cluster analysis: basic concepts and algorithms
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Cluster Analysis: Basic Concepts and Algorithms. Inter-cluster distances are maximized. Intra-cluster distances are minimized. What is Cluster Analysis?. - PowerPoint PPT PresentationTRANSCRIPT
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Cluster Analysis: Basic Concepts
and Algorithms
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What is Cluster Analysis? Finding groups of objects such that the objects in a group
will be similar (or related) to one another and different from (or unrelated to) the objects in other groups
Inter-cluster distances are maximized
Intra-cluster distances are
minimized
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Applications of Cluster Analysis Understanding
Group related documents for browsing, group genes and proteins that have similar functionality, or group stocks with similar price fluctuations
Summarization Reduce the size of large
data sets
Discovered Clusters Industry Group
1 Applied-Matl-DOWN,Bay-Network-Down,3-COM-DOWN,
Cabletron-Sys-DOWN,CISCO-DOWN,HP-DOWN, DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN,
Micron-Tech-DOWN,Texas-Inst-Down,Tellabs-Inc-Down, Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN,
Sun-DOWN
Technology1-DOWN
2 Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN,
ADV-Micro-Device-DOWN,Andrew-Corp-DOWN, Computer-Assoc-DOWN,Circuit-City-DOWN,
Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN, Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN
Technology2-DOWN
3 Fannie-Mae-DOWN,Fed-Home-Loan-DOWN, MBNA-Corp-DOWN,Morgan-Stanley-DOWN
Financial-DOWN
4 Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP,
Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP, Schlumberger-UP
Oil-UP
Clustering precipitation in Australia
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What is not Cluster Analysis?
Supervised classification Have class label information
Simple segmentation Dividing students into different registration groups
alphabetically, by last name
Results of a query Groupings are a result of an external specification
Graph partitioning Some mutual relevance and synergy, but areas are
not identical
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Notion of a Cluster can be Ambiguous
How many clusters?
Four Clusters Two Clusters
Six Clusters
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Types of Clusterings A clustering is a set of clusters
Important distinction between hierarchical and partitional sets of clusters
Partitional Clustering A division data objects into non-overlapping subsets
(clusters) such that each data object is in exactly one subset
Hierarchical clustering A set of nested clusters organized as a hierarchical tree
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Partitional Clustering
Original Points A Partitional Clustering
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Hierarchical Clustering
p4p1
p3
p2
p4 p1
p3
p2
p4p1 p2 p3
p4p1 p2 p3
Traditional Hierarchical Clustering
Non-traditional Hierarchical Clustering Non-traditional Dendrogram
Traditional Dendrogram
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Other Distinctions Between Sets of Clusters
Exclusive versus non-exclusive In non-exclusive clusterings, points may belong to
multiple clusters. Can represent multiple classes or ‘border’ points
Fuzzy versus non-fuzzy In fuzzy clustering, a point belongs to every cluster
with some weight between 0 and 1 Weights must sum to 1 Probabilistic clustering has similar characteristics
Partial versus complete In some cases, we only want to cluster some of the
data Heterogeneous versus homogeneous
Cluster of widely different sizes, shapes, and densities
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Types of Clusters Well-separated clusters
Center-based clusters
Contiguous clusters
Density-based clusters
Property or Conceptual
Described by an Objective Function
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Types of Clusters: Well-Separated
Well-Separated Clusters: A cluster is a set of points such that any point in a
cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster.
3 well-separated clusters
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Types of Clusters: Center-Based
Center-based A cluster is a set of objects such that an object in a
cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster
The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most “representative” point of a cluster
4 center-based clusters
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Types of Clusters: Contiguity-Based
Contiguous Cluster (Nearest neighbor or Transitive) A cluster is a set of points such that a point in a
cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster.
8 contiguous clusters
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Types of Clusters: Density-Based
Density-based A cluster is a dense region of points, which is
separated by low-density regions, from other regions of high density.
