Stairstep-like dendrogram cut: a permutation test approach Dario - PowerPoint PPT Presentation

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R C k L and C k R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation “ ” C 1 L ∪ C 1 h R Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R C k L and C k R C 1 C 1 L R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C 2 L ∪ C 2 h “ ” C k L ∪ C k h the height necessary to merge R R C k L and C k R C 2 C 2 L R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” “ ” C k L ∪ C k C 3 L ∪ C 3 h the height necessary to merge h R R C k L and C k R C 3 C 3 L R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R C k L and C k R “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge “ ” C 1 R h L C k L and C k R “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) C 1 L D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” “ ” C 1 C k L ∪ C k h h the height necessary to merge R R C k L and C k R “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) C 1 R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R C k L and C k “ ” C 2 h R L “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) C 2 L D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R “ ” C 2 C k L and C k h R R “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) C 2 R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R C k L and C k “ ” C 3 R h L “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) C 3 L D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? not so ?) simple idea - notation Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R “ ” C 3 h C k L and C k R R “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) C 3 R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 6 / 22

The (? simple ?) idea Input : A dataset and its related dendrogram Output : A partition of the dataset D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 7 / 22

The (? simple ?) idea Input : A dataset and its related dendrogram Output : A partition of the dataset initialization: aggregationLevelsToVisit ← h ( C 1 L ∪ C 1 R ) permClusters ← [ ] i ← 1 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 7 / 22

The (? simple ?) idea Input : A dataset and its related dendrogram Output : A partition of the dataset initialization: aggregationLevelsToVisit ← h ( C 1 L ∪ C 1 R ) permClusters ← [ ] i ← 1 repeat if C i L ≡ C i R then add C i L ∪ C i R to permClusters else add h ( C i L ) and h ( C i R ) to aggregationLevelsToVisit sort aggregationLevelsToVisit in descending order end D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 7 / 22

The (? simple ?) idea Input : A dataset and its related dendrogram Output : A partition of the dataset initialization: aggregationLevelsToVisit ← h ( C 1 L ∪ C 1 R ) permClusters ← [ ] i ← 1 repeat if C i L ≡ C i R then add C i L ∪ C i R to permClusters else add h ( C i L ) and h ( C i R ) to aggregationLevelsToVisit sort aggregationLevelsToVisit in descending order end remove the first element from aggregationLevelsToVisit i ← i+1 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 7 / 22

The (? simple ?) idea Input : A dataset and its related dendrogram Output : A partition of the dataset initialization: aggregationLevelsToVisit ← h ( C 1 L ∪ C 1 R ) permClusters ← [ ] i ← 1 repeat if C i L ≡ C i R then add C i L ∪ C i R to permClusters else add h ( C i L ) and h ( C i R ) to aggregationLevelsToVisit sort aggregationLevelsToVisit in descending order end remove the first element from aggregationLevelsToVisit i ← i+1 until aggregationLevelsToVisit is empty D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 7 / 22

The (? not so ?) simple idea in action Iteration i ← 1 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action “ ” C 1 L ∪ C 1 h R Iteration i ← 1 aggregationLevelsToVisit h ( C 1 L ∪ C 1 R ) permClusters C 1 C 1 L R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 1 aggregationLevelsToVisit h ( C 1 L ∪ C 1 R ) permClusters C 1 C 1 L R clusters to compare H 0 : C 1 L ≡ C 1 R �→ reject D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 2 aggregationLevelsToVisit h ( C 1 R ) , h ( C 1 L ) permClusters C 1 C 1 L R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 2 “ ” C 1 h aggregationLevelsToVisit R h ( C 1 R ) , h ( C 1 L ) permClusters C 1 R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 2 aggregationLevelsToVisit h ( C 1 R ) , h ( C 1 L ) permClusters C 2 C 2 L R clusters to compare H 0 : C 2 L ≡ C 2 R �→ reject D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 3 aggregationLevelsToVisit h ( C 1 L ) , h ( C 2 R ) , h ( C 2 L ) permClusters C 2 C 2 L R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 3 aggregationLevelsToVisit “ ” C 1 h L h ( C 1 L ) , h ( C 2 R ) , h ( C 2 L ) permClusters C 1 L D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 3 aggregationLevelsToVisit h ( C 1 L ) , h ( C 2 R ) , h ( C 2 L ) permClusters C 3 C 3 L R clusters to compare H 0 : C 3 L ≡ C 3 R �→ reject D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 4 aggregationLevelsToVisit h ( C 3 R ) , h ( C 2 R ) , h ( C 2 L ) , h ( C 3 L ) permClusters C 3 C 3 L R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 4 aggregationLevelsToVisit “ ” C 3 h R h ( C 3 R ) , h ( C 2 R ) , h ( C 2 L ) , h ( C 3 L ) permClusters C 3 R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 4 aggregationLevelsToVisit h ( C 3 R ) , h ( C 2 R ) , h ( C 2 L ) , h ( C 3 L ) permClusters C 4 C 4 C 4 L ∪ C 4 L R R clusters to compare H 0 : C 4 L ≡ C 4 R �→ accept D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 4 aggregationLevelsToVisit h ( C 3 R ) , h ( C 2 R ) , h ( C 2 L ) , h ( C 3 L ) permClusters C 3 R C 4 L ∪ C 4 R ⇔ C 3 R clusters to compare H 0 : C 4 L ≡ C 4 R �→ accept D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

