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Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340 1 March 4+9: Introduction to KDD...
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1Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
March 4+9: Introduction to KDD March 11: Association Rule
Mining March 23: Similarity
Assessment March 25: Clustering and
UHDM2
March 30: Data Warehouses and OLAP
KDD --- 2004 Lectures
2Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Clustering andSimilarity Assessment
©Jiawei Han and Micheline Kamberwith major Additions and Modifications by Ch.
EickOrganization for COSC 6340:
1. What is Clustering?2. Object Similarity
Assessment3. K-means/medoid
Clustering4. Grid-based Clustering5. Work at UH
4Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Motivation: Why Clustering?
Problem: Identify (a small number of) groups of similar objects in a given (large) set of object.
Goals: Find representatives for homogeneous groups Data Compression
Find “natural” clusters and describe their properties ”natural” Data Types
Find suitable and useful grouping ”useful” Data Classes
Find unusual data object Outlier Detection
5Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Examples of Clustering Applications
Plant/Animal Classification Book Ordering Cloth Sizes Fraud Detection (Find outlier)
6Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Requirements of Clustering in Data Mining
Scalability Ability to deal with different types of attributes Discovery of clusters with arbitrary shape Minimal requirements for domain knowledge to
determine input parameters Able to deal with noise and outliers Insensitive to order of input records High dimensionality Incorporation of user-specified constraints Interpretability and usability
7Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Data Structures for Clustering
Data matrix (n objects, p attributes)
(Dis)Similarity matrix (nxn)
npx...nfx...n1x
...............ipx...ifx...i1x
...............1px...1fx...11x
0...)2,()1,(
:::
)2,3()
...ndnd
0dd(3,1
0d(2,1)
0
8Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Quality Evaluation of Clusters
Dissimilarity/Similarity metric: Similarity is expressed in terms of a normalized distance function d, which is typically metric; typically: (oi, oj) = 1 - d (oi, oj)
There is a separate “quality” function that measures the “goodness” of a cluster.
The definitions of similarity functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio-scaled variables.
Weights should be associated with different variables based on applications and data semantics.
It is hard to define “similar enough” or “good enough” the answer is typically highly subjective.
9Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Challenges in Obtaining Object Similarity Measures
Many Types of Variables Interval-scaled variables Binary variables and nominal variables Ordinal variables Ratio-scaled variables
Objects are characterized by variables
belonging to different types (mixture of
variables)
10Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Case Study: Patient Similarity The following relation is given (with 10000 tuples):Patient(ssn, weight, height, cancer-sev, eye-color, age) Attribute Domains
ssn: 9 digits weight between 30 and 650; mweight=158 sweight=24.20
height between 0.30 and 2.20 in meters; mheight=1.52
sheight=19.2
cancer-sev: 4=serious 3=quite_serious 2=medium 1=minor eye-color: {brown, blue, green, grey } age: between 3 and 100; mage=45 sage=13.2
Task: Define Patient Similarity
11Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Generating a Global Similarity Measure from Single Variable Similarity Measures
Assumption: A database may contain up to six types of variables: symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio.
1. Standardize variable and associate similarity measure i with the standardized i-th variable and determine weight wi of the i-th variable.
2. Create the following global (dis)similarity measure :
f
fjif
ji
wpf
woopfoo
1
*)(1
,),(
12Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
A Methodology to Obtain a Similarity Matrix
1. Understand Variables 2. Remove (non-relevant and redundant) Variables3. (Standardize and) Normalize Variables (typically
using z-scores or variable values are transformed to numbers in [0,1])
4. Associate (Dis)Similarity Measure df/f with each Variable
5. Associate a Weight (measuring its importance) with each Variable
6. Compute the (Dis)Similarity Matrix7. Apply Similarity-based Data Mining Technique
(e.g. Clustering, Nearest Neighbor, Multi-dimensional Scaling,…)
13Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Interval-scaled Variables
Standardize data using z-scores
Calculate the mean absolute deviation:
where
Calculate the standardized measurement (z-
score)
Using mean absolute deviation is more robust than
using standard deviation
.)...21
1nffff
xx(xn m
|)|...|||(|121 fnffffff
mxmxmxns
f
fifif s
mx z
14Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Normalization in [0,1]
Problem: If non-normalized variables are used the
maximum distance between two values can be greater
than 1.
