Soft Computing

, Volume 22, Issue 9, pp 3097–3109 | Cite as

Return scaling cross-correlation forecasting by stochastic time strength neural network in financial market dynamics

Methodologies and Application
  • 90 Downloads

Abstract

A return scaling cross-correlation function of exponential parameter is introduced in the present work, and a stochastic time strength neural network model is developed to predict the return scaling cross-correlations between two real stock market indexes, Shanghai Composite Index and Shenzhen Component Index. In the proposed model, the stochastic time strength function gives a weight for each historical data and makes the model have the effect of random movement. The empirical research is performed in testing the model forecasting effect of long-term cross-correlation relationships by training short-term cross-correlations, and a corresponding comparison analysis is made to the backpropagation neural network model. The empirical results show that the proposed neural network is advantageous in increasing the forecasting precision.

Keywords

Forecast Cross-correlation Return scaling Neural network Stochastic time strength function Financial time series 

1 Introduction

Financial markets are the complex evolved dynamic systems with high volatilities and noises. The analysis and forecasting of fluctuation behaviors of financial time series have long been the focus and the challenge of financial research. In recent years, due to the flourishing development in the area of complex systems as well as data measurements and data analysis, researchers can find opportunities for examination and interpretation of financial price changes, which exhibit some interesting dynamical properties such as power-law of logarithmic returns and volumes, fat tails distribution of price changes, multifractality of volatility, scale invariance, and volatility correlation, for instances see (Cheng and Wang 2013; Fang and Wang 2012; Ilinski 2001; Lamberton and Lapeyre 2000; LeBaron et al. 1999; Liu and Wang 2012; Mantegna and Stanley 1999; Mills 1999; Niu and Wang 2013; Pei and Wang 2013; Ross 1999; Wang and Wang 2012; Wang et al. 2010, 2011; Xiao and Wang 2012; Yu and Wang 2012; Zhang and Wang 2010). Portfolio optimization is one of the basic tools of hedging in a risky and extremely complex financial environment. In the risk minimization of portfolio optimization, correlations between asset prices play a crucial role (Kullmann et al. 2002; Mantegna 1999; Onnela et al. 2002; Tsay 2005). For this purpose, the equal time cross-correlations between stocks or different financial markets have attracted much interest and play an important role in portfolio theory and financial analysis (Duan and Stanley 2011; He and Chen 2011; Wang et al. 2010; Wichard et al. 2004). Many scholars are in a great attempt to study cross-correlations between time series. For examples, Kullmann et al. (2002) proposed time-dependent correlation function to study cross-correlations of stock returns. Mantegna (1999) quantified the degree of similarity between the synchronous time evolution of a pair of stock prices by the correlation coefficient. Some literatures analyze cross-correlations between the fluctuations of different stock prices using methods of random matrix theory (Laloux et al. 2000; Plerou et al. 2002; Utsugi et al. 2004; Wilcox and Gebbie 2007). Recently, Podobnik and Stanley (Horvatic et al. 2011; Podobnik and Stanley 2008) proposed the detrended cross-correlation analysis (DCCA) to investigate the power-law cross-correlations between two simultaneously recorded time series in the presence of non-stationary. To uncover the multifractal characteristic of two cross-correlation time series, Zhou (2008) proposed the multifractal detrended cross-correlation analysis (MF-DCCA). Considering the above research work, the analysis and forecasting of cross-correlation behaviors between two stock prices or two financial markets would be an interested and meaningful research.

