Mutation rate

Journal of International Society for Environmental Management,
Printed in Japan. All right reserved
Vol.1, No.1, March 1, 2010.
Copyright© 2010 ISEM
A study on Reconstitution in the Forecasting of Insufficient
Chaotic Time Series Dataand the Application of
GA –Neuro System to the Data
Masafumi IMNAI *1
Toyohasi Sozo Senior College 20-1 Matsushita, Ushikawa-cho
Toyohasi-city 440-8511,Japan
Phone:＋81-532-54-2111, FAX:+81-532-55-0805
E-mail: [email protected]
Tomonori NISHIKAWA *2
Ryutsu-keizai University 120Hirahata,
Ryugasaki,301-8555,Japan
Phone：+81-297-297-0001, FAX: +81-297-0011
E-mail: [email protected]
Abstract
This paper deals with the forecasting of chaotic time series data in which the amount of data is insufficient for
normal reconstitution of the both the data its space. It is shown that the forecasting is possible in comparatively low
dimensional reconstitution space by adjusting the delay time, and that the GA-neuro system is very effective in
order to do the forecasting in this desirable reconstitution space.
Key Words: Chaotic time series, GA-Neuro system, forecasting, reconstitution dimension, optimization
Introduction
This method will be shown to be effective in effective in that it can both reconstitute the data in the
forecasting of chaotic time series data and do the forecasting in the reconstitution space. When
forecasting actual data in the reconstitution space, while seeing predictability, trial-and-error reconstitution dimensionality and delay time will be decided. Though also based on applying forecasting method, when the amount of data which can be especially utilized is not sufficient, the
problem arises that values in reconstitution dimensionality and delay times are increased.
This paper shows that forecasting is possible in the comparatively low-dimensional reconstitution
space by adjusting the delay time for chaotic time series data where the data is insufficient, and that
the GA-neuro system is very effective in order to do the forecasting in this desirable reconstitution
space. Sch reconstitution is comparatively possible by low-dimensional, if the correlation dimensionality is calculated through change of the delay time for the chaotic time series data, and if it
converges on the value in which the correlation dimensionality is small. Next, the optimization of the
structure of the neural network is done through the application of the GA-neuro system in the
reconstitution space, where the forecasting is done. It will be shown that the forecasting accuracy is
greatly influenced even then by delay time τ and reconstitution dimensionality. In addition, it will be
shown that the forecasting of chaotic time series data can be done including the desirable
reconstitution space by expanding the GA-neuro system in order to carry out the search including
delay time τ and reconstitution dimensionality.
------------------------------------------------------------------------------------------------------------------------------------------------------------------Rec
eived on Oct. 18, 2003. Received again on Dec. 25, 2003. Accepted on January 13,2004.
*1 Assosiate professor, The Graduate School of management infomatics, Toyohashi Sozo Senior College.
*2 Professor, The Graduate School of Logistics, Ryutsu Keizai University.
Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
1. Chaotic time series data and reconstitution
Generally for chaotic time series data, the technique based on the reconstitution theory of Takens is
effective. Concretely, though original data is mapped in delay time τ to the reconstitution space, and
the forecasting is
done in this reconstitution space, and the decision of the dimensionality of reconstitution space and
delay time τ will accomplish autocorrelation, correlation dimensions, etc. as a clue
trial-and-error[1][2]. Moreover, the number of reconstitution dimensions and the values at lag time
become problems when there are insufficient data applied, and the when the number of dimensions
cannot be properly determined.
1.1 Application of GA-neuro system to currency exchange data
The data for this sample is the day order data in t-120 terms of TTS rate ( Telegraphic Transfer
Selling Rate : Unit yen ) of Japanese yen vs. U.S. dollar in Japan from October 1, 2002 to March 31,
2003. The study data is assumed to be100 terms of the f, and 20 terms of the latter half are assumed to
be forecast data.
