Subspace clustering. Clustering. Fundamental to all clustering techniques is the choice of distance measure between data points;
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1 Subspace clusterng Clusterng Fundamental to all clusterng technques s the choce of dstance measure between data ponts; D q ( ) ( ) 2 x x = x x, j k = 1 k jk Squared Eucldean dstance Assumpton: All features are equally mportant; Such approaches fal n hgh dmensonal spaces 1
2 Clusterng: The Curse of Dmensonalty A full-dmensonal dstance s often rrelevant, as the farthest pont s expected to be almost as close as the nearest pont; In hgh dmensonal spaces, t s lkely that, for any gven par of ponts wthn the same cluster, there exst at least a few dmensons on whch the ponts are far apart from each other. Example 2
3 Example: 3 Gaussan Clusters Example: K-means result 3
4 Clusterng Clusters may exst n dfferent subspaces, comprsed of dfferent combnatons of features; Example Each dmenson s relevant to at least one cluster 4
5 Global Dmensonalty Reducton We cannot prune off dmensons wthout ncurrng a loss of crucal nformaton; Global dmensonalty reducton technques, e.g. PCA, do not handle well stuatons where dfferent clusters are dense n dfferent subspaces; The data presents local structure Local Dmensonalty Reducton To capture the local correlatons of data, a proper feature selecton procedure should operate locally; A local operaton would allow to embed dfferent dstance measures n dfferent regons; 5
6 Subspace clusterng dmensons data ponts Subspace clusterng Important problem n practce; Real lfe problems: Are hgh dmensonal; Present local structure; E.g., To compute gene smlarty: Dfferent condtons may have dfferent mportance for a gven gene; The relevance of one condton may vary from gene to gene; 6
7 Subspace clusterng smultaneous clusterng of both row and column sets n a data matrx Other terms used: 1. Bclusterng 2. Coclusterng 3. Box clusterng 4. Projectve clusterng 5. Algorthms PROCLUS (Projected Clusterng) Aggarwal et al., LAC (Locally Adaptve Clusterng) Domencon et al., Bclusterng Cheng & Church,
8 PROCLUS Projected cluster: subset of data ponts, together wth a subset of dmensons, such that the ponts are closely clustered n the correspondng subspace; Objectve: Fnd cluster centrods (medods), and set of dmensons n whch each cluster exsts. PROCLUS: Overall Approach Intalzaton: an ntal (super) set of medods s chosen; Iteratve phase: Fnd dmensons wthn a localty of each medod, and resultng clusterng. Improve the qualty of medods, Untl stop crteron satsfed. 8
9 PROCLUS Input parameters: k: the number of cluster; l: average number of dmensons n a cluster. PROCLUS Dstance measure: x, x R D = any set of dmensons, D q d 1 D 2 q 1 D ( x1, x2 ) = D x 1 x 2 Manhattan segmental dstance 9
10 PROCLUS Applcaton: collaboratve flterng Am: partton customers nto groups wth smlar nterests for target marketng; Dmensons: dfferent products or product categores; Need to be able to handle a large number of dmensons; Dstance between two customers: average dfference of preferences on dfferent products (Manhattan segmental dstance). PROCLUS Intalzaton : { m,, } M = 1 L m k Fndng dmensons : δ = mn d L = j ( m, m ) j { set of ponts wthn dstance δ from m } 10
11 L 1 L 2 11
12 Fndng dmensons : δ = mn d L = j ( m, m ) j PROCLUS { set of ponts wthn dstance δ from m } X j : averagedstance along dmenson L from m j of ponts n We wsh to assocate those dmensons j for whch the values X j Requrements : total number of are as small as possble dmensons s kl ; each medod s assgned at least 2 dmensons. For each medod : 1 Y = q σ Z j = q j= 1 X ( X j Y ) j j X j Y = σ q 1 A negatve value of Z PROCLUS 2 ndcates that along dmenson j the ponts n L are more closely correlated to the medod m. j 12
13 so that a total of PROCLUS We pck the js that gve the smallest Z kl dmensons are chosen, and at least two dmensons per cluster are chosen. Result : D, D, L, D 1 2 k Formng clusters: Gven the medods and ther assocated dmensons, Assgn each pont to the closest medod (wth respect to the average Manhattan segmental dstance relatve to D ). j PROCLUS Evaluate medods : If a "bad"medod s dentfed, sample new ones and terate the process. Untl no "bad" medods are found. 13
14 Major drawback: PROCLUS The algorthm requres the average number of dmensons per cluster as parameter n nput. The performance of PROCLUS s hghly senstve to the value of ts nput parameter. If the average number of dmensons s erroneously estmated, the performance of PROCLUS sgnfcantly worsens. Experments wth PROCLUS Data: 30 dmensons; K=2 clusters; Cluster 1: multvarate Gaussan Mean: (1,1,,1). Std: (10,5,10,5,,10,5) Cluster 2: multvarate Gaussan Mean (2,1,,1). Std: (5,10,5,10,,5,10) 14
15 Experments wth PROCLUS Error Rate Average Number of Dmensons Cluster 1: dmensons 8, 30 Cluster 2: dmensons 19, 15, 21, 1, 27, 23 Can we do better? We wsh to learn from the data the relevant features for each cluster, wthout havng to specfy the average number of features. Idea: Soft feature selecton procedure Assgn (local) weghts to features accordng to the local correlatons of data along each dmenson. 15
16 Locally Adaptve Clusterng: Example ( w1 x, w1 y ), w1 x > w1 y ( w2 x, w2 y ), w2 y > w2 x Locally Adaptve Clusterng: Example Wthn-cluster dstances between ponts are computed usng the respectve local weghts 16
17 Locally Adaptve Clusterng (LAC) Weghted cluster: subset of data ponts, together wth a weght vector, such that the ponts are closely clustered accordng to the correspondng weghted Eucldean dstance; Objectve: Fnd cluster centrods, and weght vectors. LAC: Overall Approach Intalzaton: an ntal set of centrods s chosen; Iteratve phase: Compute weghts wthn a localty of each centrod, and resultng clusterng. Update centrods and terate Untl no change occurs. 17
18 LAC Input parameter: k: the number of clusters LAC Intalzaton : c, L, c 1 w j j k = 1, for all centrods j and all features Intal partton : S D w = { x j = arg mn Dw ( cl, x) } = q = 1 w l ( c x ) l l 2 18
19 Computng the weghts : X X w j j j = = S j S l ( c j x ) j x S j e X j e X jl j LAC : averagesquared dstance along dmenson of ponts n from c 1 2 LAC Result : w, w, L, w 1 2 k Formng clusters: Gven the centrods and ther assocated weghts, Assgn each pont to the closest centrod j (wth respect to the weghted Eucldean dstance). Update centrods, Untl convergence. 19
20 Experments wth LAC Experments wth LAC lac Error rate: 7.7% K-means Error rate: 18.7% 20
21 Experments wth LAC LAC: Subspace clusterng of Mcroarray data Am: Cluster genes accordng to ther expresson levels across dfferent condtons. We can apply LAC to the gene vectors. Analyzng the dstrbuton of weght values wthn each dentfed cluster, we can determne the correlatons between genes and condtons. 21
22 LAC: lmtatons Senstve to ntal choce of centrods; Requres the value of k n nput; Extenson: allow overlappng clusters. 22
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