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Lecturer Costin-Ciprian POPESCU, PhD

Department of Mathematics

The Bucharest Academy of Economic Studies

MATHEMATICAL PROGRAMMING FOR OPTIMAL DECISION

MAKING

Abstract. Data analysis and especially forecasts are important when an

investor wants to place his capital. A useful mathematical tool in conducting

analysis and making decisions of this type is regression. In this paper we deal with

orthogonal regression, particularly focusing on two-dimensional case. On this

basis, we present a comparative analysis on the evolution of some securities listed

on stock exchange.

Key words: data analysis, orthogonal regression, forecasting, decision

making.

JEL Classification: C02, C51, C53.

1. Preliminaries

Regression, with data written in the form of real numbers or fuzzy numbers, is an

useful method in areas where data analysis plays a central role (Van Huffel et al.,

2007; Ramos, 2007; Petras and Podlubny, 2007; Popescu and Giuclea, 2007;

Bifulco et al., 2009; Giuclea and Popescu, 2009; Iacob et al., 2010). Economy is

one such research field, where various kinds of algorithms make their presence felt

(Roşca and Moldoveanu, 2009; Ruxanda, 2010; Mărăcine and Ianole, 2010). In

this article we use the orthogonal regression in the study of issues to improve

decision making on capital investment. Otherwise, orthogonal regression has been

much discussed in the literature (Van Huffel and Vanderwalle, 1991; Nievergelt,

1994; De Groen, 1996; Van Huffel et al., 2007). When it is necessary to study a

large number of interrelated (statistical) variables, the method is based on matrix

factorization (Golub and Van Loan, 1996). However, for two-dimensional or three-

dimensional case, some geometric solutions can be found (Pozzi, 2008; etc.). For

n statistical data, ( ) s skkk xxx R∈,...,, 21 , nk ,1= (representing n numerical values

which are obtained for s variables: sXX ,...,1 ), orthogonal regression in s

dimensions involves finding

( ) ( ) 

  



  

=+=Π ∑ =

0,...,,..., 1

010

jjss xaaxxaa ,

which is a solution for the problem

( ) ( )( )



  

 Π∑

=∈ +

skkk aa

xxx s

s 1

21 2

,..., ,,...,,Dmin

1 0 R

Costin-Ciprian Popescu

( )( )Π,,...,,D 21 2

skkk xxx represents the squared distance between ( )skkk xxx ,...,, 21

and Π . The hyperplane ( )saa ,...,0Π can be determined based on a sequence of

steps described below (Van Huffel and Vanderwalle, 1991; De Groen, 1996; Golub

and van Loan, 1996). First, consider the sn× matrix A with columns ja

( )sj ,1= , where ( )T CjnCj

jj xxxxa −−= ,...,1 and ∑

−= n

jkC xnx j

1 . Using singular

value decomposition approach, A can be decomposed into a matrix product as TQQA 21Σ= . The numbers saa ,...,1 will be the components of the column vector

from ss× matrix 2Q , corresponding to the smallest eigenvalue of the matrix A .

Then, it can be shown that the point  

  

 ∑∑

−

− n

k xnxnC 1

1 1 ,..., (which is called

centroid) belongs to ( )saa ,...,0Π . Finally, 0a can be determined using the fact that

( )saaC ,...,0Π∈ . Under certain conditions, the obtained solution is unique.

Because it is not necessary to know the role played by each variable

(independent/dependent), orthogonal regression can be applied in problems where

other methods do not work. Some interesting comparisons with the classical

method of least squares can be found in Nievergelt (1994) or Petras and Podlubny

(2007) (together with other important features of the method that make it unique

among regression techniques). In particular, for two variables, X and Y , we have

2=s and ( )210 ,, aaaΠ can be regarded as a line ( )*L . In this case, there are

solving methods that do not require matrix factorization, but are based on

geometric arguments. In the following lines, we will discuss such an approach.

2. A geometric approach

Consider two statistical variables, X and Y , for which we can not state the type of

dependence (which one clearly influences the other). However, similar to ordinary

regression, we wish to establish a connection that results in an equation that

includes both. Suppose that after n observations we know the numerical values

( )kk yx , , nk ,1= . Let it be ( )CC yxC , with coordinates given by ∑ =

−= n

kC xnx 1

and ∑ =

−= n

kC yny 1

1 . As we said before, C is the centroid for the set consisting of

points ( )kkk yxP , , nk ,1= . A line ( )mL (which is not vertical), with slope equal to

m , and passing through a point ( )000 , yxP has the equation

000 =+−− ymxymx .