To treat it using the neural network, data is scaled within the range of 0～1, taking the data after the
scaling as Figure1, the autocorrelation is shown in Figure 2. Delay time τ in the case of the
reconstitution from this the autocorrelation of time series data has the extreme value of t = 22, and it
can be said that under 21 may be made to be a standard. In this paper, the orbit is reconstituted based
on the embedding theorem of Takens in the state-space using the time difference axis. The state-space
vector u (t )  ( y(t ), y (t   ),  , y (t  (m  1) )) of the dimension is newly reconstituted using the fixed
difference with the embedding theorem of Takens of every delay time τ from the observed time series
data. If the dimensionality of the original object system is represented by d, and taking the
reconstituting dimensionality as m, it is m>2d+1, it is guaranteed that the embedding of the
conversion from observation time series data to the reconstitution state-space is sufficient [3].
1
0.9
0.8
0.8
0.7
0.7
autocorrelation of x(t)
1
0.9
x(t)
0.6
0.5
0.4
0.3
0.2
0.5
0.4
0.3
0.2
0.1
0
0.6
0.1
0
20
40
60
80
100
120
0
0
t
20
40
60
80
100
t
Figure1 scaled exchange data
Figure 2 autocorrelation
2
120
Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
1.2 Fractal dimensionality
The fractal dimension is the one that the concept of a usual dimension was enhanced even a region
of the noninteger, and there are a hausdorf dimension, a capacity dimension, an information
dimension, and a correlation dimension, etc.[4]. This paper discusses using the correlation dimension
of Grassberger and Procaccia from which the calculation was often used comparatively for an easy,
actual time series data proposed. The correlation dimension is obtained using the correlation integral
C defined in equation (1). Where r denotes the distance which is optionally determined, C(r) denotes
||xi-xj|| with the vector, and it requires the proportion with the whole in search of the amount under r.
The correlation dimension D can be obtained from the inclination of the graph of the plot of
log Cr  to log r to the model of C r   r D . The possibility in which the data becomes chaotic is
shown, if it shows the value of the decimal dimension especially, when it is saturated for the fixed
value with the correlation dimension.
C r  
1
N2
 H r 
n
n
i 1 j 1
xi  x j

1 x  0
H x 
0 x  0
where
(1)
Correlation dimension of the exchange data is obtained actually here. It is shown that it
reconstituted the data from the one dimension to the 20th order element at delay time τ=1 and plotted
the relationship between correlation integral log Cr  and log r in each dimension in the graph in
Figure 3. The correlation dimension though it is general that it is required from the gradient in making
the correlation integral to be a standard, in this study, log r is made to be a standard, and there are
small using data numbers, and the comparison becomes directly difficult that delay time τ is increased,
and it requires the correlation dimension from the part of 1.0  log r  1.4 , and it is shown in Figure
4.
4
2.6
3.5
2.4
2.2
correlation dimension
3
log C(r)
2.5
2
1.5
1
2
1.8
1.6
1.4
1.2
0.5
1
0
0.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
log r
Figure 3 correlation integral
2
4
6
8
10
12
14
16
18
20
reconstitution dimensionality
Figure 4 correlation dimension
（The result of requiring from log r ）
3
Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
In the same way, the range in the delay time is increased with τ=2～20, and the correlation
dimension is obtained from the part of 1.0  log r  1.4 . Then, the result is shown in Figure 5 in
respect of the x shaft in respect of the delay time τ, the y shaft in respect of making reconstitution
dimensionality and the z shaft to be the correlation dimension. Since the number of data are small,
when the delay time τ increases, it is not possible to obtain the large reconstitution dimensionality
which is displayed as a zero in the present case. It is proven that it is saturated in the decimal
dimension under three dimensions in which the numerical value of the correlation dimension is low
in Figure 5 and Figure 6. And, it is proven that the value equal to the case in which it is reconstituted
in higher dimension in small delay time, if one adjusts delay time in low dimension comparatively.
The above shows I- the possibility in which the data used in this study becomes chaos, and it can be
said that the comparatively low-dimensional reconstitution space is able to be configurative, if the
delay time is adjusted.
