Statistical Data Mining and Machine LearningHilary Term 2016

Dino SejdinovicDepartment of Statistics

Oxford

Slides and other materials available at:http://www.stats.ox.ac.uk/~sejdinov/sdmml

Visualisation and Dimensionality Reduction

Last time: Biplots, MDSBiplots: a way to visualiserelationships between principalcomponents and the original variables.In its scaled version, projected pointsare uncorrelated and equi-variant andangles between variables correspondto correlation.

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Indiana Iowa

Kansas

KentuckyLouisiana

MaineMaryland

Massachusetts

Michigan

Minnesota

Mississippi

Missouri

Montana

Nebraska

Nevada

New Hampshire

New Jersey

New Mexico

New York

North Carolina

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Oregon Pennsylvania

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Texas

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Wyoming

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Multidimensional Scaling: a class ofdimensionality reduction techniqueswhich find a representation of the dataitems x1, . . . , xn ∈ Rp in alower-dimensional spacez1, . . . , zn ∈ Rk which approximatelypreserves the inter-point(dis)similarities.

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Visualisation and Dimensionality Reduction Multidimensional Scaling

Varieties of MDSChoices of (dis)similarities and stress functions S(Z) lead to differentalgorithms.

Classical/Torgerson: preserves inner products instead - strain function(cmdscale)

S(Z) =∑i 6=j

(bij − 〈zi − z̄, zj − z̄〉)2

Metric Shephard-Kruskal: preserves distances w.r.t. squared stress

S(Z) =∑i 6=j

(dij − ‖zi − zj‖2)2

Sammon: preserves shorter distances more (sammon)

S(Z) =∑i 6=j

(dij − ‖zi − zj‖2)2

dij

Non-Metric Shephard-Kruskal: ignores actual distance values, onlypreserves ranks (isoMDS)

S(Z) = ming increasing

∑i6=j(g(dij)− ‖zi − zj‖2)

2∑i 6=j ‖zi − zj‖2

2

Visualisation and Dimensionality Reduction Multidimensional Scaling

Example: Language data

Presence or absence of 2867 homologous traits in 87 Indo-Europeanlanguages.

> X<-read.table("http://www.stats.ox.ac.uk/~sejdinov/sdmml/data/cognate.txt")> X[1:15,1:16]

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16Irish_A 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0Irish_B 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0Welsh_N 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0Welsh_C 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0Breton_List 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0Breton_SE 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0Breton_ST 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0Romanian_List 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Vlach 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Italian 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Ladin 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Provencal 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0French 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0Walloon 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0French_Creole_C 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Visualisation and Dimensionality Reduction Multidimensional Scaling

Example: Language dataUsing MDS with non-metric (Sammon) scaling.

MDS (i.e. cmdscale) which minimizes (d2ij − d̃2

ij)2. Sammon thereby puts

more weight on reproducing the separation of points which are close byforcing them apart. Projection by MDS(Jaccard/sammon) with cluster dis-covery by k-means (Jaccard): There is an obvious east to west (top-left to

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Irish_B

Welsh_NWelsh_C

Breton_List

Breton_SE

Breton_ST

Romanian_ListVlach

Italian

Ladin

Provencal

French

Walloon

French_Creole_C

French_Creole_D

Sardinian_N

Sardinian_L

Sardinian_C

Spanish Portuguese_STBrazilian

Catalan

German_STPenn_Dutch

Dutch_List

Afrikaans

Flemish

Frisian

Swedish_UpSwedish_VL

Swedish_List

Danish

Riksmal

Icelandic_ST

Faroese

English_STTakitaki

Lithuanian_O

Lithuanian_STLatvian

Slovenian

Lusatian_L

Lusatian_U

Czech

Slovak

Czech_E

Ukrainian

Byelorussian

PolishRussian

MacedonianBulgarian

Serbocroatian

Gypsy_Gk

Singhalese

Kashmiri

Marathi

Gujarati

Panjabi_ST

Lahnda

Hindi

Bengali

Nepali_List

Khaskura

Greek_ML

Greek_MD

Greek_Mod

Greek_D

Greek_K

Armenian_Mod

Armenian_List

Ossetic

Afghan

WaziriPersian_ListTadzik

Baluchi

Wakhi

Albanian_T

Albanian_Top

Albanian_G

Albanian_K

Albanian_C

HITTITE

TOCHARIAN_A

TOCHARIAN_B

bottom-right) separation of languages in the MDS and the clusters in theMDS grouping agree with the clusters discovered by agglomerative clus-tering and k-means. The two clustering methods group languages slightlydifferently with k-means splitting the Germanic languages.

