Technology and Investment, 2013, 4, 27-30
Published Online Febr uary 2013 (http://www.SciRP.org/journal/ti)
Based on ODE and ARIMA Modeling for the population
of China
Xiaohua Hu, Min Yu
School of Mathematics and Statistics, Hainan Normal University, Haikou, China
Received 2012
ABSTRACT
The economic data usually can be also composed into a deterministic part and a random part. We establish ordinary
differential equations (ODE) model for the deterministic part from a purely mathematical point of view, via the prin-
ciple of integral and difference, establishing
()ARIMAp dq,,
model for the random part, to combine the two estab-
lished models to predict and control the original series, then we apply the method to study the population of China from
1978 to 2007, establishing the corresponding mathematical model, to obtain the forecast data of the population of China
in 2008(1.3879503 billion), finally we make further stability analysis.
Keywords: Natural Asse t; Financial Value; Neural Network
1. Introduction
A lot of time series (as economic data) can be regarded
as a discretization of continuous-time process. if a time
series
Y
possesses biology back gr o und , we can establish
a differential equations model for
Y
in accordance with
the growth rate, for example, suppose that constant
L
is
the growth limit of the variable
Y
, the rate of growth
dY dt/
is proportional to
Y
or
LY−
; suppose that the
relative rate of growth
()dY dtY//
is proportional to
()LY L−/
or
ln lnLY−
; some simple growth models
can be established respectively, such as the Logistic
model, Gompertz model[1] and so on, they are called the
growth mechanism models.
In general, a time series
01
t
Qt n,=, ,...,,
can be decom-
posed into two parts[2]: a deterministic part
t
F
and a
random part
01
t
ht n,=, ,...,
, or
01
t tt
Q Fhtn=+,=, ,...,
(1)
where
t
F
is usually described by the trend, seasonal and
cyclic term, and
t
h
is described by a more complex sto-
chastic approach. In this paper, we ignore specific eco-
nomic backgrounds of variable, from a purely mathe-
matical point of view, to establish differential equations
model for the discrete-time series
t
F
on some condi-
tions, while
t
h
can be described by
()ARIMAp dq,,
model[3].
2. Principles or technology of d ifferential
equations modeling
Suppose that
()0yt t,≤< +∞
, is a continuous function,
t
F
is regarded as the discretised value of
()yt
. Set
0
() ()()
t
X tytdtytt= ≈∆ ∑
∫
()()( ()())X tytXttXtt
′=≈+∆ −/∆
, as
t∆
is small.
Now let’s consider the discrete case (time series) and let
1t∆=
(unit time). It follows that
()()()()( 1)()XtytX tytXtXt
′
≈,=≈ +− ∑
Denoting that
() ()
tt
yytXX t= ,=
, if given a original time
series
()01
t
yyt tn=,=, ,...,
,its cumulative sum series
t
X
can be generated as
0
01
t
tk
k
Xytn
=
=,=, ,...,
∑
,so that, if
we can find the relationship between the original series
t
y
and its cumulative sum series
t
X
as below,
() 001
t tt
Fy Xtn ε ,,= ,= ,,...,
where t ε is a residual term, a random variable, sati sfie s
2
() 0()0
tt
ED εε σσ =,= ,>
.We now vi e w
() ()ytX t,
as
continuous, because of
() ()X tyt
′=
, ignoring
t
ε
, cor-
responding to the one-order differential equation model
can be established as follows.
( ()())0()(())FXtX torXtf Xt
′′
,= =
(2)
We just need to solve out
()Xt
from (2), so,
0
()()( 1)()(0)(0)
t
yyt XtXtXty yX
′
==≈+−, ==,
.
1tmm n= ,...,.≥
( whe re
m
is a positive integer) .

X. HU, M. YU
In the same way,
()01
t
XXt tn=,=, ,...,.
is viewed as a
new time series, we can generate its cumulative sum se-
ries
() ()Zt Xt=∑
, or, the second cumulative sum of
t
y
,denote
()
t
Z Zt=
.
( )()(1)()Z xXtZtZt ′=≈ +−
.we consid-
er to establish the relationship of three series
t tt
yXZ
,,, if
there exists the relationship as follows
() 001
tt tt
Fy X Ztn ε ,,,= ,=,,...,
where
t
ε
is the same with the previously mentioned.
We now view
() ()ytX t,
as a continuous case, because of
()() ()ZtX tyt
′′ ′
= =
, ignoring t
ε
,the two-order differen-
tial equation model can be established.
(()() ()) 0
()( ()())
FZtZtZt
orZtfZ tZt
′′ ′
,, =
′′ ′ = ,
(3)
we just need to solve out
()Zt
from (3), so,
()()(1)()
()()(1)()
(0)(0)(0) 1
t
t
XXt ZtZtZt
yyt Xt XtXt
yXZtmm n
′ ==≈ +−,
′ ==≈+−,
==,= ,...,.≥
.