Solution: Normalize interval-scaled variables using
where minf denotes the minimum value and maxf denotes
the maximum value of the f-th attribute in the data set
and is constant that is choses depending on the
similarity measure (e.g. if Manhattan distance is used
is chosen to be 1).
)*)min/((max)min( fffifif
z x
15Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Other Normalizations
Goal:Limit the maximum distance to 1 Start using a distance measure df(x,y) Determine the maximum distance dmaxf
that can occur for two values of the f-th attribute (e.g. dmaxf=maxf-minf ).
Define f(x,y)=1- (df(x,y)/ dmaxf)
Advantage: Negative similarities cannot occur.
16Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Similarity Between Objects
Distances are normally used to measure the similarity or dissimilarity between two data objects
Some popular ones include: Minkowski distance:
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp)
are two p-dimensional data objects, and q is a positive integer
If q = 1, d is Manhattan distance
pp
jx
ix
jx
ix
jx
ixjid )||...|||(|),(
2211
||...||||),(2211 pp jxixjxixjxixjid
17Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Similarity Between Objects (Cont.)
If q = 2, d is Euclidean distance:
Properties d(i,j) 0 d(i,i) = 0 d(i,j) = d(j,i) d(i,j) d(i,k) + d(k,j)
Also one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures.
)||...|||(|),( 22
22
2
11 pp jx
ix
jx
ix
jx
ixjid
18Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Similarity with respect to a Set of Binary Variables
A contingency table for binary data
pdbcasum
dcdc
baba
sum
0
1
01
Object i
Object j
dcbada ji
cbaa ji
sym
Jaccard
),(
),(
Ignores agree-ments in O’s
Considers agree-ments in 0’s and 1’sto be equivalent.
19Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Similarity between Binary Variable Sets
Example
gender is a symmetric attribute the remaining attributes are asymmetric binary let the values Y and P be set to 1, and the value N be
set to 0
Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4
Jack M Y N P N N NMary F Y N P N P NJim M Y P N N N N
25.0211
1),(
33.0111
1),(
67.0102
2),(
maryjim
jimjack
maryjack
Jacc
Jacc
Jacc
20Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Nominal Variables
A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green
Method 1: Simple matching m: # of matches, p: total # of variables
Method 2: use a large number of binary variables creating a new binary variable for each of the M
nominal states
pmpood ji
),(
21Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Ordinal Variables
An ordinal variable can be discrete or continuous order is important (e.g. UH-grade, hotel-rating) Can be treated like interval-scaled
replacing xif by their rank:
map the range of each variable onto [0, 1] by replacing the f-th variable of i-th object by
compute the dissimilarity using methods for interval-scaled variables
11
fM
ifif
rz
}{1,...,Mif
r f
22Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Ratio-Scaled Variables
Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale,
such as AeBt or Ae-Bt Methods:
treat them like interval-scaled variables — not a good choice! (why?)
apply logarithmic transformation
yif = log(xif)
treat them as continuous ordinal data treat their rank as interval-scaled.
23Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Case Study --- NormalizationPatient(ssn, weight, height, cancer-sev, eye-color, age) Attribute Relevance: ssn no; eye-color minor; other major Attribute Normalization:
ssn remove! weight between 30 and 650; mweight=158 sweight=24.20;
transform to zweight= (xweight-158)/24.20 (alternatively,
zweight=(xweight-30)/620));
height normalize like weight! cancer_sev: 4=serious 3=quite_serious 2=medium
1=minor; transform 4 to 1, 3 to 2/3, 2 to 1/3, 1 to 0 and then normalize like weight!
age: normalize like weight!
24Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Case Study --- Weight Selection and Similarity Measure Selection
Patient(ssn, weight, height, cancer-sev, eye-color, age) For normalized weight, height, cancer_sev, age values use
Manhattan distance function; e.g.:
weight(w1,w2)= 1 | ((w1-158)/24.20 ) ((w2-158)/24.20) |
For eye-color use: eye-color(c1,c2)= if c1=c2 then 1 else 0
Weight Assignment: 0.2 for eye-color; 1 for all others
Final Solution --- chosen Similarity Measure :Let o1=(s1,w1,h1,cs1,e1,a1) and o2=(s2,w2,h2,cs2,e2,a2)
(o1,o2):= (weight(w1,w2) + height(h1,h2) + cancer-
sev(cs1,cs2) + age(a1,a2) + 0.2* eye-color(e1,e2) ) /4.2
25Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Major Clustering Approaches
Partitioning algorithms: Construct various partitions and
then evaluate them by some criterion
Hierarchy algorithms: Create a hierarchical decomposition
of the set of data (or objects) using some criterion
Density-based: based on connectivity and density
functions
Grid-based: based on a multiple-level granularity structure
Model-based: A model is hypothesized for each of the
clusters and the idea is to find the best fit of that model to
each other
26Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Partitioning Algorithms: Basic Concept
Partitioning method: Construct a partition of a database D of n objects into a set of k clusters
Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion Global optimal: exhaustively enumerate all partitions Heuristic methods: k-means and k-medoids
algorithms k-means (MacQueen’67): Each cluster is represented
by the center of the cluster k-medoids or PAM (Partition around medoids)
(Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster
27Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
The K-Means Clustering Method
Given k, the k-means algorithm is implemented in 4 steps: Partition objects into k nonempty subsets Compute seed points as the centroids of the
clusters of the current partition. The centroid is the center (mean point) of the cluster.
