Biology Reference
In-Depth Information
and are calculated as r
i
¼
Y
i
G
ð
a
;
b
X
i
Þ¼
Y
i
ð
aX
i
þ
b
Þ;
i
¼
1
;
2
;
...
;
n.
;
It is desirable to determine values for a and b that minimize the
combined deviation between model and data. The sum-squared
residuals (SSR) measure is the one most frequently used to express this
deviation:
X
X
n
2
r
1
þ
r
2
þ
r
3
þ
...
þ
r
n
¼
r
i
¼
SSR
ð
a
;
b
Þ¼
½
Y
i
G
ð
a
;
b
X
i
Þ
:
(8-1)
;
i
¼
1
i
In an attempt to simplify the notation, the initial and final value of
the summation index i are often ignored, as in the last expression of
Eq. (8-1), the understanding being that the sum is taken over the whole
range of available data points.
Notice that the measure is a function of model parameters a and b. The
least-squares data-fitting criterion can now be stated: Using the
experimental data, determine the parameter values minimizing the least-
squares measure SSR(a,b). To do this, we use a basic idea of calculus:
At a minimum value of a function, the derivative (if there is only one
variable) or all the partial derivatives (if there is more than one variable)
will be zero. The function in question is the SSR, and the variables
are the model parameters.
For the linear a model
Y
¼
aX
þ
b
(8-2)
taking the partial derivatives for the function from Eq. (8-1) and setting
them equal to zero leads to the following equations:
X
2
X
@ð
SSR
Þ
@
@
2
¼
a
½
Y
i
ð
aX
i
þ
b
Þ
¼
i
ð
Y
i
aX
i
b
Þ
X
i
¼
0
@
a
i
(8-3)
X
2
X
@ð
SSR
Þ
@
@
2
¼
b
½
Y
i
ð
aX
i
þ
b
Þ
¼
i
ð
Y
i
aX
i
b
Þ¼
0
:
@
b
i
To solve these equations, we can rewrite them in the form:
a
X
i
b
X
i
X
X
i
þ
X
i
X
i
Y
i
¼
0
a
X
i
X
i
(8-4)
X
i
þ
nb
Y
i
¼
0
;
i
and then solve the system for a and b. The following formulae give the
result and allow a direct computation of the least-squares values for
the parameters a and b of the linear model (8-1):
n
P
i
P
i
X
i
P
i
P
i
X
i
P
i
P
i
X
i
Y
i
P
i
X
i
Y
i
Y
i
Y
i
X
i
¼
;
¼
:
a
n
P
i
X
i
ð
P
i
b
n
P
i
X
i
ð
P
i
(8-5)
2
2
X
i
Þ
X
i
Þ
