About the step functional regression A lépcsősfüggvényes regresszióról

1. About the step functional regression A lépcsősfüggvényes regresszióról Dr Bánkuti Gyöngyi Kaposvári Egyetem Matematika és Fizika Tanszék, Associate Professor, Egyetemi Docens

2. The database Dependent variable: Independent variables: m – variables n – data ys1 ? ys2 Classifier method ysr

3. c1 cm x1 xm X1max. Xm max. yxm yx1 x1 xm Xm max. X1max. Thewell known regression … … +

4. c4 c1 x4 x1 Step function coefficients c3 c2 x3 x2 yx4 yx1 yx3 yx2 + + + x3 x4 x1 x2

5. c4 c1 c3 c2 x4 x1 x3 x2 X 1 X 2 X 3 X 4 The simplification idea Instead of coefficients, step function type estimation ns – number of categories in ranking ns n (number of cases) Consider: ns = n s1  s3  s2  s4 

6. c4 c1 c3 c2 x4 x1 x3 x2 X 1 X 2 X 3 X 4 The simplification idea It leads to Linear, Nonlinear or Quadratic Programming Problem s1  s3  + + + s2  s4 

7. The baseline, the impulsion: COCO Component-based Object Comparison for Objectivity or Similarity Analysis A method invented by Dr László Pitlik, (PhD, 1993, Giessen, Germany) Szent István University, Gödöllő , TATA Excellence Center and Institute of Informatics, Head of Department The investigated example: Prémium kategóriás szemes kávék összehasonlító tesztje, Gazdaságinformatika II: Egyéni feladat, Szent István University, 2009

8. Content: • Managed: • To investigate this simple example • To settle the equations for it • To understand the possible mathematical origin of COCO • To find the eigenvalues of the quadratic goal function to prove the Coffee problem has got alternative optima • An idea about step functional regression - as a datamining classification method • Based on investigation of COCO • On a simple example about Coffee Brands’ Evaluation • Goal was: • To investigate COCO generally

9. A simple example from COCO’s literature Evaluation of Coffee brands The biggest first

10. Evaluation of Coffee brands

11. Excel of Coffee brands M N =FKERES(M3,$C$16:$G$23,D$24,0) =FKERES(N5,$C$16:$G$23,E$24,0) Σ

12. Setting Solver Parameters  0 Modifing cells Subject to Goal function

13. COCO online

14. Properties of COCO Methods < 0  1  0 Linear Absolute Sum of squared

15. Options of COCO Methods

16. LP Problem of COCO Y0 (Simplex Tableau) S – vector from the columns of the staircase matrix

17. LP Problem of COCO STD (Simplex Tableau)

18. The Linear Goal Function Dynamic Binary Ranking Matrices Considering ns=n Sum of deviation = Sum of the respective stairs – Sum of yi Σ yi =min.

19. MCM with Linear Goal Function Solution by Simplex Method: only these type Solution by Excel: Linear Combination of these

20. Quadratic Goal Function

21. Matrix of Quadratic Form

22. Structure of the Matrix of Quadratic form Bigger values, sorted 1 in the main diagonal 1 below and above the main diagonal -1 out of the main diagonal 0 = zero Remark: The less category we make the less columns and rows (and zeros) we have

23. Eigenvector of the Quadratic form’s Matrix The eigenvalue vector s0 Positive semidefinit quadratic form has got alternative minima, This can be the reason why the unconstrained problem has got strong minima, Not proved for the method, just for Coffee problem

24. Step Function Coefficient Regression Method with Excel Optimal Simplex Tableau http://www.zweigmedia.com/RealWorld/simplex.html

26. Step Function Estimation Type Regression Method with Excel Optimal Simplex Tableau(It does not give the variables on the sides of the tableau) Many 0 in the goal function line -> Many alternative optima

27. Dynamic Ranking (sorting) Matrix Data1,Ranking

28. Dynamic Binary Ranking Matrix Data1,Ranking

29. LP Models of COCO Y0

30. Linear Programing Model of COCO STD

31. LP Models of COCO MCM

32. LP Model of the stepfunction coefficient regression

33. LP Models of the step function estimation regression

34. Number of classes of the classification method mn Max ( ) 8 x 8 x 8 x 8 = 4 096 As there might be equal ones

35. Summary • It managed to investigate the Coffee problem, To settle the mathematical equations, matrixes • The linear goal fiunction case has got several alternative optimum (as the optimal Simplex tableau of this problem was defined) • The sum of squared deviation goal function defines a positive semidefinit quadratic form – so the problems has gor several alternative optima • An idea about the possible mathematical origin of COCO was invented • Step functional regression methods (stepfunctional regression coefficients, and the binary ranking type) was invented, and applied for Coffee problem – succesfully • Open problem: the invariance property of the estimation for the alternative optima could not be proved jet

36. Köszönöm megtisztelő figyelmüket! Acknowledgement: Supported by Baross Gábor Program, KEITTI 009 Kaposvár University,