An often used model function in nonlinear regression is
As shown in Exercise 8.6, by using the log, the model function can be written as
(
) =
1
+
2
, where
1
= log
1
and
2
=
2. Also, a data point (
i,
i) transforms to (
i,
i
), where
i = log
i
and
i
= log
i
. The data considered in this exercise is given in Table 9.8, which was also considered in Exercise 8.6.
(a) Apply the same normalization to the transformed (
i
,
i
) data as was used in the word length example. This means you center the
and
values, and then scale the values using the maximum entry, in absolute value. Label the normalized variables as
and
. Also, make sure to give the values of
and
, as well as the values used to scale the centered values.
(b) Assuming
=
, and using the true distance, then the error function is
The minimum of
occurs when
is given in (9.10). Calculate
using your normalized data from part (a).
(c) In this problem, instead of (9.11), we have
=
+
( −
). Using the common log, show that
2
=
and
(d) Using part (c), plot the (original) data and power law curve using a log-log plot.
(e) Based on your result from part (c), what was the running speed of a Tyrannosaurus rex?
Exercise 8.6
The exercise considers fitting data using the model function
which is known as a power law function, and also as an allometric function. Two different methods are considered, one summarized in (a) and (b), and the second in (c) and (d). Assume that the data are (
1,
1), (
2,
2), ··· , (
n
,
n
), where the
i
’s and
i
’s are positive.
(a) Writing
=
1
v2
, and then taking the log of this equation, show that the transformed model function can be written as
() =
1
+
2
, where
1
= log
1
and
2
=
2. Also, show that the transformed data points (
i
,
i) are
i
= log
i
and
i
= log
i
.
(b) Continuing from part (a), using the least squares error
(
1,
2) =
(
1
+
2
i
−
i
)2, and the common log, show that
1
= 10
V1
and
2
=
2, where
1
and
2
are given in (8.31).
(c) Show that to minimize the error function
(
1,
2) =
one gets that
(d) Continuing from part (c), show that finding the minimum of
(
1,
2) reduces to solving an equation of the form
(
2) = 0. Write down the function
, and explain why the secant method might be easier to use to solve the equation than Newton’s method.