Overview of clustering analysis. Yuehua Cui

Transcription

1 Overview of clustering analysis Yuehua Cui

2 A data set with clear cluster structure How would you design an algorithm for finding the three clusters in this case?

3 Yahoo! Hierarchy isn t clustering but is the kind of output you want from clustering (30) agriculture biology physics CS space dairy crops botany cell forestry agronomy evolution magnetism relativity AI HCI courses craft missions

4 Hierarchical Clustering Dendrogram Venn Diagram of Clustered Data From

5 Hierarchical clustering may be represented by a two dimensional diagram known as dendrogram which illustrates the fusions or divisions made at each successive stage of analysis. An example of such a dendrogram is given below:

6 Nearest Neighbor Algorithm Nearest Neighbor Algorithm is an agglomerative approach (bottom-up). Starts with n nodes (n is the size of our sample), merges the 2 most similar nodes at each step, and stops when the desired number of clusters is reached. From

7 Nearest Neighbor, Level 2, k = 7 clusters. From

8 Nearest Neighbor, Level 3, k = 6 clusters.

9 Nearest Neighbor, Level 4, k = 5 clusters.

10 Nearest Neighbor, Level 5, k = 4 clusters.

11 Nearest Neighbor, Level 6, k = 3 clusters.

12 Nearest Neighbor, Level 7, k = 2 clusters.

13 Nearest Neighbor, Level 8, k = 1 cluster.

14 Hierarchical Clustering Calculate the similarity between all possible combinations of two profiles Keys Similarity Clustering Two most similar clusters are grouped together to form a new cluster Calculate the similarity between the new cluster and all remaining clusters. Adapted from P. Hong

15 Similarity Measurements Pearson Correlation Two profiles (vectors) x x 1 x N and y y 1 y N r pearson ( x, y) N i1 ( x x)( y y) i N 2 N 2 ( x ) ( ) 1 i x y i i 1 i y i x y 1 N N 1 N N n1 n1 x y n n A measure of the strength of linear dependence between two variables.

16 Similarity Measurements Spearman correlation Spearman's rank correlation coefficient or Spearman's rho is a non-parametric measure of statistical dependence between two variables. It measures how well the relationship between two variables can be described by a monotonic function. It is often thought of as being the Pearson correlation coefficient between the ranked variables. The n paired observation (X i, Y i ) are converted to ranks (R xi, R yi ), and the differences d i = R xi - R yi are calculated. If there are no tied ranks, then ρ is given by:

17 Similarity Measurements Cosine Correlation x N x x 1 y x y x N y x C N i i i 1 cosine 1 ), ( y N y y 1 y x -1 Cosine Correlation + 1 y x Adapted from P. Hong

18 Similarity Measurements Euclidean Distance Other distance measures such as the Mahalanobis distance N n n x n y y x d 1 2 ) ( ), ( x N x x 1 y N y y 1

19 Clustering Agglomerative techniques C 1 C 2 Merge which pair of clusters? C 3 Adapted from P. Hong

20 Clustering Single Linkage, also called nearest neighbor technique + Dissimilarity between two clusters = Minimum dissimilarity between the members of two clusters + C 1 C 2 D(r,s) = Min { d(i,j) : Where object i is in cluster r and object j is cluster s } At each stage of hierarchical clustering, the clusters r and s, for which D(r,s) is minimum, are merged. Tend to generate long chains Adapted from P. Hong

21 Clustering Complete Linkage: also termed farthest neighbor + + C 2 Dissimilarity between two clusters = Maximum dissimilarity between the members of two clusters D(r,s) = Max { d(i,j) : Where object i is in cluster r and object j is cluster s } C 1 At each stage of hierarchical clustering, the clusters r and s, for which D(r,s) is minimum, are merged. Tend to generate clumps Adapted from P. Hong

22 Clustering Average Linkage + Dissimilarity between two clusters = Averaged distances of all pairs of objects (one from each cluster). + C 1 C 2 D(r,s) = T rs / ( N r * N s ) Where T rs is the sum of all pairwise distances between cluster r and cluster s. N r and N s are the sizes of the clusters r and s respectively. Adapted from P. Hong

23 Clustering Average Group Linkage + Dissimilarity between two clusters = Distance between two cluster means. + Also termed centroid linkage C 2 C 1 Adapted from P. Hong

24 Single-Link Method b a c b a d c b Distance Matrix Euclidean Distance 4 5 3, c b a d c c b a d c b 4,, c b a d (1) (2) (3) a,b,c c c d a,b d d a,b,c,d

25 Complete-Link Method b a c b a d c b Distance Matrix Euclidean Distance 4 6 5, c b a d c c b a d c b 6,, b a d c (1) (2) (3) a,b c c d a,b d c,d a,b,c,d

26 Compare Dendrograms Single-Link Complete-Link a b c d a b c d

27 Other clustering method K-means clustering (a type of partitional clustering) SOM: self-organization map Principle component analysis Information visualization for exploring similarities or dissimilarities in data Multidimensional scaling (MDS)

28 K-means algorithm Given k, the k-means algorithm works as follows: 1)Randomly choose k data points (seeds) to be the initial centroids, cluster centers 2)Assign each data point to the closest centroid 3)Re-compute the centroids using the current cluster memberships. 4)If a convergence criterion is not met, go to 2). 28

29 The K-Means Clustering Method Example

30 Comments on the K-Means Method Strength Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms Weakness Applicable only when mean is defined, then what about categorical data? Need to specify k, the number of clusters, in advance Unable to handle noisy data and outliers Not suitable to discover clusters with non-convex shapes

