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p Tree- k - m eans- c lassification- s equential (pkmc-s)

P A j >c =P j,m o m ...o k+1 P j,k o i =AND iff b i =1, k is the rightmost bit pos with bit-value "0", opeations are right binding. c = b m ... b k+1 ... b 0. 1.  attr, class, calc means, mean_gaps. sLN m mg se 51 8 vi 63 7 ve 70. sWD m mg

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p Tree- k - m eans- c lassification- s equential (pkmc-s)

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  1. PAj>c=Pj,m om...ok+1Pj,koi=AND iff bi=1, k is the rightmost bit pos with bit-value "0", opeations are right binding. c = bm ... bk+1 ... b0 1.attr, class, calc means, mean_gaps. sLNmmg se 51 8 vi 63 7 ve 70 sWDmmg ve 32 1 vi 33 2 se 35 pLNmmg se 14 33 ve 47 13 vi 60 pWDmmg se 2 12 ve 14 11 vi 25 se 51 35 14 2 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 ve 70 32 47 14 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 vi 63 33 60 25 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 1 0 0 1 pTree-k-means-classification-sequential (pkmc-s) Initially, let PREMAINING be pure1. Initially from the TrainingSet, 1. For each attribute, calculate the mean for each class and sort asc on mean. Calculate all mean_gaps = difference_of_consec_means. Create MeanTable(attribute, class, mean, gapL, gapH, gapRELATIVE) sorted desc on gapRELATIVE = ( gapL + gapH)/mean ) gapL is the gap on the low side of the mean. gapH, high side. 2. Choose and remove the MT record with max gapRELATIVE. Use formula above with cL=mean-gapL/2 and with cH=mean+gapH/2 to produce PL=PA>cL and PH=P'A>cH The class mask is PCLASS = PL & PH & PREMAINING and we update PREMAINING = PREMAINING & P'CLASS 3. Repeat 2 above until all classes have a pTree mask (or until PREMAINING is empty but that's a count op.). 4. Repeat 1,2,3 until means stop changing (much). On the next two slides you will find a (partial) walk through of this algorithm for a subset of the IRIS dataset. The initial means are shown below (the clusters are color coded throughout with R,G,B for setosa,. versicolor, virginica. I also color code the features (sepal Length, sepal Width, pedal Length, pedal Width Then I take 10 samples from each class for the example. sLN sWD pLN pWD sepalLeNgth sepalWiDth pedalLeNgth pedalWiDth

