HISTORICAL BACKGROUND
The term “ regression” was named by Francis Galton in nineteenth
century to describe a biological phenomenon. The phenomenon was that the heights
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 called Equal Paring
Concept in which we can determine the central point of the ordered pair ( x, 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 ( x ,y): x 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 prediction Theory.
THEOREM:
Adongo’s Trapezium Theorem:
Ma= X
Then the Central Prediction model (or Adongo’s Trapezium Theorem) approximation line for these variable points has the model
x2 = a0+a1x1
EXAMPLE:
SOLUTION
TWO VARIABLES DATA
Suppose n averages variable point x1, x2, x3, …., xn,
xn+1 are given, where at least two or more of the x1, x2, x3,
….., xn, xn+1
are distinct. Putting
Ma= X
a=M-1X
a=( a0, a1, a2,
a3,…, an)
X=( x1, x2, x3,
…., xn, xn+1)
Where the
Trapezium Matrix M is;
1 x1
1 x1 x2
1 x1 x2 x3
………………….........
1 x1 x2 x3
x4, .., xn
|
Then the Central Prediction model (or Adongo’s Trapezium Theorem) approximation line for these variable points has the model
x2 = a0+a1x1
x3 = a0+a1x1+a2x2
x4 = a0+ a1x1+
a2x2+ a3x3
…………………………………………....
Xn+1= a0+ a1x1+
a2x2+ a3x3+…….+ anxn
EXAMPLE:
The
following table shows the yield x1
of wheat in bushels per acre, the number of days x2 of sunshine, the number of inches x3 of rain, and the number
of pounds x4 of
fertilizer applied per acre. Find the
resulting equation.
X1
|
X2
|
X3
|
X4
|
28
30
21
23
23
|
50
40
35
40
30
|
18
20
14
12
16
|
10
16
10
12
14
|
x1=125/5=25
|
x2=195/5=29
|
x3=80/5=16
|
x4=62/5=12.4
|
SOLUTION
The
resulting equation Ma=X, is given
as;
M
|
a
|
=
|
X
|
1 25
1 25 29
1 25 29 16
1 25 29 16 12.4
|
a0
a1
a2
a3
|
25
29
16
12.4
|
TWO VARIABLES DATA
The
gravitational(=central) point equation can also used to predict mean value (central
values) of two data pair (xp,yp).
This
equation is developed mainly to enhance the best accuracy for predicting the
mean value (Central values) of two data pair. For example, we can predict the value of yp
when the value xp is known.
The equation
constitutes the variables yp, xp, c and k which I named
as predictive value, suggestive value, constant and coefficient of a suggestive
value respectively. The developed formula for the equation is given as;
yp = Kxp +C
Where;
K = Σy/2Σx
C = Σy/2n
PROOF AND ANALYSIS OF THE FORMULAS:
For an
apparent relationship between x and y values from a sample or population data,
the total sum of x is apparently related to the total sum of y (i.e Σy is
relative to Σx).
This implies that
Σy = kΣx +
nc
But the relationship between k and C are given as;
Nc = Σy -
kΣx ---(1)
Or
kΣx = Σy –
nc --- (2)
Adding equation (1) and (2) together, we have
nc + kΣx =
(Σy - kΣx) + (Σy – nc)
Arranging equal pairs together, we have:
nc – (Σy -
kΣx) = (Σy – nc) - kΣx
By principle of equal pairs, we have:
Nc = Σy - nc
2nc = Σy
C = Σy/2n
---- (3)
And:
(Σy - kΣx) =
kΣx
Σy - kΣx =
kΣx
K2Σx = Σy
K = Σy/2Σx
--- (4)
EXAMPLE
The table
below contained the apparent relationship between students high school average
(HAS) and grade point average (GPA) after the freshman year of college
HAS(X) GPA(Y)
80 2.4
85 2.8
88 3.3
90 3.1
95 3.7
92 3.0
82 2.5
75 2.3
78 2.8
85 3.1
(a) If the students have average HAS of
85, find the best estimate of their average GPA
(b) If they students have average HAS of
92, find the best estimate their average GPA
SOLUTION
HAS(x) GPA(y)
80 2,4
85 2.8
88 3.3
90 3.1
95 3.7
92 3.0
82 2.5
75 2.3
78 2.8
85 3.1
ΣX= 850 ΣY= 29
K = Σy/2Σx
K = 29/2(850)
K = 0.0171
And:
C = Σy/2n
C = 29/2(10)
C = 1. 45
Hence, the predictive equation is given as:
yp = 0.0171xp + 1.45
(a) yp = 0.0171 (85) + 1.45
yp = 2.90
(b) yp = 0.0171 (92) + 1.45
yp = 3.0
THREE VARIABLES DATA
For apparent relationship
between three variables, the central pair equations are given as;
Ym=K1xm +C……….(1)
Zm=K1xm+K2ym
+C…..(2)
Where;
K1=Σy/2Σx
K2=[ΣxΣz-ΣxΣy]/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(y-cedis)
|
profit
earned(z-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 29%
for the year 2000 to 2012; find the best estimate of the corporation market
price and profit to be earned .
