Clustering

Data Mining: Concepts and Techniques

Chapter 10

Cluster Analysis

2010 Han, Kamber & Pei. All rights reserved.

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Cluster Analysis: Basic Concepts and Methods

n Cluster Analysis: Basic Concepts

n Par44oning Methods

n Hierarchical Methods

n Density-Based Methods

n Grid-Based Methods

n Evalua4on of Clustering

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What is Cluster Analysis?

n Cluster: A collection of data objects n similar (or related) to one another within the

same group n dissimilar (or unrelated) to the objects in other

groups n Cluster analysis (or clustering): Finding similarities

between data according to the characteristics found in the data and grouping similar data objects into clusters.

n Unsupervised learning: no predefined classes (i.e., learning by observations vs. learning by examples: supervised)

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Clustering Applications: Data Understanding

n Biology: taxonomy of living things: kingdom, phylum, class, order, family, genus and species

n Information retrieval: document clustering n Land use: Identification of areas of similar land use in an earth

observation database n Marketing: Help marketers discover distinct groups in their customer

bases, and then use this knowledge to develop targeted marketing programs

n City-planning: Identifying groups of houses according to their house type, value, and geographical location

n Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults

n Climate: understanding earth climate, find patterns of atmospheric and ocean

n Economic Science: market resarch

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Clustering Applications: Preprocessing Utility

n Finding K-nearest Neighbors n Localizing search to one or a small number of clusters

n Outlier detection n Outliers are often viewed as those far away from any

cluster n Summarization:

n Preprocessing for regression, PCA, classification, and association analysis

n Compression: n Image processing: vector quantization

Dissimilarity / Similarity Metric

n Similarity is expressed in terms of a distance function, typically metric: d(i, j)

n The definitions of distance functions are usually different for interval-scaled, boolean, categorical, ordinal ratio, and vector variables

n Distance-based (e.g., Euclidian, cosine) vs. connectivity-based (e.g., density or contiguity)

n Weights may be associated with different variables based on applications and data semantics

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Quality: What Is Good Clustering?

n A good clustering method will produce high quality clusters

n high intra-class similarity: cohesive within clusters

n low inter-class similarity: distinctive between clusters

n Quality function to measure the goodness

n The quality of a clustering method depends on

n the similarity measure used by the method

n its implementation, and

n Its ability to discover some or all of the hidden patterns

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Requirements and Challenges

n Scalability n Clustering all the data instead of only on samples

n Ability to deal with different types of attributes n Numerical, binary, categorical, ordinal, linked, and mixture of

these n Constraint-based clustering

n User may give inputs on constraints n Use domain knowledge to determine input parameters

n Interpretability and usability n Others

n Discovery of clusters with arbitrary shape n Ability to deal with noisy data n Incremental clustering and insensitivity to input order n High dimensionality

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Considerations for Cluster Analysis

n Partitioning criteria n Single level vs. hierarchical partitioning (often, multi-

level hierarchical partitioning is desirable)

n Separation of clusters

n Exclusive (e.g., one customer belongs to only one region) vs. non-exclusive (e.g., one document may belong to more than one class)

n Clustering space

n Full space (often when low dimensional) vs. subspaces (often in high-dimensional clustering)

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Major Clustering Approaches (I)

n Partitioning approach: n Construct various partitions and then evaluate them by some

criterion, e.g., minimizing the sum of square errors n Typical methods: k-means, k-medoids, CLARANS

n Hierarchical approach: n Create a hierarchical decomposition of the set of data (or objects)

using some criterion n Typical methods: Diana, Agnes, BIRCH, CAMELEON

n Density-based approach: n Based on connectivity and density functions n Typical methods: DBSACN, OPTICS, DenClue

n Grid-based approach: n based on a multiple-level granularity structure n Typical methods: STING, WaveCluster, CLIQUE

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Major Clustering Approaches (II)

n Model-based: n A model is hypothesized for each of the clusters and tries to find

the best fit of that model to each other n Typical methods: EM, SOM, COBWEB

n Frequent pattern-based: n Based on the analysis of frequent patterns n Typical methods: p-Cluster

n User-guided or constraint-based: n Clustering by considering user-specified or application-specific

constraints n Typical methods: COD (obstacles), constrained clustering

n Link-based clustering: n Objects are often linked together in various ways n Massive links can be used to cluster objects: SimRank, LinkClus

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Partitioning Algorithms: Basic Concept

n Partitioning method: Partitioning a database D of n objects into a set of k clusters, such that the sum of squared distances is minimized (where ci is the centroid or medoid of cluster Ci)

n Given k, find a partition of k clusters that optimizes the chosen partitioning criterion n Global optimal: exhaustively enumerate all partitions n Heuristic methods: k-means and k-medoids algorithms n k-means (MacQueen67, Lloyd57/82): Each cluster is

represented by the center of the cluster n k-medoids or PAM (Partition around medoids) (Kaufman &

Rousseeuw87): Each cluster is represented by one of the objects in the cluster

21 )( iCp

ki cpE i = =

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The K-Means Clustering Method

n Given k, the k-means algorithm is implemented in four steps:

n Partition objects into k nonempty subsets

n Compute seed points as the centroids of the clusters of the current partitioning (the centroid is the center, i.e., mean point, of the cluster)

n Assign each object to the cluster with the nearest seed point

n Go back to Step 2, stop when the assignment does not change

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An Example of K-Means Clustering

