4/8/2002 Copyright Daniel Barbara

Clustering by impact

Daniel Barbará

George Mason University

ISE Dept.

http://www.ise.gmu.edu/~dbarbara

(joint work with P. Chen, J. Couto, and Y. Li)

ProblemOrganizations are constantly acquiring and storing new data (data streams)The need to quickly extract knowledge from the newly arrived data (and compare it with the old) is pressing.Applications:

Intrusion detectionTuning Intelligence analysis

Outline

Clustering data streams

Our method

Continuous data: Fractal Clustering

Categorical (nominal) data: Entropy-based

Tracking clusters

Future work

Clustering and data streams

To cluster continuously arriving data streams a clustering algorithm should behave incrementally: make the decision based on the newly arrived point and a concise description of the clusters encountered so far.

Concise bounded amount of RAM to describe the clusters, independently of the number of data points processed so far…

Problem (cont.)

Most algorithms in the literature do not have that property:

They look at the entire set of points at once (e.g., K-means)

They cannot make decisions point by point.

The description of the clusters is usually the set of points in them.

Some of the algorithms have high complexity

Some inroadsPaper by U. Fayyad, D. Bradley and C. Reina:

“Scaling Clustering algorithms to large databases” (KDD’98)

Main idea: keep descriptions of centroids + set descriptions that are likely and unlikely to change given a new data point.Papers by Motwani, et al. Incrementally updating centroids while receiving a data stream. The goal is to have an approximation to “min squares” whose performance is bounded.

Our proposal

Find functions that naturally define clusters and that can be easily computed given a new point and a concise representation of the current clusters.

Place a new point in the cluster for which the evaluated function shows a minimum (or a maximum) – less impact---

“Impact” functions

Numerical data points: fractal dimensionMeasures the self-similarity of points.The idea is that the lower the change in the fractal dimension (when the point is included), the more self-similar the point is w/respect to the cluster

Categorical data points: entropy.Also measures similarityLower entropy means similar points.

Fractal Clustering Fractal dimension, is a (not necessarily integer) number that characterizes the number of dimensions ``filled'' by the object represented by the dataset. The object on the upper right corner, called the Menger sponge (when complete) has a F.D. equal to 2.73 (less than the embedding space, whose dimension is 3)

Conjecture: if part of a dataset brings about a change in the overall fractal dimension of the set, then this part is ``anomalous'' (exhibits different behavior) with respect to the rest of the dataset.

Fractal dimension

r = grid size

log log

1log

log

( 1) log

{i i

i

qi

i

p p

for qr

q p

otherwiseq r

D

ip Probability distribution

Box CountingCantor Dust Set

Box counting (cont.)

log 2n

D1 = - limn-> = 0.63 log ( )n

p r

2 r0

4 r0 / 3

8 r0 /9

Population vs. grid size (logxlog)

8

4

2

1

10

1 2 3

r

po

p.

Initialization Algorithm

Take an unlabelled point in the sample and start a cluster.

Find close neighbors and add them to the cluster.

Find close neighbors to points in the cluster… If you can’t go to first step.

Space management

Space management

Space in RAM is not proportional to the size of the dataset, but rather to the size of the grid and number of grid levels kept.

These vary with:

Dimensionality

Accuracy (odd-shaped clusters may require more levels).

Experiments

Dataset1

Scalability results with Dataset1

execution time

0200040006000

number of points

seco

nd

s

t

Quality of clusters (Dataset1)

Percentage of points clustered right

0

50

100

Dataset size

%

C1

C2

C3

High dimensional set

10 dimensions, 2 clusters

% of points clustered right

94.3

100

9092949698

100

C1 C2

Cluster

%

C1

C2

Results with the noisy dataset

92 % of the noise gets filtered out.

% points clustered right

99.57 100

83.62

60

80

100

C1 C2 C3

Cluster

%

C1

C2

C3

Memory usage vs. dimensions

Memory used vs. dimensions

64500

2,000

0

1000

2000

3000

1 2 3 4 5 6 7 8 9 10

dimensions

Siz

e (

Kb

.)

Size(Kb)

Memory reduction

Space taken by the boxes is small, but it grows with the number of dimensions.

Memory reduction techniques:

• Use boxes with # points > epsilon.

• Cache boxes

• Have only smallest granularity boxes and derive the rest.

None of them causes a significant degradation of quality. (2 and 3 have an impact on running time.)

