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Analog Complexing (AC) algorithm can be considered a sequential pattern recognition method for prediction. However, financial Time-series data are often nonlinear and non-stationary, which cause some difficulties when used AC algorithm in prediction. Aiming at this problem, in this paper, using Empirical Mode Decomposition (EMD) to handle original data, and we will obtain a series of stationary Intrinsic Mode Functions (IMF); then each IMF is predicted dynamically by AC. By the empirical studies on NYMEX Crude Oil Futures price show that AC algorithm based on EMD method have high precision in 1 step and 3 steps dynamically prediction.  Key words: Analog Complexing algorithm, Empirical Mode Decomposition, Intrinsic Mode Function, Dynamically predictio

HE, Ji-rui

TIAN, Yi-xiang

Management Science and Engineering            Vol.1 No.1 September 2007 
 
THE MODEL OF ANALOG COMPLEXING 
ALGORITHM BASED ON EMPIRICAL MODE 
DECOMPOSITION METHOD 
 
He Jirui1          Tian Yixiang2 
 
Abstract:  Analog Complexing (AC) algorithm can be considered a sequential 
pattern recognition method for prediction. However, financial Time-series data are 
often nonlinear and non-stationary, which cause some difficulties when used AC 
algorithm in prediction. Aiming at this problem, in this paper, using Empirical Mode 
Decomposition (EMD) to handle original data, and we will obtain a series of 
stationary Intrinsic Mode Functions (IMF); then each IMF is predicted dynamically 
by AC. By the empirical studies on NYMEX Crude Oil Futures price show that AC 
algorithm based on EMD method have high precision in 1 step and 3 steps 
dynamically prediction. 
Key words: Analog Complexing algorithm, Empirical Mode Decomposition, 
Intrinsic Mode Function, Dynamically prediction 
 
 
                                                        
1 Department of Management, UESTC, Chengdu, 610054. 
2 Department of Management, UESTC, Chengdu, 610054. 
* Received 5 July 2007;  accepted 26 August 2007 
1.  INTRODUCTION 
 
When we modeling to complex system, such as financial market and meteorological system, the 
relationship between system variables is complex, even there haven’t fixed regularity. But the 
characteristic of complex system is that a grate lot of time series patterns existed in system variables, 
researchers can describe dynamic system by utilizing these time series patterns, because different modes 
represent some periodicity, some trends and relationship existed in data. 
 
He Jirui, Tian Yixiang/Management Science and Engineering  
Vol.1 No.1 2007 75-82 
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Analog Complexing(AC) algorithm3 was developed by Lorence and was successfully applied to 
meteorological forecast first. It can be considered as a sequential pattern recognition method for 
predicting and qualitatively explaining fuzzy objects inherently. Recently, AC algorithm has been 
enhanced by an inductive self-organising approach and by an advanced selection procedure to make it 
applicable to evolutionary processes too. For example, J.-A. Muller and Guangzhong Liu (2002)4 
explained the superiority of AC algorithm, and tested 2 stocks of the German car industry (BMW, VW), 
the US-Dollar/DM exchange rate. Changzheng He (2004)5 forecasted the economic developed index in 
ChengDu used AC algorithm, Ming Huang (2005)6   used AC algorithm in forecasting stocks and got the 
turning-point of price trends accurately. AC algorithm needs neither evaluation or assumption in advance 
for imput variable, nor  priori knowledge used to choose variable for model construction, which means 
that forecasting are driven by data. It is superior than other forecasting methods that make forecasting 
based on figure patterns. 
When we use AC algorithm, the main assumptions of the method are: ○1 the system is described by a 
multidimensional process; ○2 many observations of the process are available (long time series); ○3 the 
multidimensional process is sufficiently representative, i.e., essential system variables are forming the 
data set; ○4 it is likely that any observed behaviour of the process will repeat similarly over time. 
Therefore, the prediction can be achieved by combining the known continuations of the analogous 
patterns to a continuation of the reference pattern. In general, AC algorithm is a three-step-procedure 
(see literature 3 for detail algorithm): ○1 generation of alternate patterns; ○2 transformation of analogues; 
○3 selection of most similar patterns; when using AC algorithm for prediction purposes, a fourth step is 
necessary: ○4 combining forecasting. 
Form above, the forecasting effect of AC algorithm lies on pattern similarity measures, selection, and 
similarity pattern combining. The new method of GMDH (Group Method of Data Handling)3 has solved 
similarity pattern optimized combination, but similarity pattern measures in AC algorithm based on 
Eucildean Spaces, and sensitivity to noise, so how to improve method of similarity measures need to 
discuss deeply. Then, financial time-series data are often nonlinear and non-stationary, the periods of 
volatility are different, and the trend is uncertain, which all cause some problems to choose pattern’s 
length and to measure pattern’s similarity, accordingly influence the accuracy of forecasting results. For 
that, we introduce empirical mode decomposition method to forecast. First, the original time series are 
made stationary by empirical mode decomposition method; then, all the stationary data are forecasted 
dynamically via AC algorithm. In this paper, we used a new model to mine the history information and 
understand the underlying rules for improving forecasting accuracy. 
 
