June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner...
-
Upload
pranav-tarleton -
Category
Documents
-
view
213 -
download
0
Transcript of June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner...
![Page 1: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/1.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 1
Estimating the Cluster Treeof a Density
Werner StuetzleRebecca NugentDepartment of StatisticsUniversity of Washington
![Page 2: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/2.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 2
![Page 3: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/3.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 3
![Page 4: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/4.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 4
Groups K-means with k = 2
![Page 5: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/5.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 5
![Page 6: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/6.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 6
• Detect that there are 5 or 6 distinct groups.
• Assign group labels to observations.
40 45 50 55
74
76
78
80
82
84
![Page 7: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/7.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 7
![Page 8: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/8.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 8
![Page 9: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/9.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 9
-2 0 2 4 6 8 100
10
02
00
30
0
Feature histogram
![Page 10: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/10.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 10
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
(a) (b) (c)
(d) (e) (f)
![Page 11: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/11.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 11
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
(a) (b) (c)
(d) (e) (f)
![Page 12: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/12.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 12
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 13: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/13.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 13
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 14: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/14.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 14
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 15: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/15.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 15
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 16: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/16.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 16
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 17: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/17.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 17
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 18: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/18.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 18
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 19: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/19.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 19
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 20: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/20.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 20
0 1 2 3 4
-10
12
34
rs = 1
rs = 5
rs = 2
![Page 21: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/21.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 21
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 22: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/22.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 22
Note large tree fragments in bottom left panel
-2 0 2 4 6
-3-2
-10
12
3
Weakly bimodal data
-2 0 2 4 6
-3-2
-10
12
3
MST for weakly bimodal data
-2 0 2 4 6
-3-2
-10
12
3
MST after removal of longest edges
0 20 40 60 80
01
02
03
04
0
runts .bimodal2[runts.bimodal2 > 1]
Histogram of runt sizes
![Page 23: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/23.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 23
-2 -1 0 1 2 3
-3-2
-10
12
3
Unimodal data
-2 -1 0 1 2 3
-3-2
-10
12
3
MST for unimodal data
-2 -1 0 1 2 3
-3-2
-10
12
3
MST after removal of longest edges
0 20 40 60 80
01
02
03
04
0
runts .unimodal[runts.unimodal > 1]
Histogram of runt sizes
![Page 24: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/24.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 24
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 25: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/25.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 25
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 26: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/26.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 26
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 27: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/27.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 27
leve
l
10
12
14
16
A 8
A 7 A 9
A 8
A 2
A 3A 3
A 5A 6
![Page 28: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/28.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 28
![Page 29: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/29.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 29
cc1
cc2
-4 -2 0 2 4 6
-4-2
02
4
11
1
111
1
1
11
11
1
1
1
1
1
1
1
1
1
1
1
1
1
22
22
22
2
2
2
2
2
2
2
2
2222
2 2
2 22
2
2
2
2
222
2
2
2
2222
2
2
22
2
2
2
2 22 22
22
2
2
2
2
2
33
3
3
33
3
3
3
3
33
3
3
3
3
3
33
33
3
3
3
3
3
3
3
33
33 3
3
33
3
3
3
3
33
333
33
3
3
3
3
3
3 33
3
33
3
33
3 3
3
3
3
3333
3 33
3
3
3
3
33
3
33
3
33
33
3
3
3
33
3 33
3
3
333
33
33
3
33
3
3
3
3
3
3
3 33
3
33
3
3
3
3
33
3
3
333
3
3
3
3 33
3
3
3
33
33
33 3
33
3
33
3
3
3
3
33
33
33
33
3
3
3
3
33
3
33
3
3
33
33
3
33
3
3
3
33
33
3 3 3
3
333
3
3
3
3
3
3 3
3
3
3
3
4
44
4
4
44
4
4
4
4
44
44
4
4
4
4
4
4
4
44
4
4
44
4
44
4
44
4 4
Areas 1-4, projected on crimcords
![Page 30: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/30.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 30
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 31: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/31.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 31
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 32: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/32.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 32
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 33: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/33.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 33
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 34: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/34.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 34
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 35: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/35.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 35
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 36: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/36.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 36
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 37: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/37.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 37
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 38: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/38.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 38
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 39: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/39.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 39
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 40: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/40.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 40
