBQUFS $PVQMFE $IJOFTF 1BUUFSO -FBSOFS 5IJT DIBQUFS QSPQPTFT BO BQQSPBDI XIJDI BVUPNBUJDBMMZ FYUSBDU $IJOFTF DPO DFQU LOPXMFEHF GSPN VOTUSVDUVSFE GSFF UFYU CZ UFYUVBM QBUUFSOT $IJOFTF DPODFQU LOPXMFEHF JODMVEFT WBSJPVT DBUFHPSJFT PG FOUJUJFT FYUSBDUFE GSPN VOBSZ UFYUVBM QBUUFSOT FH ֵ QBUUFSOT FH ֵ. ֶ ֵ. ֶ BOE SFMBUJPOT FOUJUJFT FYUSBDUFE GSPN CJOBSZ UFYUVBM ֶ ֵ. ֶ . -BOHVBHF QBUUFSO JT EFࠂOFE UP ࠂOE WBMJE UFYUVBM JOTUBODFT XIJDI DBO CF VTFE UP ࠂOE BEEJUJPOBM QBUUFSOT 8F TVNNBSJ[F UIF UZQFT PG DPVQMJOH UP B HFOFSBM EFࠂOJUJPO 'VSUIFSNPSF XF QSPQPTFE B BEEJUJPOBM DPVQMJOH UZQF JO BEEJUJPO UP UISFF DPVQMJOH UZQF QSPQPTFE CZ /&-- 5IF DPVQMJOH DPOTUSBJOUT DBO FࠁFDUJWFMZ EFUFDU FYUSBDUFE JOTUBODFT XIJDI BSF OPU RVJUF DPOࠂEFOU TP XF DBO JNQSPWF UIF BDDVSBDZ CZ FYDMVEJOH UIFTF JOTUBODFT. .

(39) Iteration 1. Iteration 2. Iteration 3. Iteration 4. Instances generate wildcard X. Patterns. 'JHVSF "O FYBNQMF PG CPPUTUSBQQJOH MFBSOJOH CZ UFYUVBM FYUSBDUJPO QBUUFSOT JO $IJOFTF. #PPUTUSBQQFE -FBSOJOH 5IJT BQQSPBDI BQQMJFT UFYUVBM QBUUFSOT UP FYUSBDU UIF JOTUBODFT GSPN GSFF UFYU BOE UIFO FYUSBDUT OFX QBUUFSOT CZ JOTUBODFT XIJDI BSF FYUSBDUFE NPSF UIBO DIBODF 'JOBMMZ JOUFSBDUJWFMZ DPOUJOVF UP OFYU JUFSBUJPO 5IF XPSLJOH ࠃPX JT TIPX JO 'JHVSF %FࠂOJUJPO PG UFYUVBM QBUUFSOT JT B MJUFSBM TUSJOH XJUI POF PS UXP XJMEDBSET /VNCFS PG XJMEDBSET JT CBTFE PO UIF QBUUFSO JT B DBUFHPSZ QBUUFSO PS SFMBUJPO QBUUFSO $BUFHPSZ QBUUFSOT POMZ IBWF POF XJMEDBSET CFDBVTF DBUFHPSZ QBUUFSOT BSF VTFE UP MPPL GPS DPODFQUT XIJDI BSF CFMPOH UP B DBUFHPSZ 0O UIF PUIFS IBOE SFMBUJPO QBUUFSOT IBWF UXP XJMEDBSET GPS EFTDSJCJOH B SFMBUJPO FYJTUT CFUXFFO UXP DPODFQUT *O UFYUVBM QBUUFSOT XJMEDBSET SFQSFTFOU UIF VOLOPXO UBSHFU JOTUBODFT 'PS FYBNQMF ֵX. ֶ JT B QBUUFSO XJUI MJUFSBM TUSJOH ֵ. ֶ BOE XJMEDBSE. X *G XF BQQMZ UIJT QBUUFSO UP TFOUFODF ֵ ֵ. ֶ BOE UIFO. ֶ XJMM CF FYUSBDUFE BT B DPVOUSZ OBNF 5IJT BO FYBNQMF GPS FYUSBDUJOH JOTUBODFT GSPN QBUUFSOT JO UIF PUIFS EJSFDUJPO. XF TIPX OFYU FYBNQMF GPS FYUSBDUJOH QBUUFSOT GSPN JOTUBODFT /PX XF LOPX ֵ ֶ JT BO JOTUBODF PG DPVOUSJFT BOE UIFO XF BQQMZ ֵ ֶ UP EFSJWF DBOEJEBUF QBUUFSO ֵX. ֶ . ֶ UP TFOUFODF ֵ.

(40) Iteration 1. Iteration 2. Iteration 3. Iteration 4. Instances. Patterns. 'JHVSF 4FNBOUJD ESJGU JO CPPUTUSBQQJOH MFBSOJOH 5IF TBNF TDIFNF BMTP XPSLT GPS SFMBUJPO FYUSBDUJPO 5IF SFMBUJPO QBUUFSO ֵX Y ֶ IBT UXP XJMEDBSET 8F BQQMZ UIJT QBUUFSO UP TFOUFODF ֵ ֶ BOE UIFO XF DBO PCUBJO ֵ XIJDI DBO EFTDSJCF UIF SFMBUJPO ֵ. ֶ ֵ. ֶ QBJS PG JOTUBODF. ֶ. *G QBUUFSO FYUSBDUJOH GPS DBUFHPSJFT BOE SFMBUJPOT BSF XFMM EFࠂOFE BOE USBDUBCMF UIFO UIJT CPPUTUSBQQFE MFBSOJOH GPS FYUSBDUJOH $IJOFTF LOPXMFEHF JT GFBTJCMF )PX UP FYUSBDU WBMJE JOTUBODFT BOE QBUUFSOT BSF EFQJDUFE JO 4FDUJPO 5IF QBUUFSOCBTFE BQQSPBDI JT BO FࠁFDUJWF NFUIPE GPS JOGPSNBUJPO FYUSBDUJPO *O BSUJDMFT XF OPUJDF UIBU UIF OPVO QISBTF SFQSFTFOUT DPVOUSZ OBNF JO TFOUFODF DBO BMXBZT CF SFQMBDFE CZ PUIFS DPVOUSZ OBNFT 8F DPODMVEF UIBU UIF DPVOUSZ DBUFHPSZ TIBSFT UIF TBNF QSFࠂY PS TVࠄY MJUFSBM TUSJOH PS NPSF QSFDJTFMZ UIFZ TIBSF UIF TBNF TFU PG WFSCT. 4FNBOUJD %SJGU 1SPHSFTTJPO PG CPPUTUSBQQJOH IBWF CFFO FYQFDUFE UP JNQSPWF DPSSFDUOFTT CZ QSPQBHBUJPO PG DPSSFDU MBCFMT CVU TPNFIPX XIFO FSSPS MJCFMJOH BSF QSFTFOU XJUI GFX JUFSBUJPOT QSPQBHBUJPO FYBDUMZ UIF PQQPTJUF DBTF IBQQFOFE 5IF FSSPS MBCFMJOH BDDVNVMBUFE JO SFTVMU 5IF QSPCMFN JT DBMMFE ֵTFNBOUJD ESJGUֶ<> .

