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## Clustering Algorithms

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**Clustering Algorithms**Applications Hierarchical Clustering k -Means Algorithms CURE Algorithm**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 in some sense as close to each other as possible.**Example**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 x x x x x x x x x x x x x x x**Problems With Clustering**• Clustering in two dimensions looks easy. • Clustering small amounts of data looks easy. • And in most cases, looks are not deceiving.**The Curse of Dimensionality**• 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.**Example: Curse of Dimensionality**• Assume random points within a bounding box, e.g., values between 0 and 1 in each dimension. • In 2 dimensions: a variety of distances between 0 and 1.41. • In 10,000 dimensions, the difference in any one dimension is distributed as a triangle.**Example – Continued**• The law of large numbers applies. • Actual distance between two random points is the sqrt of the sum of squares of essentially the same set of differences.**Example High-Dimension Application: SkyCat**• 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 Sky Survey is a newer, better version.**Example: Clustering CD’s (Collaborative Filtering)**• Intuitively: music divides into categories, and customers prefer a few categories. • But what are categories really? • Represent a CD by the customers who bought it. • Similar CD’s have similar sets of customers, and vice-versa.**The Space of CD’s**• Think of a space with one dimension for each customer. • Values in a dimension may be 0 or 1 only. • A CD’s point in this space is (x1, x2,…, xk), where xi = 1 iff the i th customer bought the CD. • Compare with boolean matrix: rows = customers; cols. = CD’s.**Space of CD’s – (2)**• For Amazon, the dimension count is tens of millions. • An alternative: use minhashing/LSH to get Jaccard similarity between “close” CD’s. • 1 minus Jaccard similarity can serve as a (non-Euclidean) distance.**Example: Clustering Documents**• Represent a document by a vector (x1, x2,…, xk), where xi = 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.**Aside: Cosine, Jaccard, and Euclidean Distances**• As with CD’s 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.**Example: DNA Sequences**• Objects are sequences of {C,A,T,G}. • Distance between sequences is edit distance, the minimum number of inserts and deletes needed to turn one into the other. • Note there is a “distance,” but no convenient space in which points “live.”**Methods of Clustering**• Hierarchical (Agglomerative): • Initially, each point in cluster by itself. • Repeatedly combine the two “nearest” clusters into one. • Point Assignment: • Maintain a set of clusters. • Place points into their “nearest” cluster.**Hierarchical Clustering**• Two important questions: • How do you determine the “nearness” of clusters? • How do you represent a cluster of more than one point?**Hierarchical Clustering – (2)**• Key problem: as you build 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 points. • Measure intercluster distances by distances of centroids.**Example**(5,3) o (1,2) o o (2,1) o (4,1) o (0,0) o (5,0) x (1.5,1.5) x (4.7,1.3) x (1,1) x (4.5,0.5)**And in 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: clustroid = point “closest” to other points. • Treat clustroid as if it were centroid, when computing intercluster distances.**“Closest” Point?**• Possible meanings: • Smallest maximum distance to the other points. • Smallest average distance to other points. • Smallest sum of squares of distances to other points. • Etc., etc.**Example**clustroid 1 2 6 4 3 clustroid 5 intercluster distance**Other Approaches to Defining “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.**Cohesion**• Approach 1: Use the diameter of the merged cluster = maximum distance between points in the cluster. • Approach 2: Use the average distance between points in the cluster.**Cohesion – (2)**• Approach 3: Use a density-based approach: take the diameter or average distance, e.g., and divide by the number of points in the cluster. • Perhaps raise the number of points to a power first, e.g., square-root.**k – Means Algorithm(s)**• Assumes Euclidean space. • 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.**Populating Clusters**• For each point, place it in the cluster whose current centroid it is nearest. • After all points are assigned, fix the centroids of the k clusters. • Optional: reassign all points to their closest centroid. • Sometimes moves points between clusters.**Reassigned**points Clusters after first round Example: Assigning Clusters 2 4 x 6 3 1 8 7 5 x**Best value**of k Average distance to centroid k Getting k Right • 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.**Too few;**many long distances to centroid. Example: Picking k 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 x x x x x x x x x x x x x x x**Just right;**distances rather short. Example: Picking k 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 x x x x x x x x x x x x x x x**Too many;**little improvement in average distance. Example: Picking k 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 x x x x x x x x x x x x x x x**BFR Algorithm**• BFR (Bradley-Fayyad-Reina) is a variant of k -means designed to handle very large (disk-resident) data sets. • It assumes that clusters are normally distributed around a centroid in a Euclidean space. • Standard deviations in different dimensions may vary.**BFR – (2)**• Points are read 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.**Initialization: k -Means**• Possibilities include: • 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.**Three Classes of Points**• The discard set : points close enough to a centroid to be summarized. • The compression set : groups of points that are close together but not close to any centroid. They are summarized, but not assigned to a cluster. • The retained set : isolated points.**Summarizing Sets of Points**• For each cluster, the discard set is summarized by: • The number of points, N. • The vector SUM, whose ith component is the sum of the coordinates of the points in the ith dimension. • The vector SUMSQ: ith component = sum of squares of coordinates in ith dimension.**Comments**• 2d + 1 values represent any number of points. • d = number of dimensions. • Averages in each dimension (centroid coordinates) can be calculated easily as SUMi/N. • SUMi = ith component of SUM.**Comments – (2)**• Variance of a cluster’s discard set in dimension i can be computed by: (SUMSQi /N ) – (SUMi /N )2 • And the standard deviation is the square root of that. • The same statistics can represent any compression set.**Points in**the RS Compressed sets. Their points are in the CS. A cluster. Its points are in the DS. The centroid “Galaxies” Picture**Processing a “Memory-Load” of Points**• Find those points that are “sufficiently close” to a cluster centroid; add those points to that cluster and the DS. • 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.**Processing – (2)**• Adjust statistics of the clusters to account for the new points. • Add N’s, SUM’s, SUMSQ’s. • Consider merging compressed sets in the CS. • If this is the last round, merge all compressed sets in the CS and all RS points into their nearest cluster.**A Few Details . . .**• How do we decide if a point is “close enough” to a cluster that we will add the point to that cluster? • How do we decide whether two compressed sets deserve to be combined into one?**How Close is Close Enough?**• We need a way to decide whether to put a new point into a cluster. • BFR suggest two ways: • TheMahalanobis distance is less than a threshold. • Low likelihood of the currently nearest centroid changing.**Mahalanobis Distance**• Normalized Euclidean distance from centroid. • For point (x1,…,xk) and centroid (c1,…,ck): • Normalize in each dimension: yi = (xi -ci)/i • Take sum of the squares of the yi ’s. • Take the square root.**Mahalanobis Distance – (2)**• If clusters are normally distributed in d dimensions, then after transformation, one standard deviation = d. • I.e., 70% of the points of the cluster will have a Mahalanobis distance < d. • Accept a point for a cluster if its M.D. is < some threshold, e.g. 4 standard deviations.**Should Two 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 variance is below some threshold. • Many alternatives: treat dimensions differently, consider density.**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: • Assumes a Euclidean distance. • Allows clusters to assume any shape.**Example: Stanford Faculty Salaries**h h h e e e h e e e h e e e e h e salary h h h h h h h age**Starting CURE**• Pick a random sample of points that fit in main memory. • Cluster these points hierarchically – group nearest points/clusters. • 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.