The Problem of Clustering

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1.Clustering Shannon Quinn (with thanks to J . Leskovec , A. Rajaraman , and J . Ullman of Stanford University)

2.High Dimensional Data Given a cloud of data points we want to understand its structure J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 2

3.3 The Problem of Clustering Given a set of points , with a notion of distance between points , group the points into some number of clusters , so that Members of a cluster are close/similar to each other Members of different clusters are dissimilar Usually: Points are in a high-dimensional space Similarity is defined using a distance measure Euclidean, Cosine, Jaccard , edit distance, … J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

4.4 Example: Clusters & Outliers x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x Outlier Cluster

5.Clustering is a hard problem! J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 5

6.6 Why is it hard? Clustering in two dimensions looks easy Clustering small amounts of data looks easy And in most cases, looks are not deceiving Many applications involve not 2, but 10 or 10,000 dimensions High-dimensional spaces look different: Almost all pairs of points are at about the same distance J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

7.Curse of dimensionality “Vastness” of Euclidean space http:// link.springer.com / referenceworkentry /10.1007%2F978-0-387-30164-8_192

8.Clustering Problem: Galaxies A catalog of 2 billion “sky objects” represents objects by their radiation in 7 dimensions (frequency bands) Problem: Cluster into similar objects, e.g., galaxies, nearby stars, quasars, etc. Sloan Digital Sky Survey J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 8

9.Clustering Problem: Music CDs Intuitively: Music divides into categories, and customers prefer a few categories But what are categories really ? Represent a CD by a set of customers who bought it: Similar CDs have similar sets of customers, and vice-versa 9 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

10.Clustering Problem: Music CDs Space of all CDs: Think of a space with one dim. for each customer Values in a dimension may be 0 or 1 only A CD is a point in this space ( x 1 , x 2 ,…, x k ), where x i = 1 iff the i th customer bought the CD For Amazon, the dimension is tens of millions Task: Find clusters of similar CDs J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 10

11.Clustering Problem: Documents Finding topics: Represent a document by a vector ( x 1 , x 2 ,…, x k ), where x i = 1 iff the i th word ( in some order) appears in the document It actually doesn’t matter if k is infinite; i.e., we don’t limit the set of words Documents with similar sets of words may be about the same topic 11 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

12.Cosine, Jaccard, and Euclidean As with CDs we have a choice when we think of documents as sets of words or shingles: Sets as vectors: Measure similarity by the cosine distance Sets as sets: Measure similarity by the Jaccard distance Sets as points: Measure similarity by Euclidean distance J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 12

13.13 Overview: Methods of Clustering Hierarchical: Agglomerative (bottom up): Initially, each point is a cluster Repeatedly combine the two “ nearest” clusters into one Divisive (top down): Start with one cluster and recursively split it Point assignment: Maintain a set of clusters Points belong to “nearest” cluster J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

14.Hierarchical Clustering Key operation: Repeatedly combine two nearest clusters Three important questions: 1) How do you represent a cluster of more than one point? 2) How do you determine the “nearness” of clusters? 3) When to stop combining clusters? J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 14

15.Hierarchical Clustering Key operation: Repeatedly combine two nearest clusters (1 ) How to represent a cluster of many points? Key problem: As you merge clusters, how do you represent the “location” of each cluster, to tell which pair of clusters is closest? Euclidean case: each cluster has a centroid = average of its (data)points ( 2 ) How to determine “ nearness” of clusters? Measure cluster distances by distances of centroids J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 15

16.16 Example: Hierarchical clustering (5,3) o (1,2) o o (2,1) o (4,1) o (0,0 ) o (5,0) x (1.5,1.5) x (4.5,0.5) x (1,1) x (4.7,1.3) Data: o … data point x … centroid Dendrogram J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

17.And in the Non-Euclidean Case? What about the Non-Euclidean case? The only “locations” we can talk about are the points themselves i.e ., there is no “average” of two points Approach 1: (1) How to represent a cluster of many points? clustroid = (data)point “ closest ” to other points (2 ) How do you determine the “nearness” of clusters? Treat clustroid as if it were centroid, when computing inter-cluster distances 17 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

18.“Closest” Point? (1) How to represent a cluster of many points? clustroid = point “ closest ” to other points Possible meanings of “closest”: Smallest maximum distance to other points Smallest average distance to other points Smallest sum of squares of distances to other points For distance metric d clustroid c of cluster C is: J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 18 Centroid is the avg. of all (data)points in the cluster. This means centroid is an “artificial” point. Clustroid is an existing (data)point that is “closest” to all other points in the cluster. X Cluster on 3 datapoints Centroid Clustroid Datapoint