(41) "O FYBNQMF GPS TFNBOUJD ESJGU TIPXT JO 'JHVSF *O MFBSOJOH QSPDFTTJOH UFYUVBM QBUUFSOT XIJDI MFBSOFE JO JUFSBUJPO BSF BNCJHVPVT 5IFTF QBUUFSOT IBWF PQQPSUVOJUJFT UP DPPDDVS XJUI OPVO QISBTFT XIJDI BSF OPU DPVOUSZ OBNFT *G CPPUSBQQJOH MFBSOJOH QSPDFFEFE BDDPSEJOH UP UIFTF OPODPVOUSZ OPVO QISBTFT UIFO ֵJT B DPVOUSZֶ QSFEJDBUF XJMM JOWPMWF VOSFMBUFE 5IF QSFEJDBUF JT ESJGUFE *O BO JEFBM DBTF UIF DPVOUSZ OBNF ֵ OBNF MJLF ֵ. ֶ ֵ. ֶ DBO CF SFQMBDFE UP BOZ PUIFS DPVOUSZ. ֶ JO TFOUFODF ֵ. ֶ BOE UIF TFOUFODF JT. TUJMM WBMJE BGUFS SFQMBDJOH 5IFSF JT BMXBZT BO FYDFQUJPO TPNF WFSCT BSF DPNNPO VTFE GPS NBOZ DPODFQUT 5IF MPDBUJPO IJFSBSDIZ JT UIF NPTU GBMMJCMF DBTF 1FPQMF VTVBMMZ DPOTJEFS MPDBUJPOT BT DPODFQUT GPS NBOZ EJࠁFSFOU UZQFT DPVOUSJFT TUBUFT DJUJFT BOE EJTUSJDUT 5IF WFSCT XIJDI SFMBUFE UP MPDBUJPOT BSF FBTJMZ DPOGVTJOH 'PS FYBNQMF *U JT TUJMM B GFBTJCMF $IJOFTF TFOUFODF XIFO UIF DPVOUSZ OBNF JO ֵ ֶ JT SFQMBDFE CZ DJUZ OBNF TVDI BT ֵ ֵ. ֶ PS ֵ. ֶ PS OBNF PG QSPWJODF PG DIJOB TVDI BT. ֶ CVU JO GBDU DPVOUSZ BOE DJUZ BSF EJࠁFSFOU DBUFHPSJFT 8IFO UIFTF. LJOE PG FSSPST PDDVSSFE BOE XFSF BDDFQUFE CZ TZTUFN PVS TZTUFN NBZ TFFL GPS NPSF QBUUFSOT CBTFE PO UIFTF XSPOH JOTUBODFT BT TFFE FYBNQMF JO OFYU JUFSBUJPO BOE UIFO FSSPS JO MBCFMJOH BDDVNVMBUFE. $PVQMFE $POTUSBJOUT 5P QSFWFOU TFNBOUJD ESJGU JO CPPUTUSBQQJOH MFBSOJOH QSPDFTTJOH BO FࠁFDUJWF TUSBUFHZ QSPQPTFE CZ /&-- JT BEEJOH DPOTUSBJOUT UP ࠂMUFS PVU JOBQQSPQSJBUF SFTVMUT 5IF /&-- QSPQPTFE UISFF UZQFT PG DPVQMJOH DPOTUSBJOUT 5IJT TFDUJPO TVNNBSJ[F UIF DPVQMJOH DPOTUSBJOUT UP B HFOFSBM GPSNVMB GVSUIFSNPSF JOUSPEVDF BO BEEJUJPOBM UZQF PG DPVQMJOH DPOTUSBJOU PSJHJOBMMZ 5IJT BEEJUJPOBM DPVQMJOH DPOTUSBJOU JT OPU POMZ DBO DPOTUSBJO UIF MFBSOJOH QSPCMFN CVU BMTP QSPQBHBUF SFTVMU CZ JOGFSFODF 5IF NBJO JEFB PG DPVQMFE DPOTUSBJOU JT UIFSF JT B MJNJUFE SFMBUJPOTIJQ CFUXFFO PVUQVUT PG EJࠁFSFOU GVODUJPOT XIJDI TIBSF B DFSUBJO EFHSFF PG JOQVU EPNBJO 5IF EFࠂOJUJPO PG UIF HFOFSBM UZQF PG DPVQMJOHT JT TIPXO CFMPX %F‫ז‬OJUJPO 'PS B TFU PG GVODUJPOT B DPOTUSBJOU JT EFࠂOFE BT B CJOBSZ SFMBUJPO .

(42) fv1(xv1) = fv2(xv2) = … = fvn(xvn) constraint. y. fv1(xv1) fv2(xv2). x. =⟨. xv1. xv1. ,. fvn(xvn). , …,. xvn. ⟩. 'JHVSF 5IF NVMUJWJFXBHSFFNFOU DPOTUSBJOU 5IF TJOHMF TUSPLF DJSDMFT BSF JOQVUT BOE PVUQVUT 5IF EPVCMF TUSPLF DJSDMF JT PVUQVU XIJDI JT DPOTUSBJOFE 5IF BSSPXT SFQSFTFOUT QSPDFTTJOH CZ GVODUJPO PS B UFSOBSZ SFMBUJPO BNPOH UIF SBOHFT PG GVODUJPOT 8F DBO FOGPSDF SBOHFT PG GVODUJPOT UP CFMPOH UP UIF SFMBUJPO GPS TBUJTGZJOH DPOTUSBJOU 5IF $POTUSBJOUT DBO CF EJWJEFE JOUP GPVS DBUFHPSJFT BDDPSEJOH UP PWFS MBQQJOH PG UIF JOQVUT PG GVODUJPOT XIJDI BSF SFTUSJDUFE 5IF JOUSPEVDUJPO PG DPVQMJOH DPOTUSBJOUT TUBSU GSPN UIF NPTU PWFSMBQQJOH JOQVUT PG SFTUSJDUFE GVODUJPO .VMUJ7JFX"HSFFNFOU $POTUSBJOU 'VODUJPOT BSF SFTUSJDUFE CZ .VMUJ 7JFX"HSFFNFOU $POTUSBJOU JG UIFJS JOQVUT DBO CF QBSUJUJPOFE JOUP NVMUJQMF WJFXT 'PS B JOQVU TFU X ∀x ∈ X x DBO CF SFQSFTFOUFE CZ NVMUJQMF WJFXT TVDI UIBU. x = %xv1 , xv2 , xv3 , ... & 'PS FBDI WJFX FYJTU B DPSSFTQPOEJOH GVODUJPO fvi ∈ F fvi : Vi → Y 5IF NVMUJWJFXBHSFFNFOU DPOTUSBJOU JT EFࠂOFE BT B CJOBSZ SFMBUJPO R PO Y . 'PS FYBNQMF JG CJOBSZ SFMBUJPO R JT TJNQMZ FRVBMJUZ UIFO FBDI WJFX PG UIF TBNF JOQVU x NVTU CF FRVBM GPS NVMUJWJFXBHSFFNFOU DPOTUSBJOU IPME 5IF FYBNQMF JT TIPX JO 'JHVSF 0VUQVU $POTUSBJOU 5IF PVUQVU $POTUSBJOU JT GPS GVODUJPOT XIJDI BDDFQU .