Mathematical Programming for Optimal Decision Making

The sum of squared distances from the points ( )kkk yxP , , nk ,1= to ( )mL is

calculated by the formula

( ) ( )∑∑ == +

+−− =

kk n

mk m

ymxymx LP

1 2

1 )(,D .

On the other hand, we have

( ) =+−−∑ =

kk ymxymx 1

00 ( ) ( )[ ] −−−−∑ =

CkCk yyxxm 1

( ) ( )[ ]⋅−−−− CC yyxxm 002 ( ) ( )[ ]+−−−∑ =

CkCk yyxxm 1

( ) ( )[ ] =−−−+ ∑ =

CC yyxxm 1

( ) ( )[ ] +−−−= ∑ =

CkCk yyxxm 1

2 ( ) ( )[ ]∑ =

−−− n

CC yyxxm 1

00 .

Thus

( ) =∑ =

mk LP 1

2 )(,D ( ) ( )[ ] +−−−

+ ∑ =

CkCk yyxxm m 1

2 1

( ) ( )[ ] ≥−−− +

+ ∑ =

CC yyxxm m 1

002 1

1 ( ) ( )[ ]∑ =

−−− +

CkCk yyxxm m 1

2 1

1 ,

with equality when Cxx =0 and Cyy =0 . Also, for a vertical line ( )∞L we have

( ) ( ) =−= ∑∑ ==

∞

k xxLP 1

2 )(,D ( ) =+−−∑

CCk xxxx 1

( ) ( ) ≥−+−= ∑∑ ==

Ck xxxx 1

2 ( )∑ =

− n

Ck xx 1

2 ,

with equality if Cxx =0 . In conclusion, for all line with a slope R∈m , the sum of

squared distances is minimized for that which passes through ( )CC yxC , (see also

De Groen, 1996). So the feasible set will consist only of the lines ( )L passing

through the centroid. The next step is to calculate the angle between the optimum

line, ( )*L , and one of the coordinate axis (for a Cartesian coordinate system xOy ).

Therefore, consider a line ( )L (with ( )LC∈ ), and an arbitrary point ( )PP yxP , ,

such that ( ) ( )CCPP yxCyxP ,, ≠ , ( )LP∉ (Fig. 1). Assume that α is the value of

the angle between CP and Ox . Similarly, β ( 2

,0 π

β ≠ ) is the value of the angle

between ( )L and Oy . If PT L)(pr= , then the distance from P to ( )L is equal to

the length of segment PT . For triangle QPC , we can write

Costin-Ciprian Popescu

xx CP − =αcos ,

yy CP − =αsin .

For triangle TPC we have ( ) PC

PT CPT =ˆcos . Due to the fact that ( ) PTL ⊥ and

OyPS ⊥ it results that the measure of SPT ˆ is equal to β . Also, we have

QCPCPS ˆˆ ≡ . Thus

( ) ( ) ( ) βα +=+= CPSSPTCPT ˆˆˆ .

From ( ) PC

PT =+ βαcos , it follows that

PT =− βαβα sinsincoscos .

Then the distance from P to ( )L is

=−−−= ββ sincos CPCP yyxxPT ( ) ( ) ββ sincos CPCP yyxx −−− .

Further, we make the following notation: β π

γ −= 2

. For other possible positions

of P over ( )L , the results is similar (except for a possible change of sign).

Figure 1. Graphical representation of ( )L

If ( )L is a vertical line, the distance from P to ( )L is CP xx − . Similarly, for a

horizontal line ( )L , the distance from P to ( )L is CP yy − . In conclusion, we

have

( ) ( ) ( )[ ]22 sincos)(,D ββ CPCP yyxxLP −−−= .

Mathematical Programming for Optimal Decision Making

Thus minimizing the sum of squared distances from kP ( nk ,1= ) to ( )L is

equivalent to solving the problem ( ) 

  β

β umin , where

( ) ( ) =−= ∑ =

kk bau 1

2 sincos βββ

∑∑∑ ===

−+= n

k baba 11

22 cossin2sincos ββββ .

For all nk ,1= , we used the notation: Ckk xxa −= and Ckk yyb −= . After

equating to zero the derivative of ( )βu we obtain

( )( ) 0cossinsincos 1

=+−∑ =

kkkk baba ββββ

( ) ( ) 0sincoscossin 1

22 =−+−⇒ ∑∑ ==

kk baba ββββ

( ) ( ) ( ) 02cos22sin 11

22 =+−⇒ ∑∑ ==

kk baba ββ

02cos22sin =+⇒ ββ cd ,

where ∑ =

= n

kkbac 1

and ( )∑ =

−= n

kk bad 1

22 . Consider the following two equations

02cos22sin =+ ββ cd , 12cos2sin 22 =+ ββ .