19.Defining “Nearness” of Clusters (2) How do you determine the “nearness” of clusters? Approach 2: Intercluster distance = minimum of the distances between any two points, one from each cluster Approach 3: Pick a notion of “ cohesion ” of clusters, e.g. , maximum distance from the clustroid Merge clusters whose union is most cohesive 19 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

20.Cohesion Approach 3.1: Use the diameter of the merged cluster = maximum distance between points in the cluster Approach 3.2: Use the average distance between points in the cluster Approach 3.3: Use a density-based approach Take the diameter or avg. distance, e.g., and divide by the number of points in the cluster J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 20

21.Implementation Naïve implementation of hierarchical clustering: At each step, compute pairwise distances between all pairs of clusters, then merge O( N 3 ) Careful implementation using priority queue can reduce time to O( N 2 log N ) Still too expensive for really big datasets that do not fit in memory J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 21

22.k –means Algorithm(s) Assumes Euclidean space/distance Start by picking k , the number of clusters Initialize clusters by picking one point per cluster Example: Pick one point at random, then k -1 other points, each as far away as possible from the previous points 22 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

23.Populating Clusters 1) For each point, place it in the cluster whose current centroid it is nearest 2) After all points are assigned, update the locations of centroids of the k clusters 3) Reassign all points to their closest centroid Sometimes moves points between clusters Repeat 2 and 3 until convergence Convergence: Points don’t move between clusters and centroids stabilize J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 23

24.K-Means Clustering Example

25.K-Means Clustering Example

26.K-Means Clustering Example

27.K-Means Clustering Example

28.K-Means Clustering Example

29.K-Means Clustering Example

30.K-Means Clustering Example

31.K-Means Clustering Example

32.K-Means Clustering Example

33.Getting the k right How to select k ? Try different k , looking at the change in the average distance to centroid as k increases Average falls rapidly until right k , then changes little 33 k Average distance to centroid Best value of k J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

34.Example: Picking k J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 34 x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too few; many long distances to centroid .

35.Example: Picking k J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 35 x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Just right; distances rather short.

36.Example: Picking k J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 36 x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too many; little improvement in average distance.

37.More K-means examples http:// www.naftaliharris.com /blog/visualizing-k-means- clustering/

38.BFR Algorithm BFR [Bradley-Fayyad-Reina] is a variant of k -means designed to handle very large ( disk-resident) data sets Assumes that clusters are normally distributed around a centroid in a Euclidean space Standard deviations in different dimensions may vary Clusters are axis-aligned ellipses Efficient way to summarize clusters (want memory required O(clusters) and not O(data )) 38 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

39.BFR Algorithm Points are read from disk one main-memory-full at a time Most points from previous memory loads are summarized by simple statistics To begin, from the initial load we select the initial k centroids by some sensible approach: Take k random points Take a small random sample and cluster optimally Take a sample; pick a random point, and then k–1 more points, each as far from the previously selected points as possible 39 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

40.Three Classes of Points 3 sets of points which we keep track of: Discard set (DS): Points close enough to a centroid to be summarized Compression set (CS): Groups of points that are close together but not close to any existing centroid These points are summarized, but not assigned to a cluster Retained set (RS): Isolated points waiting to be assigned to a compression set J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 40

41.BFR: “Galaxies ” Picture J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 41 A cluster. Its points are in the DS . The centroid Compressed sets. Their points are in the CS . Points in the RS Discard set (DS): Close enough to a centroid to be summarized Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points

42.Summarizing Sets of Points For each cluster, the discard set (DS) is summarized by: The number of points, N The vector SUM , whose i th component is the sum of the coordinates of the points in the i th dimension The vector SUMSQ : i th component = sum of squares of coordinates in i th dimension J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 42 A cluster. All its points are in the DS . The centroid

43.Summarizing Points: Comments 2 d + 1 values represent any size cluster d = number of dimensions Average in each dimension ( the centroid ) can be calculated as SUM i / N SUM i = i th component of SUM Variance of a cluster’s discard set in dimension i is: ( SUMSQ i / N ) – ( SUM i / N ) 2 And standard deviation is the square root of that Next step: Actual clustering 43 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org Note: Dropping the “axis-aligned” clusters assumption would require storing full covariance matrix to summarize the cluster. So, instead of SUMSQ being a d -dim vector, it would be a d x d matrix, which is too big!