The (? not so ?) simple idea in action Iteration i ← 9 aggregationLevelsToVisit aggregationLevelsToVisit h ( C 3 R ) , h ( C 2 R ) , h ( C 2 L ) , h ( C 3 L ) permClusters C 3 L , C 3 R , C 2 L , C 4 L , C 4 R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 8 / 22

La Carte A (? simple ?) idea 1 A (? not so ?) simple procedure 2 Some results 3 The Wishlist 4 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 9 / 22

The (? not so ?) simple procedure Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R C k L and C k R “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 10 / 22

The (? not so ?) simple procedure Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R max h ( C 3 j ) C k L and C k R “ ” C k the height at which C k h j has been obtained j min h ( C 3 j ) (j ∈ { L, R }) “ ” “ ” C k C k For each k , the difference between j ∈{ L , R } h max and j ∈{ L , R } h min can be considered j j as the minimum cost necessary to merge the two classes. . D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 10 / 22

The (? not so ?) simple procedure Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge h ( C 3 L ∪ C 3 R R ) C k L and C k R max h ( C 3 “ ” j ) C k the height at which C k h j has been obtained j (j ∈ { L, R }) “ ” “ ” C k C k For each k , the difference between j ∈{ L , R } h max and j ∈{ L , R } h min can be considered j j as the minimum cost necessary to merge the two classes. “ ” “ ” C k L ∪ C k C k The difference between h and j ∈{ L , R } h max can be, instead, considered as R j the cost actually incurred for merging C k L and C k R . D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 10 / 22

The (? not so ?) simple procedure Let: n the number of objects to classify; C k L and C k R the two classes merged at level k (k=1,...,n-1) “ ” C k L ∪ C k h the height necessary to merge R C k L and C k R “ ” C k the height at which C k h j has been obtained j (j ∈ { L, R }) The ratio between these two costs: “ ” “ ” C k C k j ∈{ L , R } h max − j ∈{ L , R } h min j j “ ” C k L ∪ C k C k ` ´ h − j ∈{ L , R } h max j R is thus a measure that characterizes the aggregation process resulting in the new class C k L ∪ C k R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 10 / 22

The (? not so ?) simple procedure: detail The algorithm retraces down-ward the tree, starting from the root of the dendrogram where all objects are classified in a unique cluster. D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

The (? not so ?) simple procedure: detail The algorithm retraces down-ward the tree, starting from the root of the dendrogram where all objects are classified in a unique cluster. ∀ k a permutation test is designed to test the Null Hypothesis that the two classes C k L and C k R really belong to the same cluster, i.e. : C k L ≡ C k H 0 : R C 1 L C 1 R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

The (? not so ?) simple procedure: detail The algorithm retraces down-ward the tree, starting from the root of the dendrogram where all objects are classified in a unique cluster. ∀ k a permutation test is designed to test the Null Hypothesis that the two classes C k L and C k R really belong to the same cluster, i.e. : C k L ≡ C k H 0 : R Under H 0 , mixing up ( permuting ) the statistical units of C k L and C k R should not alter the aggregation pro- cess resulting in their merging in. C 1 L C 1 R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

The (? not so ?) simple procedure: detail The algorithm retraces down-ward the tree, starting from the root of the dendrogram where all objects are classified in a unique cluster. ∀ k a permutation test is designed to test the Null Hypothesis that the two classes C k L and C k R really belong to the same cluster, i.e. : C k L ≡ C k H 0 : R Under H 0 , mixing up ( permuting ) the statistical units of C k L and C k R should not alter the aggregation pro- cess resulting in their merging in. C 1 L C 1 R m C 1 m C 1 L R Let m C k L and m C k R be the two new classes obtained by permuting the elements in C k L and C k R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

The (? not so ?) simple procedure: detail m C 1 L m C 1 R C 1 L C 1 R m C 1 m C 1 L R Let m C k L and m C k R be the two new classes obtained by permuting the elements in C k L and C k R For each of them a new dendrogram is generated. D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

The (? not so ?) simple procedure: detail h ( m C 1 L ) m C 1 L h ( m C 1 R ) m C 1 R C 1 L C 1 R m C 1 m C 1 L R Let m C k L and m C k R be the two new classes obtained by permuting the elements in C k L and C k R For each of them a new dendrogram is generated. The heights at which each of the two classes are buit up again, clearly correspond to the heights of the root nodes of the corresponding dendrograms. D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

