Environmental Engineering Reference
In-Depth Information
Appendix7
Regression Analysis
(the Linear Least-Squares
Methodology)
Regression analysis is a powerful tool for establishing a relationship between two or
more variables. There are a number of cases in environmental engineering where a
dependent variable y and an independent variable x are measured to give an array
of data points ( x i , y i ) . Examples are linear free energy relationships, and numerous
examples where adsorption data, partitioning data, reaction kinetic data, etc. are fitted
to a straight-line equation. In each case, a plot of x i versus y i is made and the best-fit
curve to the data is determined. The method is called linear least squares. Consider
the figure given below:
y = a + bx
y
b
a
x
The best fit through the data points is a straight line with a slope of b and an
intercept of a . The objective is to estimate (i) the slope b and intercept a from the data
points, and (ii) the degree of fit, that is, how well do the data fit the straight line?
We make the primary assumption that there is negligible error in the independent
variable x . For any given value of x i , there is an observed value y i and a corresponding
calculated value y i , calcd =
a
+
bx i . Thus, the residual sum of errors in the estimation
441
 
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