  2. pkmc-s PREM=pure1 1.attr, class, calc means, gaps. MT(attr,class,mean,gapL,gapH,gapREL) sorted desc on gapREL =(gapL+gapH)/2*mean) gapL=lo gap. gapH hi. 2. MT rec w max gapREL cL=mn-gapL/2 cH=mn+gapH/2 PCLASS = PA>cL & P'A>cH & PREM PREM= PREM &P'CLASS 3. Repeat 2 til all classes pTree. 4. Repeat 1,2,3 til conv Sepal LengthSepal WidthPedal LengthPedal Wth 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 se 49 30 14 2 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 0 se 47 32 13 2 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 se 46 31 15 2 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 0 se 54 36 14 2 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 se 54 39 17 4 0 1 1 0 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 se 46 34 14 3 0 1 0 1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 se 50 34 15 2 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 se 44 29 14 2 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 se 49 31 15 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 se 54 37 15 2 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 ve 64 32 45 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 ve 69 31 49 15 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 1 ve 55 23 40 13 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 ve 65 28 46 15 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 ve 57 28 45 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 ve 63 33 47 16 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 ve 49 24 33 10 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 ve 66 29 46 13 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 ve 52 27 39 14 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 ve 50 20 35 10 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1.attr, class, calc means, mean_gaps. sLNmmg se 51 8 vi 63 7 ve 70 sWDmmg ve 32 1 vi 33 2 se 35 pLNmmg se 14 33 ve 47 13 vi 60 pWDmmg se 2 12 ve 14 11 vi 25 vi 58 27 51 19 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 vi 71 30 59 21 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 vi 63 29 56 18 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 vi 65 30 58 22 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 vi 76 30 66 21 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 vi 49 25 45 17 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 vi 73 29 63 18 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 vi 67 25 58 18 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 vi 72 36 61 25 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 vi 65 32 51 20 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 MTatclmngLgHgR (not yet sorted on gR) se sSL 51 88.16( 8+ 8)/(2*51) se sWD 35 2 2.06( 2+ 2)/(2*35) x's fill ins. se pSL 14 33332.36(33+33)/(2*14) se pWD 2 1212 6(12+12)/(2* 2) PREM PA>cH vi sSL 63 8 7 .12 ( 8+ 7)/(2*63) vi sWD 33 1 2 .05 ( 1+ 2)/(2*33) vi pSL 60 1313 2.2 (13+13)/(2*60) vi pWD 25 1111 .44 (11+11)/(2*25) ve sSL 70 7 7 .1 ( 7+ 7)/(2*70) ve sWD 32 1 1 .03 ( 1+ 1)/(2*32) Pse = P'A>cH ve pSL 47 33 13 .94 (33+13)/(2*47) ve pWD 14 12 11 .82 (12+11)/(2*14) =(P4,4|(P4,3&(P4,2|(P4,1|P4,0)))) PA>cH = MTsatclmngLgHgR (sortws desc gR) se pWD 2 1212 6 se pSL 14 33332.36 vi pSL 60 1313 2.2 ve pSL 47 33 13 .94 ve pWD 14 12 11 .82 vi pWD 25 1111 .44 se sSL 51 88.16 vi sSL 63 8 7 .12 ve sSL 70 7 7 .1 se sWD 35 2 2.06 vi sWD 33 1 2 .05 ve sWD 32 1 1 .03 MTatclmngLgHgR We're separating out setosa class 2. MT rec w max gapREL cL= mean - gapL/2 cH=mean+gapH/2 se pWD 2 1212 6 PA>cL =Ppure1 = 2 - 12/2 = -4 = 2 +12/2 = 8 = 0 1 0 0 0 Psetosa =PA>cL & P'A>cH & PREM =Ppure1& P'A>cH & Ppure1 = P'A>cH PREM= PREM &P'CLASS = Ppure1 &P'setosa = P'setosa

  3. pkmc-s PREM=pure1 1.attr, class, calc means, gaps. MT(attr,class,mean,gapL,gapH,gapREL) sorted desc on gapREL =(gapL+gapH)/2*mean) gapL=lo gap. gapH hi. 2. Get MT w max gapREL cL=mn-gapL/2 cH=mn+gapH/2 PCLASS = PA>cL & P'A>cH & PREM PREM= PREM &P'CLASS 3. Repeat 2 til all classes pTree. 4. Repeat 1,2,3 til conv Sepal LengthSepal WidthPedal LengthPedal Wth se 49 30 14 2 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 0 se 47 32 13 2 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 se 46 31 15 2 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 0 se 54 36 14 2 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 se 54 39 17 4 0 1 1 0 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 se 46 34 14 3 0 1 0 1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 se 50 34 15 2 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 se 44 29 14 2 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 se 49 31 15 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 se 54 37 15 2 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 ve 64 32 45 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 ve 69 31 49 15 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 1 ve 55 23 40 13 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 ve 65 28 46 15 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 ve 57 28 45 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 ve 63 33 47 16 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 ve 49 24 33 10 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 ve 66 29 46 13 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 ve 52 27 39 14 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 ve 50 20 35 10 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1.attr, class, calc means, mean_gaps. sLNmmg se 51 8 vi 63 7 ve 70 sWDmmg ve 32 1 vi 33 2 se 35 pLNmmg se 14 33 ve 47 13 vi 60 pWDmmg se 2 12 ve 14 11 vi 25 vi 58 27 51 19 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 vi 71 30 59 21 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 vi 63 29 56 18 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 vi 65 30 58 22 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 vi 76 30 66 21 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 vi 49 25 45 17 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 vi 73 29 63 18 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 vi 67 25 58 18 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 vi 72 36 61 25 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 vi 65 32 51 20 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 MTatclmngLgHgR (not yet sorted on gR) se sLN 51 88.16( 8+ 8)/(2*51) se sWD 35 2 2.06( 2+ 2)/(2*35) x's fills. se pLN 14 33332.36(33+33)/(2*14) se pWD 2 1212 6(12+12)/(2* 2) vi sLN 63 8 7 .12 ( 8+ 7)/(2*63) vi sWD 33 1 2 .05 ( 1+ 2)/(2*33) vi pLN 60 1313 2.2 (13+13)/(2*60) vi pWD 25 1111 .44 (11+11)/(2*25) ve sLN 70 7 7 .1 ( 7+ 7)/(2*70) ve sWD 32 1 1 .03 ( 1+ 1)/(2*32) ve pLN 47 33 13 .94 (33+13)/(2*47) ve pWD 14 12 11 .82 (12+11)/(2*14) MTatclmngLgHgR (sort desc gR) se pWD 2 1212 6 se pLN 14 33332.36 vi pLN 60 1313 2.2 ve pLN 47 33 13 .94 ve pWD 14 12 11 .82 vi pWD 25 1111 .44 se sLN 51 88.16 vi sLN 63 8 7 .12 ve sLN 70 7 7 .1 se sWD 35 2 2.06 vi sWD 33 1 2 .05 ve sWD 32 1 1 .03 MTatclmngLgHgR Separating out virgininca class 2. MT rec w max gapREL cL= mean - gapL/2 etc. vi pLN 60 1313 2.2 =60-13/2=53.5 53= 0 1 1 0 1 0 1 PA>cL =P3,6|(P3,5&(P3,4&(P3,3|(P3,2&(P3,1

