about the step functional regression a lépcsősfüggvényes regresszióról

About the step functional regression

A lépcsősfüggvényes regresszióról

Dr Bánkuti Gyöngyi

Kaposvári Egyetem Matematika és Fizika Tanszék,

Associate Professor, Egyetemi Docens

Dependent variable:

m1mj21new xx,x,x,xx

The database

Independent variables:

mnmnjnnn

mmj

mmj

xxxxx

xxxxxxxxxx

)1(21

2)1(222221

1)1(111211

n

2

1

y

yy

n –

data

m – variables

Classifier method

ys1

ys2

ysr

?

The well known regression

mm1m1m

2211

^

xcxc...xcxcy

X1max.

c1

x1

X1max.

yx1

x1

…

…Xm max.

cm

xm

xmXm max.

yxm

+new

^y

x2x1

Step function coefficients

444333222111

^x)x(cx)x(cx)x(cx)x(cy

yx2

c2

x2

c4

x4

x4

yx4

c1

x1

yx1 yx3

c3

x3

x3

new

^y + + +

s2

4n3n2n1n

24232221

14131211

ssssssss

ssssssss

S

)x(s)x(s)x(s)x(s)x(y 44432211new

^

Instead of coefficients, step function type estimation

The simplification idea

ns – number of categories in ranking ns n (number of cases)

Consider: ns = n

c1

x1X 1

s1

c2

x2X 2

s3

s4

c3

x3X 3

c4

x4X 4

!!x)x(c)x(s iiii

s2

4n3n2n1n

24232221

14131211

ssssssss

ssssssss

S

)x(s)x(s)x(s)x(s)x(y 44432211new

^

The simplification idea

c1

x1X 1

s1

c2

x2X 2

s3

s4

c3

x3X 3

c4

x4X 4

)x(y new

^ + + +

It leads to

Linear, Nonlinear or Quadratic Programming Problem

!!x)x(s)x(s iiii

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

- 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

A simple example from COCO’s literature

Name: x1 x2 x3 x4 y

Coffe brands Cream Taste of harmony Firmness Price/ Value

ratePrice /

kgil Moretto Sublime 4.2 9.0 4.6 8.6 4 800Cagliari Espresso Bar 3.2 7.2 3.8 5.4 7 190Buscaglione Eurobar 4.0 6.2 3.4 4.0 7 200Bristot Espresso 4.2 6.8 3.8 7.0 3 200Giuliano Espresso Italiano 4.2 7.4 3.4 5.4 6 650Corso Verona Ristorante 0.6 1.0 0.4 0.4 4 800Tonino Lamborghini Classic 2.8 8.0 3.2 5.8 6 900Carraro Don Cortez Grandi Arabica 2.8 7.0 2.8 5.4 7 500

Original Dataű

Name: x1 x2 x3 x4

Coffe brands Cream Taste of harmony Firmness

Price/ Value rate

Coffe brands 1 1 1 1il Moretto Sublime 5 4 2 4Cagliari Espresso Bar 4 7 4 7Buscaglione Eurobar 1 6 2 2Bristot Espresso 1 3 4 4Giuliano Espresso Italiano 8 8 8 8Corso Verona Ristorante 6 2 6 3Carraro Don Cortez Grandi Arabica 6 5 7 4

Ranking

Evaluation of Coffee brands

The biggest

first

RankCream

Taste of

harm.

Firm- ness

Price/ Value rate

1 1496 2187 1461 14642 1496 2187 1461 14643 1496 2187 1461 14644 1496 2187 1461 14645 1496 2187 1461 14646 1496 779 1461 14647 1496 779 1461 1464

8 1340 779 1340 13402 3 4 5

Staircases

ci2-ci1 0 0 0 0 ci3-ci2 0 0 0 0 ci4-ci3 0 0 0 0 ci5-ci4 0 0 0 0 ci6-ci5 0 -1408 0 0 ci7-ci6 0 0 0 0 ci8-ci7 -155 0 -121 -124

Auxiliary Table

Evaluation of Coffee brands

Name: x1 x2 x3 x4

Coffe brands Cream Taste of harmony Firmness

Price/ Value rate

Coffe brands 1 1 1 1il Moretto Sublime 5 4 2 4Cagliari Espresso Bar 4 7 4 7Buscaglione Eurobar 1 6 2 2Bristot Espresso 1 3 4 4Giuliano Espresso Italiano 8 8 8 8Corso Verona Ristorante 6 2 6 3Carraro Don Cortez Grandi Arabica 6 5 7 4

Ranking

RankCream

Taste of

harm.

