LIGAND BINDING, DATA FITTING, AND LEAST-SQUARES ESTIMATES OF MODEL PARAMETERS - An Invitation to Biomathematics - page 238

As our next example shows, this non-linearity complicates the computa-
tions significantly.

Example 8-1
.......................

3 e rt , defined by Eq. (1-4) in
Chapter 1. In this model, the independent variable is the time
t (decades), and the dependent variable is the U.S. population
P (millions). The net per capita rate of U.S. population growth r is the
only parameter. In our current notation, the model can be rewritten as

P 0 e rt Recall the population model P ð t Þ¼ ¼ 5 :

3e rt

¼ ð Þ¼ : :

P G r ; t 5 (8-9)

Table 8-2 reproduces the U.S. population data from Chapter 1. We want
to find the least-squares estimate for the parameter r.

Time i
(decades)

U.S. Population P i ¼ P(i)

(millions)

0 5.3

Each column of Table 8-2 represents an experimental point of the form
(t i , P i ). The least-squares measure for this model can be written as:

1 7.2

X X

2 9.6

2 2

3e rt i

SSR ¼ SSR ð r Þ¼ ½ P i G ð r t i Þ ¼ ½ P i 5 : :

;

3 12.9

i i

4 17.1

To find the least-squares estimate for r, we need to solve the equation
@ð

5 23.2

SSR Þ

¼ 0 that is:

;

6 31.4

@ r

TABLE 8-2.
Population of the United States from 1800 to
1860.

2 X
i

6 X
i

@ð Þ

SSR

3e rt i e rt i t i ¼ 3t i e rt i e rt i ¼ ½ P i 5 : ð 5 : 3 Þ 10 : t i ½ P i 5 : ¼ 0 ;

@ r

(8-10)

or

X

3e rt i e rt i t i ½ P i 5 : ¼ 0 :

i

Using the data points (t i ,P i ) from the table above, we obtain the equation:

3e r e r 3e 2r e 2r 3e 3r e 3r

½ 7 : 2 5 : þ 2 ½ 9 : 6 5 : þ 3 ½ 12 : 9 5 :

3e 4r e 4r 3e 5r e 5r 3e 6r e 6r

þ 4 ½ 17 : 1 5 : þ 5 ½ 23 : 2 5 : þ 6 ½ 31 : 4 5 : ¼ :

(8-11)

0

Notice that this equation is not linear with respect to r. For general
nonlinear models, there are no closed-form expressions to determine the
solution, as there are for linear models. However, computers can be used
to numerically calculate the solution of the equation.

Various methods for calculating the roots of nonlinear equations have
been developed and are typically studied in courses on numerical
analysis. One of the most popular methods is Newton's method, which
provides an iterative technique for finding the roots of an algebraic