  4. PAj>c=Pj,m om...ok+1Pj,koi=AND iff bi=1, k is the rightmost bit pos with bit-value "0", operations are right binding. c = bm ... bk+1 ... b0 1.attr, class, calc means, mean_gaps. sLNmmg se 51 8 vi 63 7 ve 70 sWDmmg ve 32 1 vi 33 2 se 35 pLNmmg se 14 33 ve 47 13 vi 60 pWDmmg se 2 12 ve 14 11 vi 25 se 51 35 14 2 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 ve 70 32 47 14 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 vi 63 33 60 25 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 1 0 0 1 pTree-k-means-classification-divisive (pkmc-d) Current Cluster=CC={Class1, ...Classm} (all classes), is represented by pTree mask, PCC ( pure1 initially). From the TrainingSet, 1. For each attribute, calculate the mean for each class in CC and sort asc on mean. Calculate all mean_gaps = difference_of_consecutive_means. Create MeanTable (attribute, class, mean, gap) sorted desc on gap 2. Choose and remove the MT record with maximum gap Use PA>c (c=mean+gap/2) to separate the current cluster into two clusters. The cluster masks are PNEWCLUSTER1 = PA>c & PCC PNEWCLUSTER2= P'A>c & PCC and the new clusters then are NEWCLUSTER1= {all classes corresponding to the mean that had the max gap and those above it from CC. NEWCLUSTER2= {all other classes in CC}, also definable as {all classes below max gap class in CC) 3. Repeat 2 with CC=NEWCLUSTERi (i=1,2) until all clusters are singleton sets of classes. 4. Repeat 1,2,3 until means stop changing (much). On the next two slides you will find a (partial) walk through of this algorithm for a subset of the IRIS dataset. The initial means are shown below (the clusters are color coded throughout with R,G,B for setosa,. versicolor, virginica. I also color code the features (sepal Length, sepal Width, pedal Length, pedal Width Then I take 10 samples from each class for the example. sLN sWD pLN pWD sepalLeNgth sepalWiDth pedalLeNgth pedalWiDth