Firm- ness

Price/ Value rate

1 1496 2187 1461 14642 1496 2187 1461 14643 1496 2187 1461 14644 1496 2187 1461 14645 1496 2187 1461 14646 1496 779 1461 14647 1496 779 1461 1464

8 1340 779 1340 13402 3 4 5

Staircases

Excel of Coffee brands


rateil Moretto Sublime 1496 2187 1461 1464Cagliari Espresso Bar 1496 2187 1461 1464Buscaglione Eurobar 1496 779 1461 1464Bristot Espresso 1496 779 1461 1464Giuliano Espresso Italiano 1496 2187 1461 1464Corso Verona Ristorante 1340 779 1340 1340Tonino Lamborghini Classic 1496 2187 1461 1464Carraro Don Cortez Grandi Arabica 1496 2187 1461 1464

Coco

=FKERES(M3,$C$16:$G$23,D$24,0)

Name: x1 x2 x3 x4

Coffe brands Cream Taste of harmony Firmness Price/

Value rateCoffe brands 1 1 1 1il Moretto Sublime 5 4 2 4Cagliari Espresso Bar 4 7 4 7Buscaglione Eurobar 1 6 2 2Bristot Espresso 1 3 4 4Giuliano Espresso Italiano 8 8 8 8Corso Verona Ristorante 6 2 6 3Carraro Don Cortez Grandi Arabica 6 5 7 4

Ranking

Estima- ted Price

66086608520052006608480066086608

Σ

Price / kg4,8007,1907,2003,2006,6504,8006,9007,500

Deviation

1808-582

-20002000

-420

-292-892

Sum of Deviations: 0Sum of Absolute Deviations: 7,616Sum of squared deviation: 12,490,280

=FKERES(N5,$C$16:$G$23,E$24,0)

M N

RankCream

Taste of

harm.

Firm- ness

Price/ Value rate

1 1496 2187 1461 14642 1496 2187 1461 14643 1496 2187 1461 14644 1496 2187 1461 14645 1496 2187 1461 14646 1496 779 1461 14647 1496 779 1461 1464

8 1340 779 1340 13402 3 4 5

Staircases

Setting Solver Parameters


rateil Moretto Sublime 1496 2187 1461 1464Cagliari Espresso Bar 1496 2187 1461 1464Buscaglione Eurobar 1496 779 1461 1464Bristot Espresso 1496 779 1461 1464Giuliano Espresso Italiano 1496 2187 1461 1464Corso Verona Ristorante 1340 779 1340 1340Tonino Lamborghini Classic 1496 2187 1461 1464Carraro Don Cortez Grandi Arabica 1496 2187 1461 1464

Coco

Name: x1 x2 x3 x4



Ranking

Estima- ted Price

66086608520052006608480066086608

Price / kg4,8007,1907,2003,2006,6504,8006,9007,500

Deviation

1808-582

-20002000

-420

-292-892

Sum of Deviations: 0Sum of Absolute Deviations: 7,616Sum of squared deviation: 12,490,280


Auxiliary Table

Goal function

Modifing cells Subject to 0

COCO online


Auxiliary Table

Properties of COCO Methods


Auxiliary Table < 0

0

Name: Constrains, goalsCOCO 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 + UnconstrainedIterative step investigation

1

.min)y)s((zn

1i

2i

m

1jijSquared

.min)y)s((zn

1ii

m

1jij.)lin.(Dev

.min)ys(Abs(zn

1ii

m

1jij.Dev.Abs

Linear

Absolute

Sum of squared

Options of COCO Methods

Name:COCO Y0COCO SDTCOCO MCMCOCO MCM +

Number of stepsRanking option

Iterative

Goal functionSum of deviation

Linear (LP)

Sum of Absolute Deviation

Nonlinear

Sum of Squared Deviation

Quadratic (nonlinear)

Direction of proportionRanking option

Positivity of stepsNon-negative

No constrain

Option of conditions

RankCream

Taste of

harm.

