LIGAND BINDING, DATA FITTING, AND LEAST-SQUARES ESTIMATES OF MODEL PARAMETERS - An Invitation to Biomathematics - page 250

In this case,

X

2

SSR ð r ; c Þ¼

1 ½

Y i G ð r ; c X i Þ : (8-33)

;

i ¼

We begin with initial guesses r c 0 and the Taylor

approximation from Eq. (8-17) for the values r

¼ r 0 and c ¼

¼ r 0 and c ¼ c 0 :

Þþ @

G ð r 0 ; c 0 ; X i Þ

Þþ @

G ð r 0 ; c 0 ; X i Þ

G ð r ; c X i Þ G ð r 0 ; c 0 X i ð r r 0 ð c c 0 Þ:

; ;

@ r @ c

(8-34)

Because for the least-squares values of r and c, we want

Y i ¼ G ð r ; c X i Þþ experimental uncertainties ;

;

the equations used to find those values are

X i Þþ @

G ð r 0 ; c 0 ; X i Þ

r 0 Þþ @

G ð r 0 ; c 0 ; X i Þ

Y i ¼ G ð r 0 ; c 0 ; ð r ð c c 0 Þ:

@ r @ c

As in the one-parameter case, we have one equation of this form for
every experimental data point (X i ,Y i ) and can express this set of
equations more conveniently in matrix notation as P e ¼ Y*, where

2
4

3
5

@ G ð r 0 ; c 0 ; X 1 Þ @ G ð r 0 ; c 0 ; X 1 Þ

@ r @ c

2
4

3
5 ;

@ G ð r 0 ; c 0 ; X 2 Þ @ G ð r 0 ; c 0 ; X 2 Þ Y 1 G ð r 0 ; c 0 ; X 1 Þ

Y 2 G ð r 0 ; c 0 ; X 2 Þ

@ r @ c

Y ¼ P ¼ ; and

.

. .

Y n G ð r 0 ; c 0 ; X n Þ

@ G ð r 0 ; c 0 ; X n Þ @ G ð r 0 ; c 0 ; X n Þ

@ r @ c

r r 0

e ¼ :

c c 0

The solution can be obtained as in Eq. (8-31), with the iterative process

continuing until

e

e þ guess ) better guess returns e ¼ 0.

E XERCISE 8-4

Demonstrate that the values for parameters r and c obtained from the
Gauss-Newton method are the least-squares estimates for r and c by
showing these values provide a minimum for the SSR from Eq. (8-33).

Hint: Show that the values obtained for r and c are such that
@ SSR ð r ; c Þ @ SSR ð r ; c Þ

¼ 0 and ¼ 0 :

@ r @ c