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.