TOWARDS HIERARCHICAL CLUSTERING Mark Sh. Levin Inst. for Inform. Transm. Problems, Russian Acad. of Sci. Email: mslevin@acm.org 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=1 m ( zij – zlj )2 ) Scale for D min max

Hierarchical clustering: 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: 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: 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: 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 interval 0 interval 1 interval k max{dxy} To divide the interval [0,max{dxy}] to get an ordinal scale 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 RESULT: dab = 0 duv = 0 dpq = 1 dgh = 1 Usage of an ordinal scale: dab duv dpq dgh 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 F1 Cluster F5 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) Example from biology (evolution) Hierarchical structure

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)

Basic DM problems: choice, ranking, 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,

THUS: we have got some prospective RESEARCH RPOCEDURES 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 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