MY 2012/2013 ACADEMIC YEAR PROJECT

  "Please, forgive me if I may be offensive to you; I have to deal matters pragmatically. I am not ready to take someone project and present. I want to use my own project as the law demanded. Procrastination is dangerous"-(said by me before, presentation day of this project)


CERTIFICATION

I hereby declare that this project is my own work  presented to the Department of Statistics in partial fulfillment of the requirement for the award of BSC. degree in actuarial science.

Name                                  Student Id              Signature                             Date

Adongo, Bill      FAS/0912/06                               ………..                             ..........

          

 

Head of Department (Statistics)                        Signature                            Date

Dr. Albert Luguterah                                             …………...                     ……….


External Examiner...........................                        .................                       ............

DECLARATION                                                                                

I declare that this project was carried out by me, Bill Adongo, in the Department of Statistics, Faculty of Mathematical Sciences, University for Development Studies, independently and that no previous submission for a degree of this University or elsewhere has been made. Related work by others, which serve as source of knowledge has been duly or referenced.

ABSTRACT
Topic: The Central Point Theory and Its Effects on Predictive Theory

This work is an investigation of the central point theory which I am developing and its effects in predictive theory. Through the knowledge of central point theory, new predictive theory is devised to substitute the existing general regression theory.

This predictive theory is named as Central Prediction Theory.The work focus on using the central prediction theory to predict several mean response variables from only one mean explanatory variable that provided no room for random error(except complicate cases). This model includes the mean component only and can be used to hypothesize an exact relationship between phenomena that cannot be modeled or explained when using the existing "general regression model". The review of linear equation, central point theory and regression theory in two or more variables are discussed fundamentally. The work precisely explained the validity of the central prediction theory and its important as compared to the existing general regression theory. The conclusion is that no text-statistics is needed to evaluate the validity of the central prediction model, since it does not account for random error, (except complicated cases).

The study found that the central prediction theory can be used to construct models in econometrics, logistic prediction, survival prediction, time series prediction that is almost certainly have no less than one response variables and no variation due strictly to random phenomena that cannot be modeled or explained when using the existing general regression model.

Recommendations are made to better the development of the central prediction theory for academic research, learning reference and teaching.        

CHAPTER ONE     

1.0  Introduction

This project is to investigate the effect the central point theory of linear equations in two or more variable in predictive modeling. The review of linear equations in two or more variables will be discussed fundamentally. The review of Central Point Theory will also be discussed fundamentally.

The work focus on using the basic concept of the Central Point Theory of linear equations in two or more variables to create new concept in predictive theory. The study is to be found as whether the Central Point Theory has positive effect in predictive modeling and how useful it is as compared to the existing methods of analysis in regression.

1.1 Brief History of Me

The attainment of modern sciences, technology and economics would be impossible without an algebraic equations, theories and rules. Two statements and only two can be made about an equation; either it can be solved or else we can prove it insolvable.

At my early manhood I overturned the theories and methods of 1840’s-2006’s mathematics of linear algebraic in two or more variables and following my own revolutionize theories of algebra (linear algebra), opened the way to a mid 21^st–century mathematical analysis. Although I contributed significantly to pure mathematics, I also made practical application of importance for 21^st –century physics, economics, statistics or actuarial science, mathematical biology etc.

I am the original developing of the Multi-combinational Mathematics, Central Point Theory and Least Whole Normal mathematics. The achievement of me in pure mathematics, physics and statistics between the year 2007-2008 was to discover a unique point of a line (Or linear equation) in a plane as central point (Or point of gravity). This unique point of linear equation in a plane made it possible for me to discover the central (or Gravitation) point values interval theorem. This particular theorem helps us to understand the coordinate system very well as long as analytical geometry and astronomy is concerned. The central (gravitational) point theory also has great effect in predictive modeling and this is what we are to be considered in this project.

1.2 The Fundamental of the Central (Gravitational) Point Theory                                          (Reference: adongoayinewilliam13.blogspot.com)

1.2.1 It is characterized that at central (gravitational) point, the axial term of x is equal to the axial term of y in a line equation Ax +By = C a plane, the value of the axial coordinates x and y is calculated as;

x_g=C/2A

y_g= C/2B

1.2.2 It is characterized that at central (gravitational) point of a line equation

Ax +By=C in a plane; the slope of a line is given as;

S_g=-(y_g/x_g)
1.2.3 It is characterized that at central (gravitational) point of a line equation

Ax +By= C in a plane, the intercept of a line is given as;

(x_g; 0)Є (X_{g 0})= (2x_g, 0)

(0, x_g)Є (0, Y_g) = (0, 2y_g)

1.3 My Central (Gravitational) Point Values Interval Theorem

The theorem states that at central point x_g and y_gin the order pairs (x,y), x_g is adding itself to infinity when y_g is subtracting itself to infinity and vice-versa.

