a two-way visualization method for clustered data

Intelligent Database Systems Lab

國立雲林科技大學National Yunlin University of Science and Technology

Advisor ： Dr. Hsu

Presenter ： Keng-Wei Chang

Author: Yehuda Koren and David Harel

A Two-Way Visualization Method for Clustered Data

ACM SIGKDD international conference on Knowledge discovery and datamining


N.Y.U.S.T.

I. M.Outline

Motivation Objective Introduction Basic Notions Computing The x-Coordinates Computing The y-Coordinates Result Related Work Conclusions Personal Opinion


N.Y.U.S.T.

I. M.Motivation

A number of technological development have led to an explosion of raw data that has to be analyzed

We are especially interested in two families of tools in this domain

Clustering algorithms and data visualization methods


N.Y.U.S.T.

I. M.Objective

in this paper, we integrate the two approacheshierarchical clustering depicted as a dendrogram

low-dimensional embedding


N.Y.U.S.T.

I. M.Introduction

A number of technological development have led to an explosion of raw data that has to be analyzed

We are especially interested in two families of tools in this domain

Clustering algorithms and data visualization methods

Clustering methods can be broadly classifiedHierarchical and partitional


N.Y.U.S.T.

I. M.Introduction

Our main interest here is hierarchical clustering

The clustering hierarchy is often visualized as a dendrogram

A full binary tree

has a significant disadvantagedoes not provide exploratory visual representations of the data itself

another issue is that of cluster validity


N.Y.U.S.T.

I. M.Introduction

we are particularly interested in methods for achieving a low-dimensional embedding of data

principal component analysis (PCA)

multidimensional scaling (MDS)

force-directed placement

solve some limitations of dendrogrambut, cannot utilize external clustering information


N.Y.U.S.T.

I. M.Introduction

for a demonstration of the relative merits of the two approaches

a dendrogram vs. a low-dimensional embedding


N.Y.U.S.T.

I. M.Introduction

in this paper, we integrate the two approacheshierarchical clustering depicted as a dendrogram

low-dimensional embedding


N.Y.U.S.T.

I. M.Basic Notions

given data about n elements {1,…,n}

relationships between pairs of elements are bydistances dij ≥ 0 or

similarities wij ≥ 0

2-dimentional embedding of the dataid defined by two vectors x, y Є

the coordinates of element i are ( xi, yi)

n


N.Y.U.S.T.

I. M.Computing The x-Coordinates

The embedding must place each element exactly below its corresponding leaf in the dendrogram

this means that the x-coordinate must corresponding leaf in the dendrogram

face the problem of computing the x-coordinates of the dendrogram leaves

preserves the relationships among the data as much as possible


N.Y.U.S.T.

I. M.Computing The x-Coordinates

we exhaust all the existing methods, opting for a twofold process

find the best orientation of the dendrogramthis step determines the ordering of the leaves

decide on the exact gaps between consecutive leaves in the ordering


N.Y.U.S.T.

I. M.Dendrogram orientation

a dendrogram has 2n-1 different orientationsexample ：


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one way of defining formally what should be considered a “good” ordering

associate a cost function with the dendrogram

such that finding the best ordering is equivalent to optimizing this function

be the classical minimum linear arrangement problem

ji

jiij

def

sim xxwxLA,

.

minimizes


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in our particular problemalso faced with an ordering task

a permutation of {1, …, n}

however, here we should not consider all possible permutations, but only agree with dendrogram’s structure

n! 2n-1

using dynamic programming, running time is exponential in the dendrogram’s height not in its size


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introduce an additional form of the cost function

ji

jiij

def

dist xxdxLA,

.

maximizes


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given an ordered dendrogram T

a node v

Leaves(v) ： the set of leaves in the substree rooted by v

x be the ordering on the leaves

Let S be Leaves(v)L be the set of leaves of left of S

R be the set of leaves of right of S

if |L| = l, |S| = s, we have x(L) = {1,…,l},

x(S) = {l+1,…,l+x}, x(R) = {l+s+1,…,n}


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a key concept of the algorithm is local arrangement cost, defined as ：

RS,ji RjLiijiij

Sji LjSiiijjiij

defT

swxslw

lxwxxwvLocalLA

,

, ,

if |L| = l, |S| = s, we have x(L) = {1,…,l}, x(S) = {l+1,…,l+x}, x(R) = {l+s+1,…,n}


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two additional related terms will be used

another term that will be used in the algorithm

RjSi

ij

defT

LjSiij

defT wvRightCutwvLeftCut

,,

,

ij

rightvLeavesjleftvLeavesi

wvInnerCut

..


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I. M.


N.Y.U.S.T.

I. M.Determining coordinates of the leaves

computing the exact gaps between each two consecutive leaves

example ：


N.Y.U.S.T.

I. M.Determining coordinates of the leaves

a better approach is to take a weighted average over all influenced leaf pairs

ikij

kj

ikiji jk

d

jkgap

,

1

,

1


N.Y.U.S.T.

I. M.Computing The y-Coordinates

Principle component analysis

Classical multidimensional scaling

Eigen-projection

Stress minimization


N.Y.U.S.T.

I. M.Result

Odors datasetconsists of 30 volatile odorous pure chemicals

contains 262 elements, natural clusters : 30

use a UPGMA agglomerative clustering to construct

the dendrogram


N.Y.U.S.T.

I. M.Result

Iris datasetan example of discriminant analysis

contains 150 elements, natural clusters : 3


N.Y.U.S.T.

I. M.Result

Gene expression data ： CDC15-synchronized cell cycle

a much larger dataset of gene-expression data

contains 6113 elements


N.Y.U.S.T.

I. M.Related Work

TreeViewdendrogram over a color-coded matrix


N.Y.U.S.T.

I. M.Discussion

success for integrating two key methods in exploratory data analysis

cluster analysis and low-dimensional embedding

two unique propertiesGuaranteed separation between any kind of given clusters

The ability to deal with a predefined hierarchical clustering


N.Y.U.S.T.

I. M.Personal Opinion

Advantages─ has success for integrating two of clustering methods.─ more intuition in analyzing

Application─ Real data for clustering and analyzing.─ May solve the problem lack of clustering information

Limited ─ cannot show the real shape of clusters

a two-way visualization method for clustered data

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