## (alternative/MDS) make a field to display the clusters

## use MDS - sammon does this nicely

di.sam <- sammon(D,magic=0.20000002,niter=1000,tol=1e-8)

eqscplot(di.sam$points,pch=km$cluster,col=km$cluster)

text(di.sam$points,labels=row.names(X),pos=4,col=km$cluster)

5

Visualisation and Dimensionality Reduction Isomap

Nonlinear Dimensionality Reduction

Two aims of different varieties of MDS:To visualize the (dis)similarities among items in a dataset, where these(dis)disimilarities may not have Euclidean geometric interpretations.To perform nonlinear dimensionality reduction.

Many high-dimensional datasets exhibit low-dimensional structure (“live on alow-dimensional menifold”).

Visualisation and Dimensionality Reduction Isomap

Isomap

Isomap is a non-linear dimensional reduction technique based on classicalMDS. Differs from other MDSs as it uses estimates of geodesic distancesbetween the data points.

converts distances to inner products (17 ),which uniquely characterize the geometry ofthe data in a form that supports efficientoptimization. The global minimum of Eq. 1 isachieved by setting the coordinates yi to thetop d eigenvectors of the matrix !(DG) (13).

As with PCA or MDS, the true dimen-sionality of the data can be estimated fromthe decrease in error as the dimensionality ofY is increased. For the Swiss roll, whereclassical methods fail, the residual varianceof Isomap correctly bottoms out at d " 2(Fig. 2B).

Just as PCA and MDS are guaranteed,given sufficient data, to recover the truestructure of linear manifolds, Isomap is guar-anteed asymptotically to recover the true di-mensionality and geometric structure of astrictly larger class of nonlinear manifolds.Like the Swiss roll, these are manifolds

whose intrinsic geometry is that of a convexregion of Euclidean space, but whose ambi-ent geometry in the high-dimensional inputspace may be highly folded, twisted, orcurved. For non-Euclidean manifolds, such asa hemisphere or the surface of a doughnut,Isomap still produces a globally optimal low-dimensional Euclidean representation, asmeasured by Eq. 1.

These guarantees of asymptotic conver-gence rest on a proof that as the number ofdata points increases, the graph distancesdG(i, j) provide increasingly better approxi-mations to the intrinsic geodesic distancesdM(i, j), becoming arbitrarily accurate in thelimit of infinite data (18, 19). How quicklydG(i, j) converges to dM(i, j) depends on cer-tain parameters of the manifold as it lieswithin the high-dimensional space (radius ofcurvature and branch separation) and on the

density of points. To the extent that a data setpresents extreme values of these parametersor deviates from a uniform density, asymp-totic convergence still holds in general, butthe sample size required to estimate geodes-ic distance accurately may be impracticallylarge.

Isomap’s global coordinates provide asimple way to analyze and manipulate high-dimensional observations in terms of theirintrinsic nonlinear degrees of freedom. For aset of synthetic face images, known to havethree degrees of freedom, Isomap correctlydetects the dimensionality (Fig. 2A) and sep-arates out the true underlying factors (Fig.1A). The algorithm also recovers the knownlow-dimensional structure of a set of noisyreal images, generated by a human hand vary-ing in finger extension and wrist rotation(Fig. 2C) (20). Given a more complex dataset of handwritten digits, which does not havea clear manifold geometry, Isomap still findsglobally meaningful coordinates (Fig. 1B)and nonlinear structure that PCA or MDS donot detect (Fig. 2D). For all three data sets,the natural appearance of linear interpolationsbetween distant points in the low-dimension-al coordinate space confirms that Isomap hascaptured the data’s perceptually relevantstructure (Fig. 4).

Previous attempts to extend PCA andMDS to nonlinear data sets fall into twobroad classes, each of which suffers fromlimitations overcome by our approach. Locallinear techniques (21–23) are not designed torepresent the global structure of a data setwithin a single coordinate system, as we do inFig. 1. Nonlinear techniques based on greedyoptimization procedures (24–30) attempt todiscover global structure, but lack the crucialalgorithmic features that Isomap inheritsfrom PCA and MDS: a noniterative, polyno-mial time procedure with a guarantee of glob-al optimality; for intrinsically Euclidean man-

Fig. 2. The residualvariance of PCA (opentriangles), MDS [opentriangles in (A) through(C); open circles in (D)],and Isomap (filled cir-cles) on four data sets(42). (A) Face imagesvarying in pose and il-lumination (Fig. 1A).(B) Swiss roll data (Fig.3). (C) Hand imagesvarying in finger exten-sion and wrist rotation(20). (D) Handwritten“2”s (Fig. 1B). In all cas-es, residual variance de-creases as the dimen-sionality d is increased.The intrinsic dimen-sionality of the datacan be estimated bylooking for the “elbow”at which this curve ceases to decrease significantly with added dimensions. Arrows mark the true orapproximate dimensionality, when known. Note the tendency of PCA and MDS to overestimate thedimensionality, in contrast to Isomap.

