18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

Prediction of Time-Series Data using PSpice andRunge-Kutta Method
Kazuhisa YOSHIMATSU, Toru TANAKA
and Masayuki YAMAUCHI

Yoshifumi NISHIO

Dept. of Electronics and Computer Eng.,
Hiroshima Institute of Technology,
2-1-1, Miyake, Saeki-ku, Hiroshima 731-5193, JAPAN.
Email: yoshimatsu.k@gmail.com

Abstract— Prediction of time-series data in natural world arevery important. For example, if nonlinear time-series data are
predictable, we can take adequate action for problem of weather.
In this study, we propose two prediction methods for timeseries data using coupled oscillators system. We predict chaostime-series data like the natural phenomena by these methods.
The chaos time-sires data are made by the Chua’s circuit. The
prediction results of time-series data are compared with two our
methods. We show possibility of predictable by actual circuit.

I. introduction
Predictions of the time-series data of natural world are very
important for human, and a lot of researches are carried out
up to now[1][2]. A lot of things of the natural world suchas the atom, neurons, and planets are oscillating. These areconnected by each other, and operate as a system. Therefore,time-series data of a lot of natural things are oscillatory
solutions. Coupled oscillators system can make complex
phenomena[3][4].
The time-series data is generally predicted with statistics,
a state space model, a Karman filter and so on. We think thatthe oscillatory solution should be used for the prediction.
In our previous study, we proposed a method using
Runge-Kutta method and normalized equations of latticeoscillator[5]. However, the prediction was used numericalequations without features of the OP-Amps. Further, an input
signal source was optimized for normalization of circuit equations. Therefore, to make an actual circuit of this prediction
system is hard.
In this study, the lattice oscillators constructed by van
der Pol oscillators are simulated using Runge-Kutta method
and PSpice, and predict the time-series data of the Chua’s
circuit. Simulations by PSpice include features of OP-Amp,and include the input signal source that is constructed by only
a voltage source. Additionally, values of all elements is usedrealistic values.
The prediction results of time-series data using PSpiceare compared from the prediction results using Runge-Kutta
method.
We show possibility of predictable by the actual circuit ofour method by using PSpice.

26-28 May 2010, Dresden, Germany

106

Dept. of Electrical and Electronic Eng.,
Tokushima University,
2-1, Minami-Jyosanjima, Tokushima, JAPAN.

II. Prediction using Runge-Kutta method
A. Circuit model for Runge-Kutta method
In this study, 9 van der Pol oscillators are coupled as alattice by inductors(see Figs. 1(a) and 1(b)), and we predicttime-series data by this circuit. In this method, an arbitrary
signal f (t), that is synthesized and predicted, is presented
as a current of an inductor Lout2 , because circuit equations
should be normalized and simplified(see Fig. 1). Therefore,
an input part is substituted the inductor Lout2 and an arbitrary
voltage signal source. An output signal h(t) of the voltage
signal source is shown as follows:1h(t) =
f (t)dt.
(1)
Lout2
In this method, we can predict in ideal condition because
the measurement instrument and input power source are considered. Each oscillator are called OSC(k, l). In our method,
future time-series data are predicted from past time-series databy a circuit which is like a filter. Therefore, to input past timeseries data from outside is needed. The time-series data is
input to each oscillator through each outside inductor Lout .
Each oscillator includes a negative resistance, an inductor
L, and a capacitor C. Prediction results are gotten as timeseries data of a voltage of OSC(k, l). We want to analyze thiscircuit by using Runge-Kutta method. Therefore, normalized
equations of this circuit should be used. The parameters of
this circuit are set as Figs. 1(a) and 1(b). The all oscillators
are used same parameters, and the all coupling inductors areused same inductance. All inductors Lout2 for inputting timeseries data are used one value. The voltage of each oscillator
is called v(k,l) . Moreover, the electric current that flows to the
inductor of each oscillator is named i(k,l) . An arbitrary timeseries data are input from an inductor Lout2 and voltage signal
source. We control the voltage signal source to be a current of
Lout2 is the arbitrary time-series data synthesis and prediction
time-series data are output as the current i(1,1) of OSC(1,1).
The nonlinear resistance of each oscillator is approximated asfollows.
f (v(k,l) ) = −g1 v(k,l) + g3 v3(k,l) .