Used when the clusters are irregular or intertwined, and when noise and outliers are present.
6 density-based clusters
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Types of Clusters: Conceptual Clusters
Shared Property or Conceptual Clusters Finds clusters that share some common property or
represent a particular concept. .
2 Overlapping Circles
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Types of Clusters: Objective Function Clusters Defined by an Objective Function
Finds clusters that minimize or maximize an objective function.
Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard)
Can have global or local objectives. Hierarchical clustering algorithms typically have local
objectives Partitional algorithms typically have global objectives
A variation of the global objective function approach is to fit the data to a parameterized model.
Parameters for the model are determined from the data. Mixture models assume that the data is a ‘mixture' of a
number of statistical distributions.
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Types of Clusters: Objective Function … Map the clustering problem to a different domain
and solve a related problem in that domain Proximity matrix defines a weighted graph, where the
nodes are the points being clustered, and the weighted edges represent the proximities between points
Clustering is equivalent to breaking the graph into connected components, one for each cluster.
Want to minimize the edge weight between clusters and maximize the edge weight within clusters
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Characteristics of the Input Data Are Important Type of proximity or density measure
This is a derived measure, but central to clustering Sparseness
Dictates type of similarity Adds to efficiency
Attribute type Dictates type of similarity
Type of Data Dictates type of similarity Other characteristics, e.g., autocorrelation
Dimensionality Noise and Outliers Type of Distribution
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Clustering Algorithms K-means and its variants
Hierarchical clustering
Density-based clustering
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K-means Clustering Partitional clustering approach Each cluster is associated with a centroid (center point) Each point is assigned to the cluster with the closest centroid Number of clusters, K, must be specified The basic algorithm is very simple
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K-means Clustering – Details Initial centroids are often chosen randomly.
Clusters produced vary from one run to another. The centroid is (typically) the mean of the points in the cluster. ‘Closeness’ is measured by Euclidean distance, cosine similarity, correlation, etc. K-means will converge for common similarity measures mentioned above. Most of the convergence happens in the first few iterations.
Often the stopping condition is changed to ‘Until relatively few points change clusters’ Complexity is O( n * K * I * d )
n = number of points, K = number of clusters, I = number of iterations, d = number of attributes
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Two different K-means Clusterings
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Sub-optimal Clustering
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Optimal Clustering
Original Points
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Importance of Choosing Initial Centroids
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Importance of Choosing Initial Centroids
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Evaluating K-means Clusters Most common measure is Sum of Squared Error (SSE)
For each point, the error is the distance to the nearest cluster To get SSE, we square these errors and sum them.
x is a data point in cluster Ci and mi is the representative point for cluster Ci
can show that mi corresponds to the center (mean) of the cluster Given two clusters, we can choose the one with the smallest
error One easy way to reduce SSE is to increase K, the number of
clusters A good clustering with smaller K can have a lower SSE than a poor
clustering with higher K
K
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xmdistSSE1
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Importance of Choosing Initial Centroids …
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Importance of Choosing Initial Centroids …
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Problems with Selecting Initial Points
If there are K ‘real’ clusters then the chance of selecting one centroid from each cluster is small. Chance is relatively small when K is large If clusters are the same size, n, then
For example, if K = 10, then probability = 10!/1010 = 0.00036 Sometimes the initial centroids will readjust themselves in ‘right’ way, and sometimes they don’t Consider an example of five pairs of clusters
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10 Clusters Example
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Starting with two initial centroids in one cluster of each pair of clusters
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10 Clusters Example
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Starting with two initial centroids in one cluster of each pair of clusters
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10 Clusters Example
Starting with some pairs of clusters having three initial centroids, while other have only one.
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10 Clusters Example
Starting with some pairs of clusters having three initial centroids, while other have only one.