1.4 Research Objective:

Looking at the selected topic of this project, this research of the central point theory is examined:

1)   What is the meaning of central point theory?

(2)   What is the effect of the central point theory in predictive models?

(3)   Can new form of predictive models exist with the help of the central point theory?

(4)   Is the availability of the central point theory affect positively in predictive models?                                                                      

(5)   Can we name this form of predictive model as Central Prediction Model?

1.5 Problem Statement

Since actuaries, statisticians, economists, engineers, sociologists, and scientists are traditional gatherers for constructing predictive models for practical use; they find difficult to use a concise predictive model that can use the mean component of the dependent and the independent variables to hypothesize an exact relationship between random phenomena that cannot be modeled or explained when using the existing predictive model theme as a “regression models”. They also find difficult to relate several dependent variables with only one independent variable for predictive purpose.

The availability of the central predictive theory will truly help actuaries, statistician economists, engineers, sociologists and scientist to solve useful practical problem in real life situations.

1.6 Research Methodology

(1)   Application of the central point theory in predictive modeling

(2)   Using the central predictive model to analyze data.

(3)   Significant of the central predictive model

(4)   Comparing the central predictive model to the general regression model.

(5)   Recommendation and conclusion of the project

CHAPTER TWO

2.0 Literature Review

According to legend, the French mathematician and philosopher Rene Descartes in 1637 published the work "La Geometrie" in which he devised out the foundation for one of the most important invention of mathematics namely Cartesian – coordinate system in analytical geometry. Descartes based his system on a relationship between point in a plane and ordered pairs of real numbers.

In 1843, William Rowan Hamilton introduced quanternions that describes mechanics in three dimensional spaces which laid out one of the most important history of development of linear equations in two or more variables.

The subject of system of linear equation received a great deal of attention in nineteenth –century mathematics. However, problems of this type are very old in the history of mathematics. The solution of simultaneous systems of equations was well known in China and Japan.

The great Japanese mathematician Seki Shinsuku Kowa (1642-1708) wrote a book on the subject in 1083 that was well in advance of European work on the subject. Gauss who is one of the greatest mathematicians developed Gaussians elimination around 1800 and used it to solve least squares problem in celestial computation and later in computation to measure the earth and its surface. Even though Gauss name is credited with this method for successively eliminating variable from systems of linear equations, Chinese manuscripts from several centuries earlier have been found that explained how to solve a system of three equations in three unknown:

In the 1940’s, solution of large system of equations became a part of the new branch of mathematics called operational research. Operational research is concerned with deciding how to best design and operate man-machine system, usually under conditions requiring the allocation of scare resources.

The availability of linear equations in two or more variables open the way for linear regression and correlation analysis as long as applied statistics is concerned.

The earliest form of regression was the method of least squares which was published by Legendre in 1805 and by Gauss 1809. Legendre and Gauss both applied the method to the problem of determine, from astronomical observations, the orbits of bodies about the sun. Gauss published a further development of the theory of least squares in 1821, including a version of the Gauss- Markov theorem.

The term ‘regression’ was named by Francis Galton in the nineteenth century to describe a biological phenomenon. The phenomenon was that the heights of descendants of all ancestors tend to regress down towards a normal average. The phenomenon is also known as regression towards the mean. The term regression had only this biological meaning, but his work later extended by Udny Yule and Karl Pearson to a more general statistical contact. In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A Fisher in his works of 1922 and 1925. Fisher assumed that the conditional distribution of the response variable is closer to Gauss’s formulation of 1821.