(2)

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

The circuit equation is normalized by the following equations.


√
Cg1
g1
x(k,l) , v(k,l) =y(k,l) ,t = LCτ, i(k,l) =
L
3Lg3
3g3
(3)

L
L
.
, ε(k,l) = g1α(k,l)−(m,n) =L(k,l)−(m,n)
C

OSC(0,2)

OSC(0,1)

Lout 2

OSC(1,0)L(1, 0 ) −( 2, 0 )L( 0, 2 ) −(1, 2 )L( 0,1) −(1,1)

Lout 2

Lout 2

L(1, 0 ) −(1,1)
Lout1

Lout 2

Lout1

Lout1

Lout1

( 0 , 0 ) − (1, 0 )[corner]L( 0,1) −( 0, 2 )L( 0, 0 ) −( 0,1)
OSC(0,0)OSC(1,1)

L(1,1) −(1, 2 )OSC(1,2)

L(1, 2 ) −( 2, 2 )L(1,1) −( 2,1)

Lout1

arbitrary voltage signal source

Lout 2L( 2,1) −( 2, 2 )rdx(k0 ,l0 )OSC(2,2)
OSC(2,0)OSC(2,1)L= y(k0 ,l0 ) ,
C
L
dτ
L
L
L
dy(k0 ,l0 )rrL= −x(k0 ,l0 ) + α(k1 ,l1 )−(k2 ,l2 ) (x(k3 ,l3 ) − x(k0 ,l0 ) )L
Lout 2(4)dτ+ α(k4 ,l4 )−(k5 ,l5 ) (x(k6 ,l6 ) − x(k0 ,l0 ) ) + αout(k0 ,l0 ) (x s − x(k0 ,l0 ) )(a) Lattice circuit.
(b) Van der Pol oscillator.
1+ ε(k0 ,l0 ) (y(k0 ,l0 ) − y3(k0 ,l0 ) ).
Fig. 1. Circuit model for Runge-Kutta method.
3(k0 , l0 , k1 , l1 , k2 , l2 , k3 , l3 , k4 , l4 , k5 , l5 , k6 , l6 )=⎧part of synthesis⎪
top − le f t : (0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0),
⎪
⎪
⎪
⎪
⎪
top − right : (0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 1, 2, 1, 2),original data f(t1)⎨Time of original data t1 [τ]
⎪
⎪
25T
⎪
bottom
−
le
f
t
:
(2,
0,
2,
0,
2,
1,
2,
1,
1,
0,
2,
0,
1,
0),
17T 20T⎪⎪⎪0
10T⎩ bottom − right : (2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2).
1( k ,l )OSC(k,l)( 2 , 0 ) − ( 2 ,1)