Firm- ness

Price/ Value rate

1 1496 2187 1461 14642 1496 2187 1461 14643 1496 2187 1461 14644 1496 2187 1461 14645 1496 2187 1461 14646 1496 779 1461 14647 1496 779 1461 1464

8 1340 779 1340 13402 3 4 5

Staircases

I. S11 S21 S31 S41 S51 S61 S71 S81 S12 S22 S32 S42 S52 S62 S72 S82 S13 S23 S33 S43 S53 S63 S73 S83 S41 S42 S43 S44 S45 S46 S47 S48 bu11 -1 1 1u21 -1 1 1u31 -1 1 1u41 -1 1 1u51 -1 1 1u61 -1 1 1u71 -1 1 1u12 -1 1 1u22 -1 1 1u32 -1 1 1u42 -1 1 1u52 -1 1 1u62 -1 1 1u72 -1 1 1u13 -1 1 1u23 -1 1 1u33 -1 1 1u43 -1 1 1u53 -1 1 1u63 -1 1 1u73 -1 1 1u14 -1 1 1u24 -1 1 1u34 -1 1 1u44 -1 1 1u54 -1 1 1u64 -1 1 1u74 -1 1 1

LP Problem of COCO Y0 (Simplex Tableau)

-z 3 0 0 1 1 2 0 1 1 1 1 1 1 1 1 1 1 2 0 2 0 1 1 1 1 1 1 3 0 0 1 1 0

S – vector from the columns of the staircase matrix


Auxiliary Table

I. S11 S21 S31 S41 S51 S61 S71 S81 S12 S22 S32 S42 S52 S62 S72 S82 S13 S23 S33 S43 S53 S63 S73 S83 S41 S42 S43 S44 S45 S46 S47 S48 bu11 -1 1 1u21 -1 1 1u31 -1 1 1u41 -1 1 1u51 -1 1 1u61 -1 1 1u71 -1 1 1u12 -1 1 1u22 -1 1 1u32 -1 1 1u42 -1 1 1u52 -1 1 1u62 -1 1 1u72 -1 1 1u13 -1 1 1u23 -1 1 1u33 -1 1 1u43 -1 1 1u53 -1 1 1u63 -1 1 1u73 -1 1 1u14 -1 1 1u24 -1 1 1u34 -1 1 1u44 -1 1 1u54 -1 1 1u64 -1 1 1u74 -1 1 1

LP Problem of COCO STD (Simplex Tableau)

.minscz TDeviationofSum

-z 3 0 0 1 1 2 0 1 1 1 1 1 1 1 1 1 1 2 0 2 0 1 1 1 1 1 1 3 0 0 1 1 0

.min)y)s((zn

1i

2i

m

1jijSquared

b0000000000000000000000000000

0

The Linear Goal Function

Considering ns=n

Sum of deviation = Sum of the respective stairs – Sum of yi

.miny)s(zn

1ii

m

1jij.)lin.(Dev

S11 S21 S31 S41 S51 S61 S71 S81 S12 S22 S32 S42 S52 S62 S72 S82 S13 S23 S33 S43 S53 S63 S73 S83 S14 S24 S34 S44 S54 S64 S74 S84

y1* 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0y2* 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0y3* 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0y4* 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0y5* 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0y6* 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1y7* 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0y8* 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0

Name: x1 x2 x3 x4



Ranking

-z 3 0 0 1 1 2 0 1 1 1 1 1 1 1 1 1 1 2 0 2 0 1 1 1 1 1 1 3 0 0 1 1

y1

y2

y3

y4

y5

y6

y7

y8

Dynamic Binary Ranking Matrices

-z 3 0 0 1 1 2 0 1 1 1 1 1 1 1 1 1 1 2 0 2 0 1 1 1 1 1 1 3 0 0 1 1 Σ yi

=min.

u1 3 0 0 1 1 2 0 1 1 1 1 1 1 1 1 1 1 2 0 2 0 1 1 1 1 1 1 3 0 0 1 1 48240 48240

-z 3 0 0 1 1 2 0 1 1 1 1 1 1 1 1 1 1 2 0 2 0 1 1 1 1 1 1 3 0 0 1 1 48240

x act. 15410 200 300 400 100 50 40 530 2 2 20 20 20 40 50 50 2 40 6 40 50 60 2 60 2 60 1 60 60 70 70 80

