C. Doca, L. Doca - Extrapolation of Monotonous Functions Using the Linear Correlation Coefficient

8/3/2019 C. Doca, L. Doca - Extrapolation of Monotonous Functions Using the Linear Correlation Coefficient

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EXTRAPOLATION OF MONOTONOUS FUNCTIONS USING

THE LINEAR CORRELATION COEFFICIENT

C. DOCA and L. DOCA

ABSTRACT

The paper presents a possible method of extrapolation / prognosis for monotonous

functions based on the utilization of the linear correlation coefficient. While having the

advantage of not requiring knowledge of the analytical (explicit) expression of the function

that gave initial values, the computing formulae proposed by the authors can be

implemented in computer programs for automatically processing and interpretation of

experimental data or of those taken over from various computing tables.

Key words: prognosis, extrapolation, linear correlation coefficient

Introduction

Given the real functions, continuous and double derivable ( )...c,b,a;tfy = , of variable t andparameters a, b, c describing the evolution in time of a phenomenon (physical, economical, social

etc.) one ascertains [1], related to prognosis problems, that ( ) dt...c,b,a;tdf'y = offers information

regarding the short-duration tendency, while ( ) 22 dt...c,b,a;tfd"y = deals with the long-termtendency.

Knowing the analytical expression ( )...c,b,a;tf and with the help of a set of measurements ( )ii y,t ,N,...,,i 21= , the most probable values of parameters a, b, c are determinate, for instance, by the

least-squares method [2], solving the system of equations:

( )0=

a

...c,b,aS;

( )0=

b

...c,b,aS;

( )0=

c

...c,b,aS; (1)

where:

( ) ( )[ ]=

=

N

i

ii ...c,b,a;tfy...c,b,aS

1

2(2)


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With ( )...c,b,a;tf thus explained, in the case of extrapolation at the moment *tt= , one can define:

- short-term prognosis, if:

( ) ( )

( ) ( )

ionextrapolatleft;tttt

ionextrapolatright;tttt

N

*

NN

*

11

1

3

1

3

1

(3)

- medium-term prognosis, if:

( ) ( ) ( )

( ) ( ) ( )

ionextrapolatleft;tttt

ionextrapolatright;tttt

N

*

NN

*

11

1(5)

assuming as evaluation errors [1]: 15% for the short-term prognosis, 25% for the medium-term

prognosis and 50% for the long-term prognosis, respectively.

If the analytical expression of the function ( )...c,b,a;tf is not known, then based on the same set of

measured values ( )ii y,t , N,...,,i 21= , one can build interpolation polynomials (Newton, Lagrange,

Hermite etc.) that would allow, again, the evaluation ( )...c,b,a;tfy ** = .

In both these cases one solves systems of equations, eventually non-linear, requiring in most situations

the computer elaboration and implementation of appropriate programs.

Extrapolation Using the Linear Correlation Coefficient

It is known that for the set ofNmeasurements ( )ii y,t , N,...,,i 21= , the linear correlation coefficient[1]:

[ ]112

11

2

2

11

2

111,R;

yyNttN

ytytN

RN,t,y

N

i

i

N

i

i

N

i

i

N

i

i

N

i

i

N

i

i

N

i

ii

N,t,y

=

====

===(6)


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is a measure of deviation of the function ( )tfy = from a straight line. It is obvious that a new value

1+Ny , obtained at the moment 1+= N* tt , will lead to the amplification or diminution of this deviation

if N,t,yN,t,y RR +1 respectively.

Since:

( )

( ) ( )

+

+

+

=

+

=

+

=

+

=

+

=

+

=

+

=

+

=

+2

1

1

1

1

2

21

1

1

1

2

1

1

1

1

1

1

1

11

1

N

i

i

N

i

i

N

i

i

N

i

i

N

i

i

N

i

i

N

i

ii

N,t,y

yyNttN

ytytN

R (7)

can be re-written as:

( )

( ) ( )

+

++

+

+

++

=

+

=

+

=

+

=

+

=

+

=

+

=

++

=

+

2

1

1

2

1

1

2

21

1

1

1

2

1

1

1

1

11

1

1

11

1

N

N

i

iN

N

i

i

N

i

i

N

i

i

N

N

i

i

N

i

iNN

N

i

ii

N,t,y

yyyyNttN

yytytytN

R (18)

then, with the notations:

=

+=

N

i

iN ttNA1

1 (9)

( )

+=

=

+

==

N

i

i

N

i

i

N

i

ii ytytNB1

1

11

1 (10)

( )

+=

+

=

+

=

21

1

1

1

21

N

i

i

N

i

ittNNC (11)

( )

+

=

+

=

+

==

21

1

1

1

2

1

12N

i

i

N

i

i

N

i

i ttNyD (12)

( ) ( )