out1

out 2

out1

TL082

out1

out 2

2 ( k ,l )1( k ,l )30T(a) (b) (c) (d) (e) (f) (g) (h)[center(k = 1 and l = 1)]dx(1,1)= y(1,1) ,dτ0Time of synthesized data t2 [τ]
10T
7T
15T
dy(1,1)= −x(1,1) + α(1,0)−(1,1) (x(1,0) − x(1,1) )dτpart of prediction
prediction (synthesis) data g(t2)+ α(1,1)−(1,2) (x(1,2) − x(1,1) ) + α(0,1)−(1,1) (x(0,1) − x(1,1) ) (5)+ α(1,1)−(2,1) (x(2,1) − x(1,1) )Fig. 2. Relation between synthesis time-series data and original time-series
1data (Runge-Kutta method).
+ ε(1,1) (y(1,1) − y3(1,1) ).
3
by time period of arbitrary length T , and are predicted. Our[side]
system is predicted future time-series data from past timeseries data. The details of procedure of our method are shown
dx(k0 ,l0 )= y(k0 ,l0 )as follows, and an example of synthesis and prediction aredτdy(k0 ,l0 )shown in Fig. 2.
= −x(k0 ,l0 ) + α(k1 ,l1 )−(k2 ,l2 ) (x(k3 ,l3 ) − x(k0 ,l0 ) )dτ(6)1) The original data from f (0) to f (10T ) is added to our+ α(k4 ,l4 )−(k5 ,l5 ) (x(k6 ,l6 ) − x(k0 ,l0 ) )system, and data from g(0) to g(10T ) is synthesized at+ α(k7 ,l7 )−(k8 ,l8 ) (x(k9 ,l9 ) − x(k0 ,l0 ) + αout(k0 ,l0 ) (x s − x(k0 ,l0 ) )the same time. The original data are shown in Fig. 2.
1+ ε(k0 ,l0 ) (y(k0 ,l0 ) − y3(k0 ,l0 ) ).
The synthesis data are shown in Fig. 2. In the Fig. 2,
3time axis of synthesis data diﬀers from time axis of(k0 , l0 , k1 , l1 , k2 , l2 , k3 , l3 , k4 , l4 , k5 , l5 , k6 , l6 , k7 , l7 , k8 , l8 , k9 , l9 )=original data.
⎧⎪
top : (0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 1, 1, 1, 1),⎪
2)Thesynthesized data g(t):(7T ≤ t ≤ 10T ) are adjusted⎪⎪⎪⎪
le f t : (1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0),⎨
tovaluesclose to f (t):(17T ≤ t ≤ 20T ) by changing
⎪
⎪
⎪
1,
1,
0,
2,
1,
2,
0,
2,
1,
2,
2,
2,
2,
2),
right
:
(1,
2,
1,
1,
1,
2,
⎪
each
parameter
and eighteen initial values of oscillators.
⎪⎪⎩ bottom : (2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2).
Each initial value is set plus f (10T ) or minus f (10T )independently.
Each nonlinearity and coupling parameter must be adjusted
3) The original time-series data from f (10T ) to f (20T ) is
individually, because we synthesize and predict time-series
added to the adjusted system, and time-series data fromdata by these circuit equations. Therefore, each ε and α areg(10T ) to g(20T ) is synthesized at the same time. The
renamed as ε(k,l) and α(k,l)−(m,n) .
synthesis data g(t):(10T < t ≤ 20T ) are considered as
prediction data.
B. Prediction method using Runge-Kutta method
4)
A prediction period is fixed from 10T to 15T because
We use a time-series data of Chua’s circuit as an original
to
predict for long time is diﬃcult.
time-series data. Parameters of Chua’s circuit on pp.24 of [6]are used. The original time-series data is assumed as f (t). A
prediction of the original time-series data is assumed as g(t).
The original time-series data of continuous data are divided

26-28 May 2010, Dresden, Germany107

If the time-series data g(t):(10T < t ≤ 15T) is close tof (t):(20T < t ≤ 25T), we can consider that g(t):(10T < t ≤15T) predicts the original time-series data(see Fig. 2). We

20T

18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

R2

C1
1n

9Vdc

R7
10k

V6

U2A
3 +

C2
1n

9Vdc

V+ 8

V10

U3A
3 +

OUT1
2 4
TL082 VR16

V11

C4
1n

1

10

9Vdc

R18
10k

V12

U8B
5 +

1

C5
1n

V+ 8

OUT7

6 4
TL082 V-

R30
R28
1k

2
L13
1mH

OUT1
2 4
TL082 VR22

2
L7
1mH

9Vdc
C7
1n

9Vdc

R26
10

R32
10k
U6A
3 +

V19

2 4
TL082 V-

1

R33

10k

0

OUT1

R31
1k

2
L14
1mH

Observation part of synthesis
L19
100mH

R40
10
L12

R27

2
100mH

10

9Vdc

R35
10k
V20

U5B
5 +

OUT7
6 4
TL082 V-

V21

1

R36

10k

R34
1k

0

Fig. 3.

time-series data.

1

9Vdc
C8
1n

V25
0

C6
1n

2

1

V18

9Vdc

OUT7

4
TL082 V-

10k

R41
10

2
100mH

2
R53 R54
10
10

1

L20
100mH

V+ 8

R29
10k
U4B
5 +

U10A
3 +

V15

2
R42
10

V17

V14

0

L21
100mH

9Vdc

9Vdc

6 -

R21
10k

R23
1k

2

V16

U6B
5 +

1

1

2
R52
10

R39
10

9Vdc V24

10

0

L11

2
R51
10

0

R25

2
100mH

1

1

9Vdc

L9

10k

R20
1k

0

2
R50
10

L18
100mH
2

9Vdc

2
L6
1mH

OUT7
6 4
TL082 VR19

V13

10k

R17
1k

2

1

R38
10

R24

9Vdc

2
L5
1mH

2

c:\13.txt

L17
100mH
2

2
100mH

2

R43 R48 R49
10
10
10

V22

1

V+ 8

R15
10k

2
C3
1n

1

0

R37
10

L8

1
1
1
1
1
1
1
L22 L23 L24
L25
L26
L27
L28 L29100mH
100mH
100mH 100mH 100mH 100mH 100mH
100mH

