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1 # ### ) ))) g / gghilr g 8 d 8 > 1 > > 8 6q 8 p 6G6H8 gghil8 g 6e g gghijkl=m8ghhijkln eo> a g g " W V Z [ > 8 8 8GH=81IJ2?DE?I8J> I8\ J 7 8 ] 8 7 ^_ 8 8GH 8` > 8> 8 8 FAD Statistics 401 Lecture 40 Decemer Copyright Christopher Bingham 200 // :;9<;=8:< 0 > 7 8?@ABCDEFA GH> Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer d 16e2f16e2 > gghijkl2 gghijkl p K c 8 c1 gghil >8 Kc ] c ] 8 >8GH Z ` aa WW VV "" I8J 8K> 1 8GH2 8 K LDMNOD? EPDANQRDLFSOCN?EDL?RTU!

2 H2> < / ) ))) # ### 1] 8 2 8GH=81IJ2 : G<: H8:<\16G6H2 K DL BD8 6G6H 1 72 I7J K > e >8 8 # ### H ` 8 ]>c ce > ] 8 ` / ggg ) ))) ) ))) ))) # ###) # ### ))) # ###) H2 _ p 6G86H 87 >8 81 G G Statistics 401 Lecture 40 Decemer ` 8 2 8GH 81IJ2=`: G<: H8:< M@?E AË7 7 8GH=81IJ2 : G<: H8:< Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer <2 k1 : <2 > ]8 : 8GH=81IJ2=81 G G c 8c 8:<= : G G8 81IJ2= G H= ki1 ìg // 8 c c G8 HK>8:<=81 : <2 8 >8 8 8GH=81IJ2=81 ìh2 d 8 8q9GH;hijkl d 8 8 p8 8:<=1 e // 8 87 ` e]

3 J 8e 8GHhil H2 G8 Ï8 > 3IJ4 ] K q9GH;hijkl=18Gqhilo8Hqhil2\ K8 W # ### V a a k : W G H21 G < " = : H2\1\6Go\6H2 8:<= G ""GH 16G6H\16Go6H21 "" GH > " [ 8 > > 8 ] K > K 8 8 > > p 8GH=81IJ2=16G6H\16Go6H2f81 H2 =81 G= 8GH= G Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer Cmd> d <- matrixdmat80)\ laels:structuregetlaelsx1)getlaelsx1))) Cmd> xs <- standardizex) Cmd> fori17){forji18){ # loop over 1 <= i < j <= 8 d[ij] <- d[ji] <- \ sqrtsumvectorxs[i]-xs[j]))^2));;}} Cmd> x # row lael is population numer x1 x Cmd> printdformat:"63f") d: ! > > 8 8 p8 > WW > 8 ] : : aa 8GH=81IJ2=81 G 9GH; >> H 8 p WW VV aa H2 QFABLFN? ` > 06_J0 K> 8 > 8 K ] c c8 K > e > c c >

4 8 8 Statistics 401 Lecture 40 Decemer Cmd> neworder <- vector ) Cmd> printd[neworder neworder]format:"63f") MATRI: 8 K [ c c> 8 Cmd> temp <- xs xs Cmd> d <- sqrtdiagtemp) diagtemp) - 2temp) Cmd> printdformat:"63f") dd: dz 8 8 p8 Statistics 401 Lecture 40 Decemer Cmd> clusterx) # cluster using Average Linkage default) Case Numer of Clusters Criterion Dendrogram inverted tree) Cluster No Clusters 1 to 8 Top 7 levels of hierarchy) Clustering method: Average linkage Distance: Euclidian standardized) 7 K 13 Statistics 401 Lecture 40 Decemer Criterion 8 >> 8 0 N 0 ] Statistics 401 Lecture 40 Decemer Criterion 8AFE 87 > 0 OCN?EDL1 2 K 8 O?D 1 2 ] ]8 ] cluster) 2K> 6 _ ] >> 81 ] 8 > C@O? Criterion 1 16