  5. Sepal LengthSepal WidthPedal LengthPedal Wth 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 se 49 30 14 2 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 0 se 47 32 13 2 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 se 46 31 15 2 0 1 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 0 se 54 36 14 2 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 1 0 se 54 39 17 4 0 1 1 0 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 se 46 34 14 3 0 1 0 1 1 1 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 se 50 34 15 2 0 1 1 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 se 44 29 14 2 0 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 se 49 31 15 1 0 1 1 0 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 se 54 37 15 2 0 1 1 0 1 1 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 ve 64 32 45 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 ve 69 31 49 15 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 1 ve 55 23 40 13 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 ve 65 28 46 15 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 ve 57 28 45 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 ve 63 33 47 16 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 ve 49 24 33 10 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 ve 66 29 46 13 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 ve 52 27 39 14 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 ve 50 20 35 10 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1.attr, class, calc means, gaps. sLNmgap se 51 8 vi 63 7 ve 70 sWDmgap ve 32 1 vi 33 2 se 35 pLNmgap se 14 33 ve 47 13 vi 60 pWDmgap se 2 12 ve 14 11 vi 25 vi 58 27 51 19 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 vi 71 30 59 21 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 vi 63 29 56 18 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 vi 65 30 58 22 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 vi 76 30 66 21 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 vi 49 25 45 17 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 vi 73 29 63 18 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 vi 67 25 58 18 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 vi 72 36 61 25 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 vi 65 32 51 20 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 MTatclmngap (not yet sorted and I'm not using relative gaps this time, Note also, there is only 1 entry for each gap - not 2 for each mean) se sLN 51 8 se pLN 14 33 se pWD 2 12 vi sLN 63 7 vi sWD 33 2 PNEW1 PA>31 ve sWD 32 1 ve pLN 47 13 ve pWD 14 11 MTatclmngap (sorted desc on gap) PNEW2 P'A>31 se pLN 14 33 ve pLN 47 13 se pWD 2 12 ve pWD 14 11 P3,6 | P3,5 PA>31= se sLN 51 8 vi sLN 63 7 vi sWD 33 2 ve sWD 32 1 pkmc-d CC=all [3] classes, mask, PCC ( pure). 2.PA>c (c=mean+gap/2*mean) separate CC into 2. PNEWCLUSTER1=PA>c & PCC PNEWCLUSTER2=P'A>c & PCC 3. Repeat 2 w CC=NEWCLUSTERi (i=1,2) until all are singletons.4. Repeat 1,2,3 until means stop changing. 1.MT(attr,class,mean,gap) sorted desc on gap . MTatclmngap (separates {ve. vi} from {setosa} using pedal Length 2. MT rec w max gap c= mean + gap/2 se pLN 14 33 = 14 + 33/2 = 31 = (applied roof) 0 0 1 1 1 1 1 PNEW2 is done ( cluster is the singleton set {setosa} ) Need to further partition PNEW1 (cluster is {versicolor, virginica} )

  6. Sepal LengthSepal WidthPedal LengthPedal Wth 1.attr, class, calc means, gaps. sLNmgap se 51 8 vi 63 7 ve 70 sWDmgap ve 32 1 vi 33 2 se 35 pLNmgap se 14 33 ve 47 13 vi 60 pWDmgap se 2 12 ve 14 11 vi 25 MTatclmngap (sorted desc on gap) se pLN 14 33 ve pLN 47 13 se pWD 2 12 ve pWD 14 11 P3,6 | P3,5 =P3,6|(P3,5&(P3,4&(P3,3|(P3,2&(P3,1 &(P3,0 PA>c = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 se sLN 51 8 vi sLN 63 7 vi sWD 33 2 ve sWD 32 1 or pkmc-d CC=all [3] classes, mask, PCC ( pure). 2.PA>c (c=mean+gap/2*mean) separate CC into 2. PNEWCLUSTER1=PA>c & PCC PNEWCLUSTER2=P'A>c & PCC 3. Repeat 2 w CC=NEWCLUSTERi (i=1,2) until all are singletons.4. Repeat 1,2,3 until means stop changing. ve 64 32 45 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 ve 69 31 49 15 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 1 ve 55 23 40 13 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 ve 65 28 46 15 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 ve 57 28 45 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 ve 63 33 47 16 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 ve 49 24 33 10 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 ve 66 29 46 13 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 ve 52 27 39 14 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 ve 50 20 35 10 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1.MT(attr,class,mean,gap) sorted desc on gap . vi 58 27 51 19 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 vi 71 30 59 21 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 vi 63 29 56 18 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 vi 65 30 58 22 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 vi 76 30 66 21 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 vi 49 25 45 17 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 vi 73 29 63 18 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 vi 67 25 58 18 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 vi 72 36 61 25 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 vi 65 32 51 20 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 MTatclmngap (separates {ve} from {vi} using pedal Length 2. MT rec w max gap c= mean + gap/2 vepLN4713 = 47 + 13/2 = 54 = (applied roof) 0 1 1 0 1 1 0 This is the virginica mask. There are no mistakes on versicolor, 3 mistakes on virginica (#'s 1,6,10). With one epoch, overall accuracy is 90% 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 Ways to improve accuracy (at a slight cost in speed) include: 1. Use more than one attrubute cutpoint each time. 2. Use standard deviation calculations to optimize cutpoints. or

