Download Mining Patterns from Protein Structures

EECS 800 Research Seminar
Mining Biological Data
Instructor: Luke Huan
Fall, 2006
The UNIVERSITY of Kansas
Administrative
If you haven’t scheduled a meeting for the class project
with the instructor, please do this asap.
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide2
Overview
Hierarchical clustering
Density based clustering
Graph based clustering
Subspace clustering
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide3
Hierarchical Clustering
Use distance matrix as clustering criteria. This method does not
require the number of clusters k as an input, but needs a termination
condition
Step 0
a
Step 1
Step 2 Step 3 Step 4
agglomerative
(AGNES)
ab
b
abcde
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cde
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Step 4
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Subspace Clustering
Step 3
Step 2 Step 1 Step 0
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
divisive
(DIANA)
slide4
AGNES (Agglomerative Nesting)
Introduced in Kaufmann and Rousseeuw (1990)
Implemented in statistical analysis packages, e.g., Splus
Use the Single-Link method and the dissimilarity matrix.
Merge nodes that have the least dissimilarity
Go on in a non-descending fashion
Eventually all nodes belong to the same cluster
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KU EECS 800, Luke Huan, Fall’06
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slide5
Density-Based Clustering Methods
Clustering based on density (local cluster criterion), such as densityconnected points
Major features:
Discover clusters of arbitrary shape
Handle noise
One scan
Need density parameters as termination condition
Several studies:
DBSCAN: Ester, et al. (KDD’96)
OPTICS: Ankerst, et al (SIGMOD’99).
DENCLUE: Hinneburg & D. Keim (KDD’98)
CLIQUE: Agrawal, et al. (SIGMOD’98) (more grid-based)
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide6
Density-Based Clustering: Basic Concepts
Two parameters:
Eps: Maximum radius of the neighbourhood
MinPts: Minimum number of points in an Eps-neighbourhood of that point
NEps(p):{q belongs to D | dist(p,q) <= Eps}
Directly density-reachable: A point p is directly density-reachable
from a point q w.r.t. Eps, MinPts if
p belongs to NEps(q)
p
core point condition:
|NEps (q)| >= MinPts
q
MinPts = 5
Eps = 1 cm
Directly density-reachable is asymmetric
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide7
Density-Reachable and Density-Connected
Density-reachable:
A point p is density-reachable from a point q
w.r.t. Eps, MinPts if there is a chain of points p1,
…, pn, p1 = q, pn = p such that pi+1 is directly
density-reachable from pi
p
q
p1
Density-connected
A point p is density-connected to a point q w.r.t.
Eps, MinPts if there is a point o such that both, p
and q are density-reachable from o w.r.t. Eps and
MinPts
p
q
o
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide8
DBSCAN: Density Based Spatial Clustering
of Applications with Noise
A cluster is a maximal set of density-connected points
Discovers clusters of arbitrary shape in spatial databases with noise
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.
Outlier
Border
Eps = 1cm
Core
10/02/2006
Subspace Clustering
MinPts = 5
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide9
DBSCAN: The Algorithm
Classify all points to {core, border, noise} w.r.t. Eps and MinPts.
Select a point P arbitrarily
If p is a core point, a cluster is formed.
If p is a border point, no points are density-reachable from p and
DBSCAN visits the next point of the database.
Continue the process until all of the points have been processed.
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide10
DBSCAN Algorithm
Eliminate noise points
Perform clustering on the remaining points
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide11
DBSCAN: Core, Border and Noise Points
Original Points
Point types: core,
border and noise
Eps = 10, MinPts = 4
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide12
When DBSCAN Does NOT Work Well
(MinPts=4, Eps=9.75).
