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TOWARDS HIERARCHICAL CLUSTERING

Mark Sh. Levin

Inst. for Inform. Transm. Problems, Russian Acad. of Sci.

Email: [email protected] Http://www.iitp.ru/mslevin/

PLAN:

1.Basic agglomerative algorithm for hierarchical clustering

2.Multicriteria decision making (DM) approach

to proximity of objects

3.Integration of objects into several groups/clusters

(i.e., clustering with intersection): algorithms & applications

4.Towards resultant performance (quality of results)

5.Conclusion

CSR’2007, Ural State University, Ekaterinburg, Russia, Sept. 4, 2007

Hierarchical clustering: agglomerative algorithm

Objects (alternatives) Criteria (characteristics)

C1 … Cj … Cm

A1 = ( z11 , … , z1j , … , z1m )

… …

Ai = ( zi1 , … , zij , … , zim )

… …

An = ( zn1 , … , znj , … , znm )

Matrix

Z

“Distance” A1 … Ai … An

A1 = ( d11 , … , d1i , … , d1n )

… …

Ai = ( di1 , … , dii , … , din )

… …

An = ( dn1 , … , dni , … , dnn )

Matrix

D

Hierarchical clustering: agglomerative algorithm

Objects (alternatives) Criteria (characteristics)

C1 … Cj … Cm

A1 = ( z11 , … , z1j , … , z1m )

… …

Ai = ( zi1 , … , zij , … , zim )

… …

An = ( zn1 , … , znj , … , znm )

Matrix

Z

“Distance” A1 … Ai … An

A1 = ( 0 , … , d1i , … , d1n )

… …

Ai = ( di1 , … , 0 , … , din )

… …

An = ( dn1 , … , dni , … , 0 )

Matrix

D

Hierarchical clustering: agglomerative algorithm

Matrix D:

dil = sqrt ( ∑j=1m ( zij – zlj )2 )

Scale for D

min

max

0

Hierarchical clustering: agglomerative algorithm

Agglomerative algorithm:

Stage 1.Computing matrix D=| dil | (pair “distances”)

Stage 2.Revelation of the smallest pair “distance”

(i.e., the minimal pair “distance”,

the minimal element in matrix D)

and integration of the corresponding elements (Ax, Ay)

(objects) into a new joint (integrated) object A=Ax*Ay

Stage 3.Stopping the process or

re-computing the matrix D and GOTO Stage 2.

Hierarchical clustering: agglomerative algorithm

Pair of objects Ax and Ay

Ax = ( zx1 , … , zxj , … , zxm )

Ay = ( zy1 , … , zyj , … , zym )

Integrated object A = ( Ax * Ay )

j (j = 1,…,m) zi (A) = ( zxj + zyj ) / 2

Hierarchical clustering: agglomerative algorithm

Stage 6:

(1*2*3*4*5*6*7)

. . .

Stage 4:

2

(1*3*4)

(5*6*7)

Stage 3:

2

(1*3*4)

5

(6*7)

Stage 2:

1

2

(3*4)

5

(6*7)

Stage 1:

1

2

(3*4)

5

6

7

Stage 0:

1

2

3

4

5

6

7

Hierarchical clustering: agglomerative algorithm

ILLUSTRATIVE EXAMPLE

Cluster F4

Cluster F2

Cluster F3

Cluster F5

Cluster F1

Cluster F6

Hierarchical clustering: agglomerative algorithm

First, Complexity of agglomerative algorithm:

1.Number of stages (each stage – one integration):

(n-1) stages

2.Each stage:

(a)computing “distances” (n2 * m operations)

THUS:

Operations: O(m n3 )

Memory: O(n(n+m))

Second, we have got the TREE-LIKE STRUCTURE

Hierarchical clustering: IMPROVEMENTS (to do better)

Question 1: What we can do better

in the algorithm?

Hierarchical clustering: IMPROVEMENTS (to do better)

Question 1: What we can do better

in the algorithm?

Question 2:

What is needed in practice (e.g., applications)?

What we can do for applications?

Hierarchical clustering: IMPROVEMENTS (to do better)

Question 1: What we can do better?