  7. Sepal LengthSepal WidthPedal LengthPedal Wth 1.attr, class, calc means, gaps. sLNmgap se 51 8 vi 63 7 ve 70 sWDmgap ve 32 1 vi 33 2 se 35 pLNmgap se 14 33 ve 47 13 vi 60 pWDmgap se 2 12 ve 14 11 vi 25 MTatclmngap (sorted desc on gap) se pLN 14 33 ve pLN 47 13 se pWD 2 12 ve pWD 14 11 P3,6 | P3,5 =P3,6|(P3,5&(P4,4&(P4,3|(P4,2&(P4,1 &(P3,0 PA>c = se sLN 51 8 vi sLN 63 7 vi sWD 33 2 ve sWD 32 1 pkmc-d CC=all [3] classes, mask, PCC ( pure). 2.PA>c (c=mean+gap/2*mean) separate CC into 2. PNEWCLUSTER1=PA>c & PCC PNEWCLUSTER2=P'A>c & PCC 3. Repeat 2 w CC=NEWCLUSTERi (i=1,2) until all are singletons.4. Repeat 1,2,3 until means stop changing. ve 64 32 45 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 ve 69 31 49 15 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 1 ve 55 23 40 13 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 ve 65 28 46 15 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 ve 57 28 45 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 ve 63 33 47 16 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 ve 49 24 33 10 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 ve 66 29 46 13 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 ve 52 27 39 14 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 ve 50 20 35 10 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1.MT(attr,class,mean,gap) sorted desc on gap . vi 58 27 51 19 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 vi 71 30 59 21 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 vi 63 29 56 18 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 vi 65 30 58 22 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 vi 76 30 66 21 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 vi 49 25 45 17 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 vi 73 29 63 18 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 vi 67 25 58 18 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 vi 72 36 61 25 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 vi 65 32 51 20 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 MTatclmngap(separates {ve} from {vi} using pedal Length 2. MT rec w max gap c= mean + gap/2 vepWD1411 = 14 + 11/2 = 20 = (applied roof) 0 1 1 0 1 0 0 To improve accuracy (at a slight cost in speed) include: 1. Use more than one attribute cutpoint. Using pWD as 2nd attribute for separating ve and vi: 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 0 No improvement by including ve pWD. Next we'll try vi sLN.