Regression methods continue to be an area of active research. In recent decades, new method have been developed for robust regression, regression involving correlated responses such as time series and growth curves, regression in which the predictor or responses variable are curves, images, graphs, or other complex data objects, Bayesian methods for regression, regression in which the predictor variables are measured with error, regression with more predictor variable than observations and causal inference with regression.Even though these great mathematicians and statisticians have had contributed a lot to the development of linear equations in modern time, they failed to accomplish the true concept of the linear equation in two or more variable.From the time of Rene Descartes to present, all mathematicians hold the idea that we cannot locate a unique point in a linear equation Ax+ By =C as our central, since linear equation Ax +By =C has an infinite possible solutions or points and has no end: Unless a range of values are given that we can truly determine the end of line equation Ax+ By = C. They also hold the idea that we cannot find the possible solution of the, y- variable if the assumed values of x- variable is not given. Not until the year 2007 that I discovered that we can determine center of a line, even though a line has infinite points. I also discovered the Central Point Values Interval Theorem, which disproves the idea that we can only find the possible values of x if the values of y are given.

Through the knowledge of the central point theorem, I was able to invent a method that is used to assume the average relationship between the response variables x₂, x₃, --, x_j-- x_r and the single predictor x_1. This invention is named by me as Central Prediction Theory.

2.1 Fundamental Proof of My Central Point Theory (CPT)

Below are the presentations of the proofs of my Central Point Theory (CPT)

2.1.1 My First Proof:

At central point (x_g, y_g) in the ordered pair (x,y) of straight line equation Ax+ By = C, the formula for x_g and y_g are given as;

x_g= C/2A

y_g = C/2B

At central point (x_g,y_g),in the ordered pair (x,y) can be proved by equally pairing x to y.

i.eAx = C- By---(3)

By = C-Ax---- (4)

Equating Axin equation (1) C- Ax in equation (2), we have

x_g = C/2A

Also, equating By in equation (2) to C-By in equation (1), we have

y_g = C/2B

2.1.2 My Second Proof:

At central point, the slope of straight line equation Ax+ By =C is given as

m = - (y_g/x_g)

where x and y_g denoted as central point pairs and m denoted as slope.

The slope of the straight line can be determined by moving term Ax to right hand side and divide both side by the constant B.

y= - (A/B) x + C/B

But let m= A/B (slope) and b= C/B (intercept), hence we have

y= mx+ b --- (1)

Equally pairing m to y,we have

y=mx+b---- (2)

-mx = b- y --- (3)

Equating y in equation (2) to b-y in equation (3), we have

y= b ------ (4)

Also, equating –mx in equation (3), to mx+ b in equation (2), we have

m =-b/2x—(5)

Putting equating (4) into (6) we have

m = -y/x or m = -y_g /x_g

2.1.3 My Third Proof:

The third proof states that, at central point (x_g, y_g) in the ordered pair (x,y) of straight line equation Ax + By= C; x_gis adding itself to infinity when y_g is subtracting itself to infinity  and vice-versa.

(x_g, y_g) = [(x_g, x_g+xg, x_y+x_g+x_g...); (y_g, y_g-y_g, y_g-y_g-y_g,)]

or [(x_g, x_g-x_g, x_g-x_g-x_g,..), (y_g, y_g+y_g, y_g+y_g+y_g,..)]

(Reference: adongoayinewilliam13.blogspot.com)

CHAPTER THREE

3.0 Introduction

This chapter precisely explains the methodology used in chapter one of this project. Below, the central point theory is practically applied in predictive modeling that hypothesized the exact relationship between variables that have unexplained variation or explained variation. This predictive theory is called Central Prediction Theory.

3.1 Application of Central Point Theory

Based on the central point theory I am able to develop a theory called Central

PredictionTheory which is purposeful for constructing models that hypothesized an exact relationship between variables.

If we believe there will be unexplained variation in response variable perhaps caused by important but unincluded variables or by random phenomena we still use the central prediction model since this model does not account random error(except complicated cases). This probability model includes the mean component only and can be used to hypothesize an exact relationship between random phenomena that cannot be modeled or explained when using the existing

“General Regression Model”(except complicated cases).

3.1.1 Central Point-Line (Two Variable) exact Model

Given, x_2m = β₀ + β₁x_1m

Where x_2m= mean dependent or mean response variable

x_1m= mean independent or mean predictor variable

β₀ (beta zero) = constant = ∑x_2m/2n

β₁ (beta one) = predictive coefficient = ∑x_2m/2∑x_1m

The mean component point (-∑x/2n, ∑y_2m/4n) determines the line.

3.1Type of Statistical Data for Analyzing the Central Prediction Theory

The following table classifies the various data type that the Central Prediction

Theory can be used to analyze.