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Solutions to Initial Centroids Problem Multiple runs
Helps, but probability is not on your side Sample and use hierarchical clustering to
determine initial centroids Select more than k initial centroids and
then select among these initial centroids Select most widely separated
Postprocessing Bisecting K-means
Not as susceptible to initialization issues
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Handling Empty Clusters Basic K-means algorithm can yield empty
clusters
Several strategies Choose the point that contributes most to SSE Choose a point from the cluster with the
highest SSE If there are several empty clusters, the above
can be repeated several times.
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Updating Centers Incrementally In the basic K-means algorithm, centroids are
updated after all points are assigned to a centroid
An alternative is to update the centroids after each assignment (incremental approach) Each assignment updates zero or two centroids More expensive Introduces an order dependency Never get an empty cluster Can use “weights” to change the impact
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Pre-processing and Post-processing Pre-processing
Normalize the data Eliminate outliers
Post-processing Eliminate small clusters that may represent outliers Split ‘loose’ clusters, i.e., clusters with relatively high
SSE Merge clusters that are ‘close’ and that have relatively
low SSE Can use these steps during the clustering process
ISODATA
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Bisecting K-means Bisecting K-means algorithm
Variant of K-means that can produce a partitional or a hierarchical clustering
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Bisecting K-means Example
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Limitations of K-means K-means has problems when clusters are
of differing Sizes Densities Non-globular shapes
K-means has problems when the data contains outliers.
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Limitations of K-means: Differing Sizes
Original Points K-means (3 Clusters)
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Limitations of K-means: Differing Density
Original Points K-means (3 Clusters)
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Limitations of K-means: Non-globular Shapes
Original Points K-means (2 Clusters)
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Overcoming K-means Limitations
Original Points K-means Clusters
One solution is to use many clusters.Find parts of clusters, but need to put together.
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Overcoming K-means Limitations
Original Points K-means Clusters
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Overcoming K-means Limitations
Original Points K-means Clusters
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Hierarchical Clustering Produces a set of nested clusters
organized as a hierarchical tree Can be visualized as a dendrogram
A tree like diagram that records the sequences of merges or splits
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Strengths of Hierarchical Clustering Do not have to assume any particular
number of clusters Any desired number of clusters can be
obtained by ‘cutting’ the dendogram at the proper level
They may correspond to meaningful taxonomies Example in biological sciences (e.g., animal
kingdom, phylogeny reconstruction, …)
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Hierarchical Clustering Two main types of hierarchical clustering
Agglomerative: Start with the points as individual clusters At each step, merge the closest pair of clusters until only one
cluster (or k clusters) left
Divisive: Start with one, all-inclusive cluster At each step, split a cluster until each cluster contains a point
(or there are k clusters)
Traditional hierarchical algorithms use a similarity or distance matrix Merge or split one cluster at a time
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Agglomerative Clustering Algorithm
More popular hierarchical clustering technique
Basic algorithm is straightforward1. Compute the proximity matrix2. Let each data point be a cluster3. Repeat4. Merge the two closest clusters5. Update the proximity matrix6. Until only a single cluster remains
Key operation is the computation of the proximity of two clusters
Different approaches to defining the distance between clusters distinguish the different algorithms
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Starting Situation Start with clusters of individual points and
a proximity matrixp1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
. Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
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Intermediate Situation After some merging steps, we have some clusters
C1
C4
C2 C5
C3
C2C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
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Intermediate Situation We want to merge the two closest clusters (C2 and
C5) and update the proximity matrix.
C1
C4
C2 C5
C3
C2C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
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After Merging The question is “How do we update the proximity
matrix?”
C1
C4
C2 U C5
C3? ? ? ?
?
?
?
C2 U C5C1
C1
C3
C4
C2 U C5
C3 C4
Proximity Matrix
...p1 p2 p3 p4 p9 p10 p11 p12
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Similarity?