2.  EMD METHOD 
 
EMD (empirical mode decomposition) method7 and Hilbert transform, which is proposed by Huang et al. 
(1998), is a new method for analysing nonlinear and non-stationary data. Contrary to almost all the 
previous decomposing methods, EMD is empirical, intuitive, direct, and adaptive, with the a posteriori 
defined basis derived from the data. It has been used in many science fields successfully, such as signal 
handling, oceanography, meteorology. The decomposition is designed to seek the different simple 
intrinsic modes of oscillations in any data based on the principle of scale separation. The data, depending 
                                                        
3Mueller J A, Lemke F. Self-orgnizing data mining [M], Berlin Hamburg, 2000.114-130 
4 J.A.MULLER, Guangzhong LIU. Application of Analog Complexing Algorithm to Financial Forecasting [J]. 
Or Transactions,2002.6(3):1-16 
5 He Changzheng. Self-Organising Data Mining and Economics Forecasting [M]. Science Publishing House, 
2005. 
6 Huang Min. The Application of Self-Organising Data Mining in Stock Market [D]. Chengdu: UESTC:2006. 29-38 
7 Huang N E , Shen Z. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary 
time series analysis [J]. Proceedings of the Royal Society London, 1998 , A ,454: 903-995 
He Jirui, Tian Yixiang/Management Science and Engineering  
Vol.1 No.1 2007 75-82 
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on it complexity, may have many different coexisting modes of oscillation at the same time. Each of 
these oscillatory modes is represented by an Intrinsic Mode Function (IMF) with the following 
definitions: (a) in the whole data set, the number of extrema and the number of zero-crossings must 
either equal or differ at most by one, and (b) at any point, the mean value of the envelope defined by the 
local maxima and the envelope defined by the local minima is zero. The initial IMF by EMD method 
show the high frequency component, along the decomposed process, the IMF has lower frequency and 
longer period, When the IMFs and the residual are summed together they form the original time series.  
Pursuant to the above definition for IMF, one can implement the needed decomposition of any 
function, known as sifting, as follows: 
1）Identify all the maxima and minima of time series x(t); 
2）Generate its upper and lower envelopes ))(),(( minmax tete , with cubic spline interpolation. 
3 ） Calculate the point-by-point mean )( tm  from upper and lower envelopes:   
2))()(()( maxmin tetetm += ; 
4）Extract the mean from the time series and define the difference of x(t) and )( tm  as )( td : 
)()()( tmtxtd −= ; 
5）Check the properties of )( td : 
If it is an IMF, denote )( td as the ith IMF and replace x(t) with the residual )()()( tmtxtr −= ; 
the )( td is often denote as )( tci ; 
If it is not, replace x(t) with )( td ; 
6）Repeat steps (1)-(5) until the residual sarisfies some stopping criterion. 
The original time series could denote by: ∑
=
+=
n
i
i trtimftx
1
)()()( . 
 
3.  MODELING PROCESS 
 
According to the characteristics of AC algorithm and EMD method, we combine two methods to form a 
new prediction model, make the stationary and periodic IMF as prediction element, for solving the 
choice of pattern’s length and measurement of pattern’s similarity. Firstly, using the empirical mode 
decomposition the original sequence data are stationarized, and a variety of IMFs and a residual are 
obtained (Figure 1). We see, from figure 1, all IMFs are not only stationary and have strong periodic, so 
we can choose pattern’s length according to the IMFs’ period; secondly the IMFs and residual are 
forecasted by AC algorithm separately; finally, Summing up the predicted results of IMF and residual 
yields he predicted data of the original data. The detail algorithm flow chart showed in Figure 2. 
He Jirui, Tian Yixiang/Management Science and Engineering  
Vol.1 No.1 2007 75-82 
 78
 