(43) y. f(x) g(x) constraint. z. g(x). f(x). x 'JHVSF 5IF PVUQVU DPOTUSBJOU 5IF TJOHMF TUSPLF DJSDMFT BSF JOQVUT BOE PVUQVUT 5IF EPVCMF TUSPLF MJOF JT DPOTUSBJOU DPOOFDUJPO CFUXFFO PVUQVUT 5IF BSSPXT SFQSFTFOUT QSPDFTTJOH CZ GVODUJPO UIF TBNF JOQVU x (JWFO UXP GVODUJPOT f : X → Y BOE g : X → Z UIF PVUQVU DPOTUSBJOU JT EFࠂOFE BT B CJOBSZ SFMBUJPO R CFUXFFO Y BOE Z. 'PS FYBNQMF XIJDI JT TIPX JO 'JHVSF *G f BOE g BSF CPPMFBO GVODUJPOT BOE SFMBUJPO R JT MPHJDBM JNQMJDBUJPO TVDI UIBU ∀x ∈ X f (x) =⇒ g(x) UIFO XF DPVME FOGPSDF g(x) = 1 XIFOFWFS f (x) = 1. $PNQPTJUJPOBM $POTUSBJOU 5IF DPNQPTJUJPOBM DPOTUSBJOU JT GPS UIF GVOD UJPOT XIJDI TIBSF QBSU PG JOQVUT (JWFO UXP GVODUJPOT f : X1 → Y BOE g : X1 × X2 → Z f BOE g TIBSF POF PG JOQVU x1 5IF DPNQPTJUJPOBM DPO TUSBJOU JT EFࠂOFE BT B CJOBSZ SFMBUJPO R CFUXFFO Y BOE Z. "O FYBNQMF PG DPNQPTJUJPOBM DPOTUSBJOU JT UIBU TFU SFMBUJPO R BT MPHJDBM JNQMJDBUJPO GVODUJPO f DBO CF VTFE BT UZQF DIFDL QSFDPOEJUJPO PG GVODUJPO g CZ ∀x1 ∈ X1 ∀x2 ∈ X2 g(x1 , x2 ) =⇒ f (x1 ) 8F TIPX UIJT JO 'JHVSF )FMQJOH 'VODUJPO $POTUSBJOU 5IF IFMQJOH GVODUJPO DPOTUSBJOU JT GPS UIF GVODUJPOT UIFJS JOQVUT BSF UPUBMMZ JSSFMFWBOU CVU XF IBWF FYUFSOBM LOPXMFEHF GPS UIFTF GVODUJPOT 8F DBO BEE B BEEJUJPOBM QSFEFࠂOFE IFMQJOH GVODUJPO CBTFE PO FYUFSOBM LOPXMFEHF UP DPOOFDU UIFTF JOQVUT 'PS B GVODUJPO f : XA × XB → Y BOE IFMQJOH GVODUJPO g : XA × XA → Z UIF IFMQJOH GVODUJPO DPOTUSBJOU JT EFࠂOFE BT B UFSOBSZ SFMBUJPO T BNPOH Y BOE Z. 'PS FYBNQMF TIPX JO 'JHVSF 8F BTTVNF f JT B GVODUJPO DBO EFUFSNJOF .

(44) f(x1) constraint. g(x1, x2). y. z f(x1). g(x1, x2). x1. x2. 'JHVSF 5IF DPOQPTJUJPOBM DPOTUSBJOU 5IF TJOHMF TUSPLF DJSDMFT BSF JOQVUT BOE PVUQVUT 5IF EPVCMF TUSPLF MJOF JT DPOTUSBJOU DPOOFDUJPO CFUXFFO PVUQVUT 5IF BSSPXT SFQSFTFOUT QSPDFTTJOH CZ GVODUJPO 5IF GVODUJPO g IBT UXP JOQVUT XIFUIFS QBJS PG UXP JOQVUT xa , xb BSF WBMJE PS OPU BOE XF IBWF FYUFSOBM LOPXMFEHF UP LOPX GPS UIF JOQVU QBJS EPFT OPU FYJTU UIF TBNF xa DBO QSFTFOU JO EJࠁFSFOU GVODUJPOT UIFO g DBO CF EFࠂOFE BT B GVODUJPO XIJDI EFUFSNJOF XIFUIFS QBJS PG JOQVUT BSF OPU FRVBM UIJT DBO CF EPOF CZ FYDMVTJWF EJTKVODUJPO TVDI UIBU z = g(xa1 , xa2 ) = xa1 ⊕ xa2 #Z UIFTF BTTVNQUJPOT XF LOPX FWFSZ QBJS PG WBMJE JOQVU GPS f UIFZ IBWF EJࠁFSFOU xa UFSOBSZ SFMBUJPO T = {(y1 , y2 , z) : (y1 y2 ) =⇒ z} NBLFT UIJT DPOTUSBJOU IPME 5IF UFSOBSZ SFMBUJPO T DBO DPOTJEFS BT OPU POMZ B DPOTUSBJOU JU JT B JOGFSFODF SVMF EFQFOET PO XIBU IFMQ GVODUJPO DBO QSPWJEF 8IFO T JT TBUJTࠂFE XF DBO QSPQBHBUF JOTUBODF UP HFOFSBUF OFX JOTUBODF 'PS FYBNQMF UIF IFMQ GVODUJPO JT ֵTZNNFUSJD PS OPUֶ BOE JU JT BMXBZT CF USVF GPS TPNF SFMBUJPO TVDI MJLF ֵ ֶ 8IFO XF IBWF MFBSOFE B QBJS PG OFX JOTUBODF ֵ ֶ UIFO QBJS ֵ. ֶ ֵ. ֶ ֵ. ֶ JT BMTP B WBMJE JOTUBODF PG SFMBUJPO ֵ. ֶ. $PODFQUT &YUSBDUJPO 5IF $$1- FYUSBDUT $IJOFTF LOPXMFEHF CZ QBUUFSOCBTFE BQQSPBDI 5IFSF JT OPU BMXBZT WBMJE OPVO QISBTF JO QPTJUJPO PG XJMEDBU XIFO QBUUFSO JT NBUDIFE BOE .

(45) ( f(xa1, xb1) & f(xa2, xb2) ). g(xa1, xa2). constraint. y2. z. y1. f(xa2, xb2). g(xa1, xa2). f(xa1, xb1). xb2. xa2. xa1. xb1. 'JHVSF 5IF IFMQJOH GVODUJPO DPOTUSBJOU 5IF TJOHMF TUSPLF DJSDMFT BSF JOQVUT BOE PVUQVUT 5IF EPVCMF TUSPLF MJOF JT DPOTUSBJOU DPOOFDUJPO CFUXFFO PVUQVUT 5IF BSSPXT SFQSFTFOUT QSPDFTTJOH CZ GVODUJPO "MM GVODUJPOT JO UIJT HSBQI IBWF UXP JOQVUT BMTP OPU BMM TVSSPVOEJOH MJUFSBM TUSJOHT OFYU UP NBUDIFE JOTUBODF BSF QSPQFS DBOEJ EBUFT GPS QBUUFSOT 4P RVFTUJPO JT IPX UP SFUSJFWF WBMJE JOTUBODFT BOE QBUUFSOT $$1- SFDPHOJ[FT JOTUBODF XIFO B QBUUFSO JT NBUDIFE CZ UIFSF JT BU MFBTU POF OPVO JO UIBU QPTJUJPO PG XJME DBSE BOE SFDPHOJ[FT QBUUFSO XIFO B JOTUBODF JT NBUDIFE CZ UIFSF JT B WFSC QSFWJPVT UP PS GPMMPXJOH UIF JOTUBODF 5P ࠂOE UIF OPVO BOE WFSC QBSUPGTQFFDI UBHT DBO IFMQ UP JEFOUJGZ WBMJE JO TUBODFT BOE QBUUFSOT. %JܰDVMUJFT PG $IJOFTF $PODFQU -FBSOJOH 8JUI IFMQJOH PG QBSUPGTQFFDI UBHT $IJOFTF TUJMM TVࠁFST GSPN NBOZ EJࠄDVMUJFT DPNQBSFT XJUI PUIFS MBOHVBHFT 4FHNFOUBUJPO &OHMJTI VTF TPNF GPSN PG -BUJO BMQIBCFU UIF TQBDF JT OBUV SBMMZ B XPSE EFMJNJUFS -BOHVBHFT MJLF $IJOFTF BOE +BQBOFTF EP OPU IBWF UIJT QSPQFSUZ BEEJUJPOBM TFHNFOUBUJPO GPS QSFQSPDFTTJOH PG $IJOFTF JT JOFWJUBCMF 1BSUPGTQFFDI UBHHFS *O MJOHVJTUJD UZQPMPHZ MBOHVBHFT DBO CF TJNQMZ .

