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G lobal O ptimality of the S uccessive M ax B et A lgorithm

G lobal O ptimality of the S uccessive M ax B et A lgorithm. USC ENITIAA de NANTES France. Mohamed HANAFI and Jos M.F. TEN BERGE. Department of psychology University of Groningen The Netherlands. G lobal O ptimality of the S uccessive M ax B et A lgorithm. Summary.

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G lobal O ptimality of the S uccessive M ax B et A lgorithm

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  1. Global Optimality of the Successive MaxBet Algorithm USC ENITIAA de NANTES France Mohamed HANAFI and Jos M.F. TEN BERGE Department of psychology University of Groningen The Netherlands

  2. Global Optimality of the Successive MaxBet Algorithm Summary. 1. The Successive MaxBet Problem (SMP). 2. The MaxBet Algorithm. 3. Global Optimality : Motivation/Problems. 4. Conclusions and Open questions.

  3. 1. The Successive MaxBet Problem (S.M.P) s.p.s.d Blocks Matrix

  4. 1. The Successive MaxBet Problem (S.M.P) order 1

  5. order s { 1. The Successive MaxBet Problem (S.M.P)

  6. 2. The Successive MaxBet Algorithm Ten Berge (1986,1988) 1.Take arbitrary initial unit length vectors 2. Compute : 3. rescalevkto unit length, and setuk= vk 4. Repeat steps 2 and 3 till convergence Order 1

  7. 2. The Successive MaxBet Algorithm Ten Berge (1986,1988) 1.Take arbitrary initial unit length vectors 2. Compute : 3. rescalevkto unit length, and setuk= vk 4. Repeat steps 2 and 3 till convergence Order s

  8. Property 1 : Convergence of the MaxBet Algorithm

  9. Property 2 : Necessary Condition of Convergence

  10. 3. Motivation and results 1. MaxBet Algorithm depends on the starting vector 2. MaxBet algorithm does not guarantee the computation of the global solution of SMP

  11. 43 23 -13 0 -7 23 31 10 1 0 -13 10 64 -19 -2 0 1 -19 24 18 -7 0 -2 18 58 3. Motivation and results : an example

  12. 0.64 0.31 0.64 0.24 0.10 0.67 0.36 0.20 0.53 0.30 { Starting Vector { 0.94 0.31 -0.92 0.35 0.11 0.69 0.72 0.58 -0.43 -0.68 Solution Vector { f(u)= 10621 f(v)= 9825.1 Function value 42185 64023 3846.7 5978.4

  13. 3. Motivation and results: Two Questions Q1. How can we know that the solution computed by the Maxbet algorithm is global or not ? Q2. When the solution is not global, how can we reach using this solution the global solution ?

  14. Global solution of SMP Spectral properties (eigenvalues and eigenvectors) of 3. Motivation and Results : Proceeding

  15. RESULT 1 Result 1

  16. ELEMENTS OF PROOF (Result 1)

  17. (matrix is negative semi definite) 3. Motivation and Results

  18. then matrix is negative semi definite RESULT 2 Result 2

  19. Suppose has a positive eigenvalue 2. w is not block-normed vector 2.1. w is not block orthogonal to u 2.2. w is block orthogonal to u ELEMENTS OF PROOF (Result 2) 1. w is block-normed vector

  20. 1. w is block-normed vector w is better solution than u

  21. 2. w is not block-normed vector 2.1. w is not block orthogonal to u v is better solution than u

  22. w is not block-normed vector 2.2. w is block orthogonal to u v is better solution than u

  23. then matrix is negative semi definite RESULT 2 Result 3

  24. Suppose has a positive eigenvalue ELEMENTS OF PROOF (Result 3)

  25. w has all elements of the same sign u has all elements of the same sign ELEMENTS OF PROOF (Result 3)

  26. (matrix is negative semi definite) Result 4

  27. ELEMENTS OF PROOF (Result 4) 45 -20 5 6 16 3 -20 77 -20 -25 -8 -21 5 -20 74 47 18 -32 6 -25 47 54 7 -11 16 -8 18 7 21 -7 3 -21 -32 -11 -7 70

  28. ELEMENTS OF PROOF (Result 4) Random research with 10.000.000 starting vectors 0.49 -0.87 0.80 0.59 0.56 -0.82 F(u) =378.96 u = s =0.48

  29. - Possible Application in statistics : Multivariate Methods (Analysis of K sets of data ) 2. Rotation methods : MaxDiff, MaxBet, generalized Procrustes Analysis Gower(1975); Van de Geer(1984);Ten Berge (1986,1988) 4. General Conclusions 1. Generalized canonical correlationAnalysis: Horst (1961) 3. Soft Modeling Approach : Estimation of latent variables under mode B Wold (1984); Hanafi (2001)

  30. - Necessary condition for the case K=3 when matrix A has not all elements of the same sign? 4. Perspective and Little Open Question -

  31. Motivation: Illustration 1 MaxBet Algorithm depends on the starting vector

  32. The Successive MaxBet Problem (S.M.P) and Multivariate Methods

  33. Some multivarite methods Generalized canonical correlation methods Rotation methods(Agreement methods) SOFT MODELING APPRAOCH(Approch)

  34. S M P = MaxBet method Van de Geer (1984) Ten Berge (1986,1988) S M P = MaxBet method Van de Geer (1984) Ten Berge (1986,1988) S M P = MaxBet method Van de Geer (1984) Ten Berge (1986,1988) S M P = MaxDiff method Van de Geer (1984) Ten Berge (1986,1988) S M P = Generalized Procrustes Analysis Gower(1975), Ten Berge (1986,1988) Rotation methods

  35. Mode B soft modeling approach SMP = Horst method(1961) SMP = Soft Modeling Appraoch (Hanafi 2001) Generalized canonical correlation methods SVD

  36. Multivariate Eigenvalue Problem Watterson and Chu(1993)

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