Download Chaos Forecasting Model for GDP Based on Neural Networks Error-Correction

Chaos Forecasting Model for GDP Based on Neural Networks
Error-Correction
AO Shan, TANG Shoulian
Economics and Management School, Beijing University of Posts and Telecommunications, P.R.China,
100876
Abstract: To perform a simulation forecast about the increasing rate for 2004 -2006 annual GDP based
on chaotic attractors, aiming at the shortcoming that chaotic time series can not fit the actual fluctuation of
small sample discrete data very well, especially for long-term economic forecast errors. This paper makes
use of BP neural network to predict the fitting errors above, and to correct the final results based on the
prediction. The three-year average relative error rate is up to 0.553%, the prediction accuracy has been
improved significantly.
key words: Non-linear dynamics; Neural networks; Chaos; Logistic map; GDP
1 Introduction
Socio-economic system is a complex nonlinear system, and for many years, economists often make
use of traditional linear theory in attempting to describe nonlinear economic system, which is not so
perfect. Regarding complicated nonlinear economic system as a simple linear systems, inevitably the
linear paradigm of economic analysis results in serious deviations, such analysis is of course invalid
[Chang Zhongze,2006].As the inherent characteristics of non-linear systems, Chaos is a widespread
phenomenon of nonlinear system. Domestic and foreign scholars have carried out many meaningful
research on socio-economic phenomenon in virtue of chaos theory, especially for GDP: Wang Chunfeng
(2007), by using phase space reconstruction of chaotic time series analysis, performed an analysis on the
time series of China's GDP (1978-2004); Liu Yunzhong (2004), through the phase space reconstruction of
chaotic time series, and using local forecasting methods, established a model to predict China's GDP
(1978-2000).
Through the comparison and analysis of the related literature, Chaos using nonlinear Logistic model
to forecast the GDP based on chaos theory is a mature and feasible method. But, as the important ways of
low-order Chaos dynamics prediction theory—Logistic attractor’s identification, to a great extent, rely on
changes of its state factor K (as shown in expression 8). Especially for small samples discrete
macroeconomic data, state factor K is vulnerable to data values and the amount of data change, and once
the K is set, Chaos dynamics prediction can not conduct a good fit forecast to the actual economic data’s
fluctuation. Especially for long-term economic forecasts, it can not achieve stabile forecast results, Yu
Jinghua (2002) and Liu Yunzhong (2004) have carried out the relevant empirical research. In order to
better fit and predict the small samples data’s fluctuation, in this paper, using Neural networks’ nonlinear
approximation capability to predict the chaotic dynamics’ fitting errors, and according to these prediction
values to correct the final prediction results.
2 Chaotic dynamics calculations for GDP
2.1 Modeling and analysis
In this paper, we consider the use of Logistic equation’s general form. In the case of state factor K has
been set or found, to calculate stability chaotic attractor. Therefore, we first need to deal with discrete
annual gross domestic product data, so as to get the Logistic equation’s general form.
GDPi is established for the GDP series of calendar year, for i= 1, 2, 3, 4 ... n; by doing their ring
pretreatment, we have:
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（1）
GDP i + 1
GDP i
Pi =
Constructing nonlinear regression model:
To eliminate the constant term c, let:
（2）
2
Pi+1P= aP
= idX− bPi ++ ec
i
i
Substituting (3) into (2):
X i +1 = AX i2 − BX i + C
（（34））
Where:
A = ad ,
B = b − 2ae,
C=
ae 2 − (b + 1)e + c
d
（5）
(b + 1) + ∆
2a
（6）
（7）
Let C = 0, we can obtain:
∆ = (b + 1) 2 − 4ac ≥ 0
e=
X i +1 = AX i2 − BX i
Its transformation, in the logistic equation general form is as under:
Yi+1 = KYi (1 − Yi )
Where:
Xi =
B
Yi ,
A
K = −B
（8）
（9）
2.2 (1979-2006) GDP empirical study
To better evaluate the level of development of the GDP, in this paper, to regard the GDP growth
index (according to comparable prices, previous year as the benchmark) as test object, and to transform
according to the above analysis. This paper performs analysis and forecasting, regard (1979-2003) GDP
growth index as the experimental data, and (2004-2006) GDP data as test and verification data. At first, to
perform nonlinear autoregression analysis on the data of 1979-2003 "growth index" in Table 1(as shown
in Figure 1).