Memory reduction

19 25

75

55

0

20

40

60

80

1 2 3 4

Technique

% Memory reduction

Comparison with other algorithms

Comparison of quality

0 50 100 150

C1

C2

C1

C2

CU

RE

FC

Alg

ori

thm

%

right

outliers

Entropy-based Clustering (COOLCAT)

For Categorical dataPlace new point where it minimizes some function of the entropies of the individual clusters (e.g., min (max (entropy Ci)))Heuristic (problem is NP-Hard)Entropy of each cluster:

Minimize expected entropy 1,... 1,..

( ) ( / ) log ( / )i

ki d j j

E C P Vij Ck P Vij Ck

Initialization

Need to seed “k” clusters:Select a sample

Find 2 points that are the most dissimilar (their joint entropy is the highest).

Place them in 2 different clusters

Find another point that is the most dissimilar (pairwise) to the ones selected, and start another cluster.

Incremental phase

For a given point and k current clusters:Compute the expected entropy as the new point is placed in each cluster.Choose the one that minimizes the expected entropy After finishing with a batch of points, re-process m% of them (take the ``worse’’ fits out and re-cluster them): helps with the issue of order dependency

Conciseness

Notice that the current cluster description is concise:

Counts of Vij for every i= 1,.., d (number of attributes), and for every j (domain of each attribute)

COOLCAT and the MDL

MDL = minimum description length.

Widely used to argue about how good a classifier is: how many bits does it take to send to a receiver the description of your classifier + the exceptions (misclassifications)

MDL (cont.)

( , ) ( ) ( using )

model, D = data

K h D K h K D h

h

( ) log(| |)

( , ) ( )

K h k D

K h D E C

C clustering

Experimental results

Real and synthetic datasets

Evaluate quality and performance

Quality: Category utility function (how much “better” is the distribution probability in the individual clusters w/respect to the original distribution)

External entropy: take an attribute not used in the clustering and compute the entropy of each cluster w/respect to it, then the expected external entropy

Experimental resultsArchaeological data set

Alg. m CU Ext. E Exp E

Coolcat 0 0.7626 0 4.8599

Coolcat 10 0.7626 0 4.8599

Coolcat 20 0.7626 0 4.8599

Brute F. - 0.7626 0 4.8599

ROCK - 0.3312 0.96 -

KDD99 Cup data (intrusion detection)

k

Exp E

CU

Ext

E

Performance (synthetic data)

N x 1000

T

(sec.)

Tracking clustersClustering data streams as they come:Consider r.v X = 0 if new point is outlier; 1 otherwise.Using Chernoff bounds:Must see s “successes” – not outliers– in a window w

If you don’t, it is time for new clusters…

2

3(1 ) 2ln( )s

2

2(1 ) 2ln( )

(1 )w

p

FC, COOLCAT and Tracking

Find a good definition of outlier:FC: if the min change in FD exceeds a threshold.

COOLCAT: mutual information of new point with respect to clusters

One tracking experiment with FC

One tracking experiment with COOLCAT (intrusion detection)

Mutual Information

density

attacksNo

attacks

Hierarchical clustering

More tracking experiments

Hybrid data: numeric and categorical

Indexing based on clustering

…

Bibliography``Using the Fractal Dimension to Cluster Datasets,'' Proceedings of the the ACM-SIGKDD International Conference on Knowledge and Data Mining , Boston, August 2000. D. Barbara, P.Chen. ``Tracking Clusters in Evolving Data Sets,'' Proceedings of FLAIRS'2001, Special Track on Knowledge Discovery and Data Mining , Key West, FL, May 2001. D. Barbara, P. Chen. ``Fractal Characterization of Web Workloads,'' Proceedings of the 11th International World Wide Web Conference, May 2002. D. Menasce, V. Almeida, D. Barbara, B. Abrahao, and F. Ribeiro. ``Using Self-Similarity to Cluster Large Data Sets,’’ to appear in Journal of Data Mining and Knowledge Discovery, Kluwer Academic pub. D. Barbara, P.Chen``Requirements for Clustering Data Streams,'' SIGKDD Explorations (Special Issue on Online, Interactive, and Anytime Data Mining), Vol. 3, No. 2, Jan 2002. D. Barbara``COOLCAT: An Entropy-Based Algorithm for Categorical Clustering,’’ Submitted for publication. D. Barbara, J. Couto, Y. Li.