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 41: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/41.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 41
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 42: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/42.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 42
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 43: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/43.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 43
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 44: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/44.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 44
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 45: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/45.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 45
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 46: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/46.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 46
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 47: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/47.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 47
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 48: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/48.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 48
0 1 2 3 4
-10
12
34
rs = 1
rs = 5
rs = 2
![Page 49: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/49.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 49
Runt Pruning algorithm:
generate_cluster_tree_node (mst, runt_size_threshold) { node = new_cluster_tree_node node.leftson = node.rightson = NULL node.obs = leaves (mst) cut_edge = longest_edge_with_large_runt_size (mst, runt_size_threshold) if (cut_edge) { node.leftson = generate_cluster_tree_node (left_subtree (cut_edge), runt_size_threshold) node.rightson = generate_cluster_tree_node (right_subtree (cut_edge), runt_size_threshold) } return (node)}
0 1 2 3 4
-10
12
34
rs = 1
rs = 5
rs = 2
![Page 50: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/50.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 50
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 51: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/51.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 51
Note large tree fragments in bottom left panel
-2 0 2 4 6
-3-2
-10
12
3
Weakly bimodal data
-2 0 2 4 6
-3-2
-10
12
3
MST for weakly bimodal data
-2 0 2 4 6
-3-2
-10
12
3
MST after removal of longest edges
0 20 40 60 80
01
02
03
04
0
runts .bimodal2[runts.bimodal2 > 1]
Histogram of runt sizes
![Page 52: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/52.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 52
-2 -1 0 1 2 3
-3-2
-10
12
3
Unimodal data
-2 -1 0 1 2 3
-3-2
-10
12
3
MST for unimodal data
-2 -1 0 1 2 3
-3-2
-10
12
3
MST after removal of longest edges
0 20 40 60 80
01
02
03
04
0
runts .unimodal[runts.unimodal > 1]
Histogram of runt sizes
![Page 53: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/53.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 53
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 54: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/54.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 54
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 55: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/55.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 55
leve
l
10
12
14
16
A 8
A 7 A 9
A 8
A 2
A 3A 3
A 5A 6
![Page 56: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/56.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 56
![Page 57: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/57.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 57
cc1
cc2
-4 -2 0 2 4 6
-4-2
02
4
11
1
111
1
1
11
11
1
1
1
1
1
1
1
1
1
1
1
1
1
22
22
22
2
2
2
2
2
2
2
2
2222
2 2
2 22
2
2
2
2
222
2
2
2
2222
2
2
22
2
2
2
2 22 22
22
2
2
2
2
2
33
3
3
33
3
3
3
3
33
3
3
3
3
3
33
33
3
3
3
3
3
3
3
33
33 3
3
33
3
3
3
3
33
333
33
3
3
3
3
3
3 33
3
33
3
33
3 3
3
3
3
3333
3 33
3
3
3
3
33
3
33
3
33
33
3
3
3
33
3 33
3
3
333
33
33
3
33
3
3
3
3
3
3
3 33
3
33
3
3
3
3
33
3
3
333
3
3
3
3 33
3
3
3
33
33
33 3
33
3
33
3
3
3
3
33
33
33
33
3
3
3
3
33
3
33
3
3
33
33
3
33
3
3
3
33
33
3 3 3
3
333
3
3
3
3
3
3 3
3
3
3
3
4
44
4
4
44
4
4
4
4
44
44
4
4
4
4
4
4
4
44
4
4
44
4
44
4
44
4 4
Areas 1-4, projected on crimcords
![Page 58: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/58.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 58
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 59: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/59.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 59
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 60: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/60.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 60
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 61: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/61.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 61
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 62: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/62.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 62
![Page 63: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/63.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 63
ClusteringasastatisticalproblemAssumethatfeaturevectorsx1;:::;xniid»p(x).DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.ThereforethetotalcollectionofsetscanbearrangedintoamodetreeLeavesofmodetreecorrespondtonodesofp(x).
Considerfeaturevectorsx1;:::;xnasarandomsamplefromsome(innite)population.Letp(x)bethepopulationdensity.(Think"histogramwithinnitesimallysmallbins")DenelevelsetL(;p)byL(;p)=fxjp(x)gLetL1(;p);L2(;p);:::betheconnectedcomponentsofL(;p).If2>1thenforanyi;jeither²Li(2;p)½Lj(1;p),or²theyaredisjoint.Thereforethetotalcollectionofsetscanbearrangedintoatree,theclustertreeofthedensity.Leavesofclustertreecorrespondtomodesofp(x).
![Page 64: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/64.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 64
![Page 65: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/65.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 65
![Page 66: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/66.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 66
![Page 67: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/67.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 67
![Page 68: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/68.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 68
-10 -5 0 5 10
-50
51
0
PC1
PC
2
![Page 69: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/69.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 69
-10 -5 0 5 10
-50
51
0
PC1
PC
2
-10 -5 0 5 10
-50
51
0
PC1
PC
2
![Page 70: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/70.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 70
![Page 71: June 11, 2005Interface / CSNA Meeting St. Louis1 Estimating the Cluster Tree of a Density Werner Stuetzle Rebecca Nugent Department of Statistics University.](https://reader035.fdocuments.net/reader035/viewer/2022070307/551a9c585503466b3a8b54b0/html5/thumbnails/71.jpg)
June 11, 2005 Interface / CSNA Meeting St. Louis 71
Runt Pruning algorithm:
generate_cluster_tree_node (mst, runt_size_threshold) { node = new_cluster_tree_node node.leftson = node.rightson = NULL node.obs = leaves (mst) cut_edge = longest_edge_with_large_runt_size (mst, runt_size_threshold) if (cut_edge) { node.leftson = generate_cluster_tree_node (left_subtree (cut_edge), runt_size_threshold) node.rightson = generate_cluster_tree_node (right_subtree (cut_edge), runt_size_threshold) } return (node)}
0 1 2 3 4
-10
12
34
rs = 1
rs = 5
rs = 2