Figure 5 Delay time and correlation
Figure 6 Delay time and correlation
dimension
dimension
(It is reconstituted to the 1～5 dimensions ) (It is reconstituted to the 1～20 dimensions )
2 .forecasting by GA- neuro systems in the reconstitution space
2.1 Structural determination of neural network by the GA-neuro system
The Number of elements of input layer-intermediate layer needed to do the forecasting even
using a neural network, forecasting accuracy greatly changes by the structure of the network, if
reconstitution dimensionality and delay time τ were able to be decided. That is, being separating from
the decision problem of the reconstitution space, it is required that the decision problem of the
reconstitution space decides the network structure trial-and-error, while identification error,
forecasting error are differently observed on the basis of the autocorrelation[5][6]. Here, it is
considered that a GA-neuro system is applied for the forecasting in the reconstitution space, and that
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Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
it decides the structure of the neural network. GA-neuro system of our study decides the structure of
the neural network on the based on the gene. The predictive value of neural network after the learning
is made to be the fidelity, the structure of neuro is searched and intends to decide it. The conceptual
scheme of the GA-neuro system is shown in Figure 7.
GA Module
Gene :
Fidelity：
Input-Number of middle
layer
forecasting error
Neuro –Module n
Neuro –Module 1
Figure 7 Conceptual diagram of the GA-neuro system
By limiting in the case of the reconstitution three-dimensional, the summary of the forecasting by
the GA- neuro system is explained. The unit number of the out layers is fixed at 3 in order to estimate
the initial stage tip in the reconstitution space. The number of units of the input layer and middle layer
is coded directly to the gene. It is assumed that the first half is a number of units of input layers and
that the latter half is a number of units of middle layers. The fitness multiplies the forecasting error,
and power scaling using the reciprocal is applied, and the selection system uses the elite saving jointly
with the fitness proportion system. The learning frequency of GA-neuro - is made to be 5000 times,
and delay time τ is made to be 1～20, each 10 time forecasting is done in each every delay time
τ. The individual of the best forecasting in 5 generation is made to be a solution of each trial． The
mean value of forecasting error in the trial of 10 times, and the best forecasting error and that time
input layer and intermediate layer number are shown in Table.2. Here, it is meant that it is forecast
x(t-2τ), x(t-τ), and x(t) after one term by using the data at two periods of the past in the space when
becoming several six of the input layers for instance because it is a forecast in the space of three
dimensions, and will have the meaning only by x(t) as an actual forecast value. To represent the
average value of the forecasting error of ten times and the best forecasting error, it is shown the result
to of plot the x-axis to be a delay time and the y-axis to be a forecasting error in Figure 8
Table 1 parameter of the GA-neuro system.
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Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
Number of gene
Number of
preservatin
Mutation rate
Gene length Number of generations
20
3
0.10
8bit
5
Learning coefficient Number of
learning .
Number of the
input layer.
Number of
the middle
Stop rate
0.1
3～48
1～16
0.03
5000
Table 2 Forecasting result in the reconstitution space
Mean
value.
Input
layer
39
24
27
45
48
48
39
33
27
39
Forecasting
error
0.05558
0.05762
0.05447
0.05250
0.04426
0.04470
0.04783
0.04824
0.04829
0.04896
Middle
layer
2
7
4
8
9
13
14
14
16
5
Forecasting
error
0.05255
0.05186
0.05272
0.04871
0.04148
0.04175
0.04426
0.04402
0.04637
0.04460
The best value in the trial of Mean
10 times.
value.
Delay time
τ
11
12
13
14
15
16
17
18
19
20
Input
layer
18
18
24
18
12
6
18
15
15
18
Middle
layer
16
16
6
3
3
3
3
4
5
12
0.075
0.07
0.065
forecasting error
Delay time
τ
1
2
3
4
5
6
7
8
9
10
The best value in the trial
of 10 times.