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1.2
1.15
y = 1.7*x2 - 3.1*x + 2.5
1.1
1.05
data 9
quadratic
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
1.18
residuals
0.6
Quadratic: norm of residuals = 0.12516
0.4
0.2
0
-0.2
-0.4
-0.6
1.04
1.06
1.08
1.1
1.12
1.14
Figure 1: Quadratic nonlinear autoregression analysis for (1979-2003) growth index
We can obtain fitting parameters:
a = 1 .7 ,
b = 3 .1,
c = 2 .5
Quadratic curve fitting expression is as follows (in which residual norm for 0.12516):
2
Pi +1 = 1.7 Pi − 3.1Pi + 2.5
So we have:
∆ = (b + 1) 2 − 4ac = 0.256≥ 0
e=
B = b − 2ae = −1.506
Because:
For the general forms of Logistic
equation, when:
(b + 1) + ∆
= 1.3966
2a
K = − B = 1.506
（10）
（11）
（12）
1 < K = 1.506 < 3
1< K ≤ 3
（13）
K −1
K
（14）
Stable attraction point is:
So
Deduce from expression (3)-(9):
Yi =
K −1
= 0.33599
K
（15）
Pi 
→1.0929
（16）
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As from the above calculation: ultimate stabile attractor for Pi is 1.0929, so as to work out the error
values and the Normalized values of actual growth index for calendar year (as shown in Table 1).
Table 1: (1979 -2003) data summarization for chaotic attractor forecasting
and Neural network training
growth
index
1979
1980
1981
Pi
Error
value
Normalized
1.076
1.0929
-0.0169
0.3321
1.0781
1.0929
-0.0148
0.3503
ANN
forecast
ANN anti
normalized
Final
forecast
errors
1.0526
1.0929
-0.0403
0.1257
1982
1983
1984
1.0901
1.0929
-0.0028
0.4563
0.4355
-0.00517075
0.00237075
1.1089
1.0929
0.016
0.6225
0.6167
0.01539545
0.00060455
1.1518
1.0929
0.0589
1
0.9944
0.0582644
0.0006356
1985
1986
1987
1.1347
1.0929
0.0418
0.8496
0.8467
0.04150045
0.00029955
1.0886
1.0929
-0.0043
0.4436
0.4434
-0.0042741
-2.59E-05
1.1157
1.0929
0.0228
0.6822
0.6801
0.02259135
0.00020865
1988
1989
1.1127
1.0929
0.0198
0.6554
0.655
0.0197425
5.75E-05
1.0407
1.0929
-0.0522
0.0205
0.0174
-0.0526251
0.0004251
1990
1991
1992
1.0383
1.0929
-0.0546
0
0.0235
-0.05193275
-0.00266725
1.0919
1.0929
-0.001
0.4726
0.464
-0.001936
0.000936
1.1424
1.0929
0.0495
0.9174
0.9177
0.04955895
-5.895E-05
1993
1994
1995
1.1394
1.0929
0.0465
0.8914
0.8986
0.0473911
-0.0008911
1.1309
1.0929
0.038
0.8158
0.816
0.038016
-1.6E-05
1.1093
1.0929
0.0164
0.6258
0.6251
0.01634885
5.115E-05
1996
1997
1.1001
1.0929
0.0072
0.5449
0.5446
0.0072121
-1.21E-05
1.0928
1.0929
-0.0001
0.4805
0.4783
-0.00031295
0.00021295
1998
1999
2000
1.0783
1.0929
-0.0146
0.3526
0.358
-0.013967
-0.000633
1.0763
1.0929
-0.0166
0.3349
0.3043
-0.02006195
0.00346195
1.0842
1.0929
-0.0087
0.4042
0.4487
-0.00367255
-0.00502745
2001
1.0929
-0.0099
0.3941
0.4074
-0.0083601
-0.0015399
1.083
2002
1.0909
1.0929
-0.002
0.4633
0.4646
-0.0018679
-0.0001321
2003
1.0929
0.0073
0.5458
0.5116
0.0034666
0.0038334
1.1002
Data interpretation: GDP growth index from "China Statistical Yearbook 2006", the previous year as 100
3 Neural network to correct the chaos calculation error
3.1 Design and training for Neural network
Artificial Neural Network (ANN), as a nonlinear dynamic system, has acquired many examples of
successful application, such as the economic boom analysis, stock forecasting and other economic fields.