The (? not so ?) simple procedure: detail h ( m C 1 L ) m C 1 L h ( m C 1 R ) m C 1 R C 1 L C 1 R m C 1 m C 1 L R The ratio: “ ” “ ” m C k m C k j ∈{ L , R } h max − j ∈{ L , R } h min j j “ ” m C k L ∪ m C k cost = R “ ” C k L ∪ C k m C k ` ´ h − j ∈{ L , R } h max R j is thus a measure that characterizes the aggregation process resulting in the new ( potential ) class m C k L ∪ m C k R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

The (? not so ?) simple procedure: detail m C 1 L m C 1 R C 1 L C 1 R m C 1 m C 1 L R Under H 0 the aggregation process resulting in the new cluster C k L ∪ C k R should be very similar “ ” to the one that potentially produces m C k L ∪ m C k m C k L ∪ m C k R ; thus the two values cost and R “ ” C k L ∪ C k cost should be close enough. R D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

The (? not so ?) simple procedure: detail m C 1 L m C 1 R C 1 L C 1 R m C 1 m C 1 L R The permutation procedure is repeated M times and each time a new couple m C k L , m C k R is ob- tained. The pvalue Montecarlo is thus computed as: ˘ ` m C k L ∪ m C k ´ ` C k L ∪ C k ´¯ p = # cost ≤ cost + 1 R R M + 1 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 11 / 22

La Carte A (? simple ?) idea 1 A (? not so ?) simple procedure 2 Some results 3 The Wishlist 4 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 12 / 22

Some results The yeast galactose dataset Ideker T, Thorsson V, Ranish JA, Christmas R, Buhler J, Eng JK, Bumgarner RE, Goodlett DR, Aebersold R, Hood L Integrated genomic and proteomic analyses of a systemically perturbed metabolic network. Science 2001, 292:929-934. n = 205 p = 80 It is a subset of 205 genes that reflect four functional categories in the Gene Ontology listings. D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 13 / 22

Some results Settings distanceMethod = euclidean aggregationMethod = Ward α = 0 . 05 M = 999 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 13 / 22

Some results D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 13 / 22

Some results The diabetes dataset Banfield JD, Raftery AE Model–based Gaussian and Non–Gaussian Clustering. Biometrics, 1993, 49, 803-821. n = 145 p = 3 It contains 145 subjects divided into three groups (normal, chemical diabetes, overt diabetes) on the basis of their oral glucose tolerance descripted by three variables D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 14 / 22

Some results Settings distanceMethod = euclidean aggregationMethod = Ward α = 0 . 05 M = 999 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 14 / 22

Some results D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 14 / 22

Some results... for 5 variables genRandomCluster numClust = 2:7 numNonNoisy = 5 sepVal = 0.01 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 15 / 22

Some results... for 5 variables genRandomCluster numClust = 2:7 numNonNoisy = 5 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 15 / 22

Some results... for 5 variables genRandomCluster numClust = 2:7 numNonNoisy = 5 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 1 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 15 / 22

Some results... for 5 variables genRandomCluster numClust = 2:7 numNonNoisy = 5 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 05 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 15 / 22

Some results... for 5 variables genRandomCluster numClust = 2:7 numNonNoisy = 5 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 01 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 15 / 22

Some results... for 5 variables (100 replications) D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 16 / 22

Some results... for 10 variables genRandomCluster numClust = 2:7 numNonNoisy = 10 sepVal = 0.01 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 17 / 22

Some results... for 10 variables genRandomCluster numClust = 2:7 numNonNoisy = 10 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 17 / 22

Some results... for 10 variables genRandomCluster numClust = 2:7 numNonNoisy = 10 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 1 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 17 / 22

Some results... for 10 variables genRandomCluster numClust = 2:7 numNonNoisy = 10 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 05 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 17 / 22

Some results... for 10 variables genRandomCluster numClust = 2:7 numNonNoisy = 10 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 01 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 17 / 22

Some results... for 10 variables (100 replications) D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 18 / 22

Some results... for 15 variables genRandomCluster numClust = 2:7 numNonNoisy = 15 sepVal = 0.01 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 19 / 22

Some results... for 15 variables genRandomCluster numClust = 2:7 numNonNoisy = 15 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 19 / 22

Some results... for 15 variables genRandomCluster numClust = 2:7 numNonNoisy = 15 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 1 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 19 / 22

Some results... for 15 variables genRandomCluster numClust = 2:7 numNonNoisy = 15 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 05 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 19 / 22

Some results... for 15 variables genRandomCluster numClust = 2:7 numNonNoisy = 15 sepVal = 0.01 Settings distanceMethod = euclidean aggregationMethod = Ward M = 999 α = 0 . 01 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 19 / 22

Some results... for 15 variables (100 replications) D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 20 / 22

La Carte A (? simple ?) idea 1 A (? not so ?) simple procedure 2 Some results 3 The Wishlist 4 D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 21 / 22

The wishlist Statistical issues R issues D. Bruzzese, D. Vistocco ( ——————————————————————————————– Stairstep-like dendrogram cut Department of UseR 2009 22 / 22