  8. Sepal LengthSepal WidthPedal LengthPedal Wth 1.attr, class, calc means, gaps. sLNmgap se 51 8 vi 63 7 ve 70 sWDmgap ve 32 1 vi 33 2 se 35 pLNmgap se 14 33 ve 47 13 vi 60 pWDmgap se 2 12 ve 14 11 vi 25 MTatclmngap (sorted desc on gap) se pLN 14 33 ve pLN 47 13 se pWD 2 12 ve pWD 14 11 P3,6 | P3,5 =P1,6&(P1,5|(P1,4|(P1,3|(P1,2 PA>c = 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 se sLN 51 8 vi sLN 63 7 vi sWD 33 2 ve sWD 32 1 pkmc-d CC=all [3] classes, mask, PCC ( pure). 2.PA>c (c=mean+gap/2*mean) separate CC into 2. PNEWCLUSTER1=PA>c & PCC PNEWCLUSTER2=P'A>c & PCC 3. Repeat 2 w CC=NEWCLUSTERi (i=1,2) until all are singletons.4. Repeat 1,2,3 until means stop changing. ve 64 32 45 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 ve 69 31 49 15 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 1 ve 55 23 40 13 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 ve 65 28 46 15 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 ve 57 28 45 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 ve 63 33 47 16 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 ve 49 24 33 10 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 ve 66 29 46 13 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 ve 52 27 39 14 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 ve 50 20 35 10 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1.MT(attr,class,mean,gap) sorted desc on gap . vi 58 27 51 19 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 vi 71 30 59 21 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 vi 63 29 56 18 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 vi 65 30 58 22 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 vi 76 30 66 21 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 vi 49 25 45 17 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 vi 73 29 63 18 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 vi 67 25 58 18 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 vi 72 36 61 25 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 vi 65 32 51 20 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 MTatclmngap(separates {ve} from {vi} using pedal Length 2. MT rec w max gap c= mean + gap/2 visLN63 7 = 63 + 7/2 = 67 = (applied roof) 1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 1 No improvement by including vi sLN.

  9. Sepal LengthSepal WidthPedal LengthPedal Wth 1.attr, class, calc means, gaps. sLNmgap se 51 8 vi 63 7 ve 70 sWDmgap ve 32 1 vi 33 2 se 35 pLNmgap se 14 33 ve 47 13 vi 60 pWDmgap se 2 12 ve 14 11 vi 25 MTatclmngap (sorted desc on gap) se pLN 14 33 -------or ve pLN 47 13 se pWD 2 12 ve pWD 14 11 P3,6 | P3,5 PA>c = se sLN 51 8 vi sLN 63 7 vi sWD 33 2 ve sWD 32 1 pkmc-d CC=all [3] classes, mask, PCC ( pure). 2.PA>c (c=mean+gap/2*mean) separate CC into 2. PNEWCLUSTER1=PA>c & PCC PNEWCLUSTER2=P'A>c & PCC 3. Repeat 2 w CC=NEWCLUSTERi (i=1,2) until all are singletons.4. Repeat 1,2,3 until means stop changing. ve 64 32 45 15 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 ve 69 31 49 15 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 1 1 1 ve 55 23 40 13 0 1 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 ve 65 28 46 15 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 1 ve 57 28 45 13 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 ve 63 33 47 16 0 1 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 ve 49 24 33 10 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 ve 66 29 46 13 1 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 1 ve 52 27 39 14 0 1 1 0 1 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 ve 50 20 35 10 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1.MT(attr,class,mean,gap) sorted desc on gap . vi 58 27 51 19 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 vi 71 30 59 21 1 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 1 vi 63 29 56 18 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 vi 65 30 58 22 1 0 0 0 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 vi 76 30 66 21 1 0 0 1 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 vi 49 25 45 17 0 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 vi 73 29 63 18 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 0 vi 67 25 58 18 1 0 0 0 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 vi 72 36 61 25 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 1 vi 65 32 51 20 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 MTatclmngap(separates {ve} from {vi} using pedal Length 2. MT rec w max gap c= mean + gap/2 visWD33 2 = 33 + 2/2 = 34 = 1 0 0 0 1 0 0 =P3,6|(P2,5&(P2,4|(P2,3|(P2,2|(P2,1 &(P2,0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 0 0 Improvement by including visWD (captures 10 as virginica while missing also on 10 in versicolor.

  10. The above methods are all pkmc methods involving the distance, Lp distance in one dimension (the most relevant dimension based on mean gaps.). I say Lp because all of these distance are identical in one dimension (abs value of difference). To improve accuracy we could try using std based gap measurements and maximize the number of gap stds (using Mohammad's formula for variance), rather than gap distance and/or we could maximize the relative gap = gap/mean measure or #gap_stds/mean. We could use the L distance on all relevant dimensions. We could use the Lp distance on all relevant dimensions (L1 and L2 using Mohammad's formulas).

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