A Framework for Feature Selection in Clustering

Transcription

1 A Framework for Feature Selection in Clustering Daniela M. Witten and Robert Tibshirani October 10, 2011

2 Outline Problem Past work Proposed (sparse) clustering framework Sparse K-means clustering Sparse hierarchical clustering Applications to simulated and real data

3 Problem Clustering given data X n p identify K clusters unsupervised learning

4 Problem Clustering given data X n p identify K clusters unsupervised learning Sparse Clustering clusters only differ based on q < p features benefits: improved accuracy and interpretability cheaper prediction

5 eduction for clustering fall in this -based clustering can be found in as a model selection problem: models containing nested subsets of variables are compared. The nested models are sparse Problem - Motivating Example X 1,..., X 500 indep. bivariate normal clusters differ only in variable 1 Journal of the American Statistical Association, June 2010 l example, two classes differ only with respect to the first feature. Sparse 2-means clustering selects only the a superior result.

6 Problem - set-up Measure of dissimilarity additive assumption: d(x i, x i ) = p j=1 d i,i,j squared Euclidean distance: d i,i,j = (x ij x i j) 2

7 Past Work - matrix decomposition approximate X n p A n q B q p and cluster along rows of A examples: principal component analysis (PCA); Ghosh and Chinnaiyan (2002), Liu et al. (2003) nonnegative matrix factorization; Tamayo et al. (2007) not truly sparse - function of full p A not guaranteed to contain signal of interest PCA not justified in clustering; Chang (1983)

8 Past Work - model-based clustering model rows as independent multivariate observations from a mixture model LL = n [ K ] log π k f k (X i ; µ k, Σ k ) i=1 k=1 K: number of components f k : density, typically Gaussian use EM algorithm to fit model problems estimating Σ k when p n or p >> n

9 Past Work - model-based clustering maximize the log-likelihood subject to a sparsity inducing penalty Pan and Shen (2007): assume the features are centered at 0 PLL = n [ K ] log π k f k (X i ; µ k, Σ k ) λ i=1 k=1 K k=1 j=1 p µ kj sparsity induced when µ kj = 0 for all k

10 Past Work - model-based clustering Raftery and Dean (2006): recast variable selection problem as a model selection problem choose between models with nested subsets of variables

11 Past Work - model-based clustering Friedman and Meulman (2004): clustering objects on subsets of attributes (COSA) { K minimize C 1,...,C K,w a k k=1 i,i C k j=1 subject to p } (w j d i,i,j + λw j log w j ) p w j = 1, w j 0 j=1 C k : indices of observations in cluster k a k : function of C k w R p : feature weights λ 0 tuning parameter above is a simplified version, full COSA allows for w k

12 Proposed Framework Clustering maximize Θ D { p j=1 } f j (X j, Θ) X j R n : feature j f j : function of only the jth feature

13 Proposed Framework Sparse Clustering maximize Θ D,w { p j=1 } w j f j (X j, Θ) subject to w 2 1, w 1 s, w j 0 1 s p: tuning parameter

14 Proposed Framework maximize Θ D,w { p j=1 } w j f j (X j, Θ) subject to w 2 1, w 1 s, w j 0 Sparse Clustering: observations w 1 = = w p reduces to usual clustering L 1 penalty induces sparsity without L 2 penalty solution is trivial w j is the contribution of feature j to the clustering in general, f j (X j, Θ) > 0 for some or all j

15 Proposed Framework Sparse Clustering: optimization (iterative) 1 holding w fixed, optimize w.r.t. Θ clustering on weighted data 2 holding Θ fixed, optimize w.r.t. w rewrite as: maximize {w T a} a j = f j (X j, Θ) w proposition: by KKT, w = (a + ) + (a + ) + 2, = { 0 if it results in w 1 s > 0 otherwise to yield w 1 = s

16 K-means Clustering find C 1,, C K to... minimize the within-cluster sum of squares (WCSS) minimize C 1,,C K K 1 n k k=1 p i,i C k j=1 d i,i,j or maximize the between-cluster sum of squares (BCSS) maximize C 1,,C K maximize Θ D p ( 1 n j=1 n i=1 i =1 p f j (X j, Θ) j=1 n d i,i,j K 1 d i,i n,j k k=1 i,i C k )

