Biomedical Engineering Reference
In-Depth Information
we often need to interpret experimental or field data in terms of a given known physical
model. For example, the concentration of the reactant C can be measured at different times
t for the liquid reactant decomposing to liquid products in a well-mixed batch reactor. It might
be a perfect example to apply the linear regression model, Eqn (7.6) , to correlate the data.
However, for process engineers, the simplest regression model for the particular problem is
d C
d t ¼a 0 C a 1
(7.8a)
and for bioprocesses:
a 0 C a 1
a 2 þ C a 3
d C
d t ¼
(7.8b)
which are not only nonlinear, but is no longer an algebraic equation. Therefore, the regression
model (7.8) does not fit into the same category as we have discussed earlier. We can call this
model an ordinary differential regression model. This opens up a whole new class of regres-
sion models. Similarly, one may have partial differential regression models, differential-alge-
braic regression models, and integral regression models as well. In bioprocess engineering,
we normally encounter algebraic regression models and (ordinary or partial) differential
regression models.
7.3. CRITE RIA FOR “BEST” FIT AND SIMPLE LINEAR REG RESSIONS
The case of simple linear regression considers a single regressor or predictor x and a depen-
dent or response variable Y . Suppose that the true relationship between Y and x is a straight
line and that the observation Y at each level of x is a random variable. As noted previously,
the expected value of Y for each value of x is
EðYjxÞ¼y ¼ a 0 þ a 1 x
(7.9)
where the intercept a 0 and the slope a 1 are unknown regression coefficients. We assume that
each observation, Y , can be described by the model
Y ¼ a 0 þ a 1 x þ ε
(7.10)
2 . The random errors corresponding
to different observations are also assumed to be uncorrelated random variables.
Suppose that we have n pairs of observations ( x 1 , y 1 ), ( x 2 , y 2 )
where e is a random error with mean zero and variance
s
( x n , y n ). Figure 7.3 shows
a typical scatter plot of observed data and a candidate for the estimated regression line.
The estimates of a 0 and a 1 should result in a line that is (in some sense) a “best fit” to the data.
There are several choices for “best” fit by minimizing the norm of the error,
.
. Three of
such choices are shown in Fig. 7.4 .In Fig. 7.4 a, the dashed lines are produced by minimizing
the sum of the residuals directly, i.e.,
k ε k
minimizing X
n
minimizing X
n
ε i ¼
ðy i a 0 a 1 x i Þ
i ¼ 1
i ¼1
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