Data Type	Possible Values	Example Of Usage	Central Prediction Model
Binary	0,1 (arbitrary labels)	Binary outcome(“yes/no”, “true/false”, “success/failure”etc.)	Central Logistic Probit
Categorical	1,2,….,k(arbitrary labels)	Categorical outcome (special blood type, political party, etc.)	Central Multinomial Logic,             Central Multinomial Probit
Ordinal	Integer or Real Number(arbitrary scale)	Relative Score Significant Only For Creating Ranking	Central Ordinal Prediction(central ordinal logistic, central ordinal probit)
Binomial	0,1,..,r	Number Of Success Out Of r Possible	Central Binomial Prediction(logistic)
Count	Nonnegative Integer(0,1,…)	Number Of Items(people, births, deaths)	Central Poisson, Central Negative Binomial Prediction
Real-Value Additive	Real Number	Temperature, Relative Distance, Location Parameters	Standard Central Prediction
Real-Value Multiplicative	Positive Real Number	Price, Income, size(especially where rang over large scale)	Central Generalize Model With Logarithmic

3.2.1 Data Analysis One (Two variables)

In real life situation, suppose a fire insurance company wants to relate the amount of fire damage in major residential fires to the nearest fire station. The study is to be conducted in a large suburb of a major city; a sample of 14 recent fires in this suburb is selected. The amount of damagex_2mand the average between the fire and the nearest fire station x_1m are recorded for each fire station.

Table 1.0: Fire Damage Data

Distance from fire station x1 (miles)	Fire damage x2 (thousand of cedis )
3.4 1.8 4.6 2.3 3.1 5.5 0.7 3.0 2.6 4.3 2.1 1.1 6.1 4.8	26.2 17.8 31.3 23.1  27.5 36.0 14.1 22.3 19.6 31.3 24.0 17.3 43.2 36.4

We first, hypothesize a central prediction model to relate mean fire damage, x_2m, to mean distance from the nearest station x_1m. We hypothesize a central predictive probabilistic model:

x_2m = β₀+ β₁x_1m

Step 2

We estimate the predictive coefficient β₁ and the constant β_0. Hence, we have:

β₁ = ∑x₂/2∑x₁ = 370.1/2(45.4) = 4.00

β₀ = ∑x₂/2n= 370.1/2(14)== 13.22

and the central predictive equation is

x_2m = 13.22 + 4.08x_1m

Step 3

Use the fitted central prediction equation to predict the exact average amount of fires damage, if the exact average distance from fires stations is 3.2 (miles)

x_m = 13.22 + 40.08(3.2) = 26.28.

Hence, the exact average amount of fires damage is 26.28 thousand cedis if the exact distance from the fires stations are 3.2 miles.

3.2.2 Data Analysis Two (Two Variables)

The fertility rate of a country is defined as the number of children a woman citizen bears, on average, in her life time. Scientific American (December.1993) reported on the researchers found that family planning can have a great effect on fertility rate x₂, and contraceptive prevalence x₁(measured as the percentage of married woman who use contraceptives) for each of 27 developing countries.

Table 1.1 Contraceptive Prevalence and Fertility Rate

Country	Contraceptive Prevalence x1	Fertility Rate x2
Mauritius Thailand Colombia Costa Rica Sri Lanka Turkey Peru Mexico Jamaica Indonesia Tunisia El Salvador Morocco Zimbabwe Egypt Bangladesh Botswana Jordan Kenya Guatemala Cameroon Ghana Pakistan Senegal Sudan Yemen Nigeria	76 69 66 71 63 62 60 55 55 50 51 48 42 46 40 40 35 35 28 24 16 14 13                                        13 10 9 7	2.2 2.3 2.9 3.5 2.7 3.4 3.5 4.0 2.9 3.1 4.3 4.5 4.0 5.4 4.5 5.5 4.8 5.5 6.5 5.5 5.8 6.0 5.0 6.5 4.8 7.0 5.7

Step 1

We first, hypothesize a central prediction model relating average fertility rate, x_2m, to average contraceptive prevalence, x_2m’

x_2m =β₀+β₁x_1m

Step 2

We estimate the predictive coefficient β₁ and this content   β₀. Hence, we have

β₁= 0.058

β₀ = 2.34

and the central prediction equation is

x_2m = 2.34 + 0.058x_1m

Step 3

Using the fitted central prediction equations to estimate the exact average fertility rate of the developing countries in December 1994, if the exact average contraceptive prevalence for the developing countries in December 1994 is 41%

x_2m= 2.34 +0.058 (41) =4.72.