In a general way, suppose that the original time series is
0
()Xt
. If
0
()Xt
is not pure random data, its value ’s change
on unit time is not random, or data has a trend. its the
first cumulative sum series is
1
()Xt
, ..., the
p
-th cu-
mulative sum series is
()1
p
X ttn,= ,...,
,where
p
is a
positive integer, if we can find the relationship of
01
() ()()
p
XtXtX t, ,...,
as belo w
01
(()()())001
pt
FXtX tXttn ε
,,...,,= , =,,...,
where t ε is the same as previous, ignoring
t
ε
,the
p
-order differential equation model can be established.
1
1
()() ()
(( ))0
pp
p pp
p
pp
dXtd XtdXt
F Xt
dt dtdt
−
−
,,...,, =
1
1
()() ()
(( ))
pp
p pp
p
pp
dXtd XtdXt
orfX t
dtdt dt
−
−
=,..., ,
We just need to solve out
()
p
Xt
, so
11
()( 1)()111
ii i
Xt XtXtiptmmn
++
=+−,=,...,−,= ,...,.≥
01
(0) (0)(0)
p
XX X==... =
It is usually difficult to find
Ff,
,but, we can consider
to establish the multiple linear (or nonlinear) regression
model. Given a significance level
α
(for exam-
ple,
0 05 α = .
),if the significance test for the regression
equation can be established, we can find the correspond-
ing differential equations at 1-
α
confidence level, which
can explain the reasonable degree for the established
differential equations. The adjusted R-squared (
2
R
) or
goodness-of-fit can describe the fitting degree of good or
bad. We call
0F=
the main model. for the residual se-
ries t ε
,we test that it is or isn’t random by some test
methods, if it isn’t a pure random series(such as
white-noise series[4]), it shows that there exists some
valuable information hid in residual series, the informa-
tion should be extracted out from the residual series by
making use of the B-J method[5], at this time, we shall
establish the model for residual series, It is called as the
auxiliary model. Finally, we combine the main model
and the auxiliary model to forecast.
3. Empirical Analysis Application
We study the Chinese population data[6] from 1978 to
2007, the total number of sample observations is 21, see
Table 1 Data of the population of China(unit:ten thousand)
1978 1980 1985 1990 1991 1992 1993 1994 1995 1996 1997
9625
9
9870
5
1058
51
1143
33
1158
23
1171
71
1185
17
1198
50
1211
21
1223
89
1236
26
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
1247
61
1257
86
1267
43
1276
27
1284
53
1292
27
1299
88
1307
56
1314
48
1321
29
We use the sample data from 1978 to 2005 to model,
leaving two sample data of 2006 and 2007 as a reference
to assess the short-term forecast to see the accuracy of
the established model.
3.1. Analyse the Relationship Between the
Original Series and its Cumulative Seri es
Let the original sample time series be
()0 120
t
ytyt=,=,,...,
, the time range to model is from
1978 to 2005.that is
0 118t=, ,...,
, the first cumulative
sum series of
t
y
is
()
t
X XtX= =
, see fig.1,fig.2. We
generate new series
tt
ZR,
via
tt
yX,
as follows
()() ln()
t
tt
t
y
ZZtRRtX t
X
== ,==
The scatter plot of
t
Z
and
t
R
is such as fig.3.We es-
tablish regression model below
2
(1) (2)(3)
t tt
Zcc RcR=++.
(4)
By making use of least-squares method and EViews6.0,
it is easy to obtain the estimation value of parameters
(1)(2)(3)cc c,,
, see fig.4 and Tab.2.
2
22 19772308771470 107731
t tt
Z RR= .−.+.
(5)
Table 2. Results of estimate and test (significance level
α
=0.05)
Vari able C oef fici ent Std. Error t-Statistic Prob.
c(1) 22.19772 1.334405 16.63491 0.0000
c(2) -3.0877147 0.202384 -15.25674 0.0000
c(3) 0.107731 0.007644 14.09369 0.0000
The adjusted R2 = 0.986264.The t-statistics of three re-
gression coefficients
(1)(2)(3)cc c,,
in (4) are respec-
28

X. HU, M. YU
tively 16.64, -15.26,14.09, their Prob=0.0000<0.05,
fig.1: the original time series
y
fig.2: the cumulative sum series
X
F-statistics=647.2. Prob = 0.0000<0.05. the significant
test for coefficients and the whole regression equation
were resulted. We think that the Chinese population data
from 1978 to 2005 can be described by (5) at 95% con-
fidence level, Goodness-of-fit reaches 98.6%. However,
DW= 1.317 shows existence of autocorrelation in the
residual series, It shows that there are still some valuable
information not to be extracted out from the residual se-
ries, so we will establish the model for the residual se-
ries.
fig.3: the scatter plot of
R
and
Z
fig.4: the fitting and residuals plot of
R
and
Z
3.2. Establish the Main Model–Differential Equ-
ations Model
It follows from (2),(4)and(5)that
2
()[ (1)(2)ln()(3)ln()]()
dX X tccXtcXtXt
dt ′
==++
(6)
2
( )[221983088ln( )0108ln()]( )
dX X tXtXtXt
dt ′
== .−.+.
