INTRODUCTION

Engineers, technologists, and scientists have employed numerical methods of analysis to solve a wide range of steady and transient problems. The fundamentals are essential in the basic operations of curve fitting, approximation, interpolation, numerical solutions of simultaneous linear and nonlinear equations, numerical differentiation and integration. These requirements are greater when new processes are designed. Engineers also need theoretical information and data from published works to construct mathematical models that simulate new processes. Developing mathematical models with personal computers sometimes involves experimental programs to obtain the required information for the models. Developing an experimental program is strongly dependent on the knowledge of the process with theory, where the whole modification can be produced by some form of mathematical models or regression analyses. Figure 1-1 shows the relationship between mathematical modeling and regression analysis.

Texts [1–5] with computer programs and sometimes with supplied software are now available for scientists and engineers. They must fit a function or functions to measure data that fluctuate, which result from random error of measurement. If the number of data points equals the order of the polynomial plus one, we can exactly fit a polynomial to the data points. Fitting a function to a set of data requires more data than the order of a polynomial. The accuracy of the fitted curve depends on the large number of experimental data. In this chapter, we will develop least-squares curve fitting programs. Also, we will use linear regression analyses to develop statistical relationships involving two or more variables. The most common type of relation is a linear function. By transforming nonlinear functions, many functional relations are made linear.

For the program, we will correlate X-Y data for the following equations:

(1-1)

(1-2)

(1-3)

(1-4)

(1-5)

(1-6)

(1-7)

(1-8)

We can transform the nonlinear equations (1-5, 1-6, and 1-8) by linearizing as follows:

(1-9)

(1-10)

(1-11)

LINEAR REGRESSION ANALYSIS

Regression analysis uses statistical and mathematical methods to analyze experimental data and to fit mathematical models to these data. We can solve for the unknown parameters after fitting the model to the data. Suppose we want to find a linear function that involves paired observations on two variables, X the independent variable and Y the dependent variable. Where

We can develop a linear regression model that expresses Y as a function of X. We can further derive formulae to determine the values of a and b that give the best fit of the equations. For each experimental point corresponding to an X-Y pair, there will be an element that represents the difference between the corresponding calculated value and the original value of Y. This is expressed as:

(1-12)

ri may be either positive or negative depending on the side of the fitted curve of X-Y points. Here, we can minimize the sum of the squares of the residuals by the following expression:

(1-13)

where

n is the number of observations of X-Y points

(1-14)

(1-15)

and

(1-16)

The problem is reduced to finding the values of a and b so that the summation of Equation 1-16 is minimized. We can obtain this by taking the partial derivative of Equation 1-16 with respect to each variable a and b and set the result to zero.

(1-17)

Substituting Equation 1-16 into Equation 1-17, we obtain

(1-18)

and

(1-19)

This is equivalent to

(1-20)

and

(1-21)

Since b, X, and Y are not functions of a, and the partial derivative of a with respect to itself is unity, Equation 1-20 reduces to

(1-22)

Similarly, a, X, and Y are not functions of b. Therefore, Equation 1-21 becomes

(1-23)

where a and b are constants. Equations 1-18 and 1-19 are expressed as:

(1-24)

and

(1-25)

We have now reduced the problem of finding a straight line through a set of X–Y data points to one of solving two simultaneous equations 1-24 and 1-25. Both equations are linear in X, Y, and n, and the unknowns a and b. Using Cramer’s rule for the simultaneous equations, we have

(1-26)

and

(1-27)

Solving Equations 1-26 and 1-27 gives

(1-28)

and

(1-29)

respectively, where all the sums are taken over all experimental observations.

Also, we can construct a table with columns headed Xi, Yi, Xi2, Xi Yi as an alternative to obtain the values of a and b. The sums of the columns will then give all the values required to Equations 1-28 and 1-29.

METHODS FOR MEASURING REGRESSION

THE ANALYSIS OF VARIANCE TABLE FOR LINEAR REGRESSION

Each sum of squares...