Forecasting Parameter Estimates: A Modeling Approach Using Exponential and Linear Regression

This paper supplies a calculation method for the parameter estimates of an exponential equation through SAS algorithm. The aim of this paper is to investigate the efficiency of the gained parameter estimates through the forecasting performance. The proposed calculation method can provide a very useful technique to develop an exponential equation with better accuracy performance. This research paper illustrates a sample of the data obtained from the established study, which characterize the proliferative capacity of mesenchymal stem cells. This paper also provides the specific algorithm for the parameter estimates.


INTRODUCTION
Regression analysis is a statistical methodology that uses the relationship between two or more quantitative variables in a way that one variable can be predicted from the other, or others.This methodology is widely used in business, social, behavioral and biological sciences, including agriculture and fishery research [1].For example, fish weight at harvest can be predicted by utilizing the relationship between fish weights and other growth affecting factors like water temperature, dissolved oxygen, and free carbon dioxide.There are other situations in a fishery where relationships among variables can be exploited through regression analysis [1].Regression analysis serves three major purposes: (1) description, (2) control and (3) prediction.We frequently use equations to summarize or describe data.Regression analysis is helpful in developing such equations.For example, we may collect a considerable amount of fish growth data and a data on a number of biotic and abiotic factors and a regression model would probably be a much more convenient and useful summary of those data than a table or a graph.Besides prediction, regression models may be used for control purposes.A cause and effect relationship may not be necessary if the equation is to be used only for prediction [2].A functional relationship between two variables is expressed by a mathematical formula.If x denotes the independent variable and y the dependent variables, a functional relationship is of the form

 
x f y  .Given a particular value of x, the function indicates the corresponding value of y.A statistical relation, unlike a function, is not a perfect one.In general, the observations for a statistical relation do not fall directly on the relationship's curve.Depending on the nature of the relationship between x and y, regression approach may be classified into two categories, linear regression and nonlinear regression models.The models that are linear in these parameters are known as linear models, whereas in nonlinear models parameters show nonlinearity.Linear models are generally satisfactory approximations for most regression applications.There are occasions, however, when an empirically indicated or a theoretically justified nonlinear model is more appropriate [3].

A. Linear Regression
Linear regression is used to study the linear relationship between a dependent variable Y and one or more independent variables X.The dependent variable Y must be continuous, while the independent variables may be either continuous, binary, or categorical.The initial judgment of a possible relationship between two continuous variables should always be made on the basis of a scatter plot (scatter graph).This type of plot will show whether the relationship is linear or nonlinear.Performing a linear regression makes sense only if the relationship is linear.Other methods must be employed to study nonlinear relationships [4].A model with more than predictor variables is a straightforward one.The model can be stated as follows: where i y is the value of the response variable in the i th trial, β 0 and β 1 are parameters x i is a known constant, namely the i th value of the predictor variable and ε i is a random error term with mean zero and variance σ 2 and their covariance is zero [5].

B. History of the Exponential Function
The exponential is one of the most significant and widely occurring functions.In biology, it may depict the growth of bacteria or animal populations, the reduction of the number of bacteria in response to a sterilization process, the development of a tumor or the absorption or elimination of a drug.Exponential growth cannot go on forever because of limitations of nutrients, etc. Knowledge of the exponential function makes it more comfortable to understand birth and death rates, even when they are not perpetual.In physics, the exponential function describes the disintegration of radioactive nuclei, the emission of light by atoms, the assimilation of light as it passes through matter, the change of voltage or current in some electrical circuits, the variance of temperature with time as a warm object cools, and the rate of some chemical reactions [1].Although the exponential distribution provides a simple, elegant and closed form solution to many problems, it does not offer a reasonable parametric fit for some practical applications where the underlying failure rates are nonconstant, presenting monotone shapes.Recently, in the procedure of overcoming such problems, new categories of examples were introduced based on adjustments of the exponential distribution.Authors in [6] offered a generalized exponential distribution, which can hold data with increasing and decreasing failure rate function.Authors in [7] ushered in the exponential geometric distribution with decreasing failure rate, authors in [8] proposed a twoparameter distribution known as exponential-Poisson distribution, which takes in a decreasing failure rate and authors in [3] proposed another modification of the exponential distribution with decreasing failure rate function This model is inferred in a complementary risk scenario [9] where the lifetime associated with particular danger is not evident, rather we observe just the maximum lifetime value among all risks .

C. Exponential Growth
Exponential growth is often used to model the growth of organism populations in a resource-rich environment.Here "resource-rich" means that there is abudance of food and other resources necessary for the population to grow.For example, the initial growth of a cell bacteria in a mouth is often modeled as exponential.The justification for this model is that the rate at which a population of organisms grows should be proportional to their number, assuming that the organisms reproduce at a constant rate.For example, if you double the size of a population, then this should precisely double the rate at which the population bears an offspring, and should, therefore, double the rate at which the size of the population increases.What this means is that the population A of a given organism in a resource-rich environment should satisfy the differential equation Ax, dx dA  where x is some constant that depends on the rate of reproduction.Thus the population grows exponentially bx 0 e A A  This model predicts that the population A will grow indefinitely, which cannot be true in any real situation.Eventually, any population will run out of resources such as food or space to grow.However, the exponential model often gives fairly accurate results in cases where the short-term growth of a population is not inhibited by limited resources [10].
R 2 is frequently defined as the proportion of variance of the response that is predictable (or explained) from the regressor variables, that is the variability explained by the model.A low value of R 2 can suggest that the assumptions of linear regression are not satisfied.Plots and diagnostics will substantiate this suspicion.

II. MATERIALS AND METHODS
We used the data which characterize the proliferative capacity of mesenchymal stem cells.The data are composed of two variables which are the days of the culture (X) and population doubling level (lnY).First, we bootstrap the data in order to increase the sample size and also to optimize the parameter estimates.Then, we estimate the parameters through the exponential curve fitting and transform the nonlinear model into a linear form.This would bring a linear equation form.
From the equation, we estimate the value of the independent variable (x) and fit the data with robust weighted regression by Cauchy, robust Fair weighted regression and robust weighted regression by Huber.Then a covariate-dependent variable is used to examine the differences in performance of the model suitability.

A. The Algorithm of Exponential Calculation
The algorithm showed below is the way of inserting data in SAS algorithm and the way of calculating the bootstrapping method.The R-Squa pendent variab lue of AIC is g able V.The value of 7.98.A robu ems not to be a

Fig. 2 .
Fig. 2.obust (Weighte ves the plot m the Table VI ed by Huber as

TABLE I .
Data in SAS format.The name of the dataset is given as cell_growth.The data consist of two variables x and ln y n 36

TABLE IV .
PAR PAR