Parametric Estimation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 290

The fitted or estimated regression line is therefore

y ¼ a 0 þ a 1 x (7.17)

Note that each pair of observations satisfies the relationship

y i ¼ a 0 þ a 1 x i þ ε i ; i ¼ 1; 2; . ; n (7.18)

DEFINITION
The least squares estimates of the intercept and slope in the simple linear regression model

are

a 0 ¼ y a 1 x (7.19)

X n ! X n !

P

n

x i y i n y i x i

i ¼ 1

i ¼1 i ¼1

a 1 ¼ (7.20)

X ! 2

n

n

x i n x i

i ¼ 1

i ¼1

n X

n

n X

n

1 1

where y ¼

y i and x ¼ x i :

i ¼ 1 i ¼ 1

where y ( x i ) is called the residual. The residual describes the error in the fit of the

model to the i th observation y i . In the next section, we will use the residuals to provide infor-
mation about the adequacy of the fitted model.
We have so far discussed the regression by assuming that the data contain error in y i 's and
the error is of absolute nature as characterized by Eqn (7.20) . For a simple linear regression
model, as long as only one variable (either x i 's or y i 's) contains an error of absolute nature, the
above procedure is strictly valid. For any other regression models, if the variable x is in exper-
imental error, one will need to minimize the residual square of x . Simply rename x as the
regressor in the regression process should suffice. There can be occasions where more than
one variable in the same regression model is in experimental error, one should deal with
these types of regression with care. If at all possible, experimental technique should be
improved to eliminate the errors in the controlling variables (e.g. x ). When experimental tech-
nique fails, one needs to determine the magnitudes of the errors among different variables.
There is no general rule or simple technique to treat these types of regression problems. If the
magnitudes of the errors are consistently related with the regression model, then there is no
special attention needed and one can apply the regression technique as if only one variable
contains measurement error. Otherwise, the usual approach is to minimize the residual
square of the variable that has the largest magnitude in error.
Apart from the complicated error structures, there are cases where the regressor contains
experimental error of relative nature. That is, for the simple linear regression model as an
example,

ε i ¼ y i

y i ¼ða 0 þ a 1 x i Þþð1 þ d i Þ; i ¼ 1; 2; . ; n (7.21)