Fig. 3. The “Swiss roll” data set, illustrating how Isomap exploits geodesicpaths for nonlinear dimensionality reduction. (A) For two arbitrary points(circled) on a nonlinear manifold, their Euclidean distance in the high-dimensional input space (length of dashed line) may not accuratelyreflect their intrinsic similarity, as measured by geodesic distance alongthe low-dimensional manifold (length of solid curve). (B) The neighbor-hood graph G constructed in step one of Isomap (with K " 7 and N "

1000 data points) allows an approximation (red segments) to the truegeodesic path to be computed efficiently in step two, as the shortestpath in G. (C) The two-dimensional embedding recovered by Isomap instep three, which best preserves the shortest path distances in theneighborhood graph (overlaid). Straight lines in the embedding (blue)now represent simpler and cleaner approximations to the true geodesicpaths than do the corresponding graph paths (red).

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Tenenbaum et al. (2000)

Visualisation and Dimensionality Reduction Isomap

Isomap

Isomap

Calculate Euclidean distances dij for i, j = 1, . . . , n between all data points.Form a graph G with n samples as nodes, and edges between therespective K nearest neighbours (K-Isomap) or between i and j if dij < ε(ε-Isomap).For i, j linked by an edge, set dG

ij = dij. Otherwise, set dGij to the

shortest-path distance between i and j in G.Run classical MDS using distances dG

ij .

converts distances to inner products (17 ),which uniquely characterize the geometry ofthe data in a form that supports efficientoptimization. The global minimum of Eq. 1 isachieved by setting the coordinates yi to thetop d eigenvectors of the matrix !(DG) (13).

As with PCA or MDS, the true dimen-sionality of the data can be estimated fromthe decrease in error as the dimensionality ofY is increased. For the Swiss roll, whereclassical methods fail, the residual varianceof Isomap correctly bottoms out at d " 2(Fig. 2B).

Just as PCA and MDS are guaranteed,given sufficient data, to recover the truestructure of linear manifolds, Isomap is guar-anteed asymptotically to recover the true di-mensionality and geometric structure of astrictly larger class of nonlinear manifolds.Like the Swiss roll, these are manifolds

whose intrinsic geometry is that of a convexregion of Euclidean space, but whose ambi-ent geometry in the high-dimensional inputspace may be highly folded, twisted, orcurved. For non-Euclidean manifolds, such asa hemisphere or the surface of a doughnut,Isomap still produces a globally optimal low-dimensional Euclidean representation, asmeasured by Eq. 1.

These guarantees of asymptotic conver-gence rest on a proof that as the number ofdata points increases, the graph distancesdG(i, j) provide increasingly better approxi-mations to the intrinsic geodesic distancesdM(i, j), becoming arbitrarily accurate in thelimit of infinite data (18, 19). How quicklydG(i, j) converges to dM(i, j) depends on cer-tain parameters of the manifold as it lieswithin the high-dimensional space (radius ofcurvature and branch separation) and on the

density of points. To the extent that a data setpresents extreme values of these parametersor deviates from a uniform density, asymp-totic convergence still holds in general, butthe sample size required to estimate geodes-ic distance accurately may be impracticallylarge.

Isomap’s global coordinates provide asimple way to analyze and manipulate high-dimensional observations in terms of theirintrinsic nonlinear degrees of freedom. For aset of synthetic face images, known to havethree degrees of freedom, Isomap correctlydetects the dimensionality (Fig. 2A) and sep-arates out the true underlying factors (Fig.1A). The algorithm also recovers the knownlow-dimensional structure of a set of noisyreal images, generated by a human hand vary-ing in finger extension and wrist rotation(Fig. 2C) (20). Given a more complex dataset of handwritten digits, which does not havea clear manifold geometry, Isomap still findsglobally meaningful coordinates (Fig. 1B)and nonlinear structure that PCA or MDS donot detect (Fig. 2D). For all three data sets,the natural appearance of linear interpolationsbetween distant points in the low-dimension-al coordinate space confirms that Isomap hascaptured the data’s perceptually relevantstructure (Fig. 4).