(46) EJWJEFE JOUP UXP DBUFHPSJFT CBTF PO NPSQIFNFQFSXPSE SBUJP TZOUIFUJD MBOHVBHF BOE JTPMBUJOH MBOHVBHF 0CWJPVTMZ $IJOFTF CFMPOHT UP UIF TFDPOE POF UIBU NFBOT UIBU JOࠃFDUJPOT BSF OPU IFMQGVM UP JOEJDBUF HSBNNBUJDBM SFMBUJPOTIJQT JO $IJOFTF $IJOFTF QBSUPGTQFFDI UBHHFS TVࠁFST NPSF EJࠄDVMUJFT UIBO TZOUIFUJD MBOHVBHFT /FX /BNFE &OUJUZ *O UIF ࠂFME PG OBUVSBM MBOHVBHF QSPDFTTJOH /-1. VOLOPXO XPSET IBOEMJOH JT WFSZ DSVDJBM BOE EJࠄDVMU ,OPXMFEHF FYUSBDUJPO BSF VTVBMMZ EFBMJOH XJUI OFX XPSET UIF TZTUFN EPFT OPU LOPX JU ZFU TP OBNFE FOUJUZ SFDPHOJUJPO JO $IJOFTF JT B CJH DIBMMFOHF. 7BMJE *OTUBODF BOE 1BUUFSO 8F VTF QBSUPGTQFFDI UBH PG $IJOFTF UPLFO UP SFTUSJDU JOTUBODFT XIJDI BQQBS FOUMZ BSF OPVO QISBTFT BOE SFTUSJDU QBUUFSOT XIJDI DPNQMZ TPNF HSBNNBST XIJDI XF QSFEFࠂOFE CZ TUBUJTUJDBM SFTVMUT "GUFS TFHNFOUBUJPO BOE QBSUPGTQFFDI UBHHJOH UIF TFOUFODFT BSF DPOTJTUFE PG B TFRVFODF PG UPLFOT 5IF UPLFOT SFQSFTFOU B TJOHMF NFBOJOH BOE DPOUBJO GFX $IJOFTF XPSET *O PVS BTTVNQUJPO XF EP OPU UBLF TZOPOZN BOE WBSJBOU $IJOFTF DIBSBDUFS JOUP BDDPVOU JO PUIFS XPSET XF DPOTJEFS EJࠁFSFOUJBUJPO PG UPLFOT JO DIBSBDUFS MFWFM 8IFO BO JOTUBODFQBUUFSO JT NBUDIFE $$1- JEFOUJGZ WBMJE DPSSFTQPOEJOH QBU UFSOJOTUBODF CZ GPMMPXJOH SVMFT 5IF SVMFT BSF TVNNBSJ[FE UP ࠂOJUF TUBUF NBDIJOFT JO ࠂHVSFT ֻ $BUFHPSZ *OTUBODF 8IFO UIF QSPNPUFE DBUFHPSZ QBUUFSO NBUDIFE $$1. )PX NBOZ NPSQIFNFT JO B XPSE *O FYUSFNF DBTF UPLFO NBZ POMZ DPOUBJOT POF $IJOFTF XPSE 'PS FYBNQMF ֵ ֶ JT B BCCSFWJBUJPO PG ֵ ֶ CVU XF DPOTJEFS UIFN BT EJࠁFSFOU DPODFQUT 7BSJBOU $IJOFTF DIBSBDUFS JT B TQFDJBM QSPCMFN JO $IJOFTF MBOHVBHF XIJDI NFBOT $IJOFTF DIBSBDUFST UIBU BSF IPNPQIPOFT BOE TZOPOZNT JO DIBSBDUFS MFWFM 'PS FYBNQMF $IJOFTF QFPQMF VTVBMMZ DPOTJEFS ֵ ֶ BOE ֵ ֶ BSF UIF TBNF OP NBUUFS JO NFBOJOH PS JO QSPOPVODJOH *G XF DPOTJEFS UIF EJࠁFSFODF CFUXFFO 5SBEJUJPOBM $IJOFTF BOE 4JNQMJࠂFE $IJOFTF UIFO QSPCMFN CFDPNF NPSF DPNQMJDBUFE . .

(47) Instance. "JJ". "NN". "PU". "NR". "CC". "NT". "PP". "DEC" "VA" "AS". 'JHVSF 'JOJUFTUBUF NBDIJOF PG DBQUVSJOH PG DBUFHPSZ JOTUBODFT ֵ*OTUBODFֶ HSBZ CPY SFQSFTFOUT UIF OPVO QISBTF FYUSBDUFE JEFOUJGZ B OPVO QISBTF GSPN UIF MPDBUJPO PG XJMEDBSE PG NBUDIFE QBUUFSO 5IF TFOUFODF DPOTJTUFE PG B TFRVFODF PG UPLFOT UIFO UIF FYBDU QSPCMFN IBT USBOTGFSSFE UP ֵXIBU LJOE PG UPLFOT $$1- LFFQT UP SFBTTFNCMF UP B OPVO QISBTF ֶ PS ֵXIFO $$1- TUPQ UP SFUSJFWF NPSF UPLFOT ֶ 5IF OPVO QISBTF JT TFRVFODFT PG BEKFDUJWF UBHHFE 7" PS ++. . XJUI B DPN. QMFNFOUJ[FS UBHHFE %&$ BOE TFRVFODFT PG OPVO UBHHFE // PS /3 PS /5. FH ֵ. ֶ PS BO BTQFDU QBSUJDMF UBHHFE "4 GPMMPXFE CZ. TFRVFODFT PG OPVO FH ֵ. ֶ $$1- BMTP QFSNJUT DPPSEJOBUJOH DPOKVOD. UJPOT UBHHFE $$ PS QVODUVBUJPOT. . 'PS FYBNQMF ֵ ֶ BSF HPJOH UP FYUSBDU \. CFUXFFO OPVO QISBTFT ֶ BOE ֵ. ^ BOE \. ^ BMM OPVOT JO. TFOUFODFT "DDPSEJOH UP UIF BCPWF SVMFT XF TVNNBSJ[F UP B ࠂOJUF TUBUF NBDIJOF TIPX JO 'JHVSF ֻ $BUFHPSZ 1BUUFSO 8IFO UIF QSPNPUFE DBUFHPSZ JOTUBODF IBT CFFO GPVOE $$1- MPPLT GPSXBSE GPS DBOEJEBUF QBUUFSOT GSPN NBUDIJOH JOTUBODF JG UIFSF FYJTU BSCJUSBSZ BEWFSCT UBHHFE "% GPMMPXFE CZ BU MFBTU POF WFSC UBHHFE 77 PS 7$ BOE B PQUJPOBM QSFQPTJUJPO UBHHFE 11 FH ֵ . Xֶ . 5IF $IJOFTF UBHHFST VTF UIF 1FOO $IJOFTF 5SFFCBOL <> UBH TFU BT QBSUPGTQFFDI OBNF BC CSFWJBUJPOT 5IF EPDVNFOUBUJPO PG 1FOO $IJOFTF 5SFFCBOL ?iiT,ffrrrX+BbXmT2MMX2/mf+?BM2b2f TQb;mB/2Xj`/X+?XT/7 " TQFDJBM QVODUVBUJPO DBMMFE ֱ ֲ JO $IJOFTF JU JT QVODUVBUJPO VTFE UP TFU Pࠁ JUFNT JO B TFSJFT "OE OP DPSSFTQPOEJOH QVODUVBUJPO JO &OHMJTI 0SJHJOBMMZ QSFQPTJUJPO JT UBHHJOH CZ 1 JO 1FOO $IJOFTF 5SFFCBOL CVU 1 JT B QSFࠂY PG PUIFS QBSU. .