10k

R13
1k

L16
100mH

1

2
L4
1mH

OUT7
6 4
TL082 V-

V9

0

1

1

U9B
5 +

R12

2

9Vdc

9Vdc

R11
10k

V8

10k

R9
1k

0

10

1

R8

10k

R3
100k

R14

2
100mH

9Vdc

2
L2
1mH

OUT1
2 4
TL082 V-

V7

1

L10

V+ 8

9Vdc

2
L1
1mH

1

10

V+ 8

OUT1
2 4
TL082 V-

V2

R10

V+ 8

U1A
3 +

2
100mH

V+ 8

9Vdc

V1

V+ 8

9Vdc

L3

V+ 8

1
R1
10k

2
L15
1mH

C9
1n

1

10k

0

Circuit model for PSpice(OP-Amp:TL082).

Part of synthesis

predict long time data repeating this method.

Part of prediction

Time of original data t1[τ]III. Prediction using PSpice0

In this section, we predict the same time-series data by
PSpice.

5T

α

10T

7T

γ

A. Circuit model for PSpiceThe prediction circuit for PSpice is shown in Fig.3．This
circuit and above circuit are same circuit basically. However,in this circuit, we conscious an actual circuit. This circuit
includes registers of coupling inductors. Original time-series
data are input from an arbitrary voltage signal source. Synthesis and prediction time-series data are output as the voltage
v(1,1) of OSC(1,1). The capacitances are used from 1500nF
to 1nF. The resistances are used from 10MegΩ to 10Ω. The
inductances are used from 150mH to 0.1µH.

Prediction (synthesis) data g(t1)Original data f(t1)

0

T

2T

3T

4T

Time of synthesized data t1[τ]B. Prediction method using PSpice
We basically predict time-series data by same method,which is shown in above section. The details of procedure of
our method are shown as follows, and an example of synthesis
and prediction are shown in Fig.4.
1) The original data from f (0) to f (2T ) (α) is added to oursystem, and data from g(0) to g(2T) (β) is synthesized
at the same time(see Fig.4).
2) The synthesized data g(t) : (T ≤ t ≤ 2T ) are adjusted tovalues close to f (t) : (7T ≤ t ≤ 8T ) by changing many
parameters and many initial values of oscillators. We
adjust all parameters as like full solution search. However, when the synthesis data is very close to original
data, we stop to adjust the parameters. Therefore, we
can think a probability of global minimum, which our

26-28 May 2010, Dresden, Germany

δ

β

108

Fig. 4. Relation between synthesis time-series data and original time-series
data (PSpice).

results is, is low. Each initial values is set plus f (6T ),minus f (6T ) or 0, independently.
3) The original time-series data from f (t) : (2T < t ≤3T )(γ) is added to the adjusted system, and time-series
data from g(t) : (2T < t ≤ 3T )(δ) is synthesized at thesame time. The synthesis data g(t) : (2T < t ≤ 3T ) areconsidered as prediction data.
4) A prediction period is fixed from 2T to 3T because topredict for long time is diﬃcult.
If the time-series data g(t) : (2T < t ≤ 3T ) is close to f (t) :(8T < t ≤ 9T ), we can consider that g(t) : (2T < t ≤ 3T )predicts the original time-series data (see Fig.4). We predict18th IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010)

2.5

2.5

2

2

Original data

1

Runge-Kutta method

1.5

Voltage [V]Voltage[V]Original data
1.5

0.5
0
-0.5

0.5
0
-0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5

0

PSpice result

1

2

4

6

8

10

-2.5

12

0

1

2

3

4

5

6

7

8

9

10

Time [μs]Time[μs](a) Prediction results of Runge-Kutta method.

(b) Prediction results of PSpice.
Fig. 5.