Original Points
• Varying densities
• High-dimensional data
(MinPts=4, Eps=9.92)
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide13
DBSCAN: Sensitive to Parameters
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide14
OPTICS: A Cluster-Ordering Method (1999)
OPTICS: Ordering Points To Identify the Clustering Structure
Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99)
Produces a special order of the database w.r.t. its density-based clustering
structure
Good for both automatic and interactive cluster analysis, including finding
intrinsic clustering structure
Can be represented graphically or using visualization techniques
This cluster-ordering contains info is equiv to the density-based
clusterings that corresponds to a broad range of parameter settings
(distance parameters)
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide15
Core Distance
Core Distance
MinPts-distance(p) for an object p is the distance between p and
its MinPtrs neighbor.
Core distance(p) is MinPts-distance(p) for a core object
Core distance(p) is not defined for none core objects
D
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide16
Reachability Distance
Reachability distance:
The reachability-distance of an object p with respect to another
core object o is the smallest distance such that p is directly
density-reachable from o.
For a pair of p and o (o is a core object):
max(core-distance(o), distance(o, p))
p1
o
p2
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide17
Object Ordering
Reachabilitydistance
undefined

‘

Cluster-order
of the objects
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide18
Density-Based Clustering: OPTICS & Its Applications
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide19
DENCLUE: Using Statistical Density
Functions
Using statistical density estimations
Major features
Solid mathematical foundation
Good for data sets with large amounts of noise
Allows a compact mathematical description of arbitrarily shaped clusters in
high-dimensional data sets
Significant faster than existing algorithm (e.g., DBSCAN)
But needs a large number of parameters
An Efficient Approach to Clustering in Large Multimedia Databases
with Noise by Hinneburg & Keim (KDD’98)
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide20
Gradient: The steepness of a slope
Example
f Gaussian ( x , y )  e
f
D
Gaussian
f
( x )   i 1 e
N

d ( x , xi ) 2
2 2
( x, xi )  i 1 ( xi  x)  e
D
Gaussian
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Subspace Clustering
d ( x , y )2

2 2
N
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06

d ( x , xi ) 2
2
2
slide21
Example: Density Computation
D={x1,x2,x3,x4}
fDGaussian(x)= influence(x1) + influence(x2) + influence(x3) +
influence(x4)=0.04+0.06+0.08+0.6=0.78
x1
0.04
x3
y
x2
0.06
0.08
x
x4
0.6
Remark: the density value of y would be larger than the one for x
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide22
Denclue: Technical Essence
Uses grid cells but only keeps information about grid cells that do
actually contain data points and manages these cells in a tree-based
access structure
Influence function: describes the impact of a data point within its
neighborhood
Overall density of the data space can be calculated as the sum of
the influence function of all data points
Clusters can be determined mathematically by identifying density
attractors
Density attractors are local maximal of the overall density function
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide23
Density Attractor
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Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide24
Center-Defined and Arbitrary
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Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide25
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.
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide26
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
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide27
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
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide28
Sparsification in the Clustering Process
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide29
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
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide30
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 and noise
The clustering algorithm must address
the above characteristics and also
require minimal supervision.
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide31
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 wellconnected vertices
Each cluster should contain mostly points from one “true”
cluster, i.e., is a sub-cluster of a “real” cluster
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide32
Chameleon: Steps …
Phase 2: Use Hierarchical Agglomerative Clustering to merge subclusters
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
CHAMELEON measures the closeness of two clusters by computing the
average similarity between the points in Ci that are connected to points in
Cj
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide33
CHAMELEON (Clustering Complex Objects)
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide34
Grid-Based Clustering Method
Using multi-resolution grid data structure
Several interesting methods
STING (a STatistical INformation Grid approach) by Wang,
Yang and Muntz (1997)
WaveCluster by Sheikholeslami, Chatterjee, and Zhang
(VLDB’98)
A multi-resolution clustering approach using wavelet
method
CLIQUE: Agrawal, et al. (SIGMOD’98)
On high-dimensional data (thus put in the section of
clustering high-dimensional data
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide35
STING: A Statistical Information Grid Approach
Wang, Yang and Muntz (VLDB’97)
The spatial area area is divided into rectangular cells
There are several levels of cells corresponding to different levels of
resolution
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide36
The STING Clustering Method
Each cell at a high level is partitioned into a number of smaller cells in the next lower
level
Statistical info of each cell is calculated and stored beforehand and is used to answer
queries
Parameters of higher level cells can be easily calculated from parameters of lower
level cell
count, mean, s, min, max
type of distribution—normal, uniform, etc.