K=2

Arbitrarily partition objects into k groups

Update the cluster centroids

Update the cluster centroids

Reassign objects Loop if needed

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The initial data set

n Partition objects into k nonempty subsets

n Repeat n Compute centroid (i.e., mean

point) for each partition

n Assign each object to the cluster of its nearest centroid

n Until no change

Comments on the K-Means Method

n Strength: Efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t

Variations of the K-Means Method

n Most of the variants of the k-means which differ in

n Selection of the initial k means

n Dissimilarity calculations

n Strategies to calculate cluster means

n Handling categorical data: k-modes

n Replacing means of clusters with modes

n Using new dissimilarity measures to deal with categorical objects

n Using a frequency-based method to update modes of clusters

n A mixture of categorical and numerical data: k-prototype method

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What Is the Problem of the K-Means Method?

n The k-means algorithm is sensitive to outliers !

n Since an object with an extremely large value may substantially

distort the distribution of the data

n K-Medoids: Instead of taking the mean value of the object in a cluster

as a reference point, medoids can be used, which is the most

centrally located object in a cluster

012345678910

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The K-Medoid Clustering Method

n K-Medoids Clustering: Find representative objects (medoids) in clusters

n PAM (Partitioning Around Medoids, Kaufmann & Rousseeuw 1987) n Use real object to represent the cluster

n Starts from an initial set of medoids

n Iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering

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PAM: A Typical K-Medoids Algorithm

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Total Cost = 20

012345678910

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K=2

Arbitrary choose k object as initial medoids

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Assign each remaining object to nearest medoids

Randomly select a nonmedoid object,Oramdom

Compute total cost of swapping

012345678910

012345678910

Total Cost = 26

Swapping O and Oramdom

If quality is improved.

Do loop

Until no change

012345678910

012345678910

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PAM Algorithm

1. Select k representative objects arbitrarily 2. For each pair of non-selected object h and selected

object i, calculate the total swapping cost TCih 3. For each pair of i and h,

n If TCih < 0, i is replaced by h n Then assign each non-selected object to the

most similar representative object 4. repeat steps 2-3 until there is no change

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PAM Clustering: Finding the Best Cluster Center

n Case 1: p currently belongs to oj. If oj is replaced by orandom as a representative object and p is the closest to one of the other representative object oi, then p is reassigned to oi

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What Is the Problem with PAM?

n PAM works effectively for small data sets, but does not scale well for large data sets

n Computational complexity is O(k(n-k)2 for each iteration

n n is # of data, k is # of clusters n Sampling based Efficiency improvement on PAM

n CLARA (Clustering LARge Applications), Kaufmann & Rousseeuw, 1990

n CLARANS (Ng & Han, 1994): Randomized re-sampling

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Hierarchical Clustering n Use distance matrix as clustering criteria. This method

does not require the number of clusters k as an input, but needs a termination condition

b

d c

e

a a b

d e c d e

a b c d e

Step 0 Step 1 Step 2 Step 3 Step 4 agglomerative (AGNES)

Step 4 Step 3 Step 2 Step 1 Step 0 divisive

(DIANA)

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Dendrogram: Shows How Clusters are Merged

n Decompose data objects into a several levels of nested partitioning (tree of clusters), called a dendrogram

n A clustering of the data objects is obtained by cutting the dendrogram at the desired level, then each connected component forms a cluster

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AGNES (Agglomerative Nesting)

n Introduced in Kaufmann and Rousseeuw (1990) n Implemented in statistical packages, e.g., Splus n Use the single-link method and the dissimilarity matrix n Merge nodes that have the least dissimilarity n Go on in a non-descending fashion n Eventually all nodes belong to the same cluster

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DIANA (Divisive Analysis)

n Introduced in Kaufmann and Rousseeuw (1990)

n Implemented in many statistical analysis packages

n Inverse order of AGNES

n Eventually each node forms a cluster on its own

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Distance between Clusters

n Single link: smallest distance between an element in one cluster and an element in the other, i.e., dist(Ki, Kj) = min(tip, tjq)

n Complete link: largest distance between an element in one cluster and an element in the other, i.e., dist(Ki, Kj) = max(tip, tjq)

n Average: avg distance between an element in one cluster and an element in the other, i.e., dist(Ki, Kj) = avg(tip, tjq)

n Centroid: distance between the centroids of two clusters n dist(Ki, Kj) = dist(Ci, Cj)

n Medoid: distance between the medoids of two clusters n dist(Ki, Kj) = dist(Mi, Mj)

X X

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?

