In recent years, financial market dynamics forecasting has been a focus of economic research. To predict the price indices of stock markets, we developed an architecture which combined Elman recurrent neural networks with stochastic time effective function. By analyzing the proposed model with the linear regression, complexity invariant distance (CID), and multiscale CID (MCID) analysis methods and taking the model compared with different models such as the backpropagation neural network (BPNN), the stochastic time effective neural network (STNN), and the Elman recurrent neural network (ERNN), the empirical results show that the proposed neural network displays the best performance among these neural networks in financial time series forecasting. Further, the empirical research is performed in testing the predictive effects of SSE, TWSE, KOSPI, and Nikkei225 with the established model, and the corresponding statistical comparisons of the above market indices are also exhibited. The experimental results show that this approach gives good performance in predicting the values from the stock market indices.
Predicting stock price index is difficult due to uncertainties involved. In the past decades, the stock market prediction has played a vital role for the investment brokers and the individual investors, and the researchers are on the constant look out for a reliable method for predicting stock market trends. In recent years, the artificial neural networks (ANNs) have been applied to many areas of statistics. One of these areas is time series forecasting. References [
The backpropagation neural network (BPNN) is a neural network training algorithm for financial forecasting, which has powerful problemsolving ability. Multilayer perceptron (MLP) is one of the most prevalent neural networks, which has the capability of complex mapping between inputs and outputs that makes it possible to approximate nonlinear function. Reference [
The nonlinear and nonstationary characteristics of the stock market make it difficult and challenging for forecasting stock indices in a reliable manner. Particularly, in the current stock markets, the rapid changes of trading rules and management systems have made it difficult to reflect the markets’ development using the early data. However, if only the recent data are selected, a lot of useful information (which the early data hold) will be lost. In this research, a stochastic time effective neural network (STNN) and the corresponding learning algorithm were presented. References [
In order to display that the STERNN can provide a higher accuracy of the financial time series forecasting, we compare the forecasting performance with the BPNN model, the STNN model, and the ERNN model by employing different global stock indices. Shanghai Stock Exchange (SSE) Composite Index, Taiwan Stock Exchange Capitalization Weighted Stock Index (TWSE), Korean Stock Price Index (KOSPI), and Nikkei 225 Index (Nikkei225) are applied in this work to analyze the forecasting models by comparison.
The Elman recurrent neural network, a simple recurrent neural network, was introduced by Elman in 1990 [
In Figure
Topology of Elman recurrent neural network.
The backpropagation algorithm is a supervised learning algorithm which minimizes the global error
Intuitively, the drift function is used to model deterministic trends, the volatility function is often used to model a set of unpredictable events occurring during this motion, and Brownian motion is usually thought as random motion of a particle in liquid (where the future motion of the particle at any given time is not dependent on the past). Brownian motion is a continuous time stochastic process, and it is the limit of or continuous version of random walks. Since Brownian motion’s time derivative is everywhere infinite, it is an idealised approximation to actual random physical processes, which always have a finite time scale. We begin with an explicit definition. A Brownian motion is a realvalued, continuous stochastic process
The main objective of learning algorithm is to minimize the value of cost function
Note that the training aim of the stochastic time effective neural network is to modify the weights so as to minimize the error between the network’s prediction and the actual target. In Figure
Training algorithm procedures of STERNN.
To evaluate the performance of the proposed STERNN forecasting model, we select the daily data from Shanghai Stock Exchange (SSE) Composite Index, Taiwan Stock Exchange Capitalization Weighted Stock Index (TWSE), Korean Stock Price Index (KOSPI), and Nikkei 225 Index (Nikkei225) to analyze the forecasting models by comparison. In Table
Data selection.
Index  Date sets  Total number  Hidden number  Learning rate 

SSE  16/03/2006~19/03/2014  2000  9  0.001 
TWSE  09/02/2006~19/03/2014  2000  12  0.001 
KOSPI  20/02/2006~19/03/2014  2000  10  0.05 
Nikkei225  27/01/2006~19/03/2014  2000  10  0.01 
In the STERNN model, after we have done the experiments repeatedly on the different index data, different number of neural nodes in the hidden layer were chose as the optimal; see Table
Figure
Comparisons of the predictive data and the actual data for the forecasting models.
SSE
TWSE
KOSPI
Nikkei225
The plots of the real and the predictive data for these four price series are, respectively, shown in Figure
Comparisons and linear regressions of the actual data and the predictive values for SSE, TWSE, KOSPI, and Nikkei225.
SSE
TWSE
KOSPI
Nikkei225
A valuable numerical measure of association between two variables is the correlation coefficient
Linear regression parameters of market indices.
Parameter  SSE  TWSE  KOSPI  Nikkei225 