Artificial neural network (ANN) is a nonlinear statistical data modeling, and it is also a decision-making method. ANN can be applied to model complex relationships between inputs and outputs in an attempt to find patterns in data. Recently, ANNs have been demonstrated to be successful research models to forecast financial markets and other domains (Box et al. 1994; Lendasse et al. 2000; Liu and Wang 2011; Refenes 1994; Rumelhart and McClelland 1986; Trippi and Turban 1993). ANN models possess data-driven, self-learning and self-adaptive abilities, and have strong anti-jamming capabilities. So that they have been developed to forecast, detect, and summarize the structure of financial variables without relying too much on specific assumptions and error distributions (Azoff 1994; Demuth and Beale 1998; Refenes 1994). A key element of financial planning and financial forecasting is the ability to construct models showing the interrelatedness of financial data. Models showing correlation and causation between variables can be employed to improve financial decision-making. In the real markets, making accurate forecast is a challenging task due to inherently noisy and non-stationary nature of financial series. To improve predicting precision, some research work has been made in an attempt to hybridize several artificial intelligence techniques to improve the predictive performance of ANNs. Kim and Han (2000) proposed a genetic algorithm approach to determine the connection weights of neural networks, so as to reduce the dimensionality of the feature space and enhance the predictive performance. Liao and Wang (2010) introduced a stochastic time effective neural network model to improve some predictive results. In the present paper, a return scaling cross-correlation function of exponential parameter is introduced, and a stochastic time strength neural network model (STSNN) is developed to predict the return scaling cross-correlations between Shanghai Stock Exchange (SSE) Composite Index and Shenzhen Stock Exchange (SZSE) Component Index.
Fig. 1

