towards hierarchical clustering mark sh. levin inst. for inform. transm. problems, russian acad. of...
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TOWARDS HIERARCHICAL CLUSTERING
Mark Sh. LevinInst. for Inform. Transm. Problems, Russian Acad. of Sci.
Email: [email protected] Http://www.iitp.ru/mslevin/
CSR’2007, Ural State University, Ekaterinburg, Russia, Sept. 4, 2007
PLAN: 1.Basic agglomerative algorithm for hierarchical clustering2.Multicriteria decision making (DM) approach
to proximity of objects3.Integration of objects into several groups/clusters
(i.e., clustering with intersection): algorithms & applications4.Towards resultant performance (quality of results) 5.Conclusion
Hierarchical clustering: agglomerative algorithm
A1 = ( z11 , … , z1j , … , z1m )… …Ai = ( zi1 , … , zij , … , zim )… …An = ( zn1 , … , znj , … , znm )
Objects (alternatives) Criteria (characteristics) C1 … Cj … Cm
A1 = ( d11 , … , d1i , … , d1n )… …Ai = ( di1 , … , dii , … , din )… …An = ( dn1 , … , dni , … , dnn )
“Distance” A1 … Ai … An
Matrix Z
Matrix D
Hierarchical clustering: agglomerative algorithm
A1 = ( z11 , … , z1j , … , z1m )… …Ai = ( zi1 , … , zij , … , zim )… …An = ( zn1 , … , znj , … , znm )
Objects (alternatives) Criteria (characteristics) C1 … Cj … Cm
A1 = ( 0 , … , d1i , … , d1n )… …Ai = ( di1 , … , 0 , … , din )… …An = ( dn1 , … , dni , … , 0 )
“Distance” A1 … Ai … An
Matrix Z
Matrix D
Hierarchical clustering: agglomerative algorithm
Matrix D:
dil = sqrt ( ∑j=1 m ( zij – zlj )2 )
Scale for D
0 maxmin
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
Ax = ( zx1 , … , zxj , … , zxm ) Ay = ( zy1 , … , zyj , … , zym )
Pair of objects Ax and Ay
j (j = 1,…,m) zi (A) = ( zxj + zyj ) / 2
Integrated object A = ( Ax * Ay )
(1*2*3*4*5*6*7)
Hierarchical clustering: agglomerative algorithm
Stage 0:1 2 3 4 5 6 7
Stage 1:1 2 (3*4) 5 6 7
Stage 2:1 2 (3*4) 5 (6*7)
Stage 3:2 (1*3*4) 5 (6*7)
Stage 4:2 (1*3*4) (5*6*7)
Stage 6:
. . .
Hierarchical clustering: agglomerative algorithm
Cluster F1
Cluster F2
Cluster F3
Cluster F4
Cluster F5
Cluster F6
ILLUSTRATIVE EXAMPLE
Hierarchical clustering: agglomerative algorithm
First, Complexity of agglomerative algorithm:1.Number of stages (each stage – one integration):(n-1) stages2.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 max{dxy}
To divide the interval [0,max{dxy}] to get an ordinal scale
interval 0 interval 1 interval k
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
interval 0 interval 1 interval k
dab duv dpq dgh
Example: pairs of objects: (a,b), (u,v), (p,q), (g,h)
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
interval 0 interval 1 interval k
dab duv dpq dgh
RESULT:
dab = 0duv = 0dpq = 1dgh = 1
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
Space of the vectors =>ordinal scale&ordinal proximity
C1
C2Ideal point(equal objects)
Pareto-effectiveLayer (1)
Layer 2
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 systems2.CS: structures of software/hardware3.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?
Cluster F1
Cluster F2
Cluster F3
Cluster F4
Cluster F5Cluster F6
Clustering withintersection
Hierarchical clustering: IMPROVEMENTS (practice)
(1*2*3*4*5*6*7)
Stage 0:1 2 3 4 5 6 7
Stage 1:1 (2*3) (3*4) (5*6) (6*7)
Stage 2:1 (2*3*4) (3*4*5*6) (6*7)
Stage 3:(1*2*3*4) (3*4*5*6*7)
2
Stage 4:
Hierarchical clustering: IMPROVEMENTS (practice)
(1*2*3*4*5*6*7)
Stage 0:1 2 3 4 5 6 7
Stage 1:
Stage 2:
Stage 3:
Resultantstructure
Hierarchical clustering: IMPROVEMENTS (practice)
Example frombiology(evolution)
Traditionalevolution process as tree
Hierarchical clustering: IMPROVEMENTS (practice)
Example frombiology(evolution)
Hierarchicalstructure
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 -> edgesThus: a graphProblem: 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. Simon1.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
CONCLUSION
1.Algorithms, procedures & their analysis
2.New approaches to performance/quality for research procedures
3.Various applied examples
4.Usage in education
That’s All
Thanks!
http://www.iitp.ru/mslevin/
Mark Sh. Levin