0.06
0.055
0.05
0.045
0.04
2
4
6
8
10
12
delay time
6
14
16
18
20
Forecasting
error
0.04353
0.05001
0.04523
0.03588
0.04025
0.04783
0.05524
0.06062
0.06233
0.05633
Forecasting
error
0.04889
0.05259
0.05105
0.04130
0.04609
0.05699
0.06390
0.06898
0.07243
0.06626
Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
Figure 8 Delay time and forecasting error
It is proven that the forecasting accuracy changes from Table.2 and Figure 8 by delay time τ, x(t)
which is the predictive value shows the value which is the best in case of delay time τ =14. It is
desirable that it is set within delay time τ= 5～15, when the reconstitution dimension is
three-dimensional, in short, it can be said that there is a range of delay time τ in which the forecasting
accuracy is good. The example of x(t) which is the best forecasting value of the result in the
reconstitution space in case of delay time τ =14 in the following forecasting value of one term and
actual forecasting value is shown in Figure 9 and Figure 10.
1
1
0.8
0.8
0.7
0.6
0.6
x(t)
x(t)
0.9
0.4
0.5
0.4
0.2
0.3
0
1
0.8
1
0.6
0.8
0.6
0.4
0.1
0.4
0.2
x(t-τ)
0.2
0
0.2
0
0
x(t-2τ)
Figure 9 The Forecasting value in the
reconstitution space ( delay time τ=14 )
0
20
40
60
80
100
120
t
Figure 10 Real data and the
following forecasting value of one term
2.2 Structural determination of a neural network by the GA-neuro system in various
reconstitution spaces
The reconstitution was carried out here three-dimensional, and the forecasting was done. In
addition, the relationship between delay time τ and forecasting accuracy by the difference between
the reconstitution dimensionality is shown. The reconstitution dimensionality is made to be 2, 4 and 5
dimensions. The following forecasting of one term in the reconstitution space is done. The
parameter of GA- neuro in each reconstitution dimensionality is shown in Table 3.
Table.3 The parameter of GA- neuro of the forecasting in the reconstitution space
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Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
Number
of gene
Number of Mutation
preservation rate.
Gene
length.
20
3
0.10
8bit
Learning
coefficient
Learning
times
Number
of input
layer
3～48
Middle
of middle
layer
1～16
0.1
5000
Number
of input
layer
Numb
er of
generat
ion
5
2
dimensions
3
dimensions
4
dimensions
5
dimensions
Stop
error
0.03
2～32
Middle
of
middle
layer
1～16
Number
of
output
layer
2
3～48
1～16
3
4～64
1～16
4
5～80
1～16
5
In making delay time τ to be 1～20 in each and every reconstitution dimensionality, the each 10
time forecasting is done. The average values of the forecasting error of 10 trials and the best
forecasting error in that are shown in Figure 11-14. In Figure 11,13, the x, y, and z axes were
respectively delay time, reconstitution dimensionality, and forecasting error, and Figures 12and14
observed each graph from the top. It is proven that the forecasting accuracy of forecasting value x(t)
changes by delay time τ. The result at delay time shows that the delay time τ=10～16 in 2nd ,τ=5
～15 in 3rd ,τ=4～8 in 4th, and τ=3～8 in 5th dimensions represent a good results, respectively. It
can be said that the range in desirable delay time τ decreases, when the reconstitution dimensionality
rises. The reconstitution dimensionality in which the forecasting accuracy is the best becomes 3,
delay times of 14, number of input layers of 18, middle layer number of 3, forecasting errors of
0.03588.
5
reconstitution dimensionality
0.09
forecasting error
0.08
0.07
0.06
0.05
0.04
5
4
3
20
4
15
10
3
reconstitution dimensionality
5
2
0
delay time
2
2
4
6
8
10
12
14
16
18
20
delay time
Figure 11 Forecasting error by reconstitution Figure 12 Forecasting error and
dimensionality and delay time
reconstitution dimensionality.