17 Sparse K-means Clustering maximize C 1,,C K,w { p ( 1 w j n j=1 n n K 1 d i,i,j i=1 i =1 n k k=1 i,i C k d i,i,j )} subject to w 2 1, w 1 s, w j 0 Sparse K-means Clustering: optimization solve using earlier algorithm (1) K-means using di,i,j = w jd i,i,j to find C 1,, C K (2) proposition to find w iterate until j w j r r 1 wj r 1 j wj < 10 4 tends to converge within 5 to 10 iterations generally will not converge to global optimum (non-convex)

18 Sparse K-means Clustering Sparse K-means Clustering: tuning s objective function increases with s permutation approach is used: 1 obtain permuted datasets X 1,..., X B by independently permute the observations within each feature 2a compute O(s), the objective obtained by performing sparse K-means with tuning parameter s 2b similarly compute O b (s) for b = 1,..., B b log(o b(s)) 2c calculate Gap(s) = log(o(s)) 1 B { } 3 choose s = argmax s Gap(s) alternately, choose s to equal the smallest value for which Gap(s ) is within one standard deviation of log(o b (s )) of the largest value of Gap(s) Intuition: gap statistic measures the strength of the clustering obtained on the real data relative to the clustering obtained on null data that does not contain subgroups

19 Sparse Hierarchical Clustering produce a dendrogram that represents a nested set of clusters from dissimilarity matrix U n n and cut at desired height linkage: between cluster measure complete: d(c, C ) = max { d(x i, x i ); x i C, x i C } average: d(c, C ) = 1 C C x i,x d(x i i, x i ) single: d(c, C ) = min { d(x i, x i ); x i C, x i C }

20 Sparse Hierarchical Clustering place selection of dissimilarity matrix U in the proposed framework dissimilarity matrix U = { j d } i,i,j results in usual i,i hierarchical clustering U can be shown to be proportional to the solution of maximize U { p j=1 i,i d i,i,ju i,i subject to i,i U 2 i,i 1 } shape of dendrogram is invariant to scaling of features

21 Sparse Hierarchical Clustering maximize w,u { p w j j=1 subject to i,i U 2 i,i 1, i,i d i,i,ju i,i } w 2 1, w 1 s, w j 0 solution is proportional to U = { j w jd i,i,j } i,i clustering by reweighted dissimilarity matrix U produces sparsity

22 Spares Hierarchical Clustering maximize w,u { p w j j=1 i,i d i,i,ju i,i Sparse Hierarchical Clustering: optimization objective is bi-convex in U and w notation: D R n2 p : {d i,i,j} i,i strung out into columns u R n2 : U strung out into a vector } rewrite: maximize w,u {u T Dw}

23 Spares Hierarchical Clustering maximize w,u { p w j j=1 i,i d i,i,ju i,i Sparse Hierarchical Clustering: optimization solve using earlier algorithm (1) update u = Dw Dw 2 (2) proposition to find w iterate until j w j r r 1 wj r 1 j wj < 10 4 perform hierarchical clustering using U tends to converge within 10 iterations not guaranteed to converge to global optimum }

24 Spares Hierarchical Clustering Sparse Hierarchical Clustering: a simple underlying model consider simple model: two classes C 1, C 2 only differ with respect to first q features. X ij indep. { N(µ j + c, σ 2 ), if j q, i C 1 N(µ j, σ 2 ), otherwise for i i, X ij X i j { N(±c, 2σ 2 ), N(0, 2σ 2 ), if j q, i, i in different classes otherwise d i,i,j = { (X ij X i j) 2, 2σ 2 χ 2 ( c 2 ) 1 d i,i,j 2σ, 2 2σ 2 χ 1, if j q, i, i in different classes otherwise