Solution
Ym=K1xm
+C
K1=1.465517
C=42.5
Ym=1.465517(29)
+ 42.5
Ym=85.0
Hence, the men-profit earned is 85 million cedis.
Also;
Zm=K1xm
+K2ym +C
K2= -0.6411765
Zm=1.465517(29)
- 0.6411765(85) +42.5
Zm=30.5
Hence, the mean-market price is 30.5 million
cedis.
FOUR VARIABLE DATA
As
you may be aware: Central Prediction
(Regression) Model is a statistical model that is used for predicting a set
of mean-measurements x2m, x3m,
x4m, …, xnm called the mean-response variables from
mean-explanatory variable x1m.
For the relationship between the
variables x1, x2,
x3 and x4;
the central prediction (regression) equations are related as:
X2m=
k1x1m + c ---- (1)
X3m
= K1x1m + k2x2m + c----- (2)
X4m
= k1x1m + k2x2m + k3x3
+ C --- (3)
Where;
K1
= Σx2/2Σx1
K2
= [Σx1Σx3-Σx1Σx2] /Σx1Σx2
C=Σx2/2n
K3=
[Σx4/n-k1Σx1/n-k2Σx2/n-c]/Σx3/n
Most females of various ages are
mostly affected by emotional fidelity when disappointment come to their
relationship and may be suffered by neurotic disorder (i.e some have
unreasonable fear of being loyal to their new partners due to disappointment
from their early relationships; some have unreasonable fear of being in
relation again; some experience societal- withdrawal due to fidelity – anxiety
etc)
The table below shows the
relationship between ages of females, percentage in their relationship and
percentage of those who are
affected by neurotic disorder in one particular town in the year 2012.
Age (x1)
|
Females in relationship (%) (×2)
|
Those who are disappointed (%) (x3)
|
Those who are affected by neurotic
disorder (x4)
|
13
|
45
|
20
|
19
|
14
|
49
|
28
|
22
|
15
|
52
|
30
|
33
|
16
|
55
|
40
|
41
|
17
|
56
|
60
|
62
|
18
|
57
|
75
|
73
|
19
|
57
|
80
|
79
|
20
|
69
|
81
|
80
|
21
|
70
|
88
|
88
|
22
|
80
|
90
|
89
|
23
|
85
|
92
|
90
|
24
|
90
|
80
|
85
|
25
|
85
|
72
|
70
|
26
|
65
|
60
|
61
|
27
|
56
|
53
|
50
|
28
|
42
|
40
|
39
|
29
|
39
|
38
|
36
|
Use the table above to estimate the
average percentage of females who are in relationship, average percentage of
females who are disappointed, and average percentage of females who are
affected by neurotic disorder at an average age 21.
Solution
X2m
= k1mX1m + c
X2m
= 1.47x21 + 30.94 = 61.81
Hence, percentage female who are in
relationship in the year 2012 is 62%
X3m=
k1x1m +k2x2m + c
X3m
=1447x21- (-0.004) x 61.81 + 30.94 = 60.33
Hence, percentage of female who are
disappointed in their relationship is 60%.
X4m
= k1x1m + k2x2m + k3x3m
+c
X4m
= k1x1m + k2x2m +k3x3m
+c
X4m
= 1.47x21-0.0024x63.28 -0.00836x60.33+ 30.94
X4m=59.82
Hence, percentage of females who are
suffering neurotic disorder is 60%.
APPLICATION IN ECONOMICS
This paper presents a study of central prediction theory in economics. The study is based upon using the costs relationships and sales
relationships when one variable(output
or price) may assumed to influence
several total costs and several sales respectively.
This paper presents two main topic.
That are the central cost-volume relationships and the central price-volume
relationships. These two main topics are to be briefly explained fundamentally.
