A STUDY ON MATHEMATICAL AND STATISTICAL ASPECTS OF LINEAR MODELS

Department of Mathematics,Koneru Lakshmaiah Education Foundation,Vaddeswaram,AP,India Department of Mathematics,Audisankara College of Engineering & Technology (Autonomous) Gudur, SPSR Nellore(Dt), A.P., India Department of Mathematics, Koneru Lakshmaiah Education Foundation, Aziznagar, Hyderabad Department of Mathematics, ANURAG Engineering College, Ananthagiri (v), Kodad,Suryapet,Telangana-508 206f Department of Mathematics, SriHarsha Institute of PG Studies, SPSR Nellore(Dt),A.P., India.


1.Introduction
Model refers to a set of functional or structural relationships between two or more characteristics. These characteristics may be either measuremental or non measuremental in nature. The measuremental characteristics which assume different values in a specified range are known as variables. Generally, a set of functional relationships between two or more variables may be expressed in terms of mathematical equations, which is called a mathematical model. This model may be either in the form of a set of linear equations (linear model) or in the form of a set of nonlinear equations (nonlinear model). By introducing a random error variable or a random disturbance term, the mathematical model becomes a statistical model or a regression model. Hence one may have either linear regression model or nonlinear regression model. Regression analysis is a statistical method to establish the relationship between variables. Regression analysis has a wide number of applications in almost all fields of science, including Engineering, Physical and Chemical Sciences; Economics, Management, Social, Life and Biological Sciences. In fact, regression analysis may be the most frequently used statistical technique in practice.
Suppose that there exists a linear relationship between a dependent variable Y and an independent variable X. In the scatter diagram, if the points cluster around a straight line then the mathematical form of the linear model may be specified as where 0  is the intercept and 1  is the slope.
Generally the data points in the scatter diagram do not fall exactly on a straight line, so equation (

MATHEMATICAL ASPECTS OF LINEAR MODEL
The main purpose of mathematical modelling is to solve real practical problems. The success of mathematical modelling depends on getting things right from the start, and as in most other scientific endeavours, one is more likely to succeed if one adopts a methodical approach. In practice, it is found to complete the following steps.
(i) Clarify the problem; (ii) List the factors; (iii) List the assumptions; and (iv) Formulate a precise problem statement An essential part of the mathematical modelling technique is to translate verbal statements about variables along with assumptions into precise mathematical relationships between the variables represented by symbols. Thus, the mathematical statements become amenable to manipulation by mathematical techniques.
For instance, the simplest model is obtained by assuming that Y is proportional to X. The corresponding mathematical statement is then Y  X or as a mathematical equation 0 YX   , where β0 is the constant of proportionality. Now, the graph of Y against X shows a straight line through the origin.

Research Article
Another simplest model is the linear form 01 YX   in which Y increases by β1 units for every unit increase in X and that 0 Y   when X = 0. This also includes the situation where Y decreases as X increases. In that situation the parameter β1 is negative. Consider the situation where 'Y decreases as X increases' by inverse proportion 0 1 Y or Y XX   ………………(10) It reveals that Y decreases more steeply with X that is the situation in the linear model. One may test the validity of this assumption by examining whether XY remains nearly constant. Another way is that if the plot of Ln Y against Ln X is a straight line of slope "-1".
Thus, under mathematical modelling technique, first represent the variables by the mathematical symbols and then make the assumptions about the relationships among the variables. Further, translate the assumptions into mathematical equations or inequalities. One of the main uses of mathematical modelling is to predict the future development of the system. Such model relies on assuming that the rate of change of a variable Y is linked to or caused by some or all of the present value of Y, previous values of Y, values of other variables, the rate of change of other variables and time 't'.
Here, the mathematical model describes how Y itself varies with time 't'. There are mainly two types of such mathematical models namely, (i) Discrete Models and (ii) Continuous Models DISCRETE MODELS: For discrete models, one may write the form (ii)  (12) where M is the co-efficient matrix; n1 Y  and n Y are the vectors.
The solution can be written as no

CONTINUOUS MODELS
A variable, which is allowed to take any value within a range, is known as a continuous variable. One advantage of using continuous variables is that one may use powerful mathematical tools such as Calculus.
The linear models are the simplest continuous models. The simplest linear model relating two variables is characterized by mathematical equation of the form Y = β0 + β1X, by having a straight line graph.
Under linear interpolation, if x1 and x2 are consecutive values of X and x is some value between them, then the graph of f(x) may be approximating from X = x1 to X = x2 by a linear model.

TYPES OF LINEAR MATHEMATICAL MODEL Linear Models with Several Independent Variables:
If the value of a dependent variable Y depends on the values of other variables X1,X2,…..,Xk then a way of expressing the dependence through a linear model is of the form Here Y changes by equal amounts for equal changes in any one of the independent variables. This model can be considered the generalization of the simple two-variable linear model Simultaneous Linear Models: Sometimes there may be two or more dependent variables, all of which are modelled as linear functions of independent variables. Here, some dependent variables can be considered independent variables in some linear functions (or equations) of the simultaneous linear equations system. This system of simultaneous linear models can be solved by using matrix methods such as Cramer's Rule, Inverse Matrix method etc.

Piecewise Linear Models:
It is a model that does not have to be represented by the same single formula for all values of the independent variable X. Here, two different linear expressions agree at some value of X, so there is no sudden jump ( discontinuity) at the changeover point, usually X may be a discrete variable in this model, sometimes, one may model a non-linear function approximately by a piecewise linear function.

Transformed Linear Models:
When a dependent variable Y does not change by equal amounts for equal changes in the independent variable then a linear model may not be suitable for this situation. For instance, the quadratic function or a second degree parabola is a simple nonlinear model.     which is also the X value at which the graph has global maximum or minimum value.
The value of the parameter 0  affects the vertical position of the curve relative to the coordinate axes.
A more general higher degree polynomial model can be written as 2 3

4.FITTING OF TRANSFORMED LINEAR MODELS
In fitting of time series models or Growth curves to the time series data, the following points may be useful to specify the type of the model: (i) When the time series Yt is formed to be increasing or decreasing by equal absolute amounts, the straight line times series model is used. i.e., If ∆ is the difference operator given by , h being the interval of differencing and k t Y  is the k th difference of Ytthen for a polynomial Yt of n th degree in t, the theorem states that k t Y  constant, k = n = 0, k > n The following tests based on the calculus of finite differences may be applied in choosing approximations about the type of curve to be fitted:   (

v) Assumption of Non-Stochastic Data Matrix:
The Data matrix X is a non-stochastic matrix.
In other words, X is a fixed known coefficients matrix.

(vi) Assumption of Non-Measuremental Errors:
There are no errors involved in the explanatory variables. In other words, all the independent variables X's are measured without error. Further, X is uncorrelated with  (vii) Normality Assumption: where is the least squares estimator of  By the least squares estimation method,  minimizes the residual sum of squares ee  .
First order condition :

Conclusion and Future Research
In the above talk mathematical aspects of linear models have been extensively depicted. Different types of mathematical models are discussed here and the methods of fitting transformed models are proposed. Furthermore specific form of linear statistical model is presented and the crucial assumptions of general linear model are extensively discussed. At the last stage of this article the method of ordinary least squares estimation of parameters of a linear model has been proposed. In the context of future research one may discuss Gauss-Markoff theorem for linear estimation andmean vector and covariance matrix of blue.