A three-layer multi-input neural network with one output

2 Methodology for STSNN model

Among many neural network architectures, the three-layer feedforward backpropagation neural network (BPNN) is commonly used in financial prediction Haykin (1999), and many researchers have improved this basic model, including backpropagation neural networks, back radial basis function neural networks, generalized regression neural networks, wavelet neural networks, and dynamic artificial neural network (Azoff 1994; Ghiassi et al. 2005; Kim and Han 2000; Liao and Wang 2010; Liu and Wang 2012, 2011; Niu and Wang 2014; Wang and Wang 2012; Ao 2010; Ruan and Tan 2010). This network architecture consists of one hidden layer of neurons with nonlinear transfer functions and an output layer of linear neurons with linear transfer functions. A schematic diagram of backpropagation network is given in Fig. 1, where \(x_{it}\) \((i=1,\ldots ,n)\) represents the input variables at time t, \(z_{jt}\) \( (j=1,\ldots ,m)\) represents the outputs of neurons in the hidden layer at time t, and \(y_{t+1}\) represents the outputs of the neural network at time \(t+1\). \(w_{ij}\) is the weight that connects the node i in the input layer neurons to the node j in the hidden layer, \(v_j\) is the weight that connects the node j in the hidden layer neurons to the node in the output layer. Stochastic time strength neural network model (STSNN) follows this architecture and adds a stochastic time strength function to minimize the error between the network’s prediction and the actual target. The STSNN model’s hidden layer stage: each neuron in the hidden layer performs a weighted summation of the inputs by the following equation
$$\begin{aligned} {\text{ net }}_{jt}=\sum \limits _{i=1}^{n}w_{ij}x_{it}-\theta _j, \quad i=1,\ldots ,n. \end{aligned}$$
(1)
The output of hidden neuron is given by
$$\begin{aligned} z_{jt}=f_H({\text{ net }}_{jt})=f_H\left( \sum \limits _{i=1}^{n}w_{ij}x_{it}-\theta _j\right) ,\quad i=1,\ldots ,n\nonumber \\ \end{aligned}$$
(2)
where \(\theta _j\) is the regulated value of neuron in hidden layer, the transfer function in hidden layer is sigmoid function \(f_H(x)={1}/(1+\exp \{-x\})\). Output stage: The output of STSNN is given as follows
$$\begin{aligned} y_{t+1}=f_O\left( \sum \limits _{j=1}^{m}v_jz_{jt}-\theta _O\right) \end{aligned}$$
(3)
where \(\theta _O\) is the regulated value of neuron in output layer, and \(f_O(x)\) is an identity map as the activation function.
The backpropagation algorithm has emerged as one of the most widely used learning procedures for multilayer neural networks (Azoff 1994; Rumelhart and McClelland 1986) which is a supervised learning algorithm which minimizes the global error \(E_k\) by using the gradient descent method. For the proposed STSNN model, assuming that the error of the output is given by \(\varepsilon _{t_n}=d_{t_n}-y_{t_n}\), and the error of the sample n is defined as
$$\begin{aligned} E(t_n)=\frac{1}{2}\phi (t_n)(d_{t_n}-y_{t_n})^2 \end{aligned}$$
(4)
where \(d_{t_n}\) is the actual value, \(y_{t_n}\) is the output value, \(t_n\) is the time of the sample n \((n=1,\ldots ,N)\), \(\phi (t_n)\) is the stochastic time strength function which endows each historical data with a weight depending on the time that it occurs. Define \(\phi (t_n)\) as following
$$\begin{aligned} \phi (t_n)=\frac{1}{\beta }\exp \left\{ \int _{t_0}^{t_n}\mu (t)dt+\int _{t_0}^{t_n}\sigma (t)dB(t)\right\} \end{aligned}$$
(5)
where \(\beta >0\) is the time strength coefficient, \({t_0}\) is the newest data in the data training set, \({t_n}\) is an arbitrary time in the data training set. It is noted that only using the historical data of the past is difficult to reflect the current stock markets’ development. However, if only the recent data are selected, a lot of useful information will be lost which the early data hold. In the financial model of the present paper, a promising data mining technique in machine learning is proposed to uncover the predictive relationships of numerous financial and economic variables. This paper presents an improved neural network model, the stochastic time effective series neural network model: each historical data are given a weight depending on the time it occurs in the model, in details, the nearer the time of the historical data is to the present, the stronger impact the data has on the predictive model. \(\mu (t)\) is the drift function showing the impact of the historical data, and \(\sigma (t)\) is the volatility function showing the data fluctuation strength. B(t) is the standard Brownian motion. A Brownian motion is a real-valued,continuous stochastic process \(\left\{ X(t), t\ge 0\right\} \), on a probability space \((\Omega ,{\mathcal {A}},{\mathbb {P}})\) with independent and stationary increments, such that \(X_t-X_0\) is a normal random variable with mean rt and variance \(\sigma ^2t\), where r and \(\sigma \) are constant real numbers. A Brownian motion is standard (denoted it by B(t)) if \(B(0)=0\) \({\mathbb {P}}\)-a.s., \({\mathbb {E}}[B(t)]=0\) and \({\mathbb {E}}[B(t)]^2=t\). And the corresponding probability density function is given by \(f_t(x)=(1/\sqrt{2\pi t})e^{-x^2/2t}\). The above stochastic time strength function implies that the impact of the historical data on the stock market is a time variable function. Then, the corresponding global error of data training set at kth training iterations is defined as
$$\begin{aligned} E_k=\frac{1}{N}\sum \limits _{n=1}^{N}E(t_n). \end{aligned}$$
(6)
The training algorithm procedure of STSNN is described as follows. Step 1 Performing input data normalization and initial connective weights. At the beginning of data processing, set \(w_{ij}^0\) and \(v_{j}^0\) follow the uniform distribution on \((-1,1)\), and let the neural regulated value \(\theta _j\) and \(\theta _O\) be 0. Step 2 Determining the network structure which is \(n\times m\times 1\) three-layer network model, parameters including learning rate \(\eta \) which is between 0 and 1, the maximum training iterations number K. Step 3 Introducing the stochastic time strength function \(\phi (t_n)\) in sample n error \(E(t_n)\) and the global error \(E_k\). Choosing the drift function \(\mu (t)\) and the volatility function \(\sigma (t)\). Step 4 Setting a predefined minimum training threshold \(\zeta \). Based on network training objective \(E_k=\frac{1}{N}\sum \nolimits _{n=1}^{N}E(t_n)\), if \(E_k\) is below the \(\zeta \), go to Step 6, otherwise go to Step 5. Step 5 Updating the STSNN connective weights. Applying the error to compute the gradient of the weights \(w_{ij}\), \(\Delta w_{ij}^k\) and the gradient of the weights \(v_j\), \(\Delta v_j^k\). For the weight nodes in the input layer, the gradient of the connective weight \(w_{ij}\) is given by
$$\begin{aligned} \Delta w_{ij}=-\eta \frac{\partial E(t_n)}{\partial w_{ij}}= \eta \varepsilon _{t_n}v_j\phi (t_n)f'_H(net_{jt_n})x_{it_n} \end{aligned}$$
(7)
similarly, the gradient of the connective weight \(v_j\) in the hidden layer is given by
$$\begin{aligned} \Delta v_j=-\eta \frac{\partial E(t_n)}{\partial v_j}= \eta \varepsilon _{t_n}\phi (t_n)f_H(net_{jt_n}) \end{aligned}$$
(8)
where \(\eta \) is the learning rate, \(f'_H(net_{jt_n})\) is the derivative of the transfer function \(f_H\). Then, the update rule for the weight \(w_{ij}\) and \(v_j\) is given by
$$\begin{aligned} w_{ij}^{k+1}= & {} w_{ij}^{k}+\Delta w_{ij}^{k}= w_{ij}^{k}+\eta \varepsilon _{t_n}v_j\phi (t_n)f'_H(net_{jt_n})x_{it_n} \end{aligned}$$
(9)
$$\begin{aligned} v_j^{k+1}= & {} v_j^{k}+\Delta v_j^{k}= v_j^{k}+\eta \varepsilon _{t_n}\phi (t_n)f_H(net_{jt_n}). \end{aligned}$$
(10)
Step 6 Until the global error satisfies the predefined minimum training threshold \(\zeta \) or training times reaches the maximum iterations number, then outputting the forecasting value \(y_{t+1}=f_O\left( \sum \nolimits _{j=1}^{m}v_jf_H\left( \sum \nolimits _{i=1}^{n}w_{ij}x_{it}\right) \right) \).
Fig. 2