5 =34 8 =34 =34 = > 8 I 88 Z 81 =34> 8 =34 =34 =34 ) ))) = =34 >8 > 8k8k 8= =34 =34 =34 =348 8 =34 =34 p cluster) 8 d FACT K> > > K > K 8 > > e e 8> > > Statistics 401 Lecture 40 Decemer cluster) Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer Clus ID Stage 0 {1} {4} {2} {} {7} {3} {6} {8} Stage 1 {1} {4} {2} {8} {7} {3} {6} Stage 2 {1} {4} {2} {8} {7} {36} Stage 3 {1} {4} {236} {8} {7} Stage 4 {1} {47} {236} {8} Stage {18} {47} {236} Stage 6 {12368} {47} Stage 7 { } > ] 7 K 8 >2 = =34 =34 =34d 2 =34 = =1o2\= =1o2\= 0`8 81 =1o2\= 2= =18 o82\ 2= =18 o82\ 2= =18k o8k2\ =34 =34 =34 =34 = Z =34 =34 ] e 1

6 Ghilo6H Hhil2\16Go6H2 8q9GH;hijkl= 3 Gq8Gqhil\6Ho Hq8Hqhil\6G8GHhil\16G6H24 > Gq=\6Go\6qK 6q8 8 8 Ghilo6H 9GH;hijkl=16G ) ))) W 8GHhil K c ec 8 8 e 8GHhil 8 Statistics 401 Lecture 40 Decemer e8q9gh;hijkl= 38qGhil8qHhil4 WW 8q9GH;hijkl=`38qGhil8qHhil4 8q9GH;hijkl=16G8qGhilo6H8qHhil2\16Go6H2 K p 6G86Ḧ 8 p 6q 0] e e Statistics 401 Lecture 40 Decemer Statistics 401 Lecture 40 Decemer Hhil2\ 8q9GH;hijkl=18Gqhilo8Hqhil2\8GHhil\ K> >> 8 8 p Ghilo 8 q9gh;hijkl=18gqhilo8hqhil2\ Statistics 401 Lecture 40 Decemer cluster) x 23 "ward" "complete" 88 6 "median" 8 8 clusterdissim:d) clustersimil:s) "centroid" 24 8 c ] c ] e 8 clusterxnclust:mmethod:"average") "single" "average"18 2 6o618 > 2 8 > 6 > 162 > 8 ` "mcquitty" 9GH;hijkl=1 9GH;hijkl=16G 8q9GH;hijkl=6H8qGhilo6G8qHhil6G6H8GHhil 6 Go6H 8q9GH;hijkl Hhil2\16Go6H2

7 8 Statistics 401 Lecture 40 Decemer clusterxnclust:1method:"average"\ reorder:t) > K e > > LDFLMDLDM Cmd> clusterxreorder:ttree:f) Case Numer of Clusters No Cases are not in original order st 3 from group 2 next from group 1 last 3 from group tree:f K > Z K Z341 2 Z34 2 K > c c ] > 8 8 p standard:f 8 > Statistics 401 Lecture 40 Decemer class:f Cmd> clusterxstandard:fclass:f) Criterion Cluster No Clusters 1 to 8 Top 7 levels of hierarchy) Clustering method: Average linkage Distance: Euclidean 88 1 > 8 class:f28 8AFA?E AM LM@ DM standard:f2k 7 d ]] 8 > Statistics 401 Lecture 40 Decemer Cmd> stuff <- clusterxkeep:"all") #prints nothing y default Cmd> compnamesstuff) 1) "distances" 8 y 8 symmetric matrix 2) "classes" class tale 3) "criterion" distance apart of merged clusters Cmd> printformat:"63f" stuff) STRUCTURE: component: distances standardized distances 11) ) ) ) ) ) ) ) component: classes Same as class tale 11) ) ) ) ) ) ) ) component: criterion in reverse order of computation 1) "criterion") "distances" keep "classes" 8 >] ] "criterion" vector"classes" vector"classes" "distances") vector"criterion" "distances") Statistics 401 Lecture 40 Decemer keep tree:t print:t classes:t 8 2K 87 nclust p 7 I 1 > ] e> 8 K 88 8 euclidxy) 8 8 p standardizex) Cmd> euclid <- macro"sqrtsumvector1)-2))^2))") Cmd> xs <- standardizex) # standardize data 8 ë 8 Z Cmd> euclidxs[1]xs[2])#row 1 - row 2 standardized distance 1) > ] > 8 ] x