Centroid, Radius and Diameter of a Cluster (for numerical data sets)

n Centroid: the middle of a cluster

n Radius: square root of average distance from any point of the cluster to its centroid

n Diameter: square root of average mean squared distance between all pairs of points in the cluster

N

tNi ipmC)(1==

Nmcipt

Ni

mR2)(

1

=

=

)1(

2)(11

=

=

=

NNiqtipt

Ni

Ni

mD

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Extensions to Hierarchical Clustering

n Major weakness of agglomerative clustering methods

n Can never undo what was done previously

n Do not scale well: time complexity of at least O(n2),

where n is the number of total objects

n Integration of hierarchical & distance-based clustering

n BIRCH (1996): uses CF-tree and incrementally adjusts

the quality of sub-clusters

n CHAMELEON (1999): hierarchical clustering using

dynamic modeling 31

BIRCH (Balanced Iterative Reducing and Clustering Using Hierarchies)

n Zhang, Ramakrishnan & Livny, SIGMOD96

n Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering

n Phase 1: scan DB to build an initial in-memory CF tree (a multi-level compression of the data that tries to preserve the inherent clustering structure of the data)

n Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree

n Scales linearly: finds a good clustering with a single scan and improves the quality with a few additional scans

n Weakness: handles only numeric data, and sensitive to the order of the data record

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Clustering Feature Vector in BIRCH

Clustering Feature (CF): CF = (N, LS, SS)

N: Number of data points

LS: linear sum of N points:

SS: square sum of N points

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CF = (5, (16,30),(54,190))

(3,4)(2,6)(4,5)(4,7)(3,8)

=

N

iiX

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1=

N

iiX

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CF-Tree in BIRCH

n Clustering feature: n Summary of the sta4s4cs for a given subcluster: the 0-th, 1st, and 2nd moments of the subcluster from the sta4s4cal point of view

n Registers crucial measurements for compu4ng cluster and u4lizes storage eciently

A CF tree is a height-balanced tree that stores the clustering features for a hierarchical clustering n A nonleaf node in a tree has descendants or children n The nonleaf nodes store sums of the CFs of their children

n A CF tree has two parameters n Branching factor: max # of children n Threshold: max diameter of sub-clusters stored at the leaf nodes 34

The CF Tree Structure

CF1 child1

CF3 child3

CF2 child2

CF6 child6

CF1 child1

CF3 child3

CF2 child2

CF5 child5

CF1 CF2 CF6 prev next CF1 CF2 CF4 prev next

B = 7

L = 6

Root

Non-leaf node

Leaf node Leaf node

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The Birch Algorithm

n Cluster Diameter

n For each point in the input n Find closest leaf entry n Add point to leaf entry and update CF n If entry diameter > max_diameter, then split leaf, and possibly

parents n Algorithm is O(n) n Concerns

n Sensitive to insertion order of data points n Since we fix the size of leaf nodes, so clusters may not be so

natural n Clusters tend to be spherical given the radius and diameter

measures

2)()1(

1jxixnn

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CHAMELEON: Hierarchical Clustering Using Dynamic Modeling (1999)

n CHAMELEON: G. Karypis, E. H. Han, and V. Kumar, 1999 n Measures the similarity based on a dynamic model

n Two clusters are merged only if the interconnectivity and closeness (proximity) between two clusters are high relative to the internal interconnectivity of the clusters and closeness of items within the clusters

n Graph-based, and a two-phase algorithm 1. Use a graph-partitioning algorithm: cluster objects into

a large number of relatively small sub-clusters 2. Use an agglomerative hierarchical clustering algorithm:

find the genuine clusters by repeatedly combining these sub-clusters

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Overall Framework of CHAMELEON

Construct (K-NN)

Sparse Graph Partition the Graph

Merge Partition

Final Clusters

Data Set

K-NN Graph

P and q are connected if q is among the top k closest neighbors of p

Rela-ve interconnec-vity: connec4vity of c1 and c2 over internal connec4vity

Rela-ve closeness: closeness of c1 and c2 over internal closeness

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CHAMELEON (Clustering Complex Objects)

Probabilistic Hierarchical Clustering

n Algorithmic hierarchical clustering n Nontrivial to choose a good distance measure n Hard to handle missing attribute values n Optimization goal not clear: heuristic, local search

n Probabilistic hierarchical clustering n Use probabilistic models to measure distances between clusters n Generative model: Regard the set of data objects to be clustered as

a sample of the underlying data generation mechanism to be analyzed

n Easy to understand, same efficiency as algorithmic agglomerative clustering method, can handle partially observed data

n In practice, assume the generative models adopt common distributions functions, e.g., Gaussian distribution or Bernoulli distribution, governed by parameters

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Generative Model

n Given a set of 1-D points X = {x1, , xn} for clustering analysis & assuming they are generated by a Gaussian distribution:

n The probability that a point xi X is generated by the model

n The likelihood that X is generated by the model:

n The task of learning the generative model: find the parameters and 2 such that the maximum likelihood

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A Probabilistic Hierarchical Clustering Algorithm

n For a set of objects partitioned into m clusters C1, . . . ,Cm, the quality can be measured by,

where P() is the maximum likelihood n Distance between clusters C1 and C2: n Algorithm: Progressively merge points and clusters

Input: D = {o1, ..., on}: a data set containing n objects Output: A hierarchy of clusters Method

Create a cluster for each object Ci = {oi}, 1 i n; For i = 1 to n {

Find pair of clusters Ci and Cj such that Ci,Cj = argmaxi j {log (P(CiCj )/(P(Ci)P(Cj ))};