MIN MAX Group Average Distance Between Centroids Other methods driven by an objective
function Ward’s Method uses squared error
Proximity Matrix
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
MIN MAX Group Average Distance Between Centroids Other methods driven by an
objective function Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
MIN MAX Group Average Distance Between Centroids Other methods driven by an
objective function Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
MIN MAX Group Average Distance Between Centroids Other methods driven by an
objective function Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.Proximity Matrix
MIN MAX Group Average Distance Between Centroids Other methods driven by an
objective function Ward’s Method uses squared error
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Cluster Similarity: MIN or Single Link Similarity of two clusters is based on the
two most similar (closest) points in the different clusters Determined by one pair of points, i.e., by one
link in the proximity graph.
I1 I2 I3 I4 I5I1 1.00 0.90 0.10 0.65 0.20I2 0.90 1.00 0.70 0.60 0.50I3 0.10 0.70 1.00 0.40 0.30I4 0.65 0.60 0.40 1.00 0.80I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5
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Hierarchical Clustering: MIN
Nested Clusters Dendrogram
1
2
3
4
5
6
1
2
3
4
5
3 6 2 5 4 10
0.05
0.1
0.15
0.2
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Strength of MIN
Original Points Two Clusters
• Can handle non-elliptical shapes
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Limitations of MIN
Original Points Two Clusters
• Sensitive to noise and outliers
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Cluster Similarity: MAX or Complete Linkage Similarity of two clusters is based on the
two least similar (most distant) points in the different clusters Determined by all pairs of points in the two
clustersI1 I2 I3 I4 I5
I1 1.00 0.90 0.10 0.65 0.20I2 0.90 1.00 0.70 0.60 0.50I3 0.10 0.70 1.00 0.40 0.30I4 0.65 0.60 0.40 1.00 0.80I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5
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Hierarchical Clustering: MAX
Nested Clusters Dendrogram
3 6 4 1 2 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1
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Strength of MAX
Original Points Two Clusters
• Less susceptible to noise and outliers
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Limitations of MAX
Original Points Two Clusters
•Tends to break large clusters
•Biased towards globular clusters
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Cluster Similarity: Group Average Proximity of two clusters is the average of pairwise
proximity between points in the two clusters.
Need to use average connectivity for scalability since total proximity favors large clusters
||Cluster||Cluster
)p,pproximity(
)Cluster,Clusterproximity(ji
ClusterpClusterp
ji
jijjii
I1 I2 I3 I4 I5I1 1.00 0.90 0.10 0.65 0.20I2 0.90 1.00 0.70 0.60 0.50I3 0.10 0.70 1.00 0.40 0.30I4 0.65 0.60 0.40 1.00 0.80I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5
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Hierarchical Clustering: Group Average
Nested Clusters Dendrogram
3 6 4 1 2 50
0.05
0.1
0.15
0.2
0.25
1
2
3
4
5
6
1
2
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Hierarchical Clustering: Group Average Compromise between Single and
Complete Link
Strengths Less susceptible to noise and outliers
Limitations Biased towards globular clusters
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Cluster Similarity: Ward’s Method Similarity of two clusters is based on the
increase in squared error when two clusters are merged Similar to group average if distance between points
is distance squared
Less susceptible to noise and outliers
Biased towards globular clusters
Hierarchical analogue of K-means Can be used to initialize K-means
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Hierarchical Clustering: Comparison
Group Average
Ward’s Method
1
2
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5
61
2
5
3
4
MIN MAX
1
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61
2
5
34
1
2
3
4
5
61
2 5
3
41
2
3
4
5
6
12
3
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5
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Hierarchical Clustering: Time and Space requirements O(N2) space since it uses the proximity
matrix. N is the number of points.
O(N3) time in many cases There are N steps and at each step the size, N2,
proximity matrix must be updated and searched
Complexity can be reduced to O(N2 log(N) ) time for some approaches
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Hierarchical Clustering: Problems and Limitations Once a decision is made to combine two
clusters, it cannot be undone
No objective function is directly minimized
Different schemes have problems with one or more of the following: Sensitivity to noise and outliers Difficulty handling different sized clusters and
convex shapes Breaking large clusters
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MST: Divisive Hierarchical Clustering Build MST (Minimum Spanning Tree)
Start with a tree that consists of any point In successive steps, look for the closest pair of points (p,
q) such that one point (p) is in the current tree but the other (q) is not
Add q to the tree and put an edge between p and q
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MST: Divisive Hierarchical Clustering Use MST for constructing hierarchy of
clusters
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DBSCAN DBSCAN is a density-based algorithm.