(7)
by formula[7],when
2
40ac b−>
,
222
2
2arctan
44
ducu bC
a bu cuac bac b
+
= +,
++ −−
∫
It is easy to obtain from (6)
22
2 (3)ln(2)
2arctan
4 (1) (3)(2)4(1) (3)(2)
c XcCt
cc ccc c
++=
−−
so
2
2
1
exp{[(2)4 (1)(3)(2)]
2 (3)
1
tan[4(1) (3)(2)()]}
2
Xccc c
c
c cctC
=−+− ×
−−
or
exp{14 2961 074tan(1)}
8 624
t
XC=. +.+
.
0(0)96259169 519tX C=,=,=−.
, so
exp(14 2961 074tan(69 519))
8 624
t
X=.+.− .
.
(8)
2
0 125sec(69 519))exp(
8 624
14 2961074tan(69 519))
8 624
t
X
t
′=. −.
.
.+.−.
.
(9)
3.3. Establish the Auxiliary Model for Residual
Series
Let the residual series be
t
h
,we establish
()ARIMAp dq,,
model for
()
t
h ht=
,make 1-order,
2-order difference for
t
h
below
2
11
( )(1)
ttttt t
hhththhhhh
−−
∇=−−=−,∇=∇−∇
based on the analysis of the autocorrelation coefficients
and partial autocorrelation coefficients of
t
h
and
2-order difference of
t
h
.we take
2pqd= ==
,establish
(2 2)ARMA ,
model for
2
t
h"
.
22 2
11221122ttttt t
hh h ϕϕε θεθε
− −−−
∇ =∇+∇+−−
(10)
1 121213
2 4112 2
(2)(21)
t ttt
ttt t
h hhh
h
ϕ ϕϕϕ
ϕε θεθε
− −−
− −−
=++−− +
++− −
this is a
(422)ARIMA ,,
for
t
h
,where
t
ε
is a
white-noise series. the parameters in (10) are estimated
by EViews6.0,
12 12
ˆˆ
ˆˆ 0 84360032800 9992
ϕϕ θθ
=., =.,=,=.
.
123
42
2 84362 65440 8436
0 03280 9992
tt tt
tt t
hh hh
h εε
−− −
−−
=. −.+.
+.+− .
(11)
3.4. Forecast Based on the Main and Auxiliary
Model
By (8) and (11), as
t
=19, 20, 21,
y
=131190.5504,
136191.0595, 138795.0589;
t
h
=-0.007129, -0.0133,
-0.01979, so, by (1),the predictive data of the population
of China in 2006,2007 are 1.3119054,1.3619104 billion,
respectively, however, the actual data are 1.31448,
1.32129 billion, respectively; the absolute errors are 2.58,
40.62,respectively;the relative errors are 0.2%,
3.1%,respectively;It shows that the combination model
(1),(8) a nd (11) have a higher prediction accurac y, and
the predictive data of the population of China in 2008 is
29

X. HU, M. YU
1.3879503 billion. We also find that predictive value of
the auxiliary model (11) is little impact on the total pre-
dictive value. the total predictive value mainly depends
on the main model (8),or, mainly depends on the predic-
tive value of the differential equations model (7).So, we
can see that the short-term forecast accuracy is very high
based on differential equations modeling for the time
series on some condition.
We further consider the stability of equilibrium point of
(6) or (7). Let
2
[(1) (2)ln()(3)ln()]()()0
dX ccXtcXtXtf X
dt =++ ==
2
0[(1)(2)ln()(3)ln()]0XccXt cXt≠, ++=
, when
2
(2)4 (3) (1)0c cc−>
, there are two real roots, denoted by
12
uu
,.there are two equilibrium points
12
XX,
,
1 122
exp( )exp()X uXu= ,=
. on the other hand,
2
()(1)(2)((1)2(3)) ln(3) lnfXccccX cX
′=++ ++,
When
2(2)4 (3)(1)0c cc− >,
2
1
2
2 12
()(2)4 (3) (1)
() (2)4(3)(1)()0()0
fXccc
fXcc cfXfX
′=−,
′ ′′
=−−.>,<
The equilibrium point
2
X
of (6) is stable,
or,
2
t XX→ +∞,→
.the equilibrium point
1
X
of (6) is
unstable, or,
tX→+∞,→ +∞
. so, we must control those
factors that impact
(1)(2)(3)ccc,,
in (6),such that
2
(2)4(3) (1)0c cc−>
. otherwise,
tX→ +∞,→ +∞
However, in fact, for model (7),
2
(2)4 (3) (1)0c cc−<
,it
show that there is no equilibrium point in (7), it is ob-
viously from (9),as
/8 62469 5192 t
π
. −.→,
or
613ty→,→ +∞
. it show that China’s population will
tend to infinity after 613 years, so, the model (7) is only
suitable for short-term prediction.
4. Acknowledgemen ts
Acknowledgements: the author is grateful to the ano-
nymous referees for his helpful comments and sugges-
tions.
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30