Previous attempts to extend PCA andMDS to nonlinear data sets fall into twobroad classes, each of which suffers fromlimitations overcome by our approach. Locallinear techniques (21–23) are not designed torepresent the global structure of a data setwithin a single coordinate system, as we do inFig. 1. Nonlinear techniques based on greedyoptimization procedures (24–30) attempt todiscover global structure, but lack the crucialalgorithmic features that Isomap inheritsfrom PCA and MDS: a noniterative, polyno-mial time procedure with a guarantee of glob-al optimality; for intrinsically Euclidean man-

Fig. 2. The residualvariance of PCA (opentriangles), MDS [opentriangles in (A) through(C); open circles in (D)],and Isomap (filled cir-cles) on four data sets(42). (A) Face imagesvarying in pose and il-lumination (Fig. 1A).(B) Swiss roll data (Fig.3). (C) Hand imagesvarying in finger exten-sion and wrist rotation(20). (D) Handwritten“2”s (Fig. 1B). In all cas-es, residual variance de-creases as the dimen-sionality d is increased.The intrinsic dimen-sionality of the datacan be estimated bylooking for the “elbow”at which this curve ceases to decrease significantly with added dimensions. Arrows mark the true orapproximate dimensionality, when known. Note the tendency of PCA and MDS to overestimate thedimensionality, in contrast to Isomap.

Fig. 3. The “Swiss roll” data set, illustrating how Isomap exploits geodesicpaths for nonlinear dimensionality reduction. (A) For two arbitrary points(circled) on a nonlinear manifold, their Euclidean distance in the high-dimensional input space (length of dashed line) may not accuratelyreflect their intrinsic similarity, as measured by geodesic distance alongthe low-dimensional manifold (length of solid curve). (B) The neighbor-hood graph G constructed in step one of Isomap (with K " 7 and N "

1000 data points) allows an approximation (red segments) to the truegeodesic path to be computed efficiently in step two, as the shortestpath in G. (C) The two-dimensional embedding recovered by Isomap instep three, which best preserves the shortest path distances in theneighborhood graph (overlaid). Straight lines in the embedding (blue)now represent simpler and cleaner approximations to the true geodesicpaths than do the corresponding graph paths (red).

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R function: isomap{vegan}.

Visualisation and Dimensionality Reduction Isomap

Handwritten Characters

tion to geodesic distance. For faraway points,geodesic distance can be approximated byadding up a sequence of “short hops” be-tween neighboring points. These approxima-tions are computed efficiently by findingshortest paths in a graph with edges connect-ing neighboring data points.

The complete isometric feature mapping,or Isomap, algorithm has three steps, whichare detailed in Table 1. The first step deter-mines which points are neighbors on themanifold M, based on the distances dX (i, j)between pairs of points i, j in the input space

X. Two simple methods are to connect eachpoint to all points within some fixed radius !,or to all of its K nearest neighbors (15). Theseneighborhood relations are represented as aweighted graph G over the data points, withedges of weight dX(i, j) between neighboringpoints (Fig. 3B).

In its second step, Isomap estimates thegeodesic distances dM (i, j) between all pairsof points on the manifold M by computingtheir shortest path distances dG(i, j) in thegraph G. One simple algorithm (16 ) for find-ing shortest paths is given in Table 1.

The final step applies classical MDS tothe matrix of graph distances DG " {dG(i, j)},constructing an embedding of the data in ad-dimensional Euclidean space Y that bestpreserves the manifold’s estimated intrinsicgeometry (Fig. 3C). The coordinate vectors yi

for points in Y are chosen to minimize thecost function

E ! !#$DG% " #$DY%!L2 (1)

where DY denotes the matrix of Euclideandistances {dY(i, j) " !yi & yj!} and !A!L2

the L2 matrix norm '(i, j Ai j2 . The # operator

Fig. 1. (A) A canonical dimensionality reductionproblem from visual perception. The input consistsof a sequence of 4096-dimensional vectors, rep-resenting the brightness values of 64 pixel by 64pixel images of a face rendered with differentposes and lighting directions. Applied to N " 698raw images, Isomap (K" 6) learns a three-dimen-sional embedding of the data’s intrinsic geometricstructure. A two-dimensional projection is shown,with a sample of the original input images (redcircles) superimposed on all the data points (blue)and horizontal sliders (under the images) repre-senting the third dimension. Each coordinate axisof the embedding correlates highly with one de-gree of freedom underlying the original data: left-right pose (x axis, R " 0.99), up-down pose ( yaxis, R " 0.90), and lighting direction (slider posi-tion, R " 0.92). The input-space distances dX(i, j )given to Isomap were Euclidean distances be-tween the 4096-dimensional image vectors. (B)Isomap applied to N " 1000 handwritten “2”sfrom the MNIST database (40). The two mostsignificant dimensions in the Isomap embedding,shown here, articulate the major features of the“2”: bottom loop (x axis) and top arch ( y axis).Input-space distances dX(i, j ) were measured bytangent distance, a metric designed to capture theinvariances relevant in handwriting recognition(41). Here we used !-Isomap (with ! " 4.2) be-cause we did not expect a constant dimensionalityto hold over the whole data set; consistent withthis, Isomap finds several tendrils projecting fromthe higher dimensional mass of data and repre-senting successive exaggerations of an extrastroke or ornament in the digit.

