Parametric Estimation - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 291

where d i is the relative error or relative residual at point ( x i , y i ). The relative residual or error
d can be assumed to be distributed normally, rather than the residual or absolute error
e

y$d .
For example, in obtaining the data of Tabl e 7 .1 ,wehavemeasuredtheflowratessuchthat
we always have four significant digits irrespective of the magnitude of the flow rates. The
number of significant digits can be achieved by using a large enough beaker to collect water.
When the flow rate is very small, we collect enough water such that the amount of water can
bemeasured correctlywith four digits. When the flow rate is high, wemake sure the time lap
is long enough such that the accuracy in time has four significant digits. By generating the
data in this fashion, we know for sure that the data is accurate to within four digits.
When the relative error in measured y is normally distributed, the regression should be
carried out by minimizing the sum of d squared. That is,

¼

y i
yðx i Þ 1

2

jd i minimizing X minimizing X

n n

2

¼

i ¼ 1 i ¼ 1

or minimizing d 2 with

y i
yðx i Þ 1

2

X X

n n

1
n 1

1
n 1

d 2 ¼ d i ¼ (7.22)

i ¼1 i ¼1

In general, this approach will lead to nonlinear regression. Even for a simple linear regres-
sion model, the above regression approach will require nonlinear analysis.
In bioprocess engineering applications, the regression problems we usually encounter are
nonlinear. The minimization of the (relative or absolute) residual square sum can be achieved
by applying the optimization techniques. Therefore, there are no special discussions devoted
here in this chapter.

7.4. CORRELATION COEFFICIENT

From the perspective of the data set, y i varies with x i . The total variation of y i (a measure of
the difference of y i from the average value of y i ) can be computed from the data set as

X

n

1
n 1

ðy i yÞ 2 TV ¼ (7.23)

i ¼ 1

Using the regressionmodel, we can explainpart of the total variation. The unexplainedportion
of the variation is the estimated variance of the data around the model and is given by

X

n

1
n 1

½yðx i Þy i 2 ¼ s 2 ¼

UV (7.24)

i ¼ 1

If there is no correlation between the variables, the two variations defined above, Eqns (7.23)
and (7.24) , are identical. On the other hand, if the regression line passes through all the data
points, then the unexplained portion of variation UV or

2

s

is zero. The regression is thus