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WaveCluster

A multi-resolution clustering approachApply wavelet transformation to the feature space

Both grid-based and density-basedInput parameters:

Number of grid cells for each dimensionThe waveletThe number of applications of wavelet transform

University at Buffalo The State University of New York

What are Wavelets

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What Is Wavelet Transform?

Decomposes a signal into different frequency subbandsApplicable to n-dimensional signals

Data are transformed to preserve relative distance between objects at different levels of resolution

Allow natural clusters to become more distinguishable

University at Buffalo The State University of New York

Intuition Behind Using Wavelet Transform

Wavelet transform filters makes clusters more distinct

Effective removal of outliers

Multi-resolution property of wavelet transform can help detecting clusters at different levels of accuracy

Cost-efficiency

University at Buffalo The State University of New York

Wavelet Transformation

University at Buffalo The State University of New York

Why Is Wavelet Transform?

Use hat-shape filtersEmphasize region where points clusterSuppress weaker information in their boundaries

Effective removal of outliers Insensitive to noise, insensitive to input order

Multi-resolutionDetect arbitrary shaped clusters at different scales

EfficientComplexity O(N)

Only applicable to low dimensional data

University at Buffalo The State University of New York

WaveCluster: Method

Summarize the data by imposing a multidimensional grid structure on to data spaceMultidimensional spatial data objects are represented in

an n-dimensional feature spaceApply wavelet transform on feature space to find

the dense regions in the feature spaceApply wavelet transform multiple times

Result in clusters at different scales from fine to coarseBy Dr. Aidong Zhang’s group

University at Buffalo The State University of New York

WaveCluster

University at Buffalo The State University of New York

Shrinking: Intuition & Purpose

For data points in a data set, what if we

could make them move towards the centroid

of the natural subgroup they belong to?

Natural sparse subgroups become denser,

thus easier to be detected; noises are further

isolated.

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The Concept of Shrinking

A data preprocessing technique

It aims to optimize the inner structure of real data

sets

Each data point is “attracted” by other data points

and moves to the direction in which way the

attraction is the strongest

Can be applied in different fields

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Data Shrinking

Each data point moves along the direction of the density gradient and the data set shrinks towards the inside of the clusters.

Points are “attracted” by their neighbors and move to create denser clusters.

Proceeds iteratively; repeated until the data are stabilized or the number of iterations exceeds a threshold.

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Apply shrinking into clustering field

Multi-attribute hyperspace

Shrink the natural sparse clusters to make them much denser to facilitate further cluster-detecting process.

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Overall Structure

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Data Shrinking (Cont’d)

Space subdivisionNormalization of data spaceGiven the side length 1/k of grid cells, the normalized data

space is subdivided into kd cells.Each grid g contains the average position (grid point) and

number of data points in it.Neighboring relationship of points is grid-based.

In each iteration, data points move toward the data centroid of the neighboring grids.

Grid scale:Apply different grid scales, choose best clustering results.

University at Buffalo The State University of New York

Data Shrinking (Cont’d)

Multi-scale solution: choose multiple grids scales for data shrinking

1.Determination of a proper cell size

2.Advantages for handling clusters of various densities

University at Buffalo The State University of New York

Data Shrinking (Cont’d)

Acquirement of Multi-scale

A straightforward solution: use a sequence of grids of exponentially increasing cell sizes.

Smin, Smin*Eg, … Smin*(Eg)ŋ = Smax, for some ŋN

Disadvantage:

1) Smin depends on the granularity of data

2) Losing important grid scale candidates

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Data Shrinking (Cont’d)

A histogram-based approach to get reasonable grid scalesGet histograms for dimensions: H={h1,h2, …,hd}Density span: a combination of consecutive bins’

segments on a certain dimension in which the amount of data points exceeds a threshold.

Start from the largest bin, get density spans.Regard density spans with similar sizes as

identical ones, and choose those with largest frequencies as grid scale candidates.

University at Buffalo The State University of New York

Data Shrinking (Cont’d)

An example of density span processing

University at Buffalo The State University of New York

Data Shrinking (Cont’d)

An example of data movement

Solution:Treat the points in each cell as a rigid body which is

pulled as a unit toward the data centroid of the surrounding cells which have more points.

University at Buffalo The State University of New York

ExperimentsOriginal data set

data set after

iteration 1

data set after

iteration 3

data set after

iteration 2

2d example

University at Buffalo The State University of New York

Cluster Detection

Neighboring dense cells are connected and a neighboring graph G of the dense cells is constructed.

Use a breadth-first search algorithm to find the components of graph G. Each component is a cluster.

Label data points with cluster ids.