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Visualisation and Dimensionality Reduction Isomap

Faces

tion to geodesic distance. For faraway points,geodesic distance can be approximated byadding up a sequence of “short hops” be-tween neighboring points. These approxima-tions are computed efficiently by findingshortest paths in a graph with edges connect-ing neighboring data points.

The complete isometric feature mapping,or Isomap, algorithm has three steps, whichare detailed in Table 1. The first step deter-mines which points are neighbors on themanifold M, based on the distances dX (i, j)between pairs of points i, j in the input space

X. Two simple methods are to connect eachpoint to all points within some fixed radius !,or to all of its K nearest neighbors (15). Theseneighborhood relations are represented as aweighted graph G over the data points, withedges of weight dX(i, j) between neighboringpoints (Fig. 3B).

In its second step, Isomap estimates thegeodesic distances dM (i, j) between all pairsof points on the manifold M by computingtheir shortest path distances dG(i, j) in thegraph G. One simple algorithm (16 ) for find-ing shortest paths is given in Table 1.

The final step applies classical MDS tothe matrix of graph distances DG " {dG(i, j)},constructing an embedding of the data in ad-dimensional Euclidean space Y that bestpreserves the manifold’s estimated intrinsicgeometry (Fig. 3C). The coordinate vectors yi

for points in Y are chosen to minimize thecost function

E ! !#$DG% " #$DY%!L2 (1)

where DY denotes the matrix of Euclideandistances {dY(i, j) " !yi & yj!} and !A!L2

the L2 matrix norm '(i, j Ai j2 . The # operator

Fig. 1. (A) A canonical dimensionality reductionproblem from visual perception. The input consistsof a sequence of 4096-dimensional vectors, rep-resenting the brightness values of 64 pixel by 64pixel images of a face rendered with differentposes and lighting directions. Applied to N " 698raw images, Isomap (K" 6) learns a three-dimen-sional embedding of the data’s intrinsic geometricstructure. A two-dimensional projection is shown,with a sample of the original input images (redcircles) superimposed on all the data points (blue)and horizontal sliders (under the images) repre-senting the third dimension. Each coordinate axisof the embedding correlates highly with one de-gree of freedom underlying the original data: left-right pose (x axis, R " 0.99), up-down pose ( yaxis, R " 0.90), and lighting direction (slider posi-tion, R " 0.92). The input-space distances dX(i, j )given to Isomap were Euclidean distances be-tween the 4096-dimensional image vectors. (B)Isomap applied to N " 1000 handwritten “2”sfrom the MNIST database (40). The two mostsignificant dimensions in the Isomap embedding,shown here, articulate the major features of the“2”: bottom loop (x axis) and top arch ( y axis).Input-space distances dX(i, j ) were measured bytangent distance, a metric designed to capture theinvariances relevant in handwriting recognition(41). Here we used !-Isomap (with ! " 4.2) be-cause we did not expect a constant dimensionalityto hold over the whole data set; consistent withthis, Isomap finds several tendrils projecting fromthe higher dimensional mass of data and repre-senting successive exaggerations of an extrastroke or ornament in the digit.

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Visualisation and Dimensionality Reduction Isomap

Nonlinear Dimensionality Reduction Techniques

Kernel PCALocally Linear EmbeddingLaplacian EigenmapsMaximum Variance Unfolding

Clustering

Clustering

Clustering

Clustering

Many datasets consist of multiple heterogeneous subsets.Cluster analysis: Given an unlabelled data, want algorithms thatautomatically group the datapoints into coherent subsets/clusters.Examples:

market segmentation of shoppers based on browsing and purchase historiesdifferent types of breast cancer based on the gene expressionmeasurementsdiscovering communities in social networksimage segmentation

Clustering

Types of Clustering

Model-based clustering:Each cluster is described using a probability model.

Model-free clustering:Defined by similarity/dissimilarity among instances within clusters.

Clustering

This Lecture: Two “Model-free” Clustering Methods

K-means clustering: a partition-based method into K clusters. Findsgroups such that variation within each group is small. The number ofclusters K is usually fixed beforehand or various values of K areinvestigated as a part of the analysis.Hierarchical clustering: nearby data items are joined into clusters, thenclusters into super-clusters forming a hierarchy. Typically, the hierarchyforms a binary tree (a dendrogram) where each cluster has two“children” clusters. Dendrogram allows to view the clusterings for eachpossible number of clusters, from 1 to n (number of data items).