If log (P(CiCj )/(P(Ci)P(Cj )) > 0 then merge Ci and Cj }

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Density-Based Clustering Methods

n Clustering based on density (local cluster criterion), such as density-connected points

n Major features: n Discover clusters of arbitrary shape n Handle noise n One scan n Need density parameters as termination condition

n Several interesting studies: n DBSCAN: Ester, et al. (KDD96) n OPTICS: Ankerst, et al (SIGMOD99). n DENCLUE: Hinneburg & D. Keim (KDD98) n CLIQUE: Agrawal, et al. (SIGMOD98) (more grid-

based) 44

Density-Based Clustering: Basic Concepts

n Two parameters: n Eps: Maximum radius of the neighbourhood

n MinPts: Minimum number of points in an Eps-neighbourhood of that point

n NEps(p): {q belongs to D | dist(p,q) Eps}

n Directly density-reachable: A point p is directly density-reachable from a point q w.r.t. Eps, MinPts if

n p belongs to NEps(q)

n core point condition:

|NEps (q)| MinPts

MinPts = 5

Eps = 1 cm

p

q

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Density-Reachable and Density-Connected

n Density-reachable:

n 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

n Density-connected

n 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 p1

p q

o

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DBSCAN: Density-Based Spatial Clustering of Applications with Noise

n Relies on a density-based notion of cluster: A cluster is defined as a maximal set of density-connected points

n Discovers clusters of arbitrary shape in spatial databases with noise

Core

Border

Outlier

Eps = 1cm

MinPts = 5

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DBSCAN: The Algorithm

n Arbitrary select a point p

n Retrieve all points density-reachable from p w.r.t. Eps and MinPts

n If p is a core point, a cluster is formed

n If p is a border point, no points are density-reachable from p and DBSCAN visits the next point of the database

n Continue the process until all of the points have been processed

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DBSCAN: Sensitive to Parameters

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OPTICS: A Cluster-Ordering Method (1999)

n OPTICS: Ordering Points To Identify the Clustering Structure n Ankerst, Breunig, Kriegel, and Sander (SIGMOD99) n Produces a special order of the database wrt its

density-based clustering structure n This cluster-ordering contains info equiv to the density-

based clusterings corresponding to a broad range of parameter settings

n Good for both automatic and interactive cluster analysis, including finding intrinsic clustering structure

n Can be represented graphically or using visualization techniques

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OPTICS: Some Extension from DBSCAN

n Index-based:

n k = number of dimensions

n N = 20

n p = 75%

n M = N(1-p) = 5 n Complexity: O(NlogN)

n Core Distance: n min eps s.t. point is core

n Reachability Distance

D

p2

MinPts = 5

= 3 cm

Max (core-distance (o), d (o, p))

r(p1, o) = 2.8cm. r(p2,o) = 4cm

o

o

p1

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Reachability-distance

Cluster-order of the objects

undefined

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Density-Based Clustering: OPTICS & Its Applications

DENCLUE: Using Statistical Density Functions

n DENsity-based CLUstEring by Hinneburg & Keim (KDD98) n Using statistical density functions:

n Major features n Solid mathematical foundation

n Good for data sets with large amounts of noise

n Allows a compact mathematical description of arbitrarily shaped clusters in high-dimensional data sets

n Significant faster than existing algorithm (e.g., DBSCAN) n But needs a large number of parameters

f x y eGaussiand x y

( , )( , )

=

2

22 =

=N

i

xxdDGaussian

i

exf1

2),(

2

2

)(

=

=N

i

xxd

iiDGaussian

i

exxxxf1

2),(2

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)(),( influence of y

on x

total influence on x

gradient of x in the direction of

xi

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n 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

n Influence function: describes the impact of a data point within its neighborhood

n Overall density of the data space can be calculated as the sum of the influence function of all data points

n Clusters can be determined mathematically by identifying density attractors

n Density attractors are local maximal of the overall density function n Center defined clusters: assign to each density attractor the points

density attracted to it n Arbitrary shaped cluster: merge density attractors that are connected

through paths of high density (> threshold)

Denclue: Technical Essence

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Density Attractor

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Center-Defined and Arbitrary

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Grid-Based Clustering Method

n Using multi-resolution grid data structure n Several interesting methods

n STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997)

n WaveCluster by Sheikholeslami, Chatterjee, and Zhang (VLDB98) n A multi-resolution clustering approach

using wavelet method n CLIQUE: Agrawal, et al. (SIGMOD98)

n Both grid-based and subspace clustering

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STING: A Statistical Information Grid Approach

n Wang, Yang and Muntz (VLDB97) n The spatial area is divided into rectangular cells n There are several levels of cells corresponding to different

levels of resolution

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i-th layer

(i-1)st layer

1st layer

The STING Clustering Method

n Each cell at a high level is partitioned into a number of smaller cells in the next lower level

n Statistical info of each cell is calculated and stored beforehand and is used to answer queries

n Parameters of higher level cells can be easily calculated from parameters of lower level cell n count, mean, s, min, max n type of distributionnormal, uniform, etc.