Assign each object to the cluster with the nearest seed point.
Go back to Step 2, stop when no more new assignment.
28Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
The K-Means Clustering Method
Example
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29Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Comments on the K-Means Method Strength
Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n.
Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms
Weakness Applicable only when mean is defined, then what about
categorical data? Need to specify k, the number of clusters, in advance Unable to handle noisy data and outliers Not suitable to discover clusters with non-convex
shapes
30Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
PAM (Partitioning Around Medoids) (1987)
PAM (Kaufman and Rousseeuw, 1987), built in Splus Use real object to represent the cluster
Select k representative objects arbitrarily For each pair of non-selected object h and selected
object i, calculate the total swapping cost TCih
For each pair of i and h,
If TCih < 0, i is replaced by h
Then assign each non-selected object to the most similar representative object
repeat steps 2-3 until there is no change
31Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
PAM Clustering: Total swapping cost TCih=jCjih
0
1
2
3
4
5
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j
ih
t
Cjih = 0
0
1
2
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t
i h
j
Cjih = d(j, h) - d(j, i)
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h
i t
j
Cjih = d(j, t) - d(j, i)
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t
ih j
Cjih = d(j, h) - d(j, t)
32Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
CLARANS (“Randomized” CLARA) (1994)
CLARANS (A Clustering Algorithm based on Randomized Search) (Ng and Han’94)
CLARANS draws sample of neighbors dynamically The clustering process can be presented as searching a
graph where every node is a potential solution, that is, a set of k medoids
If the local optimum is found, CLARANS starts with new randomly selected node in search for a new local optimum
It is more efficient and scalable than both PAM and CLARA Focusing techniques and spatial access structures may
further improve its performance (Ester et al.’95)
33Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
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)
34Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
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
Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
STING: A Statistical Information Grid Approach (2)
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
Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
STING: A Statistical Information Grid Approach (3)
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
37Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
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
38Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
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
39Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Sala
ry
(10,
000)
20 30 40 50 60age
54
31
26
70
20 30 40 50 60age
54
31
26
70
Vac
atio
n(w
eek)
age
Vac
atio
n
Salary 30 50
= 3
40Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Strength and Weakness of CLIQUE
Strength It automatically finds subspaces of the highest
dimensionality such that high density clusters exist in those subspaces
It is insensitive to the order of records in input and does not presume some canonical data distribution
It 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
41Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Creating Environments for Database Clustering; problems related to Multi-relational Data Mining [ER04].
Distance Function Learning [EVR03] Supervised Clustering [EZZ04] Using Clustering to Enhance Classifiers
[ICDM03], [ECAI04], [PKDD04] not discussed
Using SQL Queries for Data Summarization [KDD96]; [RYU98]; not discussed
Work at UH related to Similarity Assessment and Clustering
Work at UH related to Similarity Assessment and Clustering
Work at UH
42Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Data Extraction Tool
DBMS
Clustering Tool
User Interface
A set of clusters
Similarity measure
Similarity Measure Tool
Default choices and domain information
Library of similarity measures
Type and weight
information
ObjectView
Library of clustering algorithms
CAL-FULL/UH Database Clustering Similarity Assessment
Environments
CAL-FULL/UH Database Clustering Similarity Assessment
Environments
LearningTool
TrainingDate
Work at UH
43Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Prototypes of Similarity Assessment Tools
Prototypes of Similarity Assessment Tools
Prototype1 (CAL State Fullerton): Supported the interactive definition of similarity measures; knowledge representation format does not rely on modular units; provides a nearest neighbor clustering algorithm for database clustering; functions were supported outside a DBMS
Prototype 2 (UH 2002): Similarity measures are defined using a special language (not interactively); tool supports modular units and functions are provided using a Java/SQL-Server 2000 framework; functions were partially moved inside a DBMS (although some are still inside Java); analysis results are stored in the database and therefore available for further analysis.