0.9873  0.9763  0.9614  0.9661 

7.582  173  38.04  653.5 

0.992  0.9952  0.9963  0.9971 
We compare the proposed and conventional forecasting approaches (BPNN, STNN, and ERNN model) on the four indices mentioned above, where STNN is based on the BPNN and combined with the stochastic effective function [
Different parameters for different models.
Index data  BPNN  STNN  ERNN  STERNN  

Hidden  L. r  Hidden  L. r  Hidden  L. r  Hidden  L. r  
SSE  8  0.01  8  0.01  10  0.001  9  0.001 
TWSE  10  0.01  10  0.01  12  0.001  12  0.001 
KOSPI  8  0.02  8  0.02  10  0.03  10  0.05 
Nikkei225  10  0.05  10  0.05  10  0.01  10  0.01 
Predictive values on the test set for SSE, TWSE, KOSPI, and Nikkei225.
SSE
TWSE
KOSPI
Nikkei225
To analyze the forecasting performance of four considered forecasting models deeply, we use the following error evaluation criteria [
Figures
Comparisons of indices’ predictions for different forecasting models.
Index errors  BPNN  STNN  ERNN  STERNN 

SSE  
MAE  45.3701  24.9687  37.262647  12.7390 
RMSE  54.4564  40.5437  49.3907  37.0693 
MAPE  20.1994  11.8947  18.2110  4.1353 
MAPE(100)  5.0644  3.6868  4.3176  2.6809 


TWSE  
MAE  252.7225  140.5971  151.2830  105.6377 
RMSE  316.8197  186.8309  205.4236  136.1674 
MAPE  3.2017  1.7303  1.8449  1.3468 
MAPE(100)  2.2135  1.1494  1.3349  1.2601 


KOSPI  
MAE  74.3073  56.3309  47.9296  18.2421 
RMSE  77.1528  58.2944  50.8174  21.0479 
MAPE  16.6084  12.4461  10.9608  4.2257 
MAPE(100)  7.4379  5.9664  4.9176  2.1788 


Nikkei225  
MAE  203.8034  138.1857  166.2480  68.5458 
RMSE  238.5933  169.7061  207.3395  89.0378 
MAPE  1.8556  1.2580  1.5398  0.6010 
MAPE(100)  0.7674  0.5191  0.4962  0.4261 
Comparisons of the actual data and the predictive data for SSE, TWSE, KOSPI, and Nikkei225.
SSE
TWSE
KOSPI
Nikkei225
In Figures
((a), (b), (c), and (d)) Relative errors of forecasting results from the STERNN model.
SSE
TWSE
KOSPI
Nikkei225
The analysis and forecast of time series have long been a focus of economic research for a more clear understanding of mechanism and characteristics of financial markets [
Complexity invariance uses information about complexity differences between two time series as a correction factor for existing distance measures. We begin by introducing Euclidean distance and use this as a starting point to bring in the definition of CID. Suppose we have two time series,
It is worth noticing that
CID distances for four network models.
Index  BPNN  STNN  ERNN  STERNN 

SSE  3052.1  1764.7  2320.2  1659.9 
TWSE  27805  9876.3  10830  6158.1 
KOSPI  3350.4  2312.8  2551.0  1006.0 
Nikkei225  44541  23726  32895  25421 
In general, the complexity of a real system is not constrained to a sole scale. In this part we consider a developed CID analysis, that is, the multiscale CID (MCID). The MCID analysis takes into account the multiple time scales while measuring the predicting results, and it is applied to the stock prices analysis for the actual data and the predicting data in this work. The MCID analysis should comprise two steps. (i) Considering onedimensional discrete time series
((a), (b), (c), and (d)) MCID values between the forecasting results and the real market prices from BPNN, ERNN, STNN, and STERNN models.
SSE
TWSE
KOSPI
Nikkei225
The aim of this research is to develop a predictive model to forecast the financial time series. In this study, we have developed a predictive model by using an Elman recurrent neural network with the stochastic time effective function to forecast the indices of SSE, TWSE, KOSPI, and Nikkei225. Through the linear regression analysis, it implies that the predictive values and the real values are not deviating too much. Then we take the proposed model compared with BPNN, STNN, and ERNN forecasting models. Empirical examinations of predicting precision for the price time series (by the comparisons of predicting measures as MAE, RMSE, MAPE, and MAPE(100)) show that the proposed neural network model has the advantage of improving the precision of forecasting, and the forecasting of this proposed model much approaches to the real financial market movements. Furthermore, from the curve of the relative error, it can make a conclusion that the large fluctuation leads to the large relative errors. In addition, by calculating CID and MCID distance the conclusion was illustrated more clearly. The study and the proposed model contribute significantly to the time series literature on forecasting.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors were supported in part by National Natural Science Foundation of China under Grant nos. 71271026 and 10971010.