Fluctuations of return scaling r(p) of SSE for different values of p

3 Empirical analysis

3.1 Return scaling cross-correlation and data selection

To investigate the cross-correlation behaviors between Shanghai Composite Index and Shenzhen Component Index, a return scaling cross-correlation function of exponential parameter is introduced in the present paper. The data cover the time period from August 01, 2002 up to September 27, 2012, which are from the website at www.sse.com.cn. Let P(t) \((t=1,2,\ldots )\) denote the daily closing price sequences of SSE and SZSE at time t, then the formula of the stock logarithmic return from \(t-1\) to t is as follows (Refenes 1994)
$$\begin{aligned} r(t)=\ln P(t+1)-\ln P(t). \end{aligned}$$
(11)
And the corresponding return scaling function of exponential parameter is defined by
$$\begin{aligned} r(p)=|r(t)|^{1/p}. \end{aligned}$$
(12)
Assume that \(\{x(i),\ i=1,2,\ldots ,L\}\) and \(\{y(i),\ i=1,2,\ldots ,L\}\) are two time series sharing the same length L. Then, the cross-correlation function is defined as
$$\begin{aligned} C_k(x,y)=\frac{\sum \nolimits _{i=k+1}^{L}x_iy_{i-k}}{\sqrt{\sum \nolimits _ {i=1}^{L}x_i^2 \sum \nolimits _{i=1}^{L}y_i^2}} \end{aligned}$$
(13)
and the return scaling cross-correlation function of exponential parameter is given by
$$\begin{aligned} C_k(p)&= C_k(r_1(p),\ r_2(p))\nonumber \\&= C_k(|r_1(t)|^{1/p},\ |r_2(t)|^{1/p}). \end{aligned}$$
(14)
Let \(|r_1(t)|\) and \(|r_2(t)|\) denote the absolute returns of SSE index and SZSE index, respectively. Figures 2 and 3 present the fluctuation patterns of return scaling r(p) for SSE and SZSE with the parameter \(p=-4,-3,-2,-1,1,2,3,4\). These two figures suggest that the value aggregation of r(p) becomes remarkable with increasing the value of |p|.
Fig. 3

Fluctuations of return scaling r(p) of SZSE for different values of p

Fig. 4

Return scaling cross-correlation \(C_k(p)\) between SSE and SZSE with different positive values of p

Fig. 5

Return scaling cross-correlation \(C_k(p)\) between SSE and SZSE with different negative values of p

Figures 4 and  5 present the fluctuation patterns of return scaling cross-correlations \(C_k(p)\) of SSE and SZSE with different exponential parameters of p. Figures 4 and  5 show the corresponding return scaling cross-correlations with positive values and negative values of p, respectively. In these two figures, the lag k is 600 and all curves show downward trend generally. They illustrate that when lag k is small such as 1 to 200, the cross-correlation values \(C_k(p)\) are larger than those when lag k values 201 to 600. They also show that the curves of \(C_k(p)\) have gradually upward trend and the local jump gradually weaken with the increase of |p|.