This is a technique I have developed
for the estimation of mean total costs c1m, c2m ,.., cj,
…, cr provided the mean output q0m of mean cost c1m is known. The
model is:
C1m=α+β0qom
C2m=α+β0q0m+β1c1m
………………………………………………………
Cjm=α+β0q0m+β1c1m+…+βj-1c(j-1)m+…+βr-1c(r-1)m
………………………………………………………
Crm=α+β0q0m+β1c1m+…….+βjcjm+………+βrcrm
The table below shows the cost-volume relationships of multi-production firm of tatal cots 1, 2,and3.
Output of c1 (q0)
5000
1000
2000
3000
5000
|
Total cost (c1)
115
151
205
212
328
|
Total cost (c2)
105
150
190
205
300
|
Total cost (c3)
151
150
204
208
308
|
We can estimate c1m, c2m, and c3m if q0m is given. To estimate, we have:
β0=0.0316
β1=- 0.0604
β2=0.0747
α =101.1
NOTE: the parameters α ,and β0 , and β1, β2 are denoted as central fixed cost, central-coefficients of mean-output q0m ,and total costs c2m, c3m espectively.
NOTE: the parameters α ,and β0 , and β1, β2 are denoted as central fixed cost, central-coefficients of mean-output q0m ,and total costs c2m, c3m espectively.
Hence, the model is given as:
C1m=101.1+0.0316q0m
C2m=101.1+0.0316q0m-0.0604c1m
C3m=101.1+0.0316q0m-0.0604c1m+0.0747c2m
This is telling us that if the mean output q0m of mean-total cost c1m is 3200, then:
C1m=202.2
C2m=190.0
C3m=204.2
1.2 Central Pricing Model
This is also a technique I have developed for the estimation of the mean-sales s1m, s2m, …, sjm, …, srm ,provided the price p0m of mean sales s1m is known. The model is:
S1m=α - β0p0m
S2m
=α - β0p0m - β1s1m
……………………………………………………………
Sjm=
α - β0p0m - β1s1m- … - β(j-1)s(j-1)m
- … - β(r-1)s(r-1)m
……………………………………………………………
Srm=
α - β0p0m - β1s1m-……..-βjsjm-……..-βrsrm
In this model, the elasticity of s1m is estimated as:
E=β0p0m/s1m
EIGHT VARIBLE DATA
Base rate as we know, is the minimum
rate of interest that a bank is allowed to charge from its customers. It is a
rate which the bank lends to the discount houses, which effectively controls
the interest rates charged throughout the banking system. A rule stipulates
that no bank can offer loans at a rate lower than the base to any of its
customers.
The question is: Is it possible to
predict the base rates of other banks if the base rate of one bank is known?
The answer depends mainly one factor which was discussed from my previous diary
(or weblog) title: “Adongo’s Central Prediction theory”. In the general regression
theory we requires 5 or more observed values to fit the regression model whereas,
in the general central prediction theory, we need one or more observed values
to fit the central prediction model.
Let us consider some Bank base rate
for 25th November, 2013 below.
Bank
|
Bank
|
Bank
|
Bank
|
Bank
|
Bank
|
Bank
|
Bank
|
|
GCB
Bank
|
HFC
Bank
|
UniBank
|
Bank of
Africa
|
Agric. Dev.Bank
|
Standard
Charted
Bank
|
GT Bank
|
Stanbic
Bank
|
|
Base Rate
|
22.27%
|
21.30%
|
24.28%
|
25.57%
|
23.91%
|
16.66%
|
18.43%
|
16.87%
|
We can use the central prediction as
a fitted model for the datum above. The model is given as:
X2=0.48x1+10.65
X3=0.48x1+0.13x2+10.65
X4=0.48x1+0.13x2+0.06x3+10.65
X5=0.48x1+0.13x2+0.06x3-0.065x4+10.65
X6=0.48x1+0.13x2+0.06x3-0.065x4-0.3x5+10.65
X7=0.48x1+0.13x2+0.06x3-0.065x4-0.3x5+10.65
X8=0.48x1+0.13x2+0.06x3-0.065x4-0.3x5+0.077x6+10.65
To determine validity of the model
let us assumed that the base rate of GCB Bank is 22.27% and the base rates for
the rest of the banks are not known. Substituting 22.27% into the model, we
have
HFC Bank (x2)=21.34%
UniBank (x3)=24.11%
Bank of Africa(x4)=25.56%
Agric.Dev.Bank (x5)=23.90%
Standard Charted Bank(x6)=16.66%
GT Bank (x7)=18.02%
Stanbic Bank (x8)=16.88%
Comparing the estimated values to
the actual values from the table above, it indicates that the model is perfect.