+

+=

+

=

+

===

21

1

1

1

2

2

11

211

N

i

i

N

i

i

N

i

i

N

i

i ttNyyNE (13)

equation (8) becomes:

EyDyC

ByAR

NN

NN,t,y

++

+=

++

+

+

1

2

1

11 (14)


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or, finally:

( ) ( ) ( ) 02 22 112

1

2

1

22

1 =++ +++++ BERyABDRyACR N,t,yNN,t,yNN,t,y (15)

In order to obtain immediate information ranging within the limits of acceptable evaluation errors

related to prognosis problems, equality (15) can be interpreted as a binomial equation in the unknown

1+Ny in the point 1+= Ntt . For this, it is sufficient to accept, in a first approximation, the equality

N,t,yN,t,yRR =

+1 and to check the compliance with the condition of existence or real solutions, i.e.:

( ) ( )( ) 042 22222 = BERACRABDR N,t,yN,t,yN,t,y (16)

Between the two solutions:

( )

( )22

2

1

2

2

ACR

ABDRy

N,t,y

N,t,y

N

=

+

(17)

one will choose the minimum or maximum value, depending on the evolution more or less

monotonous of previous observations ( )ii tfy = , N,...,,i 21= .

For monotonous increasing functions it must:

01 + + ByA N (18)

and for monotonously decreasing functions:

01 + + ByA N (19)

Because of the estimation error dR of the linear correlation coefficient 1+N,t,yR , by logarithm and

derivation of relation (17), one obtains the calculation formula of the relative error of evaluation of the

value 1+Ny :

( ) ( )22

22

2

2

1

1

2

2

ACR

ACRd

ABDR

ABDRd

y

dy

N,t,y

N,t,y

N,t,y

N,t,y

N

N

+

+=

+

+

(20)

Making the implied calculation, the result is:

( ) ( )( )

dRRABDR

ABDCBEARCED

ACR

C

ABDR

D

y

dy

Nty

Nty

Nty

NtyNtyN

N

+

++

+

=

+

+

,,2

,,

222

,,

2

22

,,2

,,1

1

2

24

22

(21)


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Starting from the fact that, in the case of a linear dependency ( )tfy = we have 1N,t,yR , and for a

non-linear monotonous function 0N,t,yR , in a first approximation, for the numerical evaluation of

relative error (19) the value:

1 N,t,yN,t,y RRdR (22)

can be admitted.

Numerical Verifications

In order to check and sustain the above assertions, Table 1 presents the results obtained in the case of

extrapolation by means of the linear correlation coefficient for several monotonous increasing

functions, strongly non-linear.

Table 1

N,...,,i;ti 21= 1+Nt ( )1+Ntf 1+Ny ( )

( )1

11

+

++

N

NN

tf

tfy

1

1

+

+

N

N

y

dy

( ) 2ttf =

1, 2, , 10 11 121 120.125 -0.723 % -0.960 %

1, 2, , 100 101 10201 10196.473 -0.044 % -0.060 %

1, 2, , 1000 1001 1002001 1001958.982 -0.004 % -0.005 %

( )3

ttf =

1, 2, , 10 11 1331 1317.899 -0.984 % -1.413 %

1, 2, , 100 101 1030301 1029826.664 -0.046 % -0.078 %

1, 2, , 1000 1001 1003003001 1002960781.302 -0.004 % -0.007 %

( ) ( )texptf =

1, 2, , 10 11 59874.141 50178.334 -16.193 % -21.847 %

1, 2, , 100 101 7.3071043

7.1541043

-2.092 % -20.267 %

1, 2, , 1000 1001 5.35510434

5.34310434

-0.214 % -20.026 %

( ) ( )tlntf =

1, 2, , 10 11 2.397895 2.445906 2.001 % 2.428 %

1, 2, , 100 101 4.615121 6.634768 0.425 % 0.343 %

1, 2, , 1000 1001 6.908755 6.914453 0.082 % 0.065 %


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Conclusions

From the assessment of these data it results that formulae (17) and (21) which helped to obtain

numerical values for 1+Ny and 11 ++ NN ydy respectively, lead to results acceptable from the point of

view of the errors assumed in the approaching short-term extrapolation and prognosis problems. The

benefits of the method presented in the paper is that it does not require knowledge / determination of

the analytical (specific) form of function ( )tf .

At least from this last perspective one may conclude that the extrapolation and prognosis method

based on the linear correlation coefficient can be used, at a more or less reliable confidence factor,

with the programs for computer automatically processing and interpretation of experimental data or of

those taken from various computing tables.

References

[1] Rafiroiu, M., Simulation models in constructing (in Romanian), Editura Facla, Timioara, 1982[2] Constantinescu, I., Experimental data analysis using numerical computers (in Romanian), EdituraTehnic Bucureti, 1980

Documents

C. Doca, L. Doca - Extrapolation of Monotonous Functions Using the Linear Correlation Coefficient