about the step functional regression a lépcsősfüggvényes regresszióról dr bánkuti gyöngyi...
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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
Dependent variable:
m1mj21new xx,x,x,xx
The database
Independent variables:
mnmnjnnn
mmj
mmj
xxxxx
xxxxx
xxxxx
)1(21
2)1(222221
1)1(111211
n
2
1
y
y
y
n –
dat
a
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
ssss
ssss
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
ssss
ssss
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 CreamTaste of harmony
FirmnessPrice/ Value
ratePrice /
kg
il Moretto Sublime 4.2 9.0 4.6 8.6 4 800
Cagliari Espresso Bar 3.2 7.2 3.8 5.4 7 190
Buscaglione Eurobar 4.0 6.2 3.4 4.0 7 200
Bristot Espresso 4.2 6.8 3.8 7.0 3 200
Giuliano Espresso Italiano 4.2 7.4 3.4 5.4 6 650
Corso Verona Ristorante 0.6 1.0 0.4 0.4 4 800
Tonino 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 CreamTaste of harmony
FirmnessPrice/ 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 1464
2 1496 2187 1461 1464
3 1496 2187 1461 1464
4 1496 2187 1461 1464
5 1496 2187 1461 1464
6 1496 779 1461 1464
7 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 CreamTaste of harmony
FirmnessPrice/ 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 1464
2 1496 2187 1461 1464
3 1496 2187 1461 1464
4 1496 2187 1461 1464
5 1496 2187 1461 1464
6 1496 779 1461 1464
7 1496 779 1461 1464
8 1340 779 1340 13402 3 4 5
Staircases
Excel of Coffee brands
Coffe brands CreamTaste of harmony
FirmnessPrice/ Value
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$24,0)
Name: x1 x2 x3 x4
Coffe brands CreamTaste of harmony
FirmnessPrice/
Value rate
Coffe brands 1 1 1 1
il Moretto Sublime 5 4 2 4
Cagliari Espresso Bar 4 7 4 7
Buscaglione Eurobar 1 6 2 2
Bristot Espresso 1 3 4 4
Giuliano Espresso Italiano 8 8 8 8
Corso Verona Ristorante 6 2 6 3Carraro Don Cortez Grandi Arabica 6 5 7 4
Ranking
Estima- ted Price
66086608520052006608480066086608
Σ
Price / kg
4,8007,1907,2003,2006,6504,8006,9007,500
Deviation
1808-582
-20002000
-420
-292-892
Sum of Deviations: 0
Sum of Absolute Deviations: 7,616Sum of squared deviation: 12,490,280
=FKERES(N5,$C$16:$G$23,E$24,0)
M N
RankCream
Taste of
harm.
Firm- ness
Price/ Value rate
1 1496 2187 1461 1464
2 1496 2187 1461 1464
3 1496 2187 1461 1464
4 1496 2187 1461 1464
5 1496 2187 1461 1464
6 1496 779 1461 1464
7 1496 779 1461 1464
8 1340 779 1340 13402 3 4 5
Staircases
Setting Solver Parameters
Coffe brands CreamTaste of harmony
FirmnessPrice/ Value
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
Coffe brands CreamTaste of harmony
FirmnessPrice/
Value rate
Coffe brands 1 1 1 1
il Moretto Sublime 5 4 2 4
Cagliari Espresso Bar 4 7 4 7
Buscaglione Eurobar 1 6 2 2
Bristot Espresso 1 3 4 4
Giuliano Espresso Italiano 8 8 8 8
Corso Verona Ristorante 6 2 6 3Carraro Don Cortez Grandi Arabica 6 5 7 4
Ranking
Estima- ted Price
66086608520052006608480066086608
Price / kg
4,8007,1907,2003,2006,6504,8006,9007,500
Deviation
1808-582
-20002000
-420
-292-892
Sum of Deviations: 0
Sum of Absolute Deviations: 7,616Sum of squared deviation: 12,490,280
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
Goal function
Modifing cells Subject to
0
COCO online
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
Properties of COCO Methods
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 < 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
.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 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
RankCream
Taste of
harm.