Hence the exact average fertility rate for the developing countries in December 1994 is 4.72%, if the exact average contraceptive prevalence for the developing countries in December 1994 is 41%.

3.3 The General Central Prediction Model

The general formula of the central prediction model is stated by relationship below.

x_2m=β₀+β₁x_1m

x_3m=β₀+β₁x_1m+β₂x_2m

x_4m=β₀+β₁x_1m+β₂x_2m+β₃x_3m

. ………………………………………………

x_jm= β₀+ β₁x_1m+β₂x_2m+β₃x_3m+…+β_(j-1)x_(j-1)m

………………………………………………………………

x_rm=β₀+β₁x_1m+β₂x_2m+β₃x_3m+…+β_(j-1)x_(j-1)m+…+β_(r-1)x_(r-1)m
Where, x_2m, x_3m, x_4m, …,x_jm,…,x_rm are dependent variables and x_1m is an independent variable.  The mean portion of the model β_(j-1) determines the contribution of the independent variable x_jm.

3.1.4 Data Analysis Three (Five Variables)

The central prediction analysis can be employed to investigate the determinants of survival size of nonprofit hospital. For a given sample of hospitals, survival size, x_5m  is defined as the largest size hospital (in terms of number of beds exhibition growth in market shore over a specific time interval. Suppose to states are randomly slated and the survival size for all nonprofit hospitals in each state is determined for two time periods five yours apart, gelding two observations per state. The 20 survival sizes are hosted, along with the following data for each state,for the second year in each time interval:

x₁= Percentage of beds that are for-profit hospitals

x₂= Ration of the number of persons enrolled in health Maintenance organizations (HMO) to the number of persons covered by hospital insurance

x₃₌state population (in thousands)

x₄=percent of state that is urban

x₅=survival size

Step 1

We first, hypothesize a central prediction model relating averages x_2m, x_3m, x_4m, x_5m, x_5m, to averagex_1m. Hence, the model is

x_2m=β₀+β₁x_1m

x_3m=β_o+β₁x_1m+β₂x_2m

x_4m=β₀+β₁x_1m+β₂x_2m+β₃x_3m

x_5m=β₀+β₁x_1m+β₂x_2m+β₃x_3m+β₄x_4m

Step 2

We estimate the predictive coefficient β₁, β_2, β_3, β₄ and the constant β₀.

Step3

Use the fitted central prediction equation to estimate the exact averages of x_2m, x_3m, x_4m and x_5m, if the exact average of x_1m is 0.12.

.

3.4Applicationofthe Central Prediction

3.4.1Introduction

The study of this project found that the central prediction theory can be applied in logistic prediction, survival prediction, time series prediction etc. Brief explanations of the topics mention above are discussed fundamentally.

3.4.2 Central Binary Prediction

Imaging Standard Charted Bank wants to determine which customers are most likely to repay their loan. Thus, they want  to record  a number  of  independent  variables  that describe  the  customer’s  reliability  and then determine whether these variables are related to the binary variables x_2m =1 if the customer repays the  loan and x_2m=0 if the customer fails to repay the loan. This simply telling us that when the mean response variable x_2m is binary, the distribution of x_2m reduces to a single value, the probability  p=pr(x_2m=1).

In this central logistic model, the natural logarithm of odds ratio is the mean explanatory variables by a central logistic model. Here, we are to consider the situation where we have a single mean independent variable, but this model can be generalized by relating the binary mean response variables x_2m,x_3m, .., x_jm, ,..,

x_rm to single mean predictor variable x_1m.Considered p(x_1m)=x_2m be the probability that x_2m equals 1 when the mean independent variable equalsx_1m.By modeling the log-odds ratio to a model in x_1m, a simple central logistic model is given as:

P(x_1m)=[℮^β0+β1X1m]/[1+℮^β0+β1X1m]

Assuming a doctor recorded the level of an enzyme, Creatinine Kinase (CK), for patients who he suspected of having a heart attack. The Objective of the study was to asses whether measuring the amount of CK on admission to the hospital was a useful diagnostic indicator of whether patients admitted with a diagnosis of a heart attack had really had a heart attack. The enzyme CK was measured in 360 patients on admission to the hospital. After a period of time a doctor review the records of these patients to decide which of the 360 patients had actually had a heart attack. The data are given in the table below with CK values given as the midpoint of the range of values in each of 13 classes of values.          

