CHAN Siu Lung, DanielCHAN Wai Kin, KenCHOW Chin Hung, VictorKOON Ping Yin, Bob

CURE: Efficient Clustering AlgorithmCURE: Efficient Clustering Algorithm for Large Databasesfor Large Databases

Content

I. Different problem in traditional clustering methodII. Basic idea of CURE clusteringIII. Improved CUREIV. SummaryV. References

I. Different problem in traditional clustering method

Partitional Clustering

• Partitional Clustering– This category of clustering method try to reduce the

data set into k clusters based on some criterion functions.

– The most common criterion is square-error criterion.

– This method favor to clusters with data points as compact and separated as possible

Different problem in traditional clustering method

Partitional Clustering

• You may find error in case the square-error is reduced by splitting some large cluster to favor some other group.

Different problem in traditional clustering method

Figure: Splitting occur in large cluster by partitional method

• Hierarchical Clustering– This category of clustering method try to merge sequences of disjoint

clusters into the target k clusters base on the minimum distance between two clusters.

– The distance between clusters can be measured as:• Distance between mean:

• Distance between average point

• Distance between two nearest point within cluster

Hierarchical Clustering

Different problem in traditional clustering method

Hierarchical Clustering

– This method favor hyper-spherical shape and uniform data.

– Let’s take some prolonged data as example:

– Result of dmean :

Different problem in traditional clustering method

Hierarchical Clustering

– Result of dmin :

Different problem in traditional clustering method

Problems summary

1. Traditional clustering mainly favors spherical shape.

2. Data in the cluster must be compact together.

3. Each cluster must separate far away enough.

4. Cluster size must be uniform.

5. Outliner will greatly disturb the cluster result.

Different problem in traditional clustering method

II. Basic idea of CURE clustering

General CURE clustering procedure.

1. It is similar to hierarchical clustering approach. But it use sample point variant as the cluster representative rather than every point in the cluster.

2. First set a target sample number c . Than we try to select c well scattered sample points from the cluster.

3. The chosen scattered points are shrunk toward the centroid in a fraction of where 0 <<1

Basic idea of CURE clustering

General CURE clustering procedure.

4. These points are used as representative of clusters and will be used as the point in dmin cluster merging approach.

5. After each merging, c sample points will be selected from original representative of previous clusters to represent new cluster.

6. Cluster merging will be stopped until target k cluster is found

Basic idea of CURE clustering

NearestMerge

Nearest

Merge

Pseudo function of CURE

Basic idea of CURE clustering

CURE efficient

• The worst-case time complexity is O(n2logn)

• The space complexity is O(n) due to the use of k-d treee and heap.

Basic idea of CURE clustering

III. Improved CURE

• In case of dealing with large database, we can’t store every data point to the memory.

• Handle of data merge in large database require very long time.

• We use random sampling to both reduce the time complexity and memory usage.

• Assume if we need to detect a cluster u present, we need to at least capture f fraction of data from this cluster f|u|

• The the required sampling data s to capture can be present as follow:

• You can refer to proof from the reference (i). Here we just want to show that we can determine a sample size smin such that the probability of get enough sample from every cluster u is 1 -

Random Sampling

Improved CURE

Partitioning and two pass clustering

• In addition, we use two-pass approach to reduce the computation time.

• First, we divide the n data point into p partition and each contain n/p data point.

• We than pre-cluster each partition until the number of cluster n/pq reached in each partition for some q > 1

• Then each cluster in the first pass result will be used as the second pass clustering input to form the final cluster.

• Each one partition’s time complexity is:

• Therefore, the first pass complexity will be:

• And the second pass complexity is:

• Overall, the time complexity will become:

Improved CURE

Partitioning and two pass clustering

• The overall improvement will be:

• Also, to maintain the quality of clustering, we must make sure n/pq must be 2 to 3 times of k.

Improved CUR

Outlier elimination

• We can introduce outliners elimination by two method.

1. Random sampling: With random sampling, most of outlier points are filtered out.

2. Outlier elimination: As outliner is not a compact group, it will grow in size very slowly during the cluster merge stage. We will then kick in the elimination procedure during the merging stage such that those cluster with 1 ~ 2 data points are removed from the cluster list.

• In order to prevent these outliners from merging into proper cluster, we must trigger the procedure in proper stage such that we can properly remove the outliners. In general, we will trigger this procedure when cluster sets reduce to 1/3 of total data sets.

Improved CURE

Data labeling

• Due to the use of random sample. We need to label back every remaining data points to the proper cluster group.

• Each data point is assigned to the cluster group with a representative point nearest to the data point.

Improved CURE

Final overview of CURE flow

Improved CURE

Data

Draw Random Sample

Partition Sample

Partially cluster partition

Elimination outliers

Cluster partial clusters

Label data in disk

Sample result with different parameter

Improved CURE

Different shrinking factor

Sample result with different parameter

Improved CURE

Different number of representatives c

Sample result with different parameter

Improved CURE

Relation of execution time, different partition number p, and different sample points s

IV. Summary

• CURE can effectively detect proper shape of the cluster with the help of scattered representative point and centroid shrinking.

• CURE can reduce computation time and memory loading with random sampling and 2 pass clustering

• CURE can effectively remove outlier.

• The quality and effectiveness of CURE can be tuned be varying different s,p,c, to adapt different input data set.

V. References

i. GRS97   Sudipto Guha, R. Rastogi, and K. Shim. CURE: A clustering algorithm for large databases. Technical report, Bell Laboratories, Murray Hill, 1997.

ii. ZRL96 Tian Zhang , Raghu Ramakrishnan , Miron Livny, BIRCH: an efficient data clustering method for very large databases, ACM SIGMOD Record, v.25 n.2, p.103-114, June 1996

iii. Sudipto Guha, Rajeev Rastogi, Kyuseok Shim: CURE: An Efficient Clustering Algorithm for Large Databases, ACM SIGMOD, 1998.