Neural networks have a good nonlinear approximation capability, which are widely used in nonlinear time
series analysis or nonlinear regression analysis. In order to improve the forecast accuracy that chaotic
dynamics predict the short-term fluctuation of discrete data, in this paper, based on the widely used and
mature BP neural network to forecast the chaos fitting errors, so as to amend the final forecast results for
GDP growth index.
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As for the Neural network input variables’ choice, adopt autoregressive time series forecasting
methods, namely:
Ti +3 = f (Ti , Ti +1 , Ti + 2 )
（17）
Therefore, BP Neural network prediction value is from the year of 1982, the input values are the
previous three-year Normalized Value of chaotic errors. The formulas for normalize are as follows:
xˆ =
x − x min
x max − x min
（18）
According to Kolmogorov theorem, BP Neural network in this study is of three-tier network with
only one hidden layer, and the hidden nodes finally confirmed through experiments are eight nodes. The
BP network makes use of Levenberg-Marquardt algorithm to train weights values, and because the
network output is one of predicted GDP growth indexes, so the network structure is:
[ 3 × 8 × 1]
To prevent over-fitting, network is set for 0.0002. Through the training putting (1979-2003)GDP
growth index as a training sample, to reach the training goal only after 51 epochs(training error curve
shown in Figure 2).
Figure 2: Neural network training iteration error curve
The final neural network weights and threshold:
IW(1,1)=[1.5314 -5.4533 -0.29842; 2.3094 6.3985 0.7702; -5.3522 4.5054 -6.3449; -9.5142 1.6748 2.3559; -5.6303
-1.336 -0.27969; -2.8929 10.554 1.6307; 5.2108 -7.2121 -1.3587; 3.0213 2.231 -2.0836]
LW(2,1)=[ -2.9771 5.5781 -3.3877 5.0333 3.5269 -4.8479 -4.1531 4.2552]
b(1)= [-4.0231; -7.5322; 4.301; 0.28366; 4.6124; -6.9275; 5.8693; -0.046569]
b(2)= [2.5956]
3.2 Neural Network Analysis and Error Correction
Using the trained Neural network to forecast the chaotic difference sequence for the years of 2004,
2005, 2006, and to carry out an anti-processing based on expression(18), the final amended forecast
absolute errors are calculated by the following formula :
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＝
（ ）
final forecast absolute error (chaotic error value) — (ANN anti Normalized Value)
19series
Through the above data processing: at first, to make use of Neural network’s nonlinear
prediction capability to predict chaotic possible errors sequence, so as to amend these errors. Related
calculated data can be obtained from Table 2.
Table 2: (2004 -2006) forecasting data after ANN error correction
2004
1.1008
1.0929
0.0079
0.551
0.6093
0.01455555
Final
forecast
error
-0.00665555
2005
1.1024
1.0929
0.0095
0.5644
0.49727
0.001840145
0.007659855
2006
1.107
1.0929
0.0141
0.6054
0.6405
0.01809675
-0.00399675
Increase
index
Error
value
Pi
Normalized
ANN
forecast
ANN anti
normalized
In this paper, to perform a horizontal comparative study about: nonlinear regression, chaotic
dynamics, Neural networks’ errors correction. Errors analysis for various methods is shown in Table 3.
The averages for absolute errors (AABE) indicate the absolute error precision, and the averages for
arithmetic errors (AARE) instruct three-year combined forecasting accuracy.