Table: 1.3 CK Level Of Patients

CK Values	Number Of Patients With Heart Attack	Number Of Patients Without Heart Attack
20	2	88
60	13	26
100	30	8
140	30	5
180	21	0
220	19	1
260	18	1
300	13	1
340	19	0
380	15	0
420	7	0
460	8	0
500	35	0

We use the formula to calculate the exact probability that a patient had a heart attack when the CK level in the patient was 160. And by calculation, we have:

β₀=230/2_*13=8.45

β₁=230/2_*3380=0.034

P(160)=[℮^{8.45+0.034*160}]/[1+℮^{8.45+0.034*160}]=1.

 Also, the actuary who concerns with the contingencies of death, retirements, sickness, withdrawals, marriage, etc. may want to know the mean (or exact) probabilities or mean (or exact) rates as a representative of individuals occurrence of such events in order to predict the exact future occurrence so as to calculate exact premiums and exact annuities for insurance and other financial operations without account of random errors.Taking into consideration, the mortality rates over certain range of ages can be fitted as Central Binary logistic prediction model to a given set of data so as to determine future exact estimates of the actual deaths d_xm, future exact crude rates q_xm, provided the exact exposed to risk E_xm, for each year of age is known.Algebraically, the central binomial logistic prediction model is given as:

q*_xm=1/[1+e^-(α+βxm)]

α=Σd/2n

β=Σd/2Σ(x)

d*_xm=E_xmq*_xm

d and (x) represent death and age respectively.

Example, mortality rates over 30-34 were estimated fitting the central binary logistic prediction model to the data below.

Table:1.4 Mortality

Ages(x)	Deaths (d)
30	335
31	391
32	428
33	436
34	458

If the mean expose to risk is 140000 we estimate:

1)      The parameters α and β.

2)      The exact crude rate of an insured of exact age 42

1)α=204.8  

   β=6.4

2) q*₄₂=1

CHAPTER FOUR

4.0 Significance of the Central Prediction Model

This chapter precisely explained the significance of the central prediction model. It precisely explained the validity of the central prediction model as compared to the general regression model.

4.1Central Prediction Model Assumptions

(1)      For any given set of values x_1, x_2, x_3,---, x_j, ---, x_r, the mean error (ME)

equal to zero,(except complicated cases).

(2)      The variance or variability of the mean error (ME) is equal to zero,(except

complicated case.

(3)      It is nonrandom model,(except complicated cases).

We already know that if there is unexplained variation in the independent variables perhaps caused by important but unincluded variables or by random phenomena, we ought to use model that accounts for this random error. One of this model is the “general regression model”. But, it is possible to use the central prediction model? Since this model contains only the mean component and does not account for random error (except complicated cases). Yes! The central prediction model is a substitute of regression model with special validity than the regression model. Under central prediction model, the mean responses must fall exactly on the line because the model leaves no room for random phenomena that cannot be modeled or explained when using the regression model, (except complicated cases):

CHAPTER FIVE

5.0Conclusion.

In this project, the central point theory which I am developing is applied in prediction theory. Through the knowledge of the central point theory, I devised a predictive theory called Central Prediction theory which can be used in logistic models, survival models,applied econometrics,time series model etc.

The work also precisely explained the validity of the central prediction theory and how important it is as compared to the general regression theory.

5.1Recommendation

This work precisely explained the important of the Central Point Theory and Central Prediction Theory which I am developing. This work revolutionized the existing linear equation in two or more variables and the general regression theory.’

Recommendations are made to better the development of the central prediction theory for academy research, learning and teaching.

Reference
.   Adongo, Bill(Me), Transcript(2008), Posted(EMS-Bolga Branch) to Mathematical . .  Association of Ghana(Tittle: New Mathematical Concept....) in the Year 2008
.   Rene Descartes.(1637). "The Geometry".
.   Galton, Francis.(1886). "Regression Towards Mediocrity in Hereditary Stature'. Volume 15.

·         John W. Brown and Donald R. Sherbert: Introduction Linear Algebra with

Application

·         Sheldon P. Gordon and Florence S. Gordon: Contemporary Statistics( A

Computer Approach)

·         Ioomfield, P. (1976) Fourier Analysis of Time Series. An Introduction.

·         Christensen, Ronald (1997). Log-Linear model and Logistics (Second

Edition)

·         Dand Collett, Modelling Survival Data In Medical Research, Second

Edition