Clustering K-means

K-meansPartition-based methods seek to divide data points into a pre-assignednumber of clusters C1, . . . ,CK where for all k, k′ ∈ {1, . . . ,K},

Ck ⊂ {1, . . . , n} , Ck ∩ Ck′ = ∅ ∀k 6= k′,K⋃

k=1

Ck = {1, . . . , n} .

For each cluster, represent it using a prototype or cluster centroid µk.We can measure the quality of a cluster with its within-cluster deviance

W(Ck, µk) =∑i∈Ck

‖xi − µk‖22.

The overall quality of the clustering is given by the total within-clusterdeviance:

W =K∑

k=1

W(Ck, µk).

The overall objective is to choose both the cluster centroids and allocation ofpoints to minimize the objective function.

Clustering K-means

K-means

W =

K∑k=1

∑i∈Ck

‖xi − µk‖22 =

n∑i=1

‖xi − µci‖22

where ci = k if and only if i ∈ Ck.Given partition {Ck}, we can find the optimal prototypes easily bydifferentiating W with respect to µk:

∂W∂µk

= 2∑i∈Ck

(xi − µk) = 0 ⇒ µk =1|Ck|

∑i∈Ck

xi

Given prototypes, we can easily find the optimal partition by assigningeach data point to the closest cluster prototype:

ci = argmink‖xi − µk‖2

2

But joint minimization over both is computationally difficult.

Clustering K-means

K-means

The K-means algorithm is a widely used method that returns a local optimumof the objective function W, using iterative and alternating minimization.

1 Randomly initialize K cluster centroids µ1, . . . , µK .2 Cluster assignment: For each i = 1, . . . , n, assign each xi to the cluster

with the nearest centroid,

ci := argmink‖xi − µk‖2

2

Set Ck := {i : ci = k} for each k.3 Move centroids: Set µ1, . . . , µK to the averages of the new clusters:

µk :=1|Ck|

∑i∈Ck

xi

4 Repeat steps 2-3 until convergence.5 Return the partition {C1, . . . ,CK} and means µ1, . . . , µK .

Clustering K-means

K-means

The algorithm stops in a finite number of iterations. Between steps 2and 3, W either stays constant or it decreases, this implies that we neverrevisit the same partition. As there are only finitely many partitions, thenumber of iterations cannot exceed this.The K-means algorithm need not converge to global optimum.K-means is a heuristic search algorithm so it can get stuck at suboptimalconfigurations. The result depends on the starting configuration. Typicallyperform a number of runs from different configurations, and pick the endresult with minimum W.

−0.5 0.0 0.5 1.0 1.5

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W= 9.184

−0.5 0.0 0.5 1.0 1.5

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1.0

x

y

W= 3.418

−0.5 0.0 0.5 1.0 1.5

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W= 9.264

Clustering K-means

K-means on Crabs

Looking at the Crabs data again.

library(MASS)library(lattice)data(crabs)

splom(~log(crabs[,4:8]),pch=as.numeric(crabs[,2]),col=as.numeric(crabs[,1]),main="circle/triangle is gender, black/red is species")

Clustering K-means

K-means on Crabscircle/triangle is gender, black/red is species

Scatter Plot Matrix

FL2.62.83.03.2

2.62.83.03.2

2.02.22.42.6

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BD2.5

3.02.5 3.0

2.0

2.5

2.0 2.5

Clustering K-means

K-means on Crabs

Apply K-means with 2 clusters and plot results.

Crabs.kmeans <- kmeans( log(crabs[,4:8]), 2, nstart=1, iter.max=10)

splom(~log(crabs[,4:8]),col=Crabs.kmeans$cluster+2,main="blue/green is cluster; finds big/small")

Clustering K-means

K-means on Crabsblue/green is cluster finds big/small

Scatter Plot Matrix

FL2.62.83.03.2

2.62.83.03.2

2.02.22.42.6

2.02.22.42.6 ●●●●●

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RW2.42.62.83.0

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BD2.5

3.02.5 3.0

2.0

2.5

2.0 2.5

Clustering K-means

K-means on Crabs

‘Whiten’ or ‘sphere’1 the data using PCA.

pcp <- princomp( log(crabs[,4:8]) )Crabs.sphered <- pcp$scores %*% diag(1/pcp$sdev)splom( ~Crabs.sphered[,1:3],

col=as.numeric(crabs[,1]),pch=as.numeric(crabs[,2]),main="circle/triangle is gender, black/red is species")

And apply K-means again.