n Use a top-down approach to answer spatial data queries n Start from a pre-selected layertypically with a small

number of cells n For each cell in the current level compute the confidence

interval 61

STING Algorithm and Its Analysis

n Remove the irrelevant cells from further consideration n When finish examining the current layer, proceed to the

next lower level n Repeat this process until the bottom layer is reached n Advantages:

n Query-independent, easy to parallelize, incremental update

n O(K), where K is the number of grid cells at the lowest level

n Disadvantages: n All the cluster boundaries are either horizontal or

vertical, and no diagonal boundary is detected

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CLIQUE (Clustering In QUEst)

n Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD98)

n Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space

n CLIQUE can be considered as both density-based and grid-based

n It partitions each dimension into the same number of equal length interval

n It partitions an m-dimensional data space into non-overlapping rectangular units

n A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter

n A cluster is a maximal set of connected dense units within a subspace

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CLIQUE: The Major Steps

n Partition the data space and find the number of points that lie inside each cell of the partition.

n Identify the subspaces that contain clusters using the Apriori principle

n Identify clusters

n Determine dense units in all subspaces of interests n Determine connected dense units in all subspaces of

interests.

n Generate minimal description for the clusters n Determine maximal regions that cover a cluster of

connected dense units for each cluster n Determination of minimal cover for each cluster

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Sala

ry

(10,

000)

20 30 40 50 60 age

5 4

3 1

2 6

7 0

20 30 40 50 60 age

5 4

3 1

2 6

7 0

Vaca

tion(

wee

k)

age

Vaca

tion

30 50

= 3

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Strength and Weakness of CLIQUE

n Strength n automatically finds subspaces of the highest

dimensionality such that high density clusters exist in those subspaces

n insensitive to the order of records in input and does not presume some canonical data distribution

n scales linearly with the size of input and has good scalability as the number of dimensions in the data increases

n Weakness n The accuracy of the clustering result may be degraded

at the expense of simplicity of the method

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n + Clustering High-Dimensional Data (from Chap. 11)


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Clustering High-Dimensional Data

n Clustering high-dimensional data n Many applications: text documents, DNA micro-array data n Major challenges:

n Many irrelevant dimensions may mask clusters n Distance measure becomes meaninglessdue to equi-distance n Clusters may exist only in some subspaces

n Methods n Feature transformation: only effective if most dimensions are

relevant n PCA & SVD useful only when features are highly correlated/

redundant n Feature selection: wrapper or filter approaches

n useful to find a subspace where the data have nice clusters n Subspace-clustering: find clusters in all the possible subspaces

n CLIQUE, ProClus, and frequent pattern-based clustering

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The Curse of Dimensionality (graphs adapted from Parsons et al. KDD Explorations 2004)

n Data in only one dimension is relatively packed

n Adding a dimension stretch the points across that dimension, making them further apart

n Adding more dimensions will make the points further aparthigh dimensional data is extremely sparse

n Distance measure becomes meaninglessdue to equi-distance

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Why Subspace Clustering? (adapted from Parsons et al. SIGKDD Explorations 2004)

n Clusters may exist only in some subspaces n Subspace-clustering: find clusters in all the subspaces

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Frequent Pattern-Based Approach

n Clustering high-dimensional space (e.g., clustering text documents, microarray data) n Projected subspace-clustering: which dimensions to be

projected on? n CLIQUE, ProClus

n Feature extraction: costly and may not be effective? n Using frequent patterns as features

n Clustering by pattern similarity in micro-array data (pClustering) [H. Wang, W. Wang, J. Yang, and P. S. Yu. Clustering by pattern similarity in large data sets, SIGMOD02]

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Clustering by Pattern Similarity (p-Clustering)

n Left figure: Micro-array raw data shows 3 genes and their values in a multi-D space: Difficult to find their patterns

n Right two: Some subsets of dimensions form nice shift and scaling patterns

n No globally defined similarity/distance measure n Clusters may not be exclusive

n An object can appear in multiple clusters

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Why p-Clustering?

n Microarray data analysis may need to n Clustering on thousands of dimensions (aWributes) n Discovery of both shiX and scaling paWerns

n Clustering with Euclidean distance measure? cannot nd shiX paWerns n Clustering on derived aWribute Aij = ai aj? introduces N(N-1) dimensions n Bi-cluster (Y. Cheng and G. Church. Biclustering of expression data. ISMB00)

using transformed mean-squared residue score matrix (I, J)

n Where n A submatrix is a -cluster if H(I, J) for some > 0

n Problems with bi-cluster n No downward closure property n Due to averaging, it may contain outliers but s4ll within -threshold

=Jj ijd

Jijd

||1

=Ii ijd

IIjd

||1

=JjIi ijd

JIIJd

,||||1

p-Clustering: Clustering by Pattern Similarity

n P-score: the similarity between 2 objects rx, ry on 2 attributes au, av

n -pCluster: If for any 2 by 2 matrix X, pScore(X) ( > 0)

n Properties of -pCluster

n Downward closure

n Clusters are more homogeneous than bi-cluster (thus the name:

pair-wise Cluster)

n MaPle (Pei et al. 2003): Efficient mining of maximum p-clusters

n For scaling patterns, taking logarithmic on will lead to

the pScore form

|)()(|)( ybyaxbxayb

xb

ya

xa dddddd

dd

pScore =

0, repeat it m times, compare the overall quality measure w.r.t. different ks, and find # of clusters that fits the data the best