(48) Pattern. "NN" "NR" "NT". "VV" "AD". "PP" "VC". "NN" "NR". "PU". "NT". 'JHVSF 'JOJUFTUBUF NBDIJOF PG DBQUVSJOH PG DBUFHPSZ QSFࠂY QBUUFSOT ֵ1BUUFSOֶ HSBZ CPY SFQSFTFOUT UIF QBUUFSO FYUSBDUFE Pattern. "NN" "VV" "AD". "PP". "NR". "VC" "NT". 'JHVSF 'JOJUFTUBUF NBDIJOF PG DBQUVSJOH PG DBUFHPSZ TVࠄY QBUUFSOT ֵ1BUUFSOֶ HSBZ CPY SFQSFTFOUT UIF QBUUFSO FYUSBDUFE $$1- BMTP BMMPXT BO PQUJPOBM OPVO QISBTF BU UIF GPSFGSPOU FH ֵ Xֶ PS BO PQUJPOBM OPVO QISBTF GPMMPXFE CZ B QVODUVBUJPO FH ֵ Xֶ $$1- MPPLT CBDLXBSE GPS DBOEJEBUF QBUUFSOT JG UIFSF BSF BSCJUSBSZ BEWFSCT WFSCT QSFQPTJUJPO BOE BO PQUJPOBM OPVO QISBTF FH ֵX. ֶ . 5IF ࠂHVSFT PG ࠂOJUF TUBUF NBDIJOF PG HSBNNBS QBUUFSO BSF TIPXO JO 'JHVSF BOE 'JHVSF ֻ 3FMBUJPO *OTUBODF 8IFO UXPXJMEDBSE QBUUFSO JT NBUDIFE FH ֵX PGTQFFDIT 8IFO XF USZ UP SFDPHOJ[F XIFUIFS QBUUFSO FYJTU JO TFOUFODF UIJT XJMM DBVTF BNCJHVPVT DJSDVNTUBODF TP 1 IBE CFFO DPOWFSUFE UP 11 5IF TBNF TJUVBUJPO XF IBWF . GPS NFBTVSF XPSE BOE XF VTF .8 JO $$1-. .

(49) Instance. "NN" "NR" "NT". 'JHVSF 'JOJUFTUBUF NBDIJOF PG DBQUVSJOH PG SFMBUJPO JOTUBODFT ֵ*OTUBODFֶ HSBZ CPY SFQSFTFOUT UIF OPVO QISBTF FYUSBDUFE $BUFHPSZ*OTUBODF $BUFHPSZ1BUUFSO 3FMBUJPO*OTUBODF 3FMBUJPO1BUUFSO. "U -FBTU 8PSET . "U .PTU 8PSET. . 5BCMF 3FTUSJDUJPOT PO MFOHUI PG UIF WBMJE JOTUBODFQBUUFSOT FNQUZ ࠂFMET SFQ SFTFOU OP SFTUSJDUJPOT. Y ֶ $$1- MPPLT GPS OPVO QISBTFT GSPN CPUI GPSXBSE CBDLXBSE EJSFDUJPOT CZ TJNQMZ DPODBUFOBUFE OPVO UPLFOT BT MPOH BT QPTTJCMF 5IF SVMF VTFE JT UIF TBNF BT DBUFHPSZ JOTUBODF DBQUVSJOH " ࠂOJUFTUBUF NBDIJOF GPS SFMBUJPO JOTUBODFT DBQUVSJOH JT TIPXO JO 'JHVSF . ֻ 3FMBUJPO 1BUUFSO 8IFO CPUI SFMBUJPO JOTUBODFT BSF GPVOE JO B TFOUFODF JOUFSWFOJOH QISBTFT CFUXFFO UXP NBUDIFE SFMBUJPO JOTUBODFT BSF FYUSBDUFE BT SFMBUJPO DBOEJEBUF QBUUFSO 5IF POMZ SFTUSJDUJPO JT UIBU MFOHUI PG QBUUFSO SFRVJSFE UP CF XJUIJO B DFSUBJO SBOHF. "GUFS TFHNFOUBUJPO OPVO QISBTFT JO $IJOFTF VTVBMMZ DPOTJTU PG B TFRVFODF PG OPVO UPLFOT 5IF UBHT GPS TJOHMF UPLFO MJLF UIF QSPQFS OPVO UBH /3 PS UIF DPNNPO OPVO UBH // EP OPU QSPWJEF BOZ JOGPSNBUJPO UP UIF XIPMF OPVO QISBTF GPS WBMJEJUZ 5IF WBMJEJUZ PG OPVO QISBTF JT TJNQMZ SFTUSJDUFE CZ UIF MFOHUI PG OPVO QISBTF 5IF SFTUSJDUJPO JT TIPXO JO 5BCMF .