Use a top-down approach to answer spatial data queries
Start from a pre-selected layer—typically with a small number of cells
For each cell in the current level compute the confidence interval
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide37
Comments on STING
Remove the irrelevant cells from further consideration
When finish examining the current layer, proceed to the next lower
level
Repeat this process until the bottom layer is reached
Advantages:
Query-independent, easy to parallelize, incremental update
O(K), where K is the number of grid cells at the lowest level
Disadvantages:
All the cluster boundaries are either horizontal or vertical, and no diagonal
boundary is detected
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide38
WaveCluster: Clustering by Wavelet Analysis (1998)
Sheikholeslami, Chatterjee, and Zhang (VLDB’98)
A multi-resolution clustering approach which applies wavelet transform to the
feature space
How to apply wavelet transform to find clusters
Summarizes the data by imposing a multidimensional grid structure onto data space
These multidimensional spatial data objects are represented in a n-dimensional feature
space
Apply wavelet transform on feature space to find the dense regions in the feature
space
Apply wavelet transform multiple times which result in clusters at different scales
from fine to coarse
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide39
Wavelet Transform
Wavelet transform: A signal processing technique that decomposes a
signal into different frequency sub-band (can be applied to ndimensional signals)
Data are transformed to preserve relative distance between objects at
different levels of resolution
Allows natural clusters to become more distinguishable
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide40
The WaveCluster Algorithm
Input parameters
# of grid cells for each dimension
the wavelet, and the # of applications of wavelet transform
Why is wavelet transformation useful for clustering?
Use hat-shape filters to emphasize region where points cluster, but simultaneously
suppress weaker information in their boundary
Effective removal of outliers, multi-resolution, cost effective
Major features:
Detect arbitrary shaped clusters at different scales
Not sensitive to noise, not sensitive to input order
Only applicable to low dimensional data
Both grid-based and density-based
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide41
Quantization
& Transformation
First, quantize data into m-D grid
structure, then wavelet transform
a) scale 1: high resolution
b) scale 2: medium resolution
c) scale 3: low resolution
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide42
Clustering High-Dimensional Data
Clustering high-dimensional data
Many applications: text documents, DNA micro-array data
Major challenges:
Many irrelevant dimensions may mask clusters
Distance measure becomes meaningless—due to equi-distance
Clusters may exist only in some subspaces
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide43
Clustering in High Dimensional Space
Methods
Feature transformation: only effective if most dimensions are relevant
PCA & SVD useful only when features are highly
correlated/redundant
Feature selection: wrapper or filter approaches
useful to find a subspace where the data have nice clusters
Subspace-clustering: find clusters in all the possible subspaces
CLIQUE, ProClus, and frequent pattern-based clustering
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide44
The Curse of Dimensionality
(graphs adapted from Parsons et al. KDD Explorations 2004)
Data in only one dimension is relatively
packed
Adding a dimension “stretch” the
points across that dimension, making
them further apart
Density decrease dramatically
Distance measure becomes
meaningless—due to equi-distance
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide45
Why Subspace Clustering?
(adapted from Parsons et al. SIGKDD Explorations 2004)
Clusters may exist only in some subspaces
Subspace-clustering: find clusters in all the subspaces
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide46
CLIQUE (Clustering In QUEst)
Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98)
Automatically identifying subspaces of a high dimensional data space that allow
better clustering than original space
CLIQUE can be considered as both density-based and grid-based
It partitions each dimension into the same number of equal length interval
It partitions an m-dimensional data space into non-overlapping rectangular units
A unit is dense if the fraction of total data points contained in the unit exceeds the input
model parameter
A cluster is a maximal set of connected dense units within a subspace
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide47
CLIQUE: The Major Steps
Partition the data space and find the number of points that lie inside
each cell of the partition.