MCM with Linear Goal Function

124048,

224048,

324048s

0

0s0

0

x0000 jiji

Solution by Simplex Method:

only these type

Solution by Excel:

Linear Combination of these

Quadratic Goal Function

.min)yssss(

)yssss(z2

244234251

2114131211Squared

Name: x1 x2 x3 x4



Ranking

.min)y)s((zn

1i

2i

m

1jijSquared

1111111111111111111111111

yssss

yssss

1

14

13

12

11

114131211

S11 S21 S31 S41 S51 S61 S71 S81 S12 S22 S32 S42 S52 S62 S72 S82 S13 S23 S33 S43 S53 S63 S73 S83 S14 S24 S34 S44 S54 S64 S74 S84 y1 y2 y3 y4 y5 y6 y7 y8S11 3 1 1 1 1 1 1 1 1 1 -1 -1 -1S21S31S41 1 1 1 1 -1S51 1 1 1 1 -1S61 2 1 1 1 1 1 1 -1 -1S71S81 1 1 1 1 -1S12 1 1 1 1 -1S22 1 1 1 1 -1S32 1 1 1 1 -1S42 1 1 1 1 -1S52 1 1 1 1 -1S62 1 1 1 1 -1S72 1 1 1 1 -1S82 1 1 1 1 -1S13 1 1 1 1 -1S23 1 1 1 1 2 1 1 -1 -1S33S43 1 1 1 1 2 1 1 -1 -1S53S63 1 1 1 1 -1S73 1 1 1 1 -1S83 1 1 1 1 -1S14 1 1 1 1 -1S24 1 1 1 1 -1S34 1 1 1 1 -1S44 1 1 1 1 1 1 1 1 1 3 -1 -1 -1S54S64S74 1 1 1 1 -1S84 1 1 1 1 -1Y1 -1 -1 -1 -1 1Y2 -1 -1 -1 -1 1Y3 -1 -1 -1 -1 1Y4 -1 -1 -1 -1 1Y5 -1 -1 -1 -1 1Y6 -1 -1 -1 -1 1Y7 -1 -1 -1 -1 1Y8 -1 -1 -1 -1 1

Matrix of Quadratic Form

1111111111111111111111111

yssss

yssss

1

14

13

12

11

114131211

Structure of the Matrix of Quadratic form

S11 S44 S61 S23 S43 S41 S51 S81 S12 S22 S32 S42 S52 S62 S72 S82 S13 S63 S73 S83 S14 S24 S34 S74 S84 y1 y2 y3 y4 y5 y6 y7 y8 S21 S31 S71 S33 S53 S54 S64S11 3 1 1 1 1 1 1 1 1 1 -1 -1 -1S44 1 3 1 1 1 1 1 1 1 1 -1 -1 -1S61 1 2 1 1 1 1 1 -1 -1S23 1 1 2 1 1 1 1 -1 -1S43 1 1 2 1 1 1 1 -1 -1S41 1 1 1 1 -1S51 1 1 1 1 -1S81 1 1 1 1 -1S12 1 1 1 1 -1S22 1 1 1 1 -1S32 1 1 1 1 -1S42 1 1 1 1 -1S52 1 1 1 1 -1S62 1 1 1 1 -1S72 1 1 1 1 -1S82 1 1 1 1 -1S13 1 1 1 1 -1S63 1 1 1 1 -1S73 1 1 1 1 -1S83 1 1 1 1 -1S14 1 1 1 1 -1S24 1 1 1 1 -1S34 1 1 1 1 -1S74 1 1 1 1 -1S84 1 1 1 1 -1Y1 -1 -1 -1 -1 1Y2 -1 -1 -1 -1 1Y3 -1 -1 -1 -1 1Y4 -1 -1 -1 -1 1Y5 -1 -1 -1 -1 1Y6 -1 -1 -1 -1 1Y7 -1 -1 -1 -1 1Y8 -1 -1 -1 -1 1S21S31S71S33S53S54S64

Bigger values, sorted

1 in the main diagonal

-1 out of the main diagonal 0 = zero

Eigenvector = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.08 3.31 4.14 5.00 5.00 5.33 6.05 8.08