Prototype 3 (UH): Learn Distance Functions for Classification Problems. Currently Investigated!Work at UH
Objectives Supervised Clustering: Maximize Cluster Purity while keeping the number of clusters low.
Work at UH
45Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Research Goals Supervised Clustering
Develop representative-based supervised clustering algorithms.
Show the benefits of supervised clustering in case studies that center on summary generation, distance function learning, and classification.
Work at UH
46Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
What is a good object distance function for supervised similarity
assessment?
Objective: Learn good distance functions for classification tasks.
Our approach: Apply a clustering algorithm with the distance function to be evaluated that returns a predetermined number of clusters k. The more pure the obtained clusters are the better is the quality of .
Our goal is to learn the weights of an object distance function such that all the clusters are pure (or as pure is possible); for more details see [ERV03] Paper.
f
fjif
ji
wpf
woopfoo
1
*)(1
,),(
Work at UH
47Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Idea: Coevolving Clusters and Similarity Functions
Clusters X SimilarityFunction
Reinforcement Learning
Clustering
Goodness of The Similarity
Function
q(X) Clustering Evaluation
Work at UH
q(X):=percentage_of_minority_examples + penalty(k)penalty(k):= If kc then 0 else sqrt((k-c)/n))withk:= number of clusters generatedn:= number of objects in the datasetc:= number of classes in the dataset
Idea CR*-ApproachLet be a clustering algorithm and Error(,O)=Error’((,O)) an error
function that measures class purity in clusters, class coverage, and assigns a penalty for large numbers of clusters.
While not done do1. Cluster with respect to (,O) receiving clusters C and report
Error’((,O))2. If Error’((,O)) is small enough stop reporting the error, C, and 3. For each cluster determine majority class4. For each cC adjust weights wj locally
x:=examples belonging to majority classo:= non-majority-class examples
x x
Clusterx
X x
xo oo
Decrease weight for modular unit
Increase weight for modular unit
Cluster
X x o
o o
Idea: Move examples of the majority class closer to each other
Work at UH
49Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Weight Adjustment within a Cluster
Let wi be the current weight of the i-th modular unit
Let i be the average absolute deviation for the examples that belong to the cluster with respect to i
Let i be the average absolute deviation for the examples of the cluster that belong to the majority class with respect to i
Learning: Then weights are adjusted as follows with respect to a particular cluster: wi’=wi+ (i – i) or betterwi’=wi+ wimin(max(i – i)
with being the learning rate and maximal adjustment (e.g. if a weight can be maximally increased/decreased by 20%) per weight per cluster.
Remark: If the cluster is ‘pure’ or does not contain 2 or more elements of a particular class, no weight adjustment takes place.
Work at UH
50Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Summary: Problems and Challenges for Clustering
Considerable progress has been made in scalable clustering methods Partitioning: k-means, k-medoid, CLARANS, EM Hierarchical: BIRCH, CURE Density-based: DBSCAN, CLIQUE, OPTICS Grid-based: STING, WaveCluster Model-based: Autoclass, Denclue, Cobweb
Current clustering techniques do not address all the requirements adequately
Constraint-based clustering analysis: Constraints exist in data space (bridges and highways) or in user queries
51Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
Summary Object Similarity & Clustering
Cluster analysis groups objects based on their similarity and has wide applications
Appropriate similarity measures have to be chosen for various types of variables and combined into a global similarity measure.
Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods
Methods to measure, compute, and learn object similarity are quite important, not only for clustering, but also for nearest neighbor approaches, information retrieval in general, and for data visualization.
52Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
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
Scietific, 1996 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 class identification. SSD'95. 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. In Proc. VLDB’98. S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for
large databases. SIGMOD'98. A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.
53Han, Kamber, Eick: Object Similarity & Clustering for COSC 6340
References (2) 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. E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very
large data sets. Proc. 1996 Int. Conf. on Pattern Recognition, 101-105. G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution
clustering approach for very large spatial databases. VLDB’98. W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to
Spatial Data Mining, VLDB’97. T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering
method for very large databases. SIGMOD'96.