77

Measuring Clustering Quality

n Two methods: extrinsic vs. intrinsic n Extrinsic: supervised, i.e., the ground truth is available

n Compare a clustering against the ground truth using certain clustering quality measure

n Ex. BCubed precision and recall metrics

n Intrinsic: unsupervised, i.e., the ground truth is unavailable

n Evaluate the goodness of a clustering by considering how well the clusters are separated, and how compact the clusters are

n Ex. Silhouette coefficient

78

Measuring Clustering Quality: Extrinsic Methods

n Clustering quality measure: Q(C, Cg), for a clustering C given the ground truth Cg.

n Q is good if it satisfies the following 4 essential criteria n Cluster homogeneity: the purer, the better n Cluster completeness: should assign objects belong to

the same category in the ground truth to the same cluster

n Rag bag: putting a heterogeneous object into a pure cluster should be penalized more than putting it into a rag bag (i.e., miscellaneous or other category)

n Small cluster preservation: splitting a small category into pieces is more harmful than splitting a large category into pieces

79

Summary n Cluster analysis groups objects based on their similarity and has

wide applications n Measure of similarity can be computed for various types of data n Clustering algorithms can be categorized into partitioning methods,

hierarchical methods, density-based methods, grid-based methods, and model-based methods

n K-means and K-medoids algorithms are popular partitioning-based clustering algorithms

n Birch and Chameleon are interesting hierarchical clustering algorithms, and there are also probabilistic hierarchical clustering algorithms

n DBSCAN, OPTICS, and DENCLU are interesting density-based algorithms

n STING and CLIQUE are grid-based methods, where CLIQUE is also a subspace clustering algorithm

n Quality of clustering results can be evaluated in various ways 80

References (1)

n R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. SIGMOD'98

n M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973. n M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points

to identify the clustering structure, SIGMOD99. n Beil F., Ester M., Xu X.: "Frequent Term-Based Text Clustering", KDD'02 n M. M. Breunig, H.-P. Kriegel, R. Ng, J. Sander. LOF: Identifying Density-Based

Local Outliers. SIGMOD 2000. n M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for

discovering clusters in large spatial databases. KDD'96. n M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial

databases: Focusing techniques for efficient class identification. SSD'95. n D. Fisher. Knowledge acquisition via incremental conceptual clustering.

Machine Learning, 2:139-172, 1987. n D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An

approach based on dynamic systems. VLDB98. n V. Ganti, J. Gehrke, R. Ramakrishan. CACTUS Clustering Categorical Data

Using Summaries. KDD'99.

81

References (2) n D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An

approach based on dynamic systems. In Proc. VLDB98. n S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for

large databases. SIGMOD'98. n 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.

n A. Hinneburg, D.l A. Keim: An Efficient Approach to Clustering in Large Multimedia Databases with Noise. KDD98.

n A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.

n G. Karypis, E.-H. Han, and V. Kumar. CHAMELEON: A Hierarchical Clustering Algorithm Using Dynamic Modeling. COMPUTER, 32(8): 68-75, 1999.

n L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, 1990.

n E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets. VLDB98.

82

References (3) n G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to

Clustering. John Wiley and Sons, 1988. n R. Ng and J. Han. Efficient and effective clustering method for spatial data mining.

VLDB'94. n L. Parsons, E. Haque and H. Liu, Subspace Clustering for High Dimensional Data: A

Review, SIGKDD Explorations, 6(1), June 2004 n E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large

data sets. Proc. 1996 Int. Conf. on Pattern Recognition,. n G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution

clustering approach for very large spatial databases. VLDB98. n A. K. H. Tung, J. Han, L. V. S. Lakshmanan, and R. T. Ng. Constraint-Based

Clustering in Large Databases, ICDT'01. n A. K. H. Tung, J. Hou, and J. Han. Spatial Clustering in the Presence of Obstacles,

ICDE'01 n H. Wang, W. Wang, J. Yang, and P.S. Yu. Clustering by pattern similarity in large

data sets, SIGMOD 02. n W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial

Data Mining, VLDB97. n T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : An efficient data clustering

method for very large databases. SIGMOD'96. n Xiaoxin Yin, Jiawei Han, and Philip Yu,

LinkClus: Efficient Clustering via Heterogeneous Semantic Links, in Proc. 2006 Int. Conf. on Very Large Data Bases (VLDB'06), Seoul, Korea, Sept. 2006.

83

Slides unused in class

84

85

CLARA (Clustering Large Applications) (1990)

n CLARA (Kaufmann and Rousseeuw in 1990)

n Built in statistical analysis packages, such as SPlus n It draws multiple samples of the data set, applies

PAM on each sample, and gives the best clustering as the output

n Strength: deals with larger data sets than PAM n Weakness:

n Efficiency depends on the sample size n A good clustering based on samples will not

necessarily represent a good clustering of the whole data set if the sample is biased

86

CLARANS (Randomized CLARA) (1994)

n CLARANS (A Clustering Algorithm based on Randomized Search) (Ng and Han94) n Draws sample of neighbors dynamically n The clustering process can be presented as searching a

graph where every node is a potential solution, that is, a set of k medoids

n If the local optimum is found, it starts with new randomly selected node in search for a new local optimum

n Advantages: More efficient and scalable than both PAM and CLARA

n Further improvement: Focusing techniques and spatial access structures (Ester et al.95)