Figure 1  Empirical Mode Decomposition  
 
 
Figure 2 Model flow chart 
Original data 
AC 
Summing up
EMD 
imf1 imf2 …… imfn r(t) 
P_imf1 P_imf2 …… P_imfn P_r(t) 
Final forecast value 
AC AC AC AC 
sign
Empirical Mode Decomposition
imf1 
imf2 
imf3 
imf4 
res. 
He Jirui, Tian Yixiang/Management Science and Engineering  
Vol.1 No.1 2007 75-82 
 79
 
4.  EMPIRICAL ANALYSIS 
 
4.1   Data 
Our data consist of daily crude oil futures closing prices traded at the Mew York Mercantile Exchange 
(NYMEX) from Jan 5, 2004 to Oct 16, 2006, constituting 698 data points. The training data set included 
638 data (from Jan 5, 2004 to Jul 21, 2004), the other 60 data (from Jul 24, 2006 to Oct 16, 2006) were 
used as a test sample. Data come from FuYuan futures software. 
 
4.2  Forecasting 
In this study, one-step-ahead forecasting policy is implemented. For example, first 638(from Jan 5, 2004 
to Jul 21, 2004) training data set was decomposed by EMD, obtained 5 IMFs and a residual (figure 2). 
We use AC algorithm to forecast every IMF and residual, summing up these predicted value get the 
predicted closing price at Jul 24, 2004. Then we add the true value at Jul 24, 2004 to the training data set 
and re-forecast the next day. Apply the same procedure to generate 60 times one-step-ahead forecasts. 
 
Figure 3 2004.1.5-2006.7.21data decomposed by EMD 
 
When we use AC algorithm to forecast IMFs, the similarity between the reference pattern and a given 
pattern reach 0.99, even 1; but forecast the original data sequence, the similarity between the reference 
pattern and a given pattern only reach 0.92, even lower. 
 
4.3  Forecasting results 
To compare the forecasting results, three forecasting error functions are calculated, they are the Root 
signal 
Empirical Mode Decomposition
imf1 
imf2 
imf3 
imf4 
imf5 
res. 
He Jirui, Tian Yixiang/Management Science and Engineering  
Vol.1 No.1 2007 75-82 
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Mean Squared Error(RMSE), Mean Absolute Error(MAE) and Mean Absolute Percentage 
Error(MAPE).  
Table 1 illustrate three error values between original AC algorithm and AC algorithm combined 
EMD method. Figures 4-5 shows actual and forecasting values of one-step-ahead and three-step-ahead 
from Jul 24, 2006 to Oct 16, 2006. 
The table and figures clearly demonstrate that the model combined EMD method is able to produce 
lower forecasting errors by all three measures, and the forecasting value are closer to true value, 
suggesting that the new model has well short-term forecasting capability. 
 
Table 1：RMSE, MAE and MAPE Value 
 
 RMSE MAE MAPE 
one-step-ahead 1.2416 1.0134 0.0148 
three-step-ahead 1.8064 1.3717 0.0195 
one-step-ahead（EMD） 1.0361 0.8278 0.0121 
three-step-ahead（EMD） 1.5833 1.2471 0.0183 
2
1
1 ˆ( )
T
t t
t
RMSE y y
T =
= −∑ ，
1
1 ˆ
T
t t
t
MAE y y
T =
= −∑ ，
1
ˆ1 T t t
t t
y yMAPE
T y=
−= ∑  
 
Figure 4 One-step-ahead forecasting (from Jul 24, 2006 to Oct 16, 2006) 
0 10 20 30 40 50 60 7060 
62 
64 
66 
68 
70 
72 
74 
76 
78 
80 
Date 
Price 
original data 
AC combined EMD forecasting
AC forecasting 
He Jirui, Tian Yixiang/Management Science and Engineering  
Vol.1 No.1 2007 75-82 
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    Figure 5 Three-step-ahead forecasting (from Jul 24, 2006 to Oct 16, 2006) 
 
5.  CONCLUSIONS 
 
This paper examines the application of AC algorithm combined EMD model for time series forecasting 
regarding to NYMEX crude oil futures closing price. First, using the empirical mode decomposition a 
variety of IMFs and a residual are obtained, which were forecasted by AC algorithm, then summing up 
these predicted IMFs’ value. Our results illustrate the new model has higher accuracy than traditional 
model, and it is feasible easily. In fact, the AC algorithm combined EMD model can forecast for a fairly 
wide time series, especially for non-stationary and nonlinear time series and complex systems, have a 
broad application. 
 
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THE MODEL OF ANALOG COMPLEXING ALGORITHM BASED ON EMPIRICAL MODE DECOMPOSITION METHOD

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THE MODEL OF ANALOG COMPLEXING ALGORITHM BASED ON EMPIRICAL MODE DECOMPOSITION METHOD

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