Firm- ness
Price/ Value rate
1 1496 2187 1461 1464
2 1496 2187 1461 1464
3 1496 2187 1461 1464
4 1496 2187 1461 1464
5 1496 2187 1461 1464
6 1496 779 1461 1464
7 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 b
u11 -1 1 1u21 -1 1 1u31 -1 1 1u41 -1 1 1u51 -1 1 1u61 -1 1 1u71 -1 1 1
u12 -1 1 1u22 -1 1 1u32 -1 1 1u42 -1 1 1u52 -1 1 1u62 -1 1 1u72 -1 1 1
u13 -1 1 1u23 -1 1 1u33 -1 1 1u43 -1 1 1u53 -1 1 1u63 -1 1 1u73 -1 1 1
u14 -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
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
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 b
u11 -1 1 1u21 -1 1 1u31 -1 1 1u41 -1 1 1u51 -1 1 1u61 -1 1 1u71 -1 1 1
u12 -1 1 1u22 -1 1 1u32 -1 1 1u42 -1 1 1u52 -1 1 1u62 -1 1 1u72 -1 1 1
u13 -1 1 1u23 -1 1 1u33 -1 1 1u43 -1 1 1u53 -1 1 1u63 -1 1 1u73 -1 1 1
u14 -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
b
0000000
0000000
0000000
0000000
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 0
y2* 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
y3* 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
y4* 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
y5* 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
y6* 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
y7* 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
y8* 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
Coffe brands CreamTaste of harmony
FirmnessPrice/
Value rate
Coffe brands 1 1 1 1
il Moretto Sublime 5 4 2 4
Cagliari Espresso Bar 4 7 4 7
Buscaglione Eurobar 1 6 2 2
Bristot Espresso 1 3 4 4
Giuliano Espresso Italiano 8 8 8 8
Corso Verona Ristorante 6 2 6 3Carraro Don Cortez Grandi Arabica 6 5 7 4
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
1
24048,
2
24048,
3
24048s
0
0
s
0
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
Coffe brands CreamTaste of harmony
FirmnessPrice/
Value rate
Coffe brands 1 1 1 1
il Moretto Sublime 5 4 2 4
Cagliari Espresso Bar 4 7 4 7
Buscaglione Eurobar 1 6 2 2
Bristot Espresso 1 3 4 4
Giuliano Espresso Italiano 8 8 8 8
Corso Verona Ristorante 6 2 6 3Carraro Don Cortez Grandi Arabica 6 5 7 4
Ranking
.min)y)s((zn
1i
2i
m
1jijSquared
11111
11111
11111
11111
11111
y
s
s
s
s
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
11111
11111
11111
11111
11111
y
s
s
s
s
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 1
S21S31S71S33S53S54S64
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
08,8
05,6
33,5
0,5
0,5
14,4
31,3
08,3
0
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 0 898.649
0 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 4800
0 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 0 862.5
0 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
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 3200
s32 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 3450
s81 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
s61 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
Sum1
1
1
11111
4.2 1
3.2 5
4.0 4
4.2 1
4.2 1
0.6 8
2.8 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 0
y8 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
1
DBRM
3 0 0 1 1 2 0 1
Sum1
1
1
11111
4.2 1
3.2 5
4.0 4
4.2 1
4.2 1
0.6 8
2.8 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
0
0
s
s
s
E00
00
000E0
0000E
m
2
1
nx)1n(
nx)1n(
nx)1n(
squaresofSumz
.minb
s
s
s
BRMD,,DBRM,DBRMAbs1z
m
2
1
m21
TAbsSTD
LP Models of COCO Y0
.minb1
s
s
s
DRM1,,DRM1,DRM1z T
m
2
1
m
T
2
T
1
TSTD
Linear Programing Model of COCO STD
0
0
0
s
s
s
E00
00
000E0
0000E
m
2
1
nx)1n(
nx)1n(
nx)1n(
squaresofSumz
.minb
s
s
s
BRMD,,DBRM,DBRMAbs1z
m
2
1
m21
TAbsSTD
.minb1
s
s
s
DRM1,,DRM1,DRM1z T
m
2
1
m
T
2
T
1
TSTD
LP Models of COCO MCM
.min)y)s((zn
1i
2i
m
1jijSquared
.minb
s
s
s
BRMD,,DBRM,DBRMAbs1z
m
2
1
m21
TAbsSTD
.minb1
s
s
s
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
s
s
DRMDRMDRM
.min)y)s((zn
1i
2i
m
1jijSquared
.minb
s
s
s
BRMD,,DBRM,DBRMAbs1z
m
2
1
m21
TAbsSTD
.minb1
s
s
s
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
s
s
DBRMDBRMDBRM
.min
s
s
s
DBRM1,,DBRM1,DBRM1z
m
2
1
m
T
2
T
1
TSTD
.minb
s
s
s
BRMD,,DBRM,DBRMAbs1z
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 1464
2 1496 2187 1461 1464
3 1496 2187 1461 1464
4 1496 2187 1461 1464
5 1496 2187 1461 1464
6 1496 779 1461 1464
7 1496 779 1461 1464
8 1340 779 1340 13402 3 4 5
Staircases
8 8 8 8x x x = 4 096snm
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,