1.Computing: pair “distance” (pair proximity)

Usage of more “correct” approaches

from multicriteria decision making, e.g.,

Revelation of Pareto-layers and usage of

an ordinal scale for pair proximity

2.Complexity: decrease the number of stages:

Integration of several pair of objects

at each stage

Hierarchical clustering: IMPROVEMENTS (to do better)

2.Complexity: decrease the number of stages:

Integration of several pair of objects

at each stage

Usage of an ordinal scale:

0

max{dxy}

Hierarchical clustering: IMPROVEMENTS (to do better)

2.Complexity: decrease the number of stages:

Integration of several pair of objects

at each stage

Usage of an ordinal scale:

0

max{dxy}

To divide the interval [0,max{dxy}]

to get an ordinal scale

Hierarchical clustering: IMPROVEMENTS (to do better)

2.Complexity: decrease the number of stages:

Integration of several pair of objects

at each stage

Usage of an ordinal scale:

0

interval 0

interval 1

interval k

max{dxy}

To divide the interval [0,max{dxy}]

to get an ordinal scale

Hierarchical clustering: IMPROVEMENTS (to do better)

2.Complexity: decrease the number of stages:

Integration of several pair of objects

at each stage

Usage of an ordinal scale:

dab

duv

dpq

dgh

0

interval 0

interval 1

interval k

max{dxy}

Example: pairs of

objects: (a,b), (u,v),

(p,q), (g,h)

To divide the interval [0,max{dxy}]

to get an ordinal scale

Hierarchical clustering: IMPROVEMENTS (to do better)

2.Complexity: decrease the number of stages:

Integration of several pair of objects

at each stage

RESULT:

dab = 0

duv = 0

dpq = 1

dgh = 1

Usage of an ordinal scale:

dab

duv

dpq

dgh

0

interval 0

interval 1

interval k

max{dxy}

To divide the interval [0,max{dxy}]

to get an ordinal scale

Hierarchical clustering: IMPROVEMENTS (to do better)

1.Computing: pair “distance” (pair proximity)

Usage of more “correct” approaches

from multicriteria decision making, e.g.,

Revelation of Pareto-layers and usage of

an ordinal scale for pair proximity

Hierarchical clustering: IMPROVEMENTS (to do better)

1.Computing: pair “distance” (pair proximity)

Usage of more “correct” approaches

from multicriteria decision making, e.g.,

Revelation of Pareto-layers and usage of

an ordinal scale for pair proximity

Objects: {A1, … , Ai, … An }

Ax -> (zx1, … , zxj , ... , zxm)

Ay -> (zy1, … , zyj , … , zym)

Hierarchical clustering: IMPROVEMENTS (to do better)

1.Computing: pair “distance” (pair proximity)

Usage of more “correct” approaches

from multicriteria decision making, e.g.,

Revelation of Pareto-layers and usage of

an ordinal scale for pair proximity

Objects: {A1, … , Ai, … An }

Ax -> (zx1, … , zxj , ... , zxm)

Ay -> (zy1, … , zyj , … , zym)

Vector of “differences” for Ax , Ay:

( (zx1 - zy1) , … , (zxj - zyj), … , (zxm - zym) )

Hierarchical clustering: IMPROVEMENTS (to do better)

1.Computing: pair “distance” (pair proximity)

Usage of more “correct” approaches

from multicriteria decision making, e.g.,

Revelation of Pareto-layers and usage of

an ordinal scale for pair proximity

Objects: {A1, … , Ai, … An }

Ax -> (zx1, … , zxj , ... , zxm)

Ay -> (zy1, … , zyj , … , zym)

Vector of “differences” for Ax , Ay:

( (zx1 - zy1) , … , (zxj - zyj), … , (zxm - zym) )

Space of the vectors

Hierarchical clustering: IMPROVEMENTS (to do better)

1.Computing: pair “distance” (pair proximity)

Usage of more “correct” approaches

from multicriteria decision making, e.g.,

Revelation of Pareto-layers and usage of

an ordinal scale for pair proximity

Objects: {A1, … , Ai, … An }

Ax -> (zx1, … , zxj , ... , zxm)

Ay -> (zy1, … , zyj , … , zym)

Vector of “differences” for Ax , Ay:

( (zx1 - zy1) , … , (zxj - zyj), … , (zxm - zym) )

Space of the vectors =>ordinal scale&ordinal proximity

Hierarchical clustering: IMPROVEMENTS (to do better)

1.Computing: pair “distance” (pair proximity)

Usage of more “correct” approaches

from multicriteria decision making, e.g.,

Revelation of Pareto-layers and usage of

an ordinal scale for pair proximity

C1

Pareto-effective

Layer (1)

Layer 2

Ideal point

(equal objects)

C2

Space of the vectors =>ordinal scale&ordinal proximity

Hierarchical clustering: IMPROVEMENTS (practice)

Question 2:

What is needed in practice (e.g., applications)?

What we can do for applications?

Hierarchical clustering: IMPROVEMENTS (practice)

Question 2:

What is needed in practice (e.g., applications)?

What we can do for applications?

Integration of objects into several groups (clusters)

to obtain more rich resultant structure

(tree => hierarchy, i.e., clusters with intersection)

Examples of applied domains:

1.Engineering: structures of complex systems

2.CS: structures of software/hardware

3.Communication networks (topology)

4.Biology 5.Others

Hierarchical clustering: IMPROVEMENTS (practice)

Question 2:

What is needed in practice (e.g., applications)?

What we can do for applications?

Clustering with

intersection

Cluster F3

Cluster F4

Cluster F2

Cluster F5

Cluster F1

Cluster F6

Hierarchical clustering: IMPROVEMENTS (practice)

Stage 4:

(1*2*3*4*5*6*7)

2

Stage 3:

(3*4*5*6*7)

(1*2*3*4)

Stage 2:

1

(2*3*4)

(6*7)

(3*4*5*6)

Stage 1:

1

(2*3)

(3*4)

(5*6)

(6*7)

Stage 0:

1

2

3

4

5

6

7

Hierarchical clustering: IMPROVEMENTS (practice)

Resultant

structure

(1*2*3*4*5*6*7)

Stage 3:

Stage 2:

Stage 1:

Stage 0:

1

2

3

4

5

6

7

Hierarchical clustering: IMPROVEMENTS (practice)

Example from

biology

(evolution)

Traditional

evolution

process

as tree

Hierarchical clustering: IMPROVEMENTS (practice)

Algorithm 1.

The number of inclusion for each object is not limited):

(i)initial set of objects -> vertices

(ii)”small” proximity -> edges

Thus: a graph

Problem: to reveal cliques in the graph

(It is NP-hard problem)

Algorithm 2.

The number of the inclusion is limited by t

(e.g., t=2/3/4). Here complexity is polynomial.

Hierarchical clustering: performance (i.e., quality)

Performance (i.e., quality) of clustering procedures:

1.Issues of complexity

2.Quality of results (??)

Some traditional approaches:

(a)computing a clustering quality

InterCluster Distance / IntraCluster Distance

(b)Coverage, Diversity

Our case: research procedure

(for investigation and problem structuring)

Hierarchical clustering: performance (i.e., quality)

Decision Making Paradigm (stages) by Herbert A. Simon

1.Analysis of an applied problem (to understand

the problem: main contradictions, etc.)

2.Structuring the problem:

2.1.Generation of alternatives

2.2.Design of criteria

2.3.Design of scales for assessment of

alternatives upon criteria

3.Evaluation of alternatives upon criteria

4.Selection of the best alternative (s)

5.Analysis of results

Basic DM problems: choice, ranking,

Hierarchical clustering: performance (i.e., quality)

FOR CLUSTERING:

1.Analysis of an applied problem (to understand

the problem: main contradictions, etc.)

2.Structuring the problem:

2.1.Generation of alternatives

2.2.Design of criteria

2.3.Design of scales for assessment of

alternatives upon criteria

3.Evaluation of alternatives upon criteria

4.Design of CLUSTERS and

STRUCTURE OF CLUSTERING PROCESS

5.Analysis of results

THUS: we have got some prospective

RESEARCH RPOCEDURES

1.Algorithms, procedures &

their analysis

2.New approaches to

performance/quality for

research procedures

3.Various applied examples

4.Usage in education

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