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In-Depth Information
r
0
e
4a
D 2:728 10
9
;
(9.13)
r
0
e
5a
D 2:780 10
9
:
(9.14)
We thus have six equations, but only two unknowns
a
and
r
0
. The number of people
on our planet did not, of course, grow precisely exponentially during this period,
and one cannot therefore expect there to exist numbers
a
and
r
0
satisfying (
9.9
)-
(
9.14
). Instead we have to be content with trying to estimate
a
and
r
0
such that these
equations are approximately satisfied.
To this end, consider the function
t D
X
1
2
.r.t I a; r
0
/ d
t
/
2
J.a; r
0
/ D
t D0
t D
X
1
2
.r
0
e
at
d
t
/
2
;
D
t D0
where
d
0
D 2:555 10
9
;
d
1
D 2:593 10
9
;
d
2
D 2:635 10
9
;
d
3
D 2:680 10
9
;
d
4
D 2:728 10
9
;
d
5
D 2:780 10
9
:
Note that
J.a; r
0
/
is a sum of quadratic terms that measure the deviation between
the output of the model and the observation data. It follows that if
J.a; r
0
/
is small,
then (
9.9
)-(
9.14
) are approximately satisfied. We thus seek to minimize
J
:
min
a;r
0
J.a; r
0
/:
The first order necessary conditions for a minimum
@J
@a
D 0;
@J
@r
0
D 0;
yield a nonlinear
2 2
system of algebraic equations for
a
and
r
0
:
t D
X
.r
0
e
at
d
t
/r
0
te
at
D 0;
(9.15)
t D0
