Information Technology Reference
In-Depth Information
Exercise 5.8.
Compute quadratic least squares approximations of the functions in
Exercise
5.7
.
˘
Exercise 5.9.
Consider
y.t/
D
e
t
for
1
t
1.
(a) Make a quadratic least squares approximation of y.
(b) Compute three terms of the Taylor series expansion of y around t
D
0.
(c) Graph y.t/ together with the approximations generated in (a) and (b) and
discuss the properties of the approximations.
˘
5.5
Project: Computing Coefficients
In Sect.
5.2
we discussed how to determine the parameters in the exponential model
y
0
.t /
D
˛y.t/
(5.148)
and in the logistic model
y
0
.t /
D
˛y.t/.1
y.t/=ˇ/ :
(5.149)
The problem is to compute ˛ in (
5.148
)and˛, ˇ in (
5.149
) based on observations
of only y (not y
0
.t /) for some values of t
D
t
n
. In the analysis of this problem, we
basically introduced the following two ideas:
(a) Approximate y
0
.t
n
/ by
y.t
nC1
/
y.t
n
/
t
:
(5.150)
Using this approximation in (
5.148
), we get
y.t
nC1
/
y.t
n
/
y.t
n
/t
˛
(5.151)
and then we can use measurements of y to get an estimate of ˛.
(b) In (
5.149
) we can use (
5.150
)toget
y.t
nC1
/
y.t
n
/
y.t
n
/t
˛
ˇ
y.t
n
/:
˛
(5.152)