Density = number of points within a specified radius (Eps)
A point is a core point if it has more than a specified number of points (MinPts) within Eps
These are points that are at the interior of a cluster
A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point
A noise point is any point that is not a core point or a border point.
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DBSCAN: Core, Border, and Noise Points
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DBSCAN Algorithm Eliminate noise points Perform clustering on the remaining points
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DBSCAN: Core, Border and Noise Points
Original Points Point types: core, border and noise
Eps = 10, MinPts = 4
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When DBSCAN Works Well
Original Points Clusters
• Resistant to Noise
• Can handle clusters of different shapes and sizes
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When DBSCAN Does NOT Work Well
Original Points
(MinPts=4, Eps=9.75).
(MinPts=4, Eps=9.92)
• Varying densities
• High-dimensional data
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DBSCAN: Determining EPS and MinPts Idea is that for points in a cluster, their kth
nearest neighbors are at roughly the same distance
Noise points have the kth nearest neighbor at farther distance
So, plot sorted distance of every point to its kth nearest neighbor
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Cluster Validity For supervised classification we have a variety of
measures to evaluate how good our model is Accuracy, precision, recall
For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters?
But “clusters are in the eye of the beholder”!
Then why do we want to evaluate them? To avoid finding patterns in noise To compare clustering algorithms To compare two sets of clusters To compare two clusters
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Clusters found in Random Data
0 0.2 0.4 0.6 0.8 10
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K-means
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Complete Link
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1. Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random structure actually exists in the data.
2. Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels.
3. Evaluating how well the results of a cluster analysis fit the data without reference to external information.
- Use only the data
4. Comparing the results of two different sets of cluster analyses to determine which is better.
5. Determining the ‘correct’ number of clusters.
For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters.
Different Aspects of Cluster Validation
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Numerical measures that are applied to judge various aspects of cluster validity, are classified into the following three types. External Index: Used to measure the extent to which cluster
labels match externally supplied class labels. Entropy
Internal Index: Used to measure the goodness of a clustering structure without respect to external information.
Sum of Squared Error (SSE)
Relative Index: Used to compare two different clusterings or clusters.
Often an external or internal index is used for this function, e.g., SSE or entropy
Sometimes these are referred to as criteria instead of indices However, sometimes criterion is the general strategy and index is the
numerical measure that implements the criterion.
Measures of Cluster Validity
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Two matrices Proximity Matrix “Incidence” Matrix
One row and one column for each data point An entry is 1 if the associated pair of points belong to the same cluster An entry is 0 if the associated pair of points belongs to different clusters
Compute the correlation between the two matrices Since the matrices are symmetric, only the correlation between
n(n-1) / 2 entries needs to be calculated.
High correlation indicates that points that belong to the same cluster are close to each other.
Not a good measure for some density or contiguity based clusters.
Measuring Cluster Validity Via Correlation
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Measuring Cluster Validity Via Correlation Correlation of incidence and proximity
matrices for the K-means clusterings of the following two data sets.
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Corr = -0.9235 Corr = -0.5810
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Order the similarity matrix with respect to cluster labels and inspect visually.
Using Similarity Matrix for Cluster Validation
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Using Similarity Matrix for Cluster Validation Clusters in random data are not so crisp
Points
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ints
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Points
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ints
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Using Similarity Matrix for Cluster Validation Clusters in random data are not so crisp
K-means
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Using Similarity Matrix for Cluster Validation Clusters in random data are not so crisp
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Complete Link
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Using Similarity Matrix for Cluster Validation
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Clusters in more complicated figures aren’t well separated Internal Index: Used to measure the goodness of a clustering
structure without respect to external information SSE
SSE is good for comparing two clusterings or two clusters (average SSE).