87

ROCK: Clustering Categorical Data

n ROCK: RObust Clustering using linKs n S. Guha, R. Rastogi & K. Shim, ICDE99

n Major ideas n Use links to measure similarity/proximity n Not distance-based

n Algorithm: sampling-based clustering n Draw random sample n Cluster with links n Label data in disk

n Experiments n Congressional voting, mushroom data

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Similarity Measure in ROCK

n Traditional measures for categorical data may not work well, e.g., Jaccard coefficient

n Example: Two groups (clusters) of transactions n C1. : {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e},

{a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e} n C2. : {a, b, f}, {a, b, g}, {a, f, g}, {b, f, g}

n Jaccard co-efficient may lead to wrong clustering result n C1: 0.2 ({a, b, c}, {b, d, e}} to 0.5 ({a, b, c}, {a, b, d}) n C1 & C2: could be as high as 0.5 ({a, b, c}, {a, b, f})

n Jaccard co-efficient-based similarity function:

n Ex. Let T1 = {a, b, c}, T2 = {c, d, e}

Sim T TT TT T

( , )1 21 2

1 2

=

2.051

},,,,{}{

),( 21 ===edcba

cTTSim

89

Link Measure in ROCK n Clusters

n C1:: {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d}, {a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}

n C2: : {a, b, f}, {a, b, g}, {a, f, g}, {b, f, g} n Neighbors

n Two transactions are neighbors if sim(T1,T2) > threshold n Let T1 = {a, b, c}, T2 = {c, d, e}, T3 = {a, b, f}

n T1 connected to: {a,b,d}, {a,b,e}, {a,c,d}, {a,c,e}, {b,c,d}, {b,c,e}, {a,b,f}, {a,b,g}

n T2 connected to: {a,c,d}, {a,c,e}, {a,d,e}, {b,c,e}, {b,d,e}, {b,c,d} n T3 connected to: {a,b,c}, {a,b,d}, {a,b,e}, {a,b,g}, {a,f,g}, {b,f,g}

n Link Similarity n Link similarity between two transactions is the # of common neighbors n link(T1, T2) = 4, since they have 4 common neighbors

n {a, c, d}, {a, c, e}, {b, c, d}, {b, c, e}

n link(T1, T3) = 3, since they have 3 common neighbors

n {a, b, d}, {a, b, e}, {a, b, g}

90

Rock Algorithm

n Method n Compute similarity matrix

n Use link similarity

n Run agglomerative hierarchical clustering n When the data set is big

n Get sample of transactions n Cluster sample

n Problems: n Guarantee cluster interconnectivity

n any two transactions in a cluster are very well connected n Ignores information about closeness of two clusters

n two separate clusters may still be quite connected

Aggregation-Based Similarity Computation

4 5

10 12 13 14

a b

ST2

ST1

11

0.2

0.9 1.0 0.8 0.9 1.0

For each node nk {n10, n11, n12} and nl {n13, n14}, their path-based similarity simp(nk, nl) = s(nk, n4)s(n4, n5)s(n5, nl).

( )( )

( )( )

171.02,

,3,

,14

13 554

12

10 4 == == l lk k

ba

nnsnns

nnsnnsim

After aggregation, we reduce quadratic time computation to linear time computation.

takes O(3+2) time

91

Computing Similarity with Aggregation

To compute sim(na,nb): n Find all pairs of sibling nodes ni and nj, so that na linked with ni and nb

with nj. n Calculate similarity (and weight) between na and nb w.r.t. ni and nj. n Calculate weighted average similarity between na and nb w.r.t. all such

pairs.

sim(na, nb) = avg_sim(na,n4) x s(n4, n5) x avg_sim(nb,n5)

= 0.9 x 0.2 x 0.95 = 0.171

sim(na, nb) can be computed from aggregated similarities

Average similarity and total weight 4 5

10 12 13 14

a b

a:(0.9,3)

b:(0.95,2)

11

0.2

92

93

Chapter 10. Cluster Analysis: Basic Concepts and Methods


n Overview of Clustering Methods





n Summary 93

Link-Based Clustering: Calculate Similarities Based On Links

Jeh & Widom, KDD2002: SimRank Two objects are similar if they are

linked with the same or similar objects

n The similarity between two objects x and y is defined as the average similarity between objects linked with x and those with y:

n Issue: Expensive to compute: n For a dataset of N objects

and M links, it takes O(N2) space and O(M2) time to compute all similarities.

Tom sigmod03

Mike

Cathy

John

sigmod04

sigmod05

vldb03 vldb04 vldb05

sigmod

vldb

Mary aaai04 aaai05

aaai

Authors Proceedings Conferences

( )( ) ( )

( ) ( )( )( )( )

= =

=aI

i

bI

jji bIaIbIaI

Cba1 1

,sim,sim

94

Observation 1: Hierarchical Structures

n Hierarchical structures often exist naturally among objects (e.g., taxonomy of animals)

All

electronics grocery apparel

DVD camera TV

A hierarchical structure of products in Walmart

Arti

cles

Words

Relationships between articles and words (Chakrabarti, Papadimitriou,

Modha, Faloutsos, 2004)

95

Observation 2: Distribution of Similarity

n Power law distribution exists in similarities n 56% of similarity entries are in [0.005, 0.015] n 1.4% of similarity entries are larger than 0.1 n Can we design a data structure that stores the significant

similarities and compresses insignificant ones?