44.The “Memory-Load” of Points Processing the “Memory-Load” of points (1): 1) Find those points that are “ sufficiently close ” to a cluster centroid and add those points to that cluster and the DS These points are so close to the centroid that they can be summarized and then discarded 2) Use any main-memory clustering algorithm to cluster the remaining points and the old RS Clusters go to the CS ; outlying points to the RS J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 44 Discard set (DS): Close enough to a centroid to be summarized. Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points

45.The “Memory-Load” of Points Processing the “Memory-Load” of points (2): 3) DS set: Adjust statistics of the clusters to account for the new points Add N s, SUM s, SUMSQ s 4) Consider merging compressed sets in the CS 5) If this is the last round, merge all compressed sets in the CS and all RS points into their nearest cluster J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 45 Discard set (DS): Close enough to a centroid to be summarized. Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points

46.BFR: “Galaxies ” Picture J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 46 A cluster. Its points are in the DS . The centroid Compressed sets. Their points are in the CS . Points in the RS Discard set (DS): Close enough to a centroid to be summarized Compression set (CS): Summarized, but not assigned to a cluster Retained set (RS): Isolated points

47.A Few Details… Q1) How do we decide if a point is “close enough” to a cluster that we will add the point to that cluster ? Q2) How do we decide whether two compressed sets (CS) deserve to be combined into one? 47 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

48.How Close is Close Enough? Q1) We need a way to decide whether to put a new point into a cluster (and discard) BFR suggests two ways: The Mahalanobis distance is less than a threshold High likelihood of the point belonging to currently nearest centroid J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 48

49.Mahalanobis Distance Normalized Euclidean distance from centroid For point (x 1 , …, x d ) and centroid (c 1 , …, c d ) Normalize in each dimension: y i = (x i - c i ) /  i Take sum of the squares of the y i Take the square root   J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 49 σ i … standard deviation of points in the cluster in the i th dimension

50.Mahalanobis Distance If clusters are normally distributed in d dimensions, then after transformation, one standard deviation = i.e ., 68% of the points of the cluster will have a Mahalanobis distance Accept a point for a cluster if its M.D. is < some threshold, e.g . 2 standard deviations   50 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

51.Picture: Equal M.D. Regions Euclidean vs. Mahalanobis distance J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 51 Contours of equidistant points from the origin Uniformly distributed points, Euclidean distance Normally distributed points, Euclidean distance Normally distributed points, Mahalanobis distance

52.Should 2 CS clusters be combined ? Q2) Should 2 CS subclusters be combined? Compute the variance of the combined subcluster N , SUM , and SUMSQ allow us to make that calculation quickly Combine if the combined variance is below some threshold Many alternatives: Treat dimensions differently, consider density J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 52

53.The CURE Algorithm Problem with BFR/ k -means: Assumes clusters are normally distributed in each dimension And axes are fixed – ellipses at an angle are not OK CURE (Clustering Using REpresentatives ): Assumes a Euclidean distance Allows clusters to assume any shape Uses a collection of representative points to represent clusters 53 Vs. J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

54.Example: Stanford Salaries 54 e e e e e e e e e e e h h h h h h h h h h h h h salary age J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

55.Starting CURE 2 Pass algorithm. Pass 1: 0) Pick a random sample of points that fit in main memory 1) Initial clusters: Cluster these points hierarchically – group nearest points/clusters 2) Pick representative points: For each cluster, pick a sample of points, as dispersed as possible From the sample, pick representatives by moving them (say) 20% toward the centroid of the cluster J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 55

56.Example: Initial Clusters 56 e e e e e e e e e e e h h h h h h h h h h h h h salary age J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

57.Example: Pick Dispersed Points 57 e e e e e e e e e e e h h h h h h h h h h h h h salary age Pick (say) 4 remote points for each cluster. J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

58.Example: Pick Dispersed Points 58 e e e e e e e e e e e h h h h h h h h h h h h h salary age Move points (say) 20% toward the centroid. J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

59.Finishing CURE Pass 2: Now , rescan the whole dataset and visit each point p in the data set Place it in the “ closest cluster ” Normal definition of “ closest ”: Find the closest representative to p and assign it to representative’s cluster 59 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org p

60.Summary Clustering: Given a set of points , with a notion of distance between points, group the points into some number of clusters Algorithms: Agglomerative hierarchical clustering : Centroid and clustroid k -means: Initialization, picking k BFR CURE J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org 60

61.Assignment 2! (to be released next week) K-Means on Hadoop !

62.Spectral clustering Tune in next week!