Download Data Mining Clustering (2)

Data Mining
Clustering (2)
Toon Calders
Sheets are based on the those provided by Tan,
Steinbach, and Kumar. Introduction to Data Mining
Outline
• Partitional Clustering
• Distance-based
− K-means, K-medoids, Bisecting K-means
• Density-based
− DBSCAN
• Hierarchical Clustering
• Cluster validity
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, …)
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
Agglomerative Clustering Algorithm
•
More popular hierarchical clustering technique
•
Basic algorithm is straightforward
1.
2.
3.
4.
5.
6.
•
Compute the proximity matrix
Let each data point be a cluster
Repeat
Merge the two closest clusters
Update the proximity matrix
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
Starting Situation
• Start with clusters of
individual points and a
proximity matrix
p1
p2
p3
p4 p5
p1
p2
p3
p4
p5
.
.
.
Proximity Matrix
...
Intermediate Situation
• After some merging steps,
we have some clusters
C1
C2
C3
C4
C1
C2
C3
C
3
C4
C
4
C5
Proximity Matrix
C
1
C
2
C
5
C5
Intermediate Situation
• We want to merge the two closest
clusters (C2 and C5) and update
the proximity matrix.
C1
C2
C3
C4
C1
C2
C3
C
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C4
C
4
C5
Proximity Matrix
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C
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C5
After Merging
• The question is “How do we update
the proximity matrix?”
C1
C1
C
3
C2 U C5
C
4
C2
U
C5
C3
C4
?
?
?
?
?
C3
?
C4
?
Proximity Matrix
C
1
C2 U
C5
How to Define Inter-Cluster Similarity
p1 p2
Similarity?
p3
p4 p5
p1
p2
p3
p4
p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
– Ward’s Method uses squared error
...
How to Define Inter-Cluster Similarity
p1 p2
p3
p4 p5
p1
p2
p3
p4
p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
– Ward’s Method uses squared error
...
How to Define Inter-Cluster Similarity
p1 p2
p3
p4 p5
p1
p2
p3
p4
p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
– Ward’s Method uses squared error
...
How to Define Inter-Cluster Similarity
p1 p2
p3
p4 p5
p1
p2
p3
p4
p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
– Ward’s Method uses squared error
...
How to Define Inter-Cluster Similarity
p1 p2
p3
p4 p5
p1
×
×
p2
p3
p4
p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
– Ward’s Method uses squared error
...
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
I5
I1
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0.90
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I2
0.90
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0.60
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I3
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0.70
1.00
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I4
0.65
0.60
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1.00
0.80
I5
0.20
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0.30
0.80
1.00
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Hierarchical Clustering: MIN
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Nested Clusters
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Dendrogram
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Strength of MIN
Original Points
• Can handle non-elliptical shapes
Two Clusters
Limitations of MIN
Original Points
• Sensitive to noise and outliers
Two Clusters
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 clusters
I1
I1 1.00
I2 0.90
I3 0.10
I4 0.65
I5 0.20
I2 I3 I4 I5
0.90 0.10 0.65 0.20
1.00 0.70 0.60 0.50
0.70 1.00 0.40 0.30
0.60 0.40 1.00 0.80
0.50 0.30 0.80 1.00
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Hierarchical Clustering: MAX
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Nested Clusters
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Dendrogram
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Strength of MAX
Original Points
• Less susceptible to noise and outliers
Two Clusters
Limitations of MAX
Original Points
•Tends to break large clusters
•Biased towards globular clusters
Two Clusters
Cluster Similarity: Group Average
• Proximity of two clusters is the average of pairwise
proximity between points in the two clusters.
∑ proximity(p , p )
i
proximity(Clusteri , Clusterj ) =
j
pi∈Clusteri
p j∈Clusterj
|Clusteri |∗|Clusterj |
• Need to use average connectivity for scalability since total
proximity favors large clusters
I1
I2
I3
I4
I5
I1
1.00
0.90
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0.65
0.20
I2
0.90
1.00
0.70
0.60
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I3
0.10
0.70
1.00
0.40
0.30
I4
0.65
0.60
0.40
1.00
0.80
I5
0.20
0.50
0.30
0.80
1.00
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Hierarchical Clustering: Group Average
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Nested Clusters
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Dendrogram
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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
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
Hierarchical Clustering: Comparison
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MIN
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MAX
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Ward’s Method
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Group Average
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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
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
Outline
• Partitional Clustering
• Distance-based
− K-means, K-medoids, Bisecting K-means
• Density-based
− DBSCAN
• Hierarchical Clustering
• Cluster validity
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
Clusters found in Random Data
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Random
Points
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DBSCAN
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Kmeans
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Complete
Link
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Different Aspects of Cluster Validation
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Determining the clustering tendency of a set of data, i.e.,
distinguishing whether non-random structure actually exists in
the data.
Comparing the results of a cluster analysis to externally known
results, e.g., to externally given class labels.
Evaluating how well the results of a cluster analysis fit the data
without reference to external information.
- Use only the data
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Comparing the results of two different sets of cluster analyses to
determine which is better.
Determining the ‘correct’ number of clusters.
Measuring Cluster Validity Via Correlation
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• Correlation of incidence and proximity matrices for
the K-means clusterings of the following two data
sets.
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Corr = -0.9235
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Corr = -0.5810
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Using Similarity Matrix for Cluster Validation
• Order the similarity matrix with respect to cluster
labels and inspect visually.
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Using Similarity Matrix for Cluster Validation
• Clusters in random data are not so crisp
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Points
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DBSCAN
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Using Similarity Matrix for Cluster Validation
• Clusters in random data are not so crisp
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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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DBSCAN
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Conclusions
• Partitional algorithms
• K-means and its variations
• Restricted to spherical clusters
• Number of clusters needs to be set by user
• DBSCAN
• Any shape of cluster
• Works only if density is uniform
• Two parameters to set …
• Hierarchical
• Merging versus splitting (bisecting K-means)
• Different criteria to determine distances between
clusters
Final Comment on Cluster Validity
Cluster validity is very hard to check!
Without a good validation, clustering is potentially just
random.
“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