Can also be used to estimate the number of clusters
Internal Measures: SSE
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Internal Measures: SSE SSE curve for a more complicated data set
1 2
3
5
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7
SSE of clusters found using K-means
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Need a framework to interpret any measure. For example, if our measure of evaluation has the value, 10, is that
good, fair, or poor?
Statistics provide a framework for cluster validity The more “atypical” a clustering result is, the more likely it represents
valid structure in the data Can compare the values of an index that result from random data or
clusterings to those of a clustering result. If the value of the index is unlikely, then the cluster results are valid
These approaches are more complicated and harder to understand.
For comparing the results of two different sets of cluster analyses, a framework is less necessary.
However, there is the question of whether the difference between two index values is significant
Framework for Cluster Validity
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Example Compare SSE of 0.005 against three clusters in random data Histogram shows SSE of three clusters in 500 sets of random
data points of size 100 distributed over the range 0.2 – 0.8 for x and y values
Statistical Framework for SSE
0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.0340
5
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SSE
Co
unt
0 0.2 0.4 0.6 0.8 10
0.1
0.2
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1
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y
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Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets.
Statistical Framework for Correlation
0 0.2 0.4 0.6 0.8 10
0.1
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x
y
Corr = -0.9235 Corr = -0.5810
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Cluster Cohesion: Measures how closely related are objects in a cluster Example: SSE
Cluster Separation: Measure how distinct or well-separated a cluster is from other clusters
Example: Squared Error Cohesion is measured by the within cluster sum of squares (SSE)
Separation is measured by the between cluster sum of squares
Where |Ci| is the size of cluster i
Internal Measures: Cohesion and Separation
i Cx
ii
mxWSS 2)(
i
ii mmCBSS 2)(
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Internal Measures: Cohesion and Separation Example: SSE
BSS + WSS = constant
1 2 3 4 5 m1 m2
m
1091
9)35.4(2)5.13(2
1)5.45()5.44()5.12()5.11(22
2222
Total
BSS
WSSK=2 clusters:
10010
0)33(4
10)35()34()32()31(2
2222
Total
BSS
WSSK=1 cluster:
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A proximity graph based approach can also be used for cohesion and separation. Cluster cohesion is the sum of the weight of all links within a
cluster. Cluster separation is the sum of the weights between nodes in
the cluster and nodes outside the cluster.
Internal Measures: Cohesion and Separation
cohesion separation
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Silhouette Coefficient combine ideas of both cohesion and separation, but for individual points, as well as clusters and clusterings
For an individual point, i Calculate a = average distance of i to the points in its cluster Calculate b = min (average distance of i to points in another cluster) The silhouette coefficient for a point is then given by
s = 1 – a/b if a < b, (or s = b/a - 1 if a b, not the usual case)
Typically between 0 and 1. The closer to 1 the better.
Can calculate the Average Silhouette width for a cluster or a clustering
Internal Measures: Silhouette Coefficient
ab
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External Measures of Cluster Validity: Entropy and Purity
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“The validation of clustering structures is the most difficult and frustrating part of cluster analysis.
Without a strong effort in this direction, cluster analysis will remain a black art accessible only to those true believers who have experience and great courage.”
Algorithms for Clustering Data, Jain and Dubes
Final Comment on Cluster Validity
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Hierarchical Clustering: Revisited Creates nested clusters
Agglomerative clustering algorithms vary in terms of how the proximity of two clusters are computed
MIN (single link): susceptible to noise/outliers MAX/GROUP AVERAGE:
may not work well with non-globular clusters
CURE algorithm tries to handle both problems
Often starts with a proximity matrix A type of graph-based algorithm
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Uses a number of points to represent a cluster
Representative points are found by selecting a constant number of points from a cluster and then “shrinking” them toward the center of the cluster
Cluster similarity is the similarity of the closest pair of representative points from different clusters
CURE: Another Hierarchical Approach
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CURE Shrinking representative points toward the
center helps avoid problems with noise and outliers
CURE is better able to handle clusters of arbitrary shapes and sizes
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Experimental Results: CURE
Picture from CURE, Guha, Rastogi, Shim.