(50) $PODFQUT 4FMFDUJPO "GUFS DPODFQUT FYUSBDUJPO $$1- PCUBJOT DBOEJEBUF JOTUBODFTQBUUFSOT WJB UIF ࠂWFUVQMF GPSNBU XIJDI DPOTJTUT PG SFMBUJPO BSJUZ QSFEJDBUF OBNF JOTUBODF PS QBUUFSO ࠃBH QSPNPUFE JOTUBODFQBUUFSO BOE MFBSOFE QBUUFSOJOTUBODF. 'JMUFSJOH *O ࠂMUFSJOH TUBHF XF TIPX GPVS DPODSFUF FYBNQMFT PG UIF PVUQVU DPOTUSBJOU UIF DPNQPTJUJPOBM DPOTUSBJOU BOE UIF IFMQJOH GVODUJPO DPOTUSBJOU .VUVBM &YDMVTJPO $$1- BQQMJFT B TQFDJBM DBTF PG UIF PVUQVU DPOTUSBJOU XIJDI TIPX NVUVBM FYDMV TJPO QSPQFSUZ &BDI QBJS PG QSFEJDBUFT BSF NVUVBMMZ FYDMVTJWF CZ EFGBVMU XF DBO SFNPWF NVUVBM FYDMVTJPO QSPQFSUZ BNPOH QSFEJDBUFT CZ BEEJOH NVUFY FYDFQUJPO QSFEJDBUF 5IF NVUVBM FYDMVTJWF QSFEJDBUFT DBO OPU CF TBUJTࠂFE CZ UIF TBNF JOQVU TJNVMUBOFPVTMZ TP XF DBO ࠂMUFS PVU DPNNPO DBOEJEBUFT XIJDI DPPDDVS XJUI NPSF UIBO POF QSFEJDBUFT .PSF QSFDJTFMZ B DBOEJEBUF JOTUBODF JT BDDFQUFE JG UIF OVNCFS PG UJNFT JU DP PDDVST XJUI B QSPNPUFE QBUUFSO JT BU MFBTU τ UJNFT NPSF UIBO UIF OVNCFS PG UJNFT JU DPPDDVST XJUI QBUUFSOT GSPN NVUVBMMZ FYDMVTJWF QSFEJDBUFT (JWFO m DPODFQUT C1 , C2 , ..., Cm TVQQPTF UIBU FWFSZ DPODFQU Ci , 1 ≤ i ≤ m. IBT n DPSSFTQPOEJOH QSPNPUFE QBUUFSOT Pi1 , Pi2 , ..., Pin BOE XF ࠂOE j DPPDDVS DBOEJEBUF JOTUBODFT I1 , I2 , ...Ij CBTFE PO m × n QSPNPUFE QBUUFSOT 5IF GPSNVMB PG BDDFQUFE DBOEJEBUF JOTUBODFT PG UIF DPODFQU Ci JT. candidates(Ci ) = {Ik : 1 ≤ k ≤ j, ∀z, z ,= i, ( . n ! x=1. co−oc(Ik , Pix ))∗τ ≥. n !. co−oc(Ik , Pzy )}. y=1. 5IF BSJUZ NFBOT FYUSBDUFE JOTUBODFQBUUFSO CFMPOHT UP VOBSZ QSFEJDBUF DBUFHPSZ PS CJOBSZ QSFEJDBUF SFMBUJPO " ࠃBH JOEJDBUFT UIF UVQMF JT MFBSOJOH JOTUBODF GSPN QBUUFSOT PS MFBSOJOH QBUUFSO GSPN JOTUBODFT 5IJT ࠃBH BࠁFDUT UIF UZQFT PG UIF SFTU FMFNFOUT JO UIF UVQMF. .

(51) 5IF candidates GVODUJPO PVUQVUT WBMJE JOTUBODFT DPSSFTQPOEJOH JOQVU DPODFQU 5IF co−oc GVODUJPO SFUVSOT JOTUBODFQBUUFSO DPPDDVS UJNFT 5IF τ JT B QBSBNFUFS XIJDI DPOUSPM IPX TUSJDU NVUVBM FYDMVTJWF DPOTUSBJOU JT BOE NVUVBM FYDMVTJPO FࠁFDU XJMM CF NPSF TUSJDU XIFO τ JODSFBTJOH 5IF TBNF NVUVBM FYDMVTJWF NBOOFS BQQMZ PO QSPDFTT XIJDI MFBSO DBOEJEBUF QBUUFSO CZ QSPNPUFE JOTUBODFT UIF GPSNVMB DBO CF PCUBJOFE CZ TJNQMZ TXBQQJOH JOTUBODF BOE QBUUFSO 5ZQF $IFDL 5ZQFDIFDLJOH JT BO BQQMJDBUJPO PG DPNQPTJUJPOBM DPOTUSBJOU 5IF UZQFDIFDLJOH DPOTUSBJOU JT BMTP BQQMJFE JO ࠂMUFSJOH QIBTF $$1- ࠂMUFS PVU JOWBMJE SFMBUJPO JO TUBODFT CZ UZQFDIFDLJOH DPOTUSBJOU 8F QSPWJEF BVYJMJBSZ BUUSJCVUFT GPS SFMBUJPO QSFEJDBUFT BVYJMJBSZ BUUSJCVUFT JODMVEF UIF EPNBJO BOE UIF SBOHF ࠂ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ࠃFYJWF SFMBUJPO R JT EFࠂOFE BT ∀x ∈ X , ∃(x, x) ∈ R ֵ. ֶ JT B JSSFࠃFYJWF SFMBUJPO CFDBVTF BO BUIMFUF JT OPU B UFBNNBUF PG IJNIFSTFMG 8F BMTP DBO JOGFS SFMBUJPOT CZ CJOBSZ SFMBUJPO QSPQFSUJFT ֵ. ֶ SFMBUJPO JT. OPU POMZ B JSSFࠃFYJWF SFMBUJPO CVU BMTP B TZNNFUSJD SFMBUJPO 8IFO $$1- MFBSOFE Bֵ ֵ. ֶ SFMBUJPO JOTUBODF TVDI BT ֵ ֶ ֵ. ֶ FYJTUT JO ֵ. ֶ ֵ. ֶ UIFSF JT B JOTUBODF. ֶ SFMBUJPO GPS TVSF 5IF PUIFS QSPQFSUJFT .

(52) TVDI BT JOWFSTF BOE USBOTJUJWF BSF BMTP IFMQGVM GPS JOGFSFODF 'PS JOWFSTF QSPQFSUZ ∀x, y ∈ X (x, y) ∈ R =⇒ (y, x) ∈ R B SFMBUJPO NBZ IBT. B DPSSFTQPOEJOH JOWFSTF SFMBUJPO ֵ. ֶ BOE ֵ. ֶ BSF JOWFSTF SFMBUJPOTIJQ CFDBVTF UIFZ EFTDSJCF. UIF TBNF JEFB 5IF SFBTPO PG VTJOH UXP SFMBUJPOT UP EFTDSJCF UIF TBNF JEFB JT UIBU QFPQMF DBO FYQSFTT UIF TBNF QSFEJDBUF JO EJࠁFSFOU PSEFS 'PS FYBNQMF TFOUFODFT ֵ. ֶ BOE ֵ. ֶ BSF CPUI SFMBUFE UP UIF TBNF TFU PG. QSFEJDBUFT ֵ. ֶ. 8IFO $$1- MFBSOFE SFMBUJPO JOTUBODF ֵ. ֶ ֵ. ֶ JO B SFMBUJPO XIJDI IBT. JOWFSTF QSPQFSUZ XF HFOFSBUF DPSSFTQPOEJOH SFMBUJPO JOTUBODF ֵ. ֶ ֵ. ֶ JO. UIF JOWFSTF SFMBUJPO 5IJT JT XBZ UP QSPQBHBUF JOTUBODFT CZ UIF IFMQ GVODUJPO DPOTUSBJOU 'PS B DPODSFUF FYBNQMF PG USBOTJUJWF SFMBUJPO USBOTJUJWF QSPQFSUZ JT EFࠂOFE BT ∀x, y, z ∈ X (x, y), (y, z) ∈ R =⇒ (x, z) ∈ R 5IF SFMBUJPO ֵ. ֶ. JOEJDBUFT UIF PSEFS PG EZOBTUJFT 8IFO $$1- MFBSOFE ֵ. ֶ ֵ. ֶ BOE ֵ. ֵ. ֶ ֵ. ֶ JOTUBODF GPS. ֶ BMSFBEZ FYJTUT JO LOPXMFEHF CBTF XF DBO JOGFS ֵ. SFMBUJPO ֵ. ֶ. ֶ. .BQQJOH 5IF MBTU POF JT UIF NBQQJOH UZQF DPOTUSBJOU CZ UIF IFMQJOH GVODUJPO 5IF QVSQPTF PG UIJT DPOTUSBJOU JT UIBU CBTFE PO FYUFSOBM LOPXMFEHF UP SFTUSJDU SFMBUJPO QSPQBHBUJPO 5IF NBQQJOH UZQFT JODMVEF POFUPPOF POFUPNBOZ NBOZUPPOF BOE NBOZUPNBOZ ֻ 0OFUPPOF " SFMBUJPO XJUI POFUPPOF NBQQJOH UZQF NFBOT UIBU GPS BMM JOTUBODFT PG UIF SFMBUJPO EP OPU FYJTU UIF TBNF DBUFHPSZ JOTUBODF JO EPNBJO PS SBOHF 'PS UIF SFMBUJPO ֵ ֶ ֵ. ֶ ֵ. JOTUBODF ֵ. ֶ UIF WBMJE SFMBUJPO JOTUBODFT TVDI BT ֵ. ֶ ֵ ֶ ֵ. ֶ ֵ. ֶ *G $$1- MFBSOFE B SFMBUJPO. ֶ UIFO XF LOPX UIBU POF PG ֵ . ֶ ֵ. ֶ ֵ. ֶ BOE ֵ.