（The mean value of the best individual）
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Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
5
reconstitution dimensionality
0.08
forecasting error
0.07
0.06
0.05
0.04
0.03
5
4
3
20
4
15
10
3
5
reconstitution dimensionality
2
0
delay time
2
2
4
6
8
10
12
14
16
18
20
delay time
Figure 13 Forecasting error by reconstitution
reconstitution dimensionality and delay time
（The best value of the best individual）
Figure 14 Forecasting error and
reconstitution dimensionality and delay
time
2.3 Extended forecasting by GA-neuro for which searches and including reconstitution
dimensionalty and delay time
In the previous section, it becomes clear that one can decide the number of elements of input
layers-middle layers by GA- neuro in various reconstitution spaces, namely the structure of neural
network, and that delay time and reconstitution dimension in which the forecasting of which the
accuracy is good as a result is possible do exist. Then, in this section we forecast using expanded an
GA-neuro in order to carry out the search by adding reconstitution dimensionality and delay time τ
in this knot in the structure of the neural network by GA for the action. As a result, it is shown that
GA- neuro which was expanded for the optimization of structure of the neural network, reconstitution
dimensionality, delay time τ is effective. In the difference between GA- neuro in the previous section ,
we did direct coding of reconstitution dimensionality and delay time τ in addition to the unit number
of input layer and middle layer to the gene and used the uniform cross propodite GA- neuro. The
parameter of expanded GA- neuro is shown in Table4 and the conceptual diagram is shown in Fig.15.
However, the unit number of the output layer is assumed to be same with the reconstitution
dimensionality in order to estimate the initial stage tip in the reconstitution space. In making the
learning frequency of GA-neuro to be 5000 times, each 10 timesforecasting is done. The average
value of forecasting error in the trial of 10 times, the best forecasting error and the reconstitution
dimensionality at that time and delay time τ, input layers and intermediate layers are shown in Table 5.
In comparison with expanded GA- neuro to be searched including result of the GA- neuro of only
structural determination in the previous section and reconstitution dimensionality and delay time in
this section, it is proven that both the reconstitution dimensionality and delay time τ and structure of
the neural network which the forecasting error is small can be almost searched. In short, it can be said
that simultaneously, production of the reconstitution space and structure of the neural network can be
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Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
optimized by expanding the GA- neuro in order to carry out the search including the reconstitution
dimensionality and delay time τ.
Table4 The parameter of expanded GA-neuro system
Number of
gene
500
Number of
preservation
20
Mutation rate
Length of gene
0.10
14bit
Number of
generation
5
Learning
coefficient
0.1
Learning
frequency.
5000
Number of
input Layer
2～80
Number of
middle layer
1～16
Number of
output layer
2～5
10
Stop rate
0.03
Delay time
τ
1～16
Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
Gene. ：
Input number of
middle layer
GA Module
Delay timeτ
Fidelity：
reconstitution
dimensionalty
Forecasting error
Neuro―Module 1
Neuro―Module n
Figure 15 Conceptual diagram of expanded GA-neuro system
( Search including reconstitution dimensionality and delay time τ )
Table.5
Result of average value and best individual of the
forecasting error in the trial of 10 times
best individual in the trial of 10 times.
average value
reconstitution
dimensionality
delay time τ
Number of
input layer
Number of Forecasting
out put layer error
Forecasting
error
3
11
24
4
0.04245
0.03769
3 Conclusions
The consideration was carried out on the decision in respect of reconstitution dimensionality and
delay time as a problem in the case of the forecasting of the chaotic time series data in the
reconstitution space, especially, the case in which possible utilizing data number was not sufficient
was examined. First, we calculated the correlation dimensionality by the change of the delay time for
chaotic time series data in which the data number is not sufficient, and it was shown that the
forecasting is possible in the comparatively low-dimensional reconstitution space by reconstitution
dimensionality and that size itself in the delay time becomes a problem and that it adjusts the delay
time. Next, it was shown that the structure of the neural network was optimized by doing the
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Journal of International Society for Environmental Management, Vol.1, No.1, March 1, 2010
forecasting using the GA-neuro system in the reconstitution space. However, it was shown that the
range of τ that delay time τ and that it influences forecasting accuracy by the reconstitution
dimensionality and good forecasting value are shown even in it existed. In addition, it was clarified
that by expanding the GA-neuro system in order to carry out the search including delay time τ and
reconstitution dimensionality, it constituted the desirable reconstitution space for the chaotic time
series data, and that it could be forecast.
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