Biomedical Engineering Reference
In-Depth Information
2
with mean 0.0661
þ
0.05842
x
and variance
s
¼
0.001554. For example, if
x
¼
22, then
Y
has
a mean value
E
(
Y
1.351 and variance 0.001554.
In most real-world problems, the values of the intercept and slope (
a
0
,
a
1
) and the error
variance
j
x
)
¼
0.0661
þ
0.05842(22)
¼
2
will not be known, and they must be estimated from sample data. Regression
analysis is a collection of statistical and numerical tools for finding estimates of the parame-
ters in the regression model. Then this fitted regression equation or model is typically used in
prediction of future observations of
Y
or for estimating the mean response at a particular level
of
x
. To illustrate, a lab technician might be interested in estimating the mean water flow rate
when the rotameter reading is
x
s
33%. This chapter discusses such procedures and applica-
tions from simple linear to differential regression models.
¼
7
.2. CLASSIFICATION OF REGRESSION MODEL
S
The water flow rate is linearly related to the rotameter readings. This is the simplest regres-
sion model in that only two variables are involved: one independent variable (the rotameter
reading) and one dependent variable (the water flow rate). The fundamental mathematical
model is algebraically linear in the parameters to be determined:
y ¼ a
0
þ a
1
x
(7.5)
In this case, we have enough background to know the exact qualitative functional form. If one
is to correlate or regress the data set without any prior knowledge, one may try a more
general form:
y ¼ a
0
þ a
1
x þ a
2
x
2
þ
/
þ a
n
x
n
(7.6)
which is a polynomial model. The polynomial model is a linear regression model because
Eqn (7.6)
is linear with respect to the undetermined parameters:
a
0
,
a
1
,...,
a
n
. Linear regres-
sion models are highly desirable in that there is abundance of literature on it and easy to
perform as we will show in the next section.
The regressionmodel need not be linear. For example, rational expressions may be preferred
over polynomial expressions. If one chooses, in place of
Eqn (7.6)
, a rational expression,
a
0
þ
a
1
x
þ
a
2
x
2
þ
/
þ
a
n
x
n
1 þ b
1
x þ b
2
x
2
þ
/
þ b
m
x
m
y ¼
(7.7)
then it is no longer linear in the undetermined parameters:
a
0
,
a
1
,...,
a
n
and
b
0
,
b
1
,...,
b
m
.In
this case, we call it a nonlinear regression model. Because
Eqns (7.6) and (7.7)
can usually
correlate a set of data equally well, one can imagine that using correlation to extrapolate
the underlying physics or fundamental relationship is not always reliable. Coupled with
the unknown nature of the experimental error, extrapolating fundamental relations are
even more adventurous if one is looking at the data alone.
If you were an applied mathematician, you would probably stop here. As it is already
cumbersome (or even may be unnecessary, as most nonlinear regression models could be
linearized by regrouping the variables) to study nonlinear regression models, other more
complicated regression models are surely not welcome. However, as an engineer, we do
not have the luxury to ignore other more complicated regression models. In engineering,
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