Remark:

The less category we make

the less columns and rows (and zeros) we have

1 below and above the main diagonal

Eigenvector of the Quadratic form’s Matrix

08,805,633,50,50,514,431,308,30

0

s

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

y1 4.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 9.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 8.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 >= 4800y2 0.0 0.0 0.0 0.0 3.2 0.0 0.0 0.0 0.0 0.0 0.0 7.2 0.0 0.0 0.0 0.0 0.0 3.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.4 0.0 0.0 0.0 0.0 >= 7190y3 0.0 0.0 0.0 4.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.2 0.0 0.0 0.0 0.0 3.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.0 0.0 >= 7200y4 4.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.8 0.0 0.0 0.0 3.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.0 0.0 0.0 0.0 0.0 0.0 0.0 >= 3200y5 4.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.4 0.0 0.0 0.0 0.0 >= 6650y6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 >= 4800y7 0.0 0.0 0.0 0.0 0.0 2.8 0.0 0.0 0.0 8.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.2 0.0 0.0 0.0 0.0 5.8 0.0 0.0 0.0 0.0 0.0 >= 6900y8 0.0 0.0 0.0 0.0 0.0 2.8 0.0 0.0 0.0 0.0 0.0 0.0 7.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.8 0.0 0.0 0.0 0.0 5.4 0.0 0.0 0.0 0.0 >= 7500

-z 12.6 0.0 0.0 4.0 3.2 5.6 0.0 0.6 9.0 8.0 7.4 7.2 7.0 6.8 6.2 1.0 4.6 7.6 0.0 6.8 0.0 3.2 2.8 0.4 8.6 7.0 5.8 16.2 0.0 0.0 4.0 0.4 0

Step Function Coefficient Regression Method with Excel

c14 c24 c34 c44 c54 c64 c74 c84 c13 c23 c33 c43 c53 c63 c73 c83 c12 c22 c32 c42 c52 c62 c72 c82 c11 c21 c31 c41 c51 c61 c71 c81 b

0.96 0 0 0 0 0 0 0 0.51 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0.4667 0 0 0 0 0 0 0 533.333

0 0 0 0.75 0 0 0 0 0 0.53 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0.44 0 0 0 998.611

0 0 0 0 0 0 0.65 0 0 0 0 0.55 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0.65 0 0 0 0 1161.29

0 1 0 0 0 0 0 0 0 0.54 0 0 0 0 0 0 0 0 0 0 0 0.97 0 0 0.6 0 0 0 0 0 0 0 457.143

0 0 0 0.73 0 0 0 0 0 0 0 0.46 0 0 0 0 0 0 1 0 0 0 0 0 0.5676 0 0 0 0 0 0 0 898.6490 0 0 0 0 0 0 0.4 0 0 0 0 0 0 0 0.4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0.6 48000 0 0.73 0 0 0 0 0 0 0 0 0 0 0.4 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0.35 0 0 862.50 0 0 0.771 0 0 0 0 0 0 0 0 0 0 0.4 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0.4 0 0 1071.43

z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 48240

x 0 0 0 0 0 0 0 0 0 0 898.649, 0 998.611, 0 0 0 0 0 0 0 0 0 2678.57, 0 558.14, 457.143,1189.66, 0 0 0 1800, 12000,

Optimal Simplex Tableau http://www.zweigmedia.com/RealWorld/simplex.html

http://www.zweigmedia.com/RealWorld/simplex.html

http://www.zweigmedia.com/RealWorld/simplex.html

S11 S21 S31 S41 S51 S61 S71 S81 S12 S22 S32 S42 S52 S62 S72 S82 S13 S23 S33 S43 S53 S63 S73 S83 S41 S42 S43 S44 S45 S46 S47 S48 b b

y1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4800 4800y2 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 7190 7190y3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 7200 7200y4 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3200 3200y5 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 6650 6650y6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 4800 4800y7 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 6900 6900y8 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 7500 7500

-z 3 0 0 1 1 2 0 1 1 1 1 1 1 1 1 1 1 2 0 2 0 1 1 1 1 1 0 3 0 0 1 1 48240

x act. 1330 0 0 0 0 2180 0 4800 3470 4720 0 0 0 0 0 0 0 1870 0 0 0 0 0 0 0 0 0 5320 0 0 7200 0

