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

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Dependent variable: The database Independent variables: n – data m – variables Classifier method y s1 y s2 y sr ?

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The well known regression X 1max. c1c1 x1x1 y x1 x1x1 … … X m max. cmcm xmxm xmxm y xm +

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x2x2 x1x1 Step function coefficients y x2 c2c2 x2x2 c4c4 x4x4 x4x4 y x4 c1c1 x1x1 y x1 y x3 c3c3 x3x3 x3x3 + + +

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s 2 Instead of coefficients, step function type estimation The simplification idea n s – number of categories in ranking n s n ( number of cases) Consider: n s = n c1c1 x1x1 X 1 s 1 c2c2 x2x2 X 2 s 3 s 4 c3c3 x3x3 X 3 c4c4 x4x4 X 4

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s 2 The simplification idea c1c1 x1x1 X 1 s 1 c2c2 x2x2 X 2 s 3 s 4 c3c3 x3x3 X 3 c4c4 x4x4 X 4 ++ + It leads to Linear, Nonlinear or Quadratic Programming Problem

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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 The baseline, the impulsion: COCO Component-based Object Comparison for Objectivity or Similarity Analysis

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- An idea about step functional regression - as a datamining classification method - Based on investigation of COCO - On a simple example about Coffee Brands’ Evaluation 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 Goal was: - To investigate COCO generally

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

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Evaluation of Coffee brands

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

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Setting Solver Parameters Goal function Modifing cellsSubject to 0 0

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COCO online

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Properties of COCO Methods < 0 0 0 Name:Constrains, goals COCO Y0 (No real, measured Y0) Difference of stairs < 0 (Pratically < 1) COCO SDT (Standard) Difference of stairs 0 COCO MCM (Monte Carlo Method) Unconstrained (Just goal function) COCO MCM +Unconstrained Iterative step investigation 1 Linear Absolute Sum of squared

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Options of COCO Methods Name: COCO Y0 COCO SDT COCO MCM COCO MCM + Number of steps Ranking option Iterative Goal function Sum of deviation Linear (LP) Sum of Absolute Deviation Nonlinear Sum of Squared Deviation Quadratic (nonlinear) Direction of proportion Ranking option Positivity of steps Non- negative No constrain Option of conditions

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

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LP Problem of COCO STD (Simplex Tableau)

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The Linear Goal Function Considering n s =n Sum of deviation = Sum of the respective stairs – Sum of y i Dynamic Binary Ranking Matrices Σ y i =min.

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

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Quadratic Goal Function

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Matrix of Quadratic Form

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Structure of the Matrix of Quadratic form Bigger values, sorted 1 in 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 1 below and above the main diagonal

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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

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

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

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Dynamic Ranking (sorting) Matrix Data 1,Ranking

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Dynamic Binary Ranking Matrix Data 1,Ranking

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LP Models of COCO Y0

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Linear Programing Model of COCO STD

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LP Models of COCO MCM

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LP Model of the stepfunction coefficient regression

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LP Models of the step function estimation regression

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Max ( ) Number of classes of the classification method mnmn 8888xxx= 4 096 As there might be equal ones

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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

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

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