31 How to choose a clustering algorithm Choosing the best algorithm is a challenge. Every algorithm has limitations and works well with certain data distributions. It is very hard, if not impossible, to know what distribution the application data follow. The data may not fully follow any ideal structure or distribution required by the algorithms. One also needs to decide how to standardize the data, to choose a suitable distance function and to select other parameter values. Due to these complexities, the common practice is to run several algorithms using different distance functions and parameter settings, and then carefully analyze and compare the results. The interpretation of the results must be based on insight into the meaning of the original data together with knowledge of the algorithms used. Clustering is highly application dependent and to certain extent subjective (personal preferences). 31

32 Principal Component Analysis

33 Problem: Too much data! Pat1 Pat2 Pat3 Pat4 Pat5 Pat6 Pat7 Pat8 Pat _at _at _s_at _at _at _at _at _x_at _at _s_at _s_at _at _s_at _s_at _x_at _at _x_at _s_at _s_at _at _s_at _at _at _at _at _s_at _s_at _s_at _at _at _at _at _at

34 Why Dimension Reduction Computation: The complexity grows exponentially with the dimension. Visualization: projection of high-dimensional data to 2D or 3D. Interpretation: the intrinsic dimension maybe small.

35 Dimensional Reduction vs feature selection Reduction of dimensions Principle Component Analysis (PCA) Feature selection (gene selection) Significant genes: t-test Selection of a limited number of genes

36 Principal Component Analysis (PCA) Used for visualization of complex data Developed to capture as much of the variation in data as possible Generic features of principal components summary variables linear combinations of the original variables uncorrelated with each other capture as much of the original variance as possible

37 Principal components 1. principal component (PC1) the direction along which there is greatest variation 2. principal component (PC2) the direction with maximum variation left in data, orthogonal to the direction (i.e. vector) of PC1 3. principal component (PC3) the direction with maximal variation left in data, orthogonal to the plane of PC1 and PC2 (Rarely used) etc...

38 Philosophy of PCA A PCA is concerned with explaining the variancecovariance sturcture of a set of variables through a few linear combinations. We typically have a data matrix of n observations on p correlated variables x 1,x 2, x p PCA looks for a transformation of the x i into p new variables y i that are uncorrelated. Want to present x 1,x 2, x p with a few y i s without lossing much information.

39 PCA Looking for a transformation of the data matrix X (nxp) such that Y= T X= 1 X X p X p Where =( 1, 2,.., p ) T is a column vector of wheights with 1 ²+ 2 ²+..+ p ²=1

40 Maximize the variance of the projection of the observations on the Y variables Find so that Var( T X)= T Var(X) is maximal Var(X) is the covariance matrix of the X i variables

41 Eigen Vector and Eigen Value A square matrix A is said to have eigenvalue l, with corresponding eigenvector x ¹ 0, if æ è ç ö ø æ è ç 3 2 Ax = lx. ö ø = æ 12 è ç 8 ö ø = 4 æ 3 è ç 2 Only square matrices have eigenvectors, but not all square matrices have eigen vectors. All eigen vectors are orthogonal to each other. ö ø

42 PCA Result: let be the covariance matix associated with the random vector X ' [ X, X,... X ]. 1 2 p Let have the eigenvalue-eigenvector pairs (, e ),...,(, e ) where p p 1 2 p (, e 1 1 ), Then the ith principle component is given by Y e' X e X e X... e X, i=1,2,...,p. i i i1 1 i 2 2 ip p

43 And so.. We find that The direction of is given by the eigenvector 1 correponding to the largest eigenvalue of matrix Σ. The second vector that is orthogonal (uncorrelated) to the first is the one that has the second highest variance which comes to be the eignevector corresponding to the second eigenvalue And so on

44 So PCA gives New variables Y i that are linear combination of the original variables (x i ): Y i = e i1 x 1 +e i2 x 2 + e ip x p ; i=1..p The new variables Y i are derived in decreasing order of importance; they are called principal components

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46 Scale before PCA PCA is sensitive to scale PCA should be applied on data that have approximately the same scale in each variable

47 Result: Let X T = é ëx 1, X 2,..., X p eigenvalue-eigen vector pairs ( l 1,e 1 ),..., l p,e p Let Y i = e i ' X be the PCs. Then s 11 + s s pp = æ ç ç ç è ç p åvar X i i=1 Proportion of total population variance due to the k th principle component ù û have covariance matrix S, with ( ), where l 1 ³ l 2 ³... ³ l p ³ 0. ( ) = l 1 + l l p = ö = l k p, k = 1,.., p. ål i ø i=1 p åvar Y i i=1 ( )

48 How many PCAs to keep

49 PCA application: genomic study Population stratification: allele frequency differences between cases and controls due to systematic ancestry differences can cause spurious associations in disease studies. PCA could be used to infer underlying population structure.

50 Figure 2 Nature Genetics 38, (2006) Principal components analysis corrects for stratification in genome-wide association studies Alkes L Price, Nick J Patterson, Robert M Plenge, Michael E Weinblatt, Nancy A Shadick & David Reic

51 Chao Tian, Peter K. Gregersen and Michael F. Seldin. (2008) Accounting for ancestry: population substructure and genomewide association studies.

52 Example: 3 dimensions 2 dimensions

53 PCA on all Genes Leukemia data, precursor B and T Plot of 34 patients, 8973 dimensions (genes) reduced to 2

54 Variance (%) Variance retained, variance lost PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10

55 PCA on 100 top significant genes after DE gene selection Plot of 34 patients, 100 dimensions (genes) reduced to 2