25 Spares Hierarchical Clustering Sparse Hierarchical Clustering: a simple underlying model consider overall dissimilarity j d i,i {,j, j d 2σ 2 χ 2 ( qc 2 ) p i,i,j 2σ, if j q, i, i in different classes 2 2σ 2 χ p, otherwise consider weighted dissimilarity j w jd i,i,j under ideal situation where { w j 1 j q, j w 2σ 2 χ 2 ( qc 2 ) q jd i,i,j 2σ, j q, i, i in different classes 2 2σ 2 χ q, otherwise dissimilarity matrix for sparse clustering is a denoised version of the dissimilarity matrix used for standard clustering

26 Spares Hierarchical Clustering Sparse Hierarchical Clustering: a simple underlying model standard clustering, E ( j d i,i,j) = { 2σ 2 p + c 2 q, if j q, i, i in different classes 2σ 2 p, otherwise sparse clustering, E ( j w jd i,i,j) = { 2σ 2 j w j + c 2 j q w j, 2σ 2 j w j, j q, i, i in different classes otherwise note: 1 w taken to be first SPC of E(D), not D 2 w 1 = = w q > w q+1 = = w p

27 Spares Hierarchical Clustering Sparse Hierarchical Clustering: tuning s same as sparse K-means approach: permutation method

28 Spares Hierarchical Clustering Sparse Hierarchical Clustering: complementary clustering finding a secondary clustering after removing the signal found in primary clustering identify U 1, w 1 solve for U 2, w 2 : maximize u 2,w 2 {u T 2 Dw 2 } subject to u 2 2 1, u T 1 u 2 = 0, w 2 2 1, w 2 1 s, w j 0

29 Sparse K-means - simulated data simulation 1: comparison of sparse and standard K-means K = 3 q = 50 X ij N(µ ij, 1), µ ij = µ(1 i C1,j q 1 i C2,j q) 20 observations per class vary values of p and µ classification error rate (CER): i>i 1 P(i,i ) 1 Q(i,i ) / ( ) n 2

30 cribed in Section 3.2. Sparse K-means - simulated data simulation 1: comparison of sparse et al. 2007). and standard K-means imulation 2: A Comparison With Other Approaches. re the performance of sparse K-means to a number itors: sidered the first principal component contained the si This is similar to several proposals in the literature e.g., Ghosh and Chinnaiyan 2002; Liu et al. 2003; Tam Table 1. Standard 3-means results for Simulation 1. The reported values are the mean (and standard error) of the CER over 20 simulations. The µ/p combinations for which the CER of standard 3-means is significantly less than that of sparse 3-means (at level α = 0.05) are shown in bold p = 50 p = 200 p = 500 p = 1000 µ = (0.01) (0.015) 0.22 (0.009) (0.006) µ = (0.005) (0.009) 0.16 (0.012) (0.01) µ = (0.004) (0.007) 0.08 (0.005) (0.01) µ = (0.001) (0.005) (0.008) (0.013) µ = (0.002) 0.004(0.002) (0.004) 0.05(0.006) Journal of the American Statistical Association, June Table 2. Sparse 3-means results for Simulation 1. The reported values are the mean (and standard error) of the CER over 20 simulations. The µ/p combinations for which the CER of sparse 3-means is significantly less than that of standard 3-means (at level α = 0.05) are shown in bold p = 50 p = 200 p = 500 p = 1000 µ = (0.014) (0.016) (0.015) (0.017) µ = (0.011) (0.008) (0.013) (0.013) µ = (0.008) (0.007) (0.005) (0.006) µ = (0.006) (0.003) (0.004) (0.004) µ = (0.004) (0.002) (0.001) (0.002)

