Algorithms
Applications Clustering Hierarchical Clustering k -Means Algorithms CURE Algorithm
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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.
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Example
x x x x x x x x xx x x x x x x x x xx x x x x x x x xx x x
x x x x x x x x x x
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Problems With Clustering
Clustering in two dimensions looks easy. Clustering small amounts of data looks easy. And in most cases, looks are not deceiving.
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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.
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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.
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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.
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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.
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Example: Clustering CDs (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 CDs have similar sets of customers, and vice-versa.
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The Space of CDs
Think of a space with one dimension for each customer.
Values in a dimension may be 0 or 1 only.
A CDs point in this space is (x1, x2,, xk), where xi = 1 iff the i customer bought the CD.
th
Compare with boolean matrix: rows = customers; cols. = CDs.
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Space of CDs (2)
For Amazon, the dimension count is tens of millions. An alternative: use minhashing/LSH to get Jaccard similarity between close CDs. 1 minus Jaccard similarity can serve as a (non-Euclidean) distance.
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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 doesnt matter if k is infinite; i.e., we dont limit the set of words.
Documents with similar sets of words may be about the same topic.
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Aside: Cosine, Jaccard, and Euclidean Distances
As with CDs we have a choice when we think of documents as sets of words or shingles:
1. Sets as vectors: measure similarity by the cosine distance. 2. Sets as sets: measure similarity by the Jaccard distance. 3. Sets as points: measure similarity by Euclidean distance.
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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.
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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.
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Hierarchical Clustering
Two important questions:
1. How do you determine the nearness of clusters? 2. How do you represent a cluster of more than one point?
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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.
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Example
(5,3) o (1,2) o x (1.5,1.5) x (1,1) o (2,1) o (0,0) x (4.7,1.3) o (4,1) x (4.5,0.5) o (5,0)
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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.
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Closest Point?
Possible meanings:
1. Smallest maximum distance to the other points. 2. Smallest average distance to other points. 3. Smallest sum of squares of distances to other points. 4. Etc., etc.
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Example
clustroid 1 3 intercluster distance
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2 6 5 4 clustroid
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.
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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.
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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.
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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.
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Populating Clusters
1. For each point, place it in the cluster whose current centroid it is nearest. 2. After all points are assigned, fix the centroids of the k clusters. 3. Optional: reassign all points to their closest centroid.
Sometimes moves points between clusters.
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Example: Clusters
Reassigned Assigning points x 6 7 5 x 3 1 8 2 4
Clusters after first round
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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.
Average distance to centroid Best value of k
k
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Example: Picking k
Too few; many long distances to centroid. x x x x x x x x xx x x x x x x x x xx x x x x x x x xx x x
x x x x x x x x x x
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Example: Picking k
Just right; distances rather short. x x x x x x x x xx x x x x x x x x xx x x x x x x x xx x x
x x x x x x x x x x
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Example: Picking k
Too many; little improvement in average x distance. x x x x xx x x x x x x x xx x x x x x x xx x x x x x x x
x x x x x x x x x x
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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.
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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.
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Initialization: k -Means
Possibilities include:
1. Take a small random sample and cluster optimally. 2. Take a sample; pick a random point, and then k 1 more points, each as far from the previously selected points as possible.
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Three Classes of Points
1. The discard set : points close enough to a centroid to be summarized. 2. 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. 3. The retained set : isolated points.
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Summarizing Sets of Points
For each cluster, the discard set is summarized by:
1. The number of points, N. 2. The vector SUM, whose i th component is the sum of the coordinates of the points in the i th dimension. 3. The vector SUMSQ: i th component = sum of squares of coordinates in i th dimension.
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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 = i
th
component of SUM.
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Comments (2)
Variance of a clusters 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.
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Galaxies Picture
Points in the RS Compressed sets. Their points are in the CS.
A cluster. Its points are in the DS.
The centroid
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Processing a Memory-Load of Points
1. Find those points that are sufficiently close to a cluster centroid; add those points to that cluster and the DS. 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.
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Processing (2)
3. Adjust statistics of the clusters to account for the new points.
Add Ns, SUMs, SUMSQs.
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.
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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?
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How Close is Close E...