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Experimental Results: CURE
Picture from CURE, Guha, Rastogi, Shim.
(centroid)
(single link)
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CURE Cannot Handle Differing Densities
Original Points CURE
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Graph-Based Clustering Graph-Based clustering uses the proximity
graph Start with the proximity matrix Consider each point as a node in a graph Each edge between two nodes has a weight which is
the proximity between the two points Initially the proximity graph is fully connected MIN (single-link) and MAX (complete-link) can be
viewed as starting with this graph
In the simplest case, clusters are connected components in the graph.
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Graph-Based Clustering: Sparsification
The amount of data that needs to be processed is drastically reduced Sparsification can eliminate more than 99% of
the entries in a proximity matrix The amount of time required to cluster the data
is drastically reduced The size of the problems that can be handled is
increased
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Graph-Based Clustering: Sparsification …
Clustering may work better Sparsification techniques keep the connections to the
most similar (nearest) neighbors of a point while breaking the connections to less similar points.
The nearest neighbors of a point tend to belong to the same class as the point itself.
This reduces the impact of noise and outliers and sharpens the distinction between clusters.
Sparsification facilitates the use of graph partitioning algorithms (or algorithms based on graph partitioning algorithms.
Chameleon and Hypergraph-based Clustering
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Sparsification in the Clustering Process
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Limitations of Current Merging Schemes Existing merging schemes in hierarchical
clustering algorithms are static in nature MIN or CURE:
merge two clusters based on their closeness (or minimum distance)
GROUP-AVERAGE: merge two clusters based on their average
connectivity
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Limitations of Current Merging Schemes
Closeness schemes will merge (a) and (b)
(a)
(b)
(c)
(d)
Average connectivity schemes will merge (c) and (d)
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Chameleon: Clustering Using Dynamic Modeling
Adapt to the characteristics of the data set to find the natural clusters
Use a dynamic model to measure the similarity between clusters Main property is the relative closeness and relative inter-connectivity of
the cluster Two clusters are combined if the resulting cluster shares certain properties
with the constituent clusters The merging scheme preserves self-similarity
One of the areas of application is spatial data
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Characteristics of Spatial Data Sets• Clusters are defined as densely
populated regions of the space
• Clusters have arbitrary shapes, orientation, and non-uniform sizes
• Difference in densities across clusters and variation in density within clusters
• Existence of special artifacts (streaks) and noise
The clustering algorithm must address the above characteristics
and also require minimal supervision.
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Chameleon: Steps Preprocessing Step:
Represent the Data by a Graph Given a set of points, construct the k-nearest-neighbor
(k-NN) graph to capture the relationship between a point and its k nearest neighbors
Concept of neighborhood is captured dynamically (even if region is sparse)
Phase 1: Use a multilevel graph partitioning algorithm on the graph to find a large number of clusters of well-connected vertices Each cluster should contain mostly points from one
“true” cluster, i.e., is a sub-cluster of a “real” cluster
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Chameleon: Steps … Phase 2: Use Hierarchical Agglomerative
Clustering to merge sub-clusters Two clusters are combined if the resulting cluster shares
certain properties with the constituent clusters
Two key properties used to model cluster similarity: Relative Interconnectivity: Absolute interconnectivity of
two clusters normalized by the internal connectivity of the clusters
Relative Closeness: Absolute closeness of two clusters normalized by the internal closeness of the clusters
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Experimental Results: CHAMELEON
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Experimental Results: CHAMELEON
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Experimental Results: CURE (10 clusters)
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Experimental Results: CURE (15 clusters)
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Experimental Results: CHAMELEON
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Experimental Results: CURE (9 clusters)
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Experimental Results: CURE (15 clusters)
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i j i j4
SNN graph: the weight of an edge is the number of shared neighbors between vertices given that the vertices are connected
Shared Near Neighbor Approach
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Creating the SNN Graph
Sparse Graph
Link weights are similarities between neighboring points
Shared Near Neighbor Graph
Link weights are number of Shared Nearest Neighbors
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ROCK (RObust Clustering using linKs) Clustering algorithm for data with categorical and
Boolean attributes A pair of points is defined to be neighbors if their similarity is
greater than some threshold Use a hierarchical clustering scheme to cluster the data.