If 0≠c , 0≠d , we get that

( ) 12222 442sin −

+= dccβ , ( ) 12222 42cos −

+= dcdβ ,

( ) ( )cdsgn2cos2sinsgn −=ββ .

Thus 122tan −−= cdβ . Using the relationship ( ) γγπβ 2tan2tan2tan −=−= we

obtain 122tan −= cdγ . Since ( ) 12tan1tan22tan −

−= γγγ then ( ) 1121 −− =− cdmm .

Thus 02 =−+ cdmcm . For this last equation, we have 04 22 >+=∆ cd . This

means that we get two distinct real solutions, 1m and 2m . Also using the second

derivative of ( )δu , it can be shown that one and only one of the two solutions

gives a minimum point. For its determination, we can actually compare the values

of the sum of squared distances in the two cases and with those given by the

horizontal and vertical lines through centroid. The line leading to a lower value of

the mentioned sum is chosen as the optimal solution, ( )*L .

Costin-Ciprian Popescu

3. Numerical application

Models that allow analysis of developments in the stock markets are a subject of

great interest in Financial Mathematics (Ruxanda and Stoenescu, 2009; Fulga et

al., 2009; Kavkler and Festić, 2010; Shavandi and Alizadeh, 2010; Popescu and

Fulga, 2011). In this paper we apply the method discussed in previous sections, for

the study of some stock exchange listed assets. Consider two securities (which will

be formally marked by S1 and S2) whose yields are given in Table 1 (the displayed

data were taken from the source: Deutsche Bundesbank, Prices and Yields of

Listed Federal Securities, January 2010).

Table 1. Prices and yields of listed securities

S1 S2 Date

Price 1 Yield 1 ( )ix Price 2 Yield 2 ( )iy

04.01.2010 100.477 0.31 100.750 0.30

05.01.2010 100.472 0.30 100.746 0.28

06.01.2010 100.472 0.26 100.745 0.25

07.01.2010 100.440 0.31 100.715 0.28

08.01.2010 100.435 0.30 100.690 0.34

11.01.2010 100.432 0.27 100.690 0.31

12.01.2010 100.430 0.24 100.690 0.28

13.01.2010 100.420 0.25 100.660 0.37

14.01.2010 100.400 0.24 100.650 0.31

15.01.2010 100.390 0.25 100.650 0.28

18.01.2010 100.385 0.24 100.635 0.31

19.01.2010 100.375 0.25 100.625 0.32

20.01.2010 100.360 0.31 100.615 0.32

21.01.2010 100.345 0.25 100.590 0.33

22.01.2010 100.330 0.31 100.585 0.31

25.01.2010 100.330 0.25 100.570 0.35

26.01.2010 100.320 0.28 100.565 0.33

27.01.2010 100.315 0.25 100.561 0.31

28.01.2010 100.290 0.28 100.535 0.33

29.01.2010 100.290 0.21 100.528 0.32

These data can be represented in three space dimensions (where, for a more

illustrative representation, Yield 1 and Yield 2 are multiplied by 210 ) (Fig. 2).

Such a representation is useful because it outlines a pattern of simultaneous

evolution. But a rigorous decision by an investor requires an analysis based on an

equation established between the two financial instruments. Such a mathematical

relationship can be established, for example, between Yield 1 (represented by

variable X ) and Yield 2 (represented by Y ), using orthogonal regression.

Mathematical Programming for Optimal Decision Making

Figure 2. The initial values depending on time

Based on data from Table 1, we find that the centroid for ( )ii yx , , 20,1=i is the

point ( )3115.0,268.0C . Finally, we obtain the regression line

( ):*L 040265.087297.048777.0 =−+ yx (Fig. 3).

Figure 3. The datapoints ( )ii yx , , 20,1=i and the line ( )*L

Remark The least squares approach leads to the following two solutions:

( ):YL 3403.010748.0 +−= xy (if Y is the dependent variable),

Costin-Ciprian Popescu

( ):XL 30658.012386.0 +−= yx (when X is the dependent variable).

Instead, for the method discussed in this paper, regardless of dependency, the result

is the same, ( )*L . The lines ( )*L , ( )YL and ( )XL are plotted in Figure 4. Note that

all three pass through the centroid, the first being included within the angle formed

by the last two. These results can be added to those obtained by Nievergelt (1994),

or Petras and Podlubny (2007). There, ( )YL and ( )XL were called conjugate lines.

Figure 4. ( )XL , ( )YL and ( )*L (the last as a thick line)

4. Conclusions

Unlike other methods of regression, orthogonal regression allow the development

of a model even when the direction of causality is not known with precision.