Identify the subspaces that contain clusters using the Apriori principle
Identify clusters
Determine dense units in all subspaces of interests
Determine connected dense units in all subspaces of interests.
Generate minimal description for the clusters
Determine maximal regions that cover a cluster of connected dense units for
each cluster
Determination of minimal cover for each cluster
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide48
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Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
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slide49
Strength and Weakness of CLIQUE
Strength
automatically finds subspaces of the highest dimensionality such that high
density clusters exist in those subspaces
insensitive to the order of records in input and does not presume some
canonical data distribution
scales linearly with the size of input and has good scalability as the number of
dimensions in the data increases
Weakness
The accuracy of the clustering result may be degraded at the expense of
simplicity of the method
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide50
Frequent Pattern-Based Approach
Clustering high-dimensional space (e.g., clustering text documents, microarray
data)
Projected subspace-clustering: which dimensions to be projected on?
CLIQUE, ProClus
Feature extraction: costly and may not be effective?
Using frequent patterns as “features”
“Frequent” are inherent features
Mining freq. patterns may not be so expensive
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide51
Summary
Cluster analysis groups objects based on their similarity and has
wide applications
Measure of similarity can be computed for various types of data
Clustering algorithms can be categorized into partitioning methods,
hierarchical methods, density-based methods, grid-based methods,
and model-based methods
Outlier detection and analysis are very useful for fraud detection,
etc. and can be performed by statistical, distance-based or deviationbased approaches
There are still lots of research issues on cluster analysis
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide52
Problems and Challenges
Considerable progress has been made in scalable clustering
methods
Partitioning: k-means, k-medoids, CLARANS
Hierarchical: BIRCH, ROCK, CHAMELEON
Density-based: DBSCAN, OPTICS, DenClue
Grid-based: STING, WaveCluster, CLIQUE
Model-based: EM, Cobweb, SOM
Frequent pattern-based: pCluster
Current clustering techniques do not address all the requirements
adequately, still an active area of research
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide53
References (1)
R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for
data mining applications. SIGMOD'98
M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973.
M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to identify the clustering structure,
SIGMOD’99.
P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scientific, 1996
Beil F., Ester M., Xu X.: "Frequent Term-Based Text Clustering", KDD'02
M. M. Breunig, H.-P. Kriegel, R. Ng, J. Sander. LOF: Identifying Density-Based Local Outliers. SIGMOD 2000.
M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial
databases. KDD'96.
M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases: Focusing techniques for efficient
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D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning, 2:139-172, 1987.
D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based on dynamic systems.
VLDB’98.
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Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide54
References (2)
V. Ganti, J. Gehrke, R. Ramakrishan. CACTUS Clustering Categorical Data Using Summaries. KDD'99.
D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based on dynamic systems. In
Proc. VLDB’98.
S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large databases. SIGMOD'98.
S. Guha, R. Rastogi, and K. Shim. ROCK: A robust clustering algorithm for categorical attributes. In ICDE'99, pp.
512-521, Sydney, Australia, March 1999.
A. Hinneburg, D.l A. Keim: An Efficient Approach to Clustering in Large Multimedia Databases with Noise. KDD’98.
A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.
G. Karypis, E.-H. Han, and V. Kumar. CHAMELEON: A Hierarchical Clustering Algorithm Using Dynamic
Modeling. COMPUTER, 32(8): 68-75, 1999.
L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons,
1990.
E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets. VLDB’98.
G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to Clustering. John Wiley and Sons,
1988.
P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997.
R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94.
10/02/2006
Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
slide55
References (3)
L. Parsons, E. Haque and H. Liu, Subspace Clustering for High Dimensional Data: A Review , SIGKDD
Explorations, 6(1), June 2004
E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data sets. Proc. 1996 Int.
Conf. on Pattern Recognition,.
G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution clustering approach for very large
spatial databases. VLDB’98.
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Subspace Clustering
Mining Biological Data
KU EECS 800, Luke Huan, Fall’06
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