Crabs.kmeans <- kmeans(Crabs.sphered, 2, nstart=1, iter.max=20)splom( ~Crabs.sphered[,1:3],

col=Crabs.kmeans$cluster+2, main="blue/green is cluster")

1Apply a linear transformation so that the covariance matrix is identity.

Clustering K-means

K-means on Crabs

circle/triangle is gender, black/red is species

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Discovers gender difference...But the result depends crucially on sphering the data first!

Clustering K-means

K-means on Crabs with K = 4circle/triangle is gender, black/red is species

Scatter Plot Matrix

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colors are clusters

Scatter Plot Matrix

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●●●●

●

V30

1

2 0 1 2

−2

−1

0

−2 −1 0

> table(Crabs.kmeans$cluster,Crabs.class)Crabs.classBF BM OF OM

1 3 0 41 02 39 8 6 03 8 42 0 04 0 0 3 50

Clustering K-means

K-means Additional Comments

Good practice initialization. Randomly pick K training examples(without replacement) and set µ1, µ2, . . . , µK equal to those examplesSensitivity to distance measure. Euclidean distance can be greatlyaffected by measurement unit and by strong correlations. Can useMahalanobis distance instead:

‖x− y‖M =√(x− y)>M−1(x− y)

where M is positive semi-definite matrix, e.g. sample covariance.Determination of K. The K-means objective will always improve withlarger number of clusters K. Determination of K requires an additionalregularization criterion. E.g., in DP-means2, use

W =

K∑k=1

∑i∈Ck

‖xi − µk‖22 + λK

2DP-means paper.

Clustering K-means

Other partition based methods

Other partition-based methods with related ideas:K-medoids3: requires cluster centroids µi to be an observation xj

K-medians: cluster centroids represented by a median in eachdimensionK-modes: cluster centroids represented by a mode estimated from acluster

3See also Affinity propagation.

Clustering Hierarchical Clustering

Hierarchical Clustering

Hierarchically structured data is ubiquitous (genus, species, subspecies,individuals...)There are two general strategies for generating hierarchical clusters.Both proceed by seeking to minimize some measure of overalldissimilarity.

Agglomerative / Bottom-Up / MergingDivisive / Top-Down / Splitting

Higher level clusters are created by merging clusters at lower levels. Thisprocess can easily be viewed by a tree/dendrogram.Avoids specifying how many clusters are appropriate.

hclust, agnes{cluster}

Clustering Hierarchical Clustering

EU Indicators Data

6 Economic indicators for EU countries in 2011.

> eu<-read.csv(’http://www.stats.ox.ac.uk/~sejdinov/sdmml/data/eu_indicators.csv’,sep=’ ’)

> eu[1:15,]Countries abbr CPI UNE INP BOP PRC UN_perc

1 Belgium BE 116.03 4.77 125.59 908.6 6716.5 -1.62 Bulgaria BG 141.20 7.31 102.39 27.8 1094.7 3.53 CzechRep. CZ 116.20 4.88 119.01 -277.9 2616.4 -0.64 Denmark DK 114.20 6.03 88.20 1156.4 7992.4 0.55 Germany DE 111.60 4.63 111.30 499.4 6774.6 -1.36 Estonia EE 135.08 9.71 111.50 153.4 2194.1 -7.77 Ireland IE 106.80 10.20 111.20 -166.5 6525.1 2.08 Greece EL 122.83 11.30 78.22 -764.1 5620.1 6.49 Spain ES 116.97 15.79 83.44 -280.8 4955.8 0.710 France FR 111.55 6.77 92.60 -337.1 6828.5 -0.911 Italy IT 115.00 5.05 87.80 -366.2 5996.6 -0.512 Cyprus CY 116.44 5.14 86.91 -1090.6 5310.3 -0.413 Latvia LV 144.47 12.11 110.39 42.3 1968.3 -3.614 Lithuania LT 135.08 11.47 114.50 -77.4 2130.6 -4.315 Luxembourg LU 118.19 3.14 85.51 2016.5 10051.6 -3.0

Data from Greenacre (2012)

Clustering Hierarchical Clustering

EU Indicators Data

dat<-scale(eu[,3:8])rownames(dat)<-eu$Countriesbiplot(princomp(dat))

−0.2 0.0 0.2 0.4

−0.

20.

00.

20.

4

Comp.1

Com

p.2

Belgium

Bulgaria

CzechRep.