31 lted in a low CER and sparsity. The tuning parameter to maximize the gap statistic; however, greater sparhave been achieved by choosing the smallest tuning value within one standard deviation of the maximal c, as described in Section 3.2. Ourproposalalsohas age of generalizing to other types of clustering, as n the next section. subject to Sparse K-means - simulated data simulation 1: comparison of sparse and standard K-means i,i U 2 i,i 1. Let U optimize (14). It is not hard to show that Ui,i j d and so performing hierarchical clustering on U results in dard hierarchical clustering. So we can think of standard hi chical clustering as resulting from the criterion (14). To ob Table 3. Sparse 3-means results for Simulation 1. The mean number of nonzero feature weights resulting from the method for tuning parameter selection of Section 3.2 is shown; standard errors are given in parentheses. Note that 50 features differ between the three classes p = 50 p = 200 p = 500 p = 1000 µ = (0.895) (7.147) (31.726) (41.259) µ = (0.642) (2.514) (19.995) (17.007) µ = (0.651) (0.654) (13.491) (10.988) µ = (0.719) 200 (0) (19.96) 83.7 (9.271) µ = (0.478) 200 (0) (20.247) (14.573) fewer features with nonzero weights would result from selecting the tuning parameter at the smallest value that is within one standard deviation of the maximal gap statistic

32 Sparse K-means - simulated data simulation 2: comparison of sparse and other approaches K = 3 X ij N(µ ij, 1), µ ij = µ(1 i C1,j q 1 i C2,j q) two scenarios: small simulation: p = 25, q = 5, 10 observations per class larger simulation: p = 500, q = 50, 20 observations per class other methods: 1 COSA + PAM/heirarchical clustering, Friedman and Meulman (2004) 2 model-based clustering, Raftery and Dean (2006) 3 penalized log-likelihood, Pan and Sen (2007) 4 PCA + 3-means clustering

33 Sparse K-means - simulated data simulation 2: comparison of sparse and other approaches ibshirani: A Framework for Feature Selection in Clustering Table 4. Results for Simulation 2. The quantities reported are the mean and standard error (given in parentheses) of the CER, and of the number of nonzero coefficients, over 25 simulated datasets Simulation Method CER Num. nonzero coef. Small simulation: Sparse K-means (0.019) 8.2 (0.733) p = 25, q = 5, K-means (0.011) 25 (0) 10 obs. per class Pan and Shen (0.017) 6.72 (0.334) COSA w/hier. Clust (0.016) 25 (0) COSA w/k-medoids (0.012) 25 (0) Raftery and Dean (0.031) 22 (0.86) PCA w/k-means 0.16 (0.012) 25 (0) Large simulation: Sparse K-means (0.019) (9.561) p = 500, q = 50, K-means (0.011) 500 (0) 20 obs. per class Pan and Shen (0.013) 76 (3.821) COSA w/hier. Clust (0.011) 500 (0) COSA w/k-medoids (0.004) 500 (0) PCA w/k-means (0.006) 500 (0) the features, we modify (14) bymultiplyingeach the summation over j by a weight w j,subjecttocon- 2. Iterate until convergence: (a) Update u = Dw Dw 2.

34 Sparse Hierarchical - breast cancer data, Perou et al. (2000) 65 surgical specimens of human breast tumors 1753 genes, 496 intrinsic genes hierarchical clustering using Eigen linkage 4 classes identified, 3 samples unassigned Journal of the American Statistical Association, June Using the intrinsic gene set, hierarchical clustering was performed on all 65 observations (left panel) and on only the 62 observa ssigned to one of the four classes (right panel). Note that the classes identified using all 65 observations are largely lost i obtained using just 62 observations. The four classes are basal-like (dashed gray), Erb-B2 (solid gray), normal breast-like (d

35 mor before and after chemotherapy. The dendrogram obtained using the intrinsic gene set was used to identify four classes basal-like, Sparse Erb-B2, Hierarchical normal breast-like, and ER+ to - breast which 62 assigned cancer to the four data, classes: Perou et al. of the 65 samples belong. It was determined that the remaining three observations did not belong to any of the four classes. (2000) These four classes are not visible in the dendrogram obtained using the full set of genes, and the authors concluded that the the intrinsic genes. We performed four versions of hierarchical clustering with Eisen linkage on the 62 observations that were 1. Sparse hierarchical clustering of all 1753 genes, with the tuning parameter chosen to yield 496 nonzero genes. 2. Standard hierarchical clustering using all 1753 genes. Figure 5. Four hierarchical clustering methods were used to cluster the 62 observations that were assigned to one of four classes in Perou et al. (2000). Sparse clustering results in the best separation between the four classes. The classes are indicated as in Figure 4. The online version of this figure is in color.