Opt. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? bs21 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 1 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1600s51 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 7190s41 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 7200s11 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3200s32 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 -1 0 1 0 0 0 0 0 -1 0 1 0 0 0 0 3450s81 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 4800s61 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 6900 ? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0s52 0 0 0 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 -1 1 0 0 0 0 600 -z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -48240

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)

Dynamic Ranking (sorting) Matrix

Data1,Ranking

1

DRM

3 0 0 1 1 2 0 1

Sum11111111

4.2 13.2 54.0 44.2 14.2 10.6 82.8 62.8 6

1

T DRM1

1n1211T1 ssss

y1 4.2 0 0 0 0 0 0 0

y2 0 0 0 0 3.2 0 0 0

y3 0 0 0 4.0 0 0 0 0

y4 4.2 0 0 0 0 0 0 0

y5 4.2 0 0 0 0 0 0 0

y6 0 0 0 0 0 0 0 0.6

y7 0 0 0 0 0 2.8 0 0y8 0 0 0 0 0 2.8 0 0

12.6 0.0 0.0 4.0 3.2 5.6 0.0 0.6

Dynamic Binary Ranking Matrix

Data1,Ranking

1DBRM

3 0 0 1 1 2 0 1

Sum11111111

4.2 13.2 54.0 44.2 14.2 10.6 82.8 62.8 6

1

T DBRM1

1n1211T1 ssss

00100000y8

00100000y7

10000000y6

00000001y5

00000001y4

00001000y3

00010000y2

00000001y1

S81S71S61S51S41S31S21S11

00100000y8

00100000y7

10000000y6

00000001y5

00000001y4

00001000y3

00010000y2

00000001y1

S81S71S61S51S41S31S21S11

0

00

s

ss

E0000

000E0

0000E

m

2

1

nx)1n(

nx)1n(

nx)1n(

squaresofSumz

.minb

s

ss

BRMD,,DBRM,DBRMAbs1z

m

2

1

m21

TAbsSTD

LP Models of COCO Y0

.minb1

s

ss

DRM1,,DRM1,DRM1z T

m

2

1

m

T

2

T

1

TSTD

Linear Programing Model of COCO STD

0

00

s

ss

E0000

000E0

0000E

m

2

1

nx)1n(

nx)1n(

nx)1n(

squaresofSumz

.minb

s

ss


m

2

1

m21

TAbsSTD

.minb1

s

ss

DRM1,,DRM1,DRM1z T

m

2

1

mT

2T

1T

STD

LP Models of COCO MCM

.min)y)s((zn

1i

2i

m

1jijSquared

.minb

s

ss


m

2

1

m21

TAbsSTD

.minb1

s

ss

DRM1,,DRM1,DRM1z T

m

2

1

m

T

2

T

1

TSTD

LP Model of the stepfunction coefficient regression

fact

m

2

1

m21b

s

ss

DRMDRMDRM

.min)y)s((zn

1i

2i

m

1jijSquared

.minb

s

ss


m

2

1

m21

TAbsSTD

.minb1

s

ss

DRM1,,DRM1,DRM1z T

m

2

1

m

T

2

T

1

TSTD

LP Models of the step function estimation regression

fact

m

2

1

m21b

s

ss

DBRMDBRMDBRM

.min

s

ss

DBRM1,,DBRM1,DBRM1z

m

2

1

m

T

2

T

1

TSTD

.minb

s

ss


m

2

1

m21

TAbsSTD

.min)y)s((zn

1i

2i

m

1jijSquared

Max ( )

Number of classes of the classification method

mn

RankCream

Taste of

harm.

Firm- ness

Price/ Value rate

1 1496 2187 1461 14642 1496 2187 1461 14643 1496 2187 1461 14644 1496 2187 1461 14645 1496 2187 1461 14646 1496 779 1461 14647 1496 779 1461 1464

8 1340 779 1340 13402 3 4 5

Staircases

8 8 8 8x x x = 4 096 snm

As there might be equal ones

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

Köszönöm megtisztelő figyelmüket!

Acknowledgement: Supported by Baross Gábor Program, KEITTI 009

Kaposvár University,

about the step functional regression a lépcsősfüggvényes regresszióról

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