Denmark

Germany

Estonia

Ireland

Greece

Spain

FranceItaly

Cyprus

LatviaLithuania

Luxembourg

HungaryMalta

NetherlandsAustria

Poland

Portugal

Romania

Slovenia

Slovakia

Finland

Sweden

UnitedKingdom

−4 −2 0 2 4 6

−4

−2

02

46

CPI

UNE

INPBOP

PRC

UN_perc

Clustering Hierarchical Clustering

Visualising Hierarchical Clustering

> hc<-hclust(dist(dat))> plot(hc,hang=-1)

Luxe

mbo

urg

Bel

gium

Ger

man

yA

ustr

iaN

ethe

rland

sD

enm

ark

Sw

eden

Cze

chR

ep.

Mal

taS

love

nia

Fra

nce

Fin

land

Italy

Uni

tedK

ingd

omC

ypru

sP

ortu

gal

Gre

ece

Irel

and

Spa

inR

oman

iaB

ulga

riaH

unga

ryP

olan

dS

lova

kia

Est

onia

Latv

iaLi

thua

nia

01

23

45

6

Cluster Dendrogram

hclust (*, "complete")dist(dat)

Hei

ght

> library(ape)> plot(as.phylo(hc), type = "fan")

Belgium

Bul

garia

CzechR

ep.

Den

mar

k

Germany

Estonia

Irelan

d

Greece

Spai

n

France

Italy

Cyprus

Latvia

Lithuania

Luxembourg

Hungary

Malta

Net

herla

nds

Austria

Poland

Portugal

Rom

ania

Slovenia

Slovakia

Finland

Sw

eden

UnitedKingdom

Clustering Hierarchical Clustering

Visualising Hierarchical Clustering

Levels in the dendrogram represent a dissimilarity between examples.Tree dissimilarity dT

ij = minimum height in the tree at which examples iand j belong to the same cluster.ultrametric (stronger than triangle) inequality:

dTij ≤ max{dT

ik, dTkj}.

Hierarchical clustering can be interpreted as an approximation of a givendissimilarity dij by an ultrametric dissimilarity.

Clustering Hierarchical Clustering

Measuring Dissimilarity Between ClustersTo join clusters Ci and Cj into super-clusters, we need a way to measure thedissimilarity D(Ci,Cj) between them.

b

b

b

b

bx1

x2

x3

x4

x5

single linkage: d(x1, x3)

b

b

b

b

bx1

x2

x3

x4

x5

complete linkage: d(x2, x5)

b

b

b

b

bx1

x2

x3

x4

x5

average linkage: 1

6

∑2

i=1

∑5

j=3d(xi, xj)

Clustering Hierarchical Clustering

Measuring Dissimilarity Between ClustersTo join clusters Ci and Cj into super-clusters, we need a way to measure thedissimilarity D(Ci,Cj) between them.(a) Single Linkage: elongated, loosely connected clusters

D(Ci,Cj) = minx,y

(d(x, y)|x ∈ Ci, y ∈ Cj)

(b) Complete Linkage: compact clusters, relatively similar objects canremain separated at high levels

D(Ci,Cj) = maxx,y

(d(x, y)|x ∈ Ci, y ∈ Cj)

(c) Average Linkage: tries to balance the two above, but affected by thescale of dissimilarities

D(Ci,Cj) = avgx,y (d(x, y)|x ∈ Ci, y ∈ Cj)

Clustering Hierarchical Clustering

Hierarchical Clustering on Artificial Dataset

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3000

Clustering Hierarchical Clustering

Hierarchical Clustering on Artificial Dataset

dat=xclara #3000 x 2library(cluster)

#plot the dataplot(dat,type="n")text(dat,labels=row.names(dat))

plot(agnes(dat,method="single"))plot(agnes(dat,method="complete"))plot(agnes(dat,method="average"))

Clustering Hierarchical Clustering

Hierarchical Clustering on Artificial Dataset

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Dendrogram of agnes(x = dat, method = "single")

Agglomerative Coefficient = 0.93dat

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Clustering Hierarchical Clustering

Hierarchical Clustering on Artificial Dataset

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Dendrogram of agnes(x = dat, method = "average")

Agglomerative Coefficient = 0.99dat

Hei

ght

Clustering Hierarchical Clustering

Using Dendrograms

Different ways of measuring dissimilarity result in different trees.Dendrograms are useful for getting a feel for the structure ofhigh-dimensional data though they don’t represent distances betweenobservations well.Dendrograms show hierarchical clusters with respect to increasing valuesof dissimilarity between clusters, cutting a dendrogram horizontally at aparticular height partitions the data into disjoint clusters which arerepresented by the vertical lines it intersects. Cutting horizontallyeffectively reveals the state of the clustering algorithm when thedissimilarity value between clusters is no more than the value cut at.Despite the simplicity of this idea and the above drawbacks, hierarchicalclustering methods provide users with interpretable dendrograms thatallow clusters in high-dimensional data to be better understood.