(53) ֶ ֵ. ֶ NVTU CF XSPOH CFDBVTF POMZ POF OBUJPOBM DBQJUBM GPS DPVOUSJFT. $$1- DPOTJEFS IJHIFS DPOࠂEFODF JOTUBODF BT DPSSFDU POF BOE SFKFDU UIF PUIFST ֻ 0OFUPNBOZ BOE NBOZUPPOF 5IF POFUPNBOZ NBQQJOH UZQF SFMBUJPO TVDI BT ֵ. ֶ " DPVOUSZ NBZ DPOUBJOT NBOZ DJUJFT JO PUIFS XPSET B. DJUZ EPFT OPU CFMPOH UP NVMUJQMF DPVOUSJFT $$1- BQQMJFT UIF TBNF SFKFDUJPO QSJODJQMF XIFO JU MFBSOFE SFMBUJPO JOTUBODFT UIFJS SBOHF ࠂFME BSF EFEVQMJDBU JOH "O FYBNQMF TVDI BT ֵ. ֶ ֵ. ֶ BOE ֵ. ֶ ֵ. ֶ $$1- SFKFDUT. MPXFS DPOࠂEFODF JOTUBODF 5IF EJࠁFSFODF CFUXFFO POFUPNBOZ BOE NBOZUPPOF NBQQJOH UZQF JT UIBU UIF SFTUSJDUFE ࠂFME JT PO EPNBJO ֻ .BOZUPNBOZ .BOZUP.BOZ NBQQJOH UZQF JT UIF TJNQMFTU XIJDI NFBOT OP SFTUSJDUJPO PO CPUI EPNBJO BOE SBOHF. 3BOLJOH "GUFS BQQMZJOH DPVQMFE DPOTUSBJOUT $$1- SBOLT DBOEJEBUFT CZ GSFRVFODZ PG DPPDDVST 'JSTU $$1- JHOPSFT DBOEJEBUFT XIJDI POMZ DPPDDVS XJUI POMZ POF QSPNPUFS 5IF DBOEJEBUFQSPNPUFS SFMBUJPOTIJQ JT TVDI BT JOTUBODFQBUUFSO SFMB UJPOTIJQ 5IF DBOEJEBUF JT B QBUUFSO XIFO UIF QSPNPUFS JT B JOTUBODF WJDF WFSTB 5IF DBOEJEBUFT BSF SBOLFE CZ EJࠁFSFOU TUSBUFHJFT CBTFE PO DBOEJEBUFT BSF JOTUBODFT PS QBUUFSOT 3BOLJOH *OTUBODF $BOEJEBUFT $$1- JT SBOLJOH CZ NVMUJQMF ࠂFMET TPSUJOH 5IF NVMUJQMF ࠂFMET TPSUJOH DPO TJEFST ࠂSTU ࠂFME GPS TPSUJOH XIFO OVNCFS PG ࠂSTU ࠂFMET BSF UIF TBNF DPOTJEFST TFDPOE ࠂFMET BOE TP PO . &WFSZ JOTUBODFT JO $$1- IBWF B DPOࠂEFODF WBMVF CFUXFFO UP 5IF EFUBJM PG DPOࠂEFODF WBMVF JT TIPXO JO 4FDUJPO . .

(54) 5IF ࠂSTU ࠂFME JT EJWFSTJUZ 'PS B DBOEJEBUF JOTUBODF i $$1- DBMDVMBUFT UIF EJWFSTJUZ BT GPMMPXJOH. diversity(i) = |{p : p ∈ Pi }| 5IF Pi JT TFU PG QBUUFSOT XIJDI DPPDDVST XJUI JOTUBODF i GPS UIF QSFEJDBUF VOEFS DPOTJEFSBUJPO 5IF TFDPOE ࠂFME JT UIF TVN PG OVNCFS PG QBUUFSOT XIJDI UIF JOTUBODF i DPPDDVST XJUI BT GPMMPXJOH. f requency(i) =. !. count(i, p). p∈Pi. 5IF GVODUJPO count(i, p) JT UIF OVNCFS PG UJNFT UIF QBUUFSO p DPPDDVST XJUI JOTUBODF i JO DPSQVT 8F DPOTJEFS UIBU UIF EJWFSTJUZ PG QSPNPUFST JT NPSF JNQPSUBOU UIBO UIF DP PDDVST UJNFT 5IF SFBTPO JT UIBU XIFO B JOTUBODF XJUI IJHI DPPDDVST UJNFT BOE GSPN WFSZ GFX QBUUFSOT JU NFBOT POMZ GFX QBUUFSOT BHSFF XJUI UIJT JOTUBODF CVU BHSFFNFOU JT TUSPOH 5IF JOTUBODFQBUUFSO DPPDDVS QBJS NJHIU CF B DPOWFOUJPO VTBHF PS VOࠂMUFSFE UFNQMBUF MJUFSBM TUSJOHT JO UIJT DBTF 5IF EJWFSTJUZ BMTP SFQSFTFOUT B FYBNQMF PG NVMUJWJFXBHSFFNFOU DPOTUSBJOU. 3BOLJOH 1BUUFSO $BOEJEBUFT 5IFSF BSF UISFF ࠂFMET BSF VTFE GPS SBOLJOH QBUUFSOT JO $$1- 5IF ࠂSTU BOE UIF MBTU ࠂFMET BSF UIF TBNF GPSNVMBT diversity(i) BOE f requency(i) XIJDI BSF JO SBOLJOH DBOEJEBUF JOTUBODFT 5IF NPEJࠂDBUJPO DBO CF EPOF CZ TXBQQJOH WBSJBCMF JOTUBODF i XJUI QBUUFSO p 5IF GPSNVMB BSF MJTUFE BT GPMMPXJOH. diversity(p) = |{i : i ∈ Ip }| .