0

0.1

0.2

0.3

0.40

0.02

0.04

0.06

0.08 0.1

0.12

0.14

0.16

0.18 0.2

0.22

0.24

similarity value

porti

on o

f ent

ries Distribution of SimRank similarities

among DBLP authors

96

A Novel Data Structure: SimTree Each leaf node

represents an object Each non-leaf node represents a group

of similar lower-level nodes

Similarities between siblings are stored

Consumer electronics

Apparels

Canon A40 digital camera

Sony V3 digital camera

Digital Cameras

TVs

97

Similarity Defined by SimTree

n Path-based node similarity n simp(n7,n8) = s(n7, n4) x s(n4, n5) x s(n5, n8)

n Similarity between two nodes is the average similarity between objects linked with them in other SimTrees

n Adjust/ ratio for x =

n1 n2

n4 n5 n6

n3

0.9 1.0

0.9 0.8

0.2

n7 n9

0.3

n8

0.8

0.9

Similarity between two sibling nodes n1 and n2

Adjustment ratio for node n7

Average similarity between x and all other nodes Average similarity between xs parent and all other nodes

98

LinkClus: Efficient Clustering via Heterogeneous Semantic Links

Method n Initialize a SimTree for objects of each type n Repeat until stable

n For each SimTree, update the similarities between its nodes using similarities in other SimTrees n Similarity between two nodes x and y is the average

similarity between objects linked with them n Adjust the structure of each SimTree

n Assign each node to the parent node that it is most similar to

For details: X. Yin, J. Han, and P. S. Yu, LinkClus: Efficient Clustering via Heterogeneous Semantic Links, VLDB'06

99

Initialization of SimTrees

n Initializing a SimTree n Repeatedly find groups of tightly related nodes, which

are merged into a higher-level node n Tightness of a group of nodes

n For a group of nodes {n1, , nk}, its tightness is defined as the number of leaf nodes in other SimTrees that are connected to all of {n1, , nk}

n1 1 2 3 4 5

n2 The tightness of {n1, n2} is 3

Nodes Leaf nodes in another SimTree

100

Finding Tight Groups by Freq. Pattern Mining

n Finding tight groups Frequent pattern mining

n Procedure of initializing a tree n Start from leaf nodes (level-0) n At each level l, find non-overlapping groups of similar

nodes with frequent pattern mining

Reduced to

g1

g2

{n1} {n1, n2} {n2} {n1, n2} {n1, n2} {n2, n3, n4} {n4} {n3, n4} {n3, n4}

Transactions

n1 1 2 3 4 5 6 7 8 9

n2

n3

n4

The tightness of a group of nodes is the support of a frequent pattern

101

Adjusting SimTree Structures

n After similarity changes, the tree structure also needs to be changed n If a node is more similar to its parents sibling, then move

it to be a child of that sibling n Try to move each node to its parents sibling that it is

most similar to, under the constraint that each parent node can have at most c children

n1 n2

n4 n5 n6

n3

n7 n9 n8

0.8 0.9

n7

102

Complexity

Time Space Updating similarities O(M(logN)2) O(M+N)

Adjusting tree structures O(N) O(N)

LinkClus O(M(logN)2) O(M+N) SimRank O(M2) O(N2)

For two types of objects, N in each, and M linkages between them.

103

Experiment: Email Dataset n F. Nielsen. Email dataset. www.imm.dtu.dk/

~rem/data/Email-1431.zip n 370 emails on conferences, 272 on jobs,

and 789 spam emails n Accuracy: measured by manually labeled

data n Accuracy of clustering: % of pairs of objects

in the same cluster that share common label

Approach Accuracy time (s) LinkClus 0.8026 1579.6 SimRank 0.7965 39160 ReCom 0.5711 74.6

F-SimRank 0.3688 479.7 CLARANS 0.4768 8.55

n Approaches compared: n SimRank (Jeh & Widom, KDD 2002): Computing pair-wise similarities n SimRank with FingerPrints (F-SimRank): Fogaras & Racz, WWW 2005

n pre-computes a large sample of random paths from each object and uses samples of two objects to estimate SimRank similarity

n ReCom (Wang et al. SIGIR 2003) n Iteratively clustering objects using cluster labels of linked objects

104

WaveCluster: Clustering by Wavelet Analysis (1998)

n Sheikholeslami, Chatterjee, and Zhang (VLDB98) n A multi-resolution clustering approach which applies wavelet transform

to the feature space; both grid-based and density-based n Wavelet transform: A signal processing technique that decomposes a

signal into different frequency sub-band n Data are transformed to preserve relative distance between objects

at different levels of resolution n Allows natural clusters to become more distinguishable

105

The WaveCluster Algorithm

n How to apply wavelet transform to find clusters n Summarizes the data by imposing a multidimensional grid

structure onto data space n These multidimensional spatial data objects are represented in a

n-dimensional feature space n Apply wavelet transform on feature space to find the dense

regions in the feature space n Apply wavelet transform multiple times which result in clusters at

different scales from fine to coarse n Major features:

n Complexity O(N) n Detect arbitrary shaped clusters at different scales n Not sensitive to noise, not sensitive to input order n Only applicable to low dimensional data

106

107

Quantization & Transformation

n 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

Clustering

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