Information Technology Reference
In-Depth Information
Let
D
˛=ˇ.Then(
5.90
) reads
p
0
.t /
p.t/
D
˛
C
p.t /:
(5.91)
Hence, we can determine constants ˛ and by fitting a linear function to the
observations of
p
0
.t /
p.t/
:
(5.92)
As above, we let
D
100
p.n
C
1/
p.n/
p.n/
b
n
:
(5.93)
The values of the data set .n; b
n
/ for n
D
0;1;:::;n are given in Table
5.4
.We
now want to determine two constants A and B, such that the data are modeled as
accurately as possible by a linear function
b
n
A
C
Bp.n/;
(5.94)
in the sense of least squares. It is important to note that we model the data .n; b
n
/ as
a function of p, not as a function of time. We see this from (
5.91
), where it is clear
that if we can make a linear model of p
0
=p of the form (
5.94
), we have a logistic
model.
By using the standard formulas for approximation by a linear function,
see page
162
,wefindthatA and B are determined by the following 2
2 linear
system:
0
1
0
1
9
X
9
X
0
1
@
10
p.n/
A
@
b
n
A
A
@
A
nD0
nD0
D
:
(5.95)
9
X
9
X
9
X
p
2
.n/
B
p.n/
p.n/b
n
nD0
nD0
nD0
Here
9
X
9
X
p
2
.n/
319:5;
p.n/
56:5;
nD0
nD0
9
X
9
X
b
n
14:1;
p.n/b
n
79:6;
nD0
nD0