(55) f requency(p) =. !. count(i, p). i∈Ip. "O BEEJUJPOBM ࠂFME precision(p) JT TFDPOE DPOTJEFSBCMF ࠂFME GPS SBOLJOH QBU UFSOT 8F TIPX UIF GPSNVMB BT GPMMPXJOH. precision(p) =. ". count(i, p) count(p). i∈I. 5IF I JT B TFU PG QSPNPUFE JOTUBODF GPS UIF QSFEJDBUF VOEFS DPOTJEFSBUJPO BOE UIF GVODUJPO count(p) JT UIF OVNCFS PG UJNFT QBUUFSO p PDDVST JO UIF DPSQVT 5IF GVODUJPO precision(p) JT QSPQPTFE CZ /&-- PSJHJOBMMZ GPS DBMDVMBUJOH IPX JNQPSUBOU UIJT QBUUFSO p SFMBUFE UP DPOTJEFSBCMF QSFEJDBUF 0VS QSJPSJUZ JT TUJMM UIF TBNF diversity > precision > f requency UIF NPTU JNQPSUBOU JT UIF EJWFSTJUZ BOE UIF JNQPSUBODF ࠂOBMMZ UIF GSFRVFODZ. 1SPNPUJPO $$1- TJNQMZ QSPNPUFT UIF UPQ n DBUFHPSZ DBOEJEBUFT BOE m QBUUFSO DBOEJ EBUFT CBTFE PO DPSQVT TJ[F 8F DBO EZOBNJDBMMZ BEKVTU UIJT UISFTIPME 5IF SFB TPOBCMF OVNCFS GPS PVS UFTUJOH EBUB BSF n = 20 BOE m = 5 $$1- POMZ QSPNPUF DBOEJEBUFT XIJDI EP OPU BMSFBEZ FYJTU JO NVUVBM FYDMVTJWF QSFEJDBUFT JO LOPXMFEHF CBTF 5IFSF JT B TDPSF GPS FWFSZ QSPNPUFE JOTUBODF BOE QBUUFSO XIJDI JOEJDBUFT MFWFM PG USVTU 8F GPMMPX UIF GPSNVMB GSPN /&-- DPOࠂEFODF = 1 − 0.5c XIFSF c JT UIF OVNCFS PG EJࠁFSFOU QSPNPUFST DPPDDVS XJUI UIJT DBOEJEBUF. $$1- "MHPSJUIN *O UIJT TFDUJPO XF TIPX $PVQMFE $IJOFTF 1BUUFSO -FBSOFS BMHPSJUIN JO "MHP SJUIN . .

(56) "MHPSJUIN $PVQMFE $IJOFTF 1BUUFSO -FBSOFS "MHPSJUIN ˆ BOE QSFQSPDFTTJOH DPSQVT C ˆ *OQVU "O POUPMPHZ O 0VUQVU &YQBOEFE POUPMPHZ GPS JUFSBUJPO i = 1, 2, 3, ..., ∞ EP GPS TFOUFODF s ∈ Cˆ EP ˆ EP GPS QSFEJDBUF p ∈ O &YUSBDU OFX DBOEJEBUF JOTUBODFTQBUUFSOT GSPN s CZ QSPNPUFE TFFET JO p FOE GPS FOE GPS 'JMUFS PVU DBOEJEBUFT UIBU WJPMBUF UIF DPVQMJOH DPOTUSBJOUT 3BOL DBOEJEBUF JOTUBODFTQBUUFSOT ˆ new 1SPNPUF UIF UPQ DBOEJEBUF JOTUBODFTQBUUFSOT BT OFX POUPMPHZ O ˆ ˆ ˆ O = O ∪ Onew FOE GPS. .

(57) .

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(59) 4&"4FU &YQBOEFS GPS "OZ -BOHVBHF 4&"- JT B UFYU NJOJOH DPNQPOFOU CBTFE PO XSBQQFS JOEVDUJPO 4&"- BDDFQUT TFFET BOE TFNJTUSVDUVSF UFYU EPDVNFOUT BT JOQVU BOE UP FYUSBDU UBSHFUT XIJDI XJUI UIF TJNJMBS TUSVDUVSF BSPVOE 4&"- JT MBOHVBHFJOEFQFOEFOU CFDBVTF JU CFOFࠂUT PO UFYUT XIJDI BSF FNCFEEFE JO TJNJ MBS TUSVDUVSF UIF TUSVDUVSF JT OPU POMZ BCPVU TFOUFODF MFWFM CVU BMTP BMM UFYU JO EPDVNFOUT FWFO NBSLVQ MBOHVBHF " XSBQQFS JT B UFNQMBUF GPS HFOFSBUJOH TFNJTUSVDUVSF EPDVNFOU 1FPQMF VTV BMMZ VTF UFNQMBUF UP IBOEMF WBSJBOU EBUB GPS HFOFSBUJOH EPDVNFOUT BOE JO UIFTF EPDVNFOUT UFNQMBUF XSBQT QSFTFOUJOH EBUB 4&"- QSPDFTTFT UIF ࠃPX SFWFSTFMZ 4&"- USJFT UP ࠂOE PVU IJEEFO XSBQQFST XIJDI BSF SFMBUFE UP TFFET 5IF XSBQQFST XIJDI IBWF CFFO GPVOE BSF EPDVNFOU TQFDJࠂD CFDBVTF UIFSF JT OP HVBSBOUFF UIBU UXP EJࠁFSFOU EPDVNFOUT BSF HFOFSBUFE CZ UIF TBNF UFNQMBUF #BTFE PO TFNJTUSVDUVSF QSPQFSUZ PG JOQVU $4&"- JT TVJUBCMF GPS QSPDFTTJOH XFC QBHFT /PXBEBZT QFPQMF TUPSF WBMVBCMF JOGPSNBUJPO JO EBUBCBTFT BOE SFOEFS CZ )5.- 4&"- DBO EFUFDU UIFTF JOGPSNBUJPO FࠄDJFOUMZ FTQFDJBMMZ GPS JOGPSNB UJPO JO B UBCMF PS JO B MJTU 'PS FYBNQMF JG XF XBOU UP ࠂOE UIF NFNCFST PG DBUFHPSZ ֵ LOPX TFWFSBM NFNCFS JOTUBODFT ֵ ֶ BOE ֵ. ֶ ֵ. ֶ BOE XF BMSFBEZ ֶ ֵ. ֶ 5IF ࠂSTU TUFQ JT UP ࠂOE UIF XFC QBHFT XIJDI DPOUBJO UIFTF. LFZXPSET 0OF PG TJNQMF XBZT GPS HFUUJOH SFMBUFE XFC QBHFT JT RVFSZJOH TFBSDI FOHJOF 5IF DPODBUFOBUJPO PG LOPXO TFFET TVDI BT ֵ ֶ JT VTFE BT TFBSDI LFZXPSET 5IF ࠂSTU MJOL 63- ?iiT, ffrrrXMimX2/mXirf+/2KB+bf+/2KB+bX?iKH PG SFTVMU PG TFBSDIJOH JT UIF NPTU SFMFWBOU XFC QBHF XF VTF JU BT FYBNQMF 5IF QBHF DPOUBJOT B MJTU PG BMM EFQBSUNFOUT JO /BUJPOBM 5BJXBO 6OJWFSTJUZ 8F TIPX UIF TFBSDI SFTVMU BOE UIF QBHF JO 'JHVSF BOE . )ZQFS 5FYU .BSLVQ -BOHVBHF JU JT UIF NBJO NBSLVQ MBOHVBHF GPS EJTQMBZJOH TUSVDUVSF DPOUFOU JO XFC QBHF. .

(60) 'JHVSF 5IF SFTVMU GSPN (PPHMF TFBSDI FOHJOF XJUI LFZXPSET ֵ ֶ. .

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