Moreover, the discussion may be taken for an arbitrary number of parameters that

characterize a particular process. Thus, bringing a unified formula of the various

variables allows the possibility of the analysis, making predictions and optimal

decision-making.

Acknowledgements

This work was supported by CNCSIS-UEFISCSU, project number 844 PN II-

IDEI, code 1778/2008.

Mathematical Programming for Optimal Decision Making

REFERENCES

[1] Bifulco, I; Raiconi, G; Scarpa R. (2009), Computer algebra software for

least squares and and total least norm inversion of geophysical models,

Computers & Geosciences 35, 1427-1438;

[2] De Groen, P. (1996), An introduction to total least squares, Nieuw Archief

voor Wiskunde, Vierde serie 14, 237-253;

[3] Fulga, C.; Dedu, S.; Şerban, F. (2009), Portfolio Optimization with Prior

Stock Selection, Economic Computation and Economic Cybernetics Studies

and Research 43(4), ASE Publishing House, Bucharest, 157-172;

[4] Giuclea, M.; Popescu, C. C. (2009), On statistical pattern with fuzzy data,

Economic Computation and Economic Cybernetics Studies and Research, 4,

ASE Publishing House, Bucharest, 187-198;

[5] Golub, G. H; Van Loan, C. H. (1996), Matrix Computations, Johns Hopkins

University Press;

[6] Iacob, A. I.; Popescu, C. C.; Dimian, G. C. (2010), Tracking changes in the

regional social-economic activity by a geometric fitting model: the Romanian

case, Selected topics in Economy & Management Transformation, WSEAS

Press, 428-433;

[7] Kavkler, A.; Festić, M (2010), Smooth transition regression model for

Slovene Stock Exchange index returns, Economic Computation and

Economic Cybernetics Studies and Research 4, ASE Publishing House,

Bucharest, 147-164;

[8] Mărăcine, V.; Ianole, R. (2010), How knowledge determines demand

dynamics-a new perspective for evaluation. Study case for healthcare

services, Economic Computation and Economic Cybernetics Studies and

Research 4, ASE Publishing House, Bucharest, 5-22;

[9] Nievergelt, Y. (1994), Total least squares: state of the art regression in

numerical analysis, SIAM Review 36, 258-264;

[10] Petras, I.; Podlubny, I. (2007), State space description of national

economies: The V4 countries, Computational Statistics & Data Analysis 52,

1223-1233;

[11] Popescu, C; Giuclea, M. (2007), A model of multiple linear regression,

Proceedings of the Romanian Academy Series A: Mathematics, Physics,

Technical Sciences, Information Science, 2 (8), 137-144;

[12] Popescu, C. C.; Fulga, C. (2011), Possibilistic Optimization with

Applications to Portfolio Selection, Proceedings of the Romanian Academy

Series A: Mathematics, Physics, Technical Sciences, Information Science 2,

88-94;

[13] Pozzi F. (2008), Orthogonal linear least squares on a two-dimensional

plane, Matlab Exchange Files;

[14] Ramos, J. A. (2007), Applications of TLS and related methods in the

environmental sciences, Computational Statistics & Data Analysis 52, 1234-

1267;

Costin-Ciprian Popescu

[15] Roşca, I., G.; Moldoveanu, G. (2009), Management in turbulent

conditions, Economic Computation and Economic Cybernetics Studies and

Research 2, ASE Publishing House, Bucharest, 5-12;

[16] Ruxanda, G.; Stoenescu, S. (2009), Bivariate and multivariate

cointegration and their application in stock markets, Economic Computation

and Economic Cybernetics Studies and Research 4, ASE Publishing House,

Bucharest, 17-32;

[17] Ruxanda, G. (2010), Learning perceptron neural network with

backpropagation algorithm, Economic Computation and Economic

Cybernetics Studies and Research 4, ASE Publishing House, Bucharest, 37-54;

[18] Shavandi, H.; Alizadeh, P. (2010), A hybrid intelligent model using

technical and fundamental analysis to forecasting stock price index,

Economic Computation and Economic Cybernetics Studies and Research 2,

ASE Publishing House, Bucharest, 95-112;

[19] Van Huffel, S; Vanderwalle, J. (1991), The total least squares problem:

Computational aspects and analysis, SIAM, Philadelphia;

[20] Van Huffel, S.; Cheng, C. L.; Mastronardi, L.; Paige , C.; Kukush A. (2007), Total least squares and errors-in-variables modeling, Computational

Statistics & Data Analysis 52, 1076-1080.

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