Table 3: (2004 -2006) error data analysis for various forecasting methods
Nonlinear
Chaotic
Neural networks’
Final relative
regression
dynamics
correction
error
0.0079
-0.00665555
-0.006046103
0.046328068
Errors in 2004
Errors in 2005
0.045113088
0.0095
0.007659855
0.006948345
Errors in 2006
0.041545792
0.0141
-0.00399675
-0.003610434
0.00248569
0.003218695
0.007614421
0.006907435
AABE
0.044328983
0.0105
0.006104052
0.0055349
AARE
0.044328983
0.0105
-0.000997482
-0.000902731
2004
2005
STD EV
0.05
0.04
0.03
0.02
0.01
0
-0.01
-0.02
Nonlinear regression
ANN correction
2006
Chaotic dynamics
0.05
0.04
0.03
0.02
0.01
0
-0.01
Figure 3:
2004
2005
2006
Error Comparison for various forecasting methods
The above analysis shows that: forecast accuracy of nonlinear regression has a big difference form
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the other two (see Figure 3). After Neural network’s amendment, prediction accuracy is up to a relatively
high level for the absolute errors and arithmetic errors, and its average of absolute errors and arithmetic
absolute errors is of 0.0061 and -0.000997. As compared with Chaos forecasting methods, according to
that in 2004, 2005, 2006, prediction accuracy is improved by 15.75%, 19.37%, 71.65%. Especially, it
should be noted: the increase for improvement degree is mainly due to that, the chaotic forecasting can not
produce very good results for long-term prediction. In this paper, its absolute prediction error is from
0.0079 in 2004 to 0.0141 in 2006 and this is in line with YU Jinghua (2002)and LIU Yunzhong (2004)’s
empirical research.
4 Conclusion
This paper forecasts and analyzes the (1979- 2003) GDP growth rate through the chaotic time series
forecasting methods. Based on this, this paper amends the errors of the GDP growth rate during 2004 and
2006 by means of BP Neural network, so as to improve the three-year average relative error rate to 0.553%,
which is more precise comparing with( WANG Chunfeng 2007 3.38 [1998 2004 mean], LIU
Yunzhong 2004 1.21 [1998-2002 mean], SU Wenli 2003 1.42 [1999-2000 mean]. In conclusion,
the forecasting precision has been enhanced largely compared with the only chaos forecasting and
nonlinear regression methods.
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References
[1] Chang Zhongze. Application of Non-Linear Dynamics in the Field of Macroeconomics: A Survey. Economic
Research Journal, 2006, (09) 117~128(in Chinese).
[2] Wang Chunfeng, Song Hui. GDP Forecasting Based on Chaotic Time Series Analyses. Journal of Tianjin
University (Social Sciences), 2007, (02): 137~139(in Chinese).
[3] Liu Yunzhong, Xuan Huiyu. Chaotic Time Series and Its Application of GDP (1978 2000) Forecasting in
China. Journal of Industrial Engineering and Engineering Management, 2004 (02) 8~10(in Chinese).
[4] Su Wenli. Nonlinear chaotic prediction for GDP. Quantitative & Technica Economics, 2003(2):127~130( in
Chinese).
[5] Yu Jinghua, Tian Lixin. Chaotic Time Series and Application of Energy System. Journal of Jiangsu University
of Science and Technology, 2002 23(4) 84~86( in Chinese)
[6] Hornik K M, Stinchcombe M, White H. Multilayer feed-forward networks are universal
approximators Neural Networks. 1989, 2 (5):359~366.
[7] Hinton G E Connectionist learning procedures Artificial Intelligence, 1989, (40):185~234.
[8] Garcia C E, prettt D M, Morari M. Model Predictive control. Theory and practice-A survey (S1368-5562),
1989, 25(3): 335-348.
[9] Epaminondas Panas, Vassilia Ninni. Are oil markets chaotic? Anon-linear dynamic analysis. Energy
Economics (S0140-9883), 2000, 22: 549-568.
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The Author can be contacted from Email: [email protected]
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