36 Sparse Hierarchical - breast cancer data, Perou et al. (2000) Witten and Tibshirani: A Framework for Feature Selection in Clustering 723 Figure 6. The gap statistic was used to determine the optimal value of the tuning parameter for sparse hierarchical clustering. Left: The largest value of the gap statistic corresponds to 93 genes with nonzero weights. Right: The dendrogram corresponding to 93 nonzero weights. The classes are indicated as in Figure 4.Theonlineversionofthisfigureisincolor. 3. Standard hierarchical clustering using the 496 genes with highest marginal variance. 4. COSA hierarchical clustering using all 1753 genes. The resulting dendrograms are shown in Figure 5. Sparseclustering of all 1753 genes with the tuning parameter chosen to the high-variance genes (Figure 5). The reason for this lies in the form of the criterion (15). Though the nonzero w j s tend to correspond to genes with high marginal variances, sparse clustering does not simply cluster the genes with highest marginal variances. Rather, it weights each gene-wise dissimilarity ma-

37 abitbetterthanclusteringbasedontheintrinsicgenesonly! Figure 6 displays the result of performing the automated tuning parameter selection method. This resulted in 93 genes having Sparse Hierarchical nonzero weights. - breast cancer data, Perou et al. (2000) Figure 7 shows that the gene weights obtained using sparse clustering are highly correlated with the marginal variances of the genes. However, the results obtained from sparse clustering are different from the results obtained by simply clustering on ing parameters for the initial and complem terings were selected to yield 496 genes wi The complementary sparse clustering dend Figure 8, alongwithaplotofw 1 and w 2 ( for the initial and complementary cluster gram obtained using complementary sparse apreviouslyunknownpatterninthedata. drogram for the initial sparse clustering ca ure 5. In response to a reviewer s inquiry abou sparse hierarchical clustering, we repeated observations that belong to one of the fo formed sparse hierarchical clustering on the Asmalljitterwasaddedtotheresampledo to avoid samples having correlation one w found that for the most part, the resulting reflected the true class labels. Figure 7. For each gene, the sparse clustering weight is plotted against the marginal variance. 6. SNP DATA EXAMPL We wondered whether one could use spa der to identify distinct populations in sing morphism (SNP) data, and also to identify fer between the populations. A SNP is a in a DNA sequence at which genetic vari population. We used the publicly availab ( HapMap ) data of the International H (International HapMap Consortium 2005, Phase III SNP data for chromosome 22. We sis to three of the populations: African an U.S.A., Utah residents with European anc nese from Beijing. We used the SNPs for w

38 Sparse Hierarchical - SNP data Phase III SNP data from chromosome 22 3 populations: African ancestry in southwest U.S.A. Utah residents with European ancestry Han Chinese from Beijing n = 315, p = 17, 026

39 198 and 2809 SNPs with nonzero weights, sparse clustering resulted in slightly lower CER than standard 3-means clustering. The Sparse main improvement Hierarchical of sparse clustering over - standard SNPclus-data tering is in interpretability, since the nonzero elements of w determine the SNPs involved in the sparse clustering. We can use deviation rule described in Section 3.2 results in a tuning parameter that yields 7160 genes with nonzero weights. The fact that the gap statistic seemingly overestimates the number of features with nonzero weights may reflect the need for a more accurate method for tuning parameter selection, or it may suggest Figure 9. Left: The gap statistics obtained as a function of the number of SNPs with nonzero weights.center: The CERs obtained using sparse and standard 3-means clustering, for a range of values of the tuning parameter. Right: Sparse clustering was performed using the tuning parameter that yields 198 nonzero SNPs (this is the smallest number of SNPs that resulted in minimal CER in the center panel). Chromosome 22 was split into 500 segments of equal length. The average weights of the SNPs in each segment are shown, as a function of the nucleotide position of the segments.