06.04.1
Chapter 06.04
Nonlinear Models for Regression
After reading this chapter, you should be able to
1.
derive constants of nonlinear regression models,
2.
use in examples, the derived formula for the constants of the nonlinear regression
model, and
3.
linearize (transform) data to find constants of some nonlinear regression models.
From fundamental theories, we may know the relationship between two variables.
An example in chemical engineering is the Clausius-Clapeyron equation that relates vapor
pressure
P
of a vapor to its absolute temperature,
T
.
(
)
T
B
A
P
+
=
log
(1)
where
A
and
B
are the unknown parameters to be determined.
The above equation is not
linear in the unknown parameters.
Any model that is not linear in the unknown parameters is
described as a nonlinear regression model.
Nonlinear models using least squares
The development of the least squares estimation for nonlinear models does not
generally yield equations that are linear and hence easy to solve.
An example of a nonlinear
regression model is the exponential model.
Exponential model
Given
(
)
1
1
,y
x
,
(
)
2
2
,y
x
, . . .
(
)
n
n
,y
x
, best fit
bx
ae
y
=
to the data.
The variables
a
and
b
are the constants of the exponential model.
The residual at each data point
i
x
is
i
bx
i
i
ae
y
E
−
=
(2)
The sum of the square of the residuals is
∑
=
=
n
i
i
r
E
S
1
2
(
)
∑
=
−
=
n
i
bx
i
i
ae
y
1
2
(3)

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