Prediction results

TABLE I
Comparison of mean absolute errors and error rates.
Runge-KuttaPSpiceMean absolute errors
Error rate[%]Mean absolute errors
Error rate[%]0-T
0.010766
0.54090.032161
1.5909T-2T
0.005188
0.85520.0371125.2871

2T-3T
0.009479
0.98610.035688
4.31393T-4T
0.049336
2.55430.006525
0.2968long time data repeating this method.
IV. Comparison between prediction results of Runge-Kutta
method and PSpice.
The composite prediction result using Runge-Kutta method
and the original data are shown in Fig.5(a). The composite
prediction result using PSpice and the original data are shownin Fig.5(b). We can confirm that the result of PSpice is better
than the result of Runge-Kutta method. However, the RungeKutta method synthesizes during 3T , and predicts during 5T .
In other hand, PSpice synthesizes during T , and predicts
during T . We can think that to obtain good results by using
PSpice are easier than to obtain good results by using RungeKutta method. However, the adjustment of our method using
PSpice is diﬃcult.
The error rates are shown in Table I.
The mean error rate is calculated as follows:
Mean absolute errors
T he error rate =×100 [%](7)Maximum amplitude
The result of PSpice is better than the result of RungeKutta method in 7 domains of 10 domains. The mean error
rates of PSpice results are large in the domain where change
is sharp. Because the adjustment of our method using PSpiceis diﬃcult. In the Runge-Kutta method, a mean error rate
between the prediction time-series data and the original timeseries data is 3.42%. In PSpice, a mean error rate is 1.80%.
V. Conclusion
In this study, the lattice oscillator constructed by van der
Pol oscillators was simulated using Runge-Kutta method andPSpice, and we predicted the time-series data of the Chua’s
circuit using the lattice oscillator. Moreover, the prediction

26-28 May 2010, Dresden, Germany

109

4T-5T
0.024742
1.25190.001489
0.0749

5T-6T
0.043291
3.05640.028941
2.18906T-7T
0.131468
6.49870.012496
0.57787T-8T
0.017386
0.81300.006792
0.3102

8T-9T
0.084017
6.64710.040905
3.0528

9T-10T
0.054813
11.0220.001294
0.3540results of time-series data using PSpice were compared withresults using Runge-Kutta method. The mean error rates inour methods using Runge-Kutta method and using PSpicewere 3.42% and 1.80%, respectively. Therefore, we can saythat we were able to predict well. In other words, we canthink that possibility of predictable by the actual circuit ofour method is shown. However, synthesis using Runge-Kutta
method was easier than synthesis using PSpice, because thechanged parameter is few, and the time eﬃciency is good.
The mean error rate in our system using Runge-Kutta method
enough is small. Therefore, we can think that prediction
eﬃciency is good when the Runge-Kutta method is used.
VI. Acknowledgements
This research is supported by the Grants-in-Aid for Young
Scientific Research (B) (No. 19760270) from the Japan Society for the Promotion of Science.
References[1] Sapankevych. N and Sankar. R, “Time Series Prediction Using Support
Vector Machines: A Survey,” IEEE COMPUTATIONAL INTELLIGENCEMAGAZINE, May 2009.
[2] T. Matsumoto, Y. Nakajima, M. Saito, J. Sugi and H. Hamagishi,“Reconstructions and Predictions of Nonlinear Dynamical Systems: A
Hierarchical Bayesian Approach,” IEEE TRANSACTIONS ON SIGNAL
PROCESSING, vol. 49, no. 9, Sept. 2001.
[3] L. L. Bonilla, C. J. Pérez Vicente and R. Spigler, “Time-Periodic
Phases in Populations of Nonlinearly Coupled Oscillators with Bimodal
Frequency Distributions,” Physica D: Nonlinear Phenomena vol. 113,
issues 1, pp. 79–97, Feb. 1998.
[4] M. Yamauchi, M. Okuda, Y. Nishio and A. Ushida, “Analysis of PhaseInversion Waves in Coupled Oscillators Synchronizing at In-and-AntiPhase,” IEICE Trans. Fundamentals, vol.E86-A, no.7, pp.1799-1806,
July 2003.
[5] T.Imoto, S.Yamane and M.yamauchi“Prediction of Chaos Time-series
Data using Coupled Oscillators System,” Proc. of SSJW’08, pp.32-35,
2008.
[6] K. Aihara, Chaos, Saiensu–sha Co., Ltd. Publishers, 1990(in Japanese).