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