1. Obtain a sample of points from the data set2. Compute the link value for each set of points, i.e., transform
the original similarities (computed by Jaccard coefficient) into similarities that reflect the number of shared neighbors between points
3. Perform an agglomerative hierarchical clustering on the data using the “number of shared neighbors” as similarity measure and maximizing “the shared neighbors” objective function
4. Assign the remaining points to the clusters that have been found
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Jarvis-Patrick Clustering First, the k-nearest neighbors of all points are found
In graph terms this can be regarded as breaking all but the k strongest links from a point to other points in the proximity graph
A pair of points is put in the same cluster if any two points share more than T neighbors and the two points are in each others k nearest neighbor list
For instance, we might choose a nearest neighbor list of size 20 and put points in the same cluster if they share more than 10 near neighbors
Jarvis-Patrick clustering is too brittle
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When Jarvis-Patrick Works Reasonably Well
Original Points Jarvis Patrick Clustering
6 shared neighbors out of 20
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Smallest threshold, T, that does not merge clusters.
Threshold of T - 1
When Jarvis-Patrick Does NOT Work Well
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SNN Clustering Algorithm1. Compute the similarity matrix
This corresponds to a similarity graph with data points for nodes and edges whose weights are the similarities between data points
2. Sparsify the similarity matrix by keeping only the k most similar neighborsThis corresponds to only keeping the k strongest links of the similarity graph
3. Construct the shared nearest neighbor graph from the sparsified similarity matrix. At this point, we could apply a similarity threshold and find the connected components to obtain the clusters (Jarvis-Patrick algorithm)
4. Find the SNN density of each Point.Using a user specified parameters, Eps, find the number points that have an SNN similarity of Eps or greater to each point. This is the SNN density of the point
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SNN Clustering Algorithm …5. Find the core points
Using a user specified parameter, MinPts, find the core points, i.e., all points that have an SNN density greater than MinPts
6. Form clusters from the core points If two core points are within a radius, Eps, of each other they are place in the same cluster
7. Discard all noise pointsAll non-core points that are not within a radius of Eps of a core point are discarded
8. Assign all non-noise, non-core points to clusters This can be done by assigning such points to the nearest core point
(Note that steps 4-8 are DBSCAN)
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SNN Density
a) All Points b) High SNN Density
c) Medium SNN Density d) Low SNN Density
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SNN Clustering Can Handle Differing Densities
Original Points SNN Clustering
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SNN Clustering Can Handle Other Difficult Situations
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Finding Clusters of Time Series In Spatio-Temporal Data26 SLP Clusters via Shared Nearest Neighbor Clustering (100 NN, 1982-1994)
longitude
latit
ud
e
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180
90
60
30
0
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-90
13 26
24 25
22
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16 20 17 18
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6 4
7 10 12 11
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5 2
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21
SNN Clusters of SLP.
SNN Density of SLP Time Series Data
longitudela
titu
de
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180
90
60
30
0
-30
-60
-90
SNN Density of Points on the Globe.
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Features and Limitations of SNN Clustering
Does not cluster all the points
Complexity of SNN Clustering is high O( n * time to find numbers of neighbor within Eps) In worst case, this is O(n2) For lower dimensions, there are more efficient ways
to find the nearest neighbors R* Tree k-d Trees