MODEFICATION OF THE CENTRAL PREDICTION THEORY

HISTORICAL BACKGROUND

The term “ regression” was named by Francis Galton in nineteenth century to describe a biological phenomenon. The phenomenon was that the hights of descendants of tall ancestors tend to regress down towards a normal average(a phenomenon also known as regression towards the mean).

For Galton regression had only this biological meaning, but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context. In the work of Yule and Pearson the joint distribution of the response and explanatory variables is assumed to be Gaussian (i.e the earliest form of regression was the method of least squares which was published by Legendre in 1805 and by gauss in 1809). This assumption was weakened by R.F fisher in his works of 1922 and 1925. Fisher assumed that the conditional distribution of the response variable is Gaussian , but the joint distribution need not be in respect , fisher’s assumption is closer to Gauss’s formulation of 1821. In the year 2007, I was able to formulate a concept call Equal Paring Concept in which we can determine the central point of the ordered pair ( ×, y) of a straight line equations. This concept was later translated by me in the year 2010 as Gravitational (Central) Point Values Theory of a straight line equation.

This theorem states that at central point in the ordered pair ( × ,y),  ×  is adding itself to infinity when  y  is subtracting itself to infinity and vice-versal.

This concept was later applied in statistical form of analysis which I named Central Prediction (Regression) Theory in which we can predict the mean value of one or more variables if the mean value of only one variable is known . This concept is total different from the Galton’s or Gaussian's concept and proved useful and accurate than theirs. Below is the continuous development of the Central Regression Theory. 

PREDICTING TWO VARIABLES FROM ONE VARIABLE

 For apparent relationship between three variables, the central pair equations are given as;

Y_m=K₁X_m +C……….(1)

Z_m=K₁X_m+K₂Y_m +C…..(2)

Where;

K₁=ΣY/2ΣX

K₂=[ΣXΣZ-ΣXΣX]/ΣxΣy

C=ΣY/2N

The table below contained the apparent relationship between three variables (i.e inflation rate, market price and profit earned ) from the year 2000 to 2009 for certain business corporation in million of cedis.

                                

Year	inflation rate(×%)	market price(z-cedis)	profit earned(y-cedis)
2000	24	80	35
2001	28	85	39
200	33	88	41
2003	31	90	29
2004	37	95	40
2005	30	92	28
2006	25	82	20
2007	23	75	19
2008	28	78	24
2009	31	85	30

From the table above we know mean inflation rate of the existing data from 2000-2009 to be 29%. Empirical speaking, the inflation rate to continuous years from 2009 may vary from year to year and the mean –inflation rate may change to continuous years from 2009 and the mean-market price and mean-profit earned may change with changing mean-inflation rates to continuous years from 2009.

Example.

If the mean- inflation rate is 26% for the year 2000 to 2012; find the best estimate of the corporation market price and profit to be earned .

Solution

 Y_m=K₁X_m +C

K₁=0.526

C=15.25

Ym=0.526(26) + 15.25

Y_m=28.93

Hence, the men-profit earned is 28.93million cedis.

Also;

Z_m=K₁X_m +K₂Y_m +C

K₂=1.79

Z_m=0.53(26) + 1.79(28.93) +15.25

Z_m=80.62

Hence, the mean-market price is 80.62million cedis.





REFERENCE
Galton, Francis.(1886). 'Regression Towards Mediocrity in Hereditary Stature'. Volume 15.