In the following, we will investigate the behaviors of return scaling cross-correlation \(C_k(p)\) for SSE and SZSE, and forecast the long-term cross-correlations by training short-term cross-correlations through using STSNN and BPNN. In this process, we attempt to train the cross-correlations \(C_k(p)\) with lag 1 to 400 to forecast the cross-correlations with lag 401 to 600. To reduce the impact of noise in the data set and finally lead to a better prediction, the collected data should be properly adjusted and normalized at the beginning of modeling. To let the normalized data in the range of [0, 1], the following formula is adopted in this work
$$\begin{aligned} {\hat{{\mathscr {S}}}}(t)=\frac{{\mathscr {S}}(t)-\min {\mathscr {S}}(t)}{\max {\mathscr {S}}(t)-\min {\mathscr {S}}(t)} \end{aligned}$$
(15)
where \({\mathscr {S}}(t)\) is the data in the neural network data sets. In order to obtain the true value after the forecasting, we can revert the output variables as \({\mathscr {S}}(t)= \hat{{\mathscr {S}}}(t) (\max {\mathscr {S}}(t)-\min {\mathscr {S}}(t))+\min {\mathscr {S}}(t)\). Figure 6 is the graphical procedures of the long-term cross-correlations \(C_k(p)\) prediction by STSNN model.
Fig. 6

Graphical procedures of \(C_k(p)\) prediction by STSNN model

3.2 Parameter determining of STSNN model

Following the procedure of three-layer STSNN and BPNN models introduced in Sect. 2, the appropriate input variable numbers should be identified. Then, the autocorrelation function (ACF) and the partial autocorrelation function (PACF) are employed to study the statistical properties of \(C_k(p)\). The ACF is a useful statistical tool that measures if earlier values in the series have some relation to later values, and the PACF is the amount of correlation between a variable and a lag of itself that is not explained by correlations at all low order lags (Mishra and Desai 2005). Considering ACF and PACF graphs of \(C_k(p)\) series, the appropriate input numbers are identified to model selection (Faruk 2010). For example, Figures 7 and  8 present ACF and PACF plots when \(p=-2.5, 2.5\), respectively. From the empirical results of ACF and PACF of \(C_k(-2.5)\) series displayed in Figs. 7, 10 input layer nodes (the input neurons sets \(x_{it}\) \((i=1,\ldots ,10)\)) are considered, which means that the historical 10 lags \(C_{k-10}(-2.5), C_{k-9}(-2.5)\ldots , C_{k-1}(-2.5)\) are the input variables and \(C_k(-2.5)\) is the output variable. In the same way, 8 input layer nodes are selected in the forecasting of \(C_k(2.5)\).
Fig. 7

ACF and PACF for the 1st–600th ones of the \({C_k(-2.5)}\) series

Fig. 8

ACF and PACF for the 1st–600th ones of the \({C_k(2.5)}\) series

Table 1

Predicting metrics of return scaling cross-correlations between SSE and SZSE by STSNN with different values of p

\(C_k(p)\)

Numbers of nodes

Training set

Testing set

p

Input

Hidden

RMSE

MAE

MAPE

R

RMSE

MAE

MAPE

R

\(-\)1.5

7

11

0.0196

0.0131

9.4027

0.5531

0.0164

0.0126

13.4479

0.2370

\(-\)1.75

10

19

0.0179

0.0127

5.3168

0.8097

0.0127

0.0095

5.3873

0.5845

\(-\)2

8

12

0.0146

0.0106

3.0577

0.9092

0.0102

0.0074

2.7445

0.7593

\(-\)2.25

10

19

0.0124

0.0093

2.0740

0.9585

0.0080

0.0060

1.6850

0.8812

\(-\)2.5

10

19

0.0108

0.0082

1.5251

0.9778

0.0065

0.0050

1.1571

0.9350

\(-\)2.75

10

19

0.0093

0.0071

1.1694

0.9869

0.0055

0.0042

0.8701

0.9620

\(-\)3

10

19

0.0090

0.0069

1.0373

0.9919

0.0044

0.0034

0.6365

0.9763

\(-\)3.25

10

19

0.0080

0.0061

0.8633

0.9944

0.0039

0.0030

0.5276

0.9844

\(-\)3.5

10

19

0.0080

0.0061

0.8574

0.9944

0.0039

0.0031

0.5379

0.9844

\(-\)3.75

10

19

0.0071

0.0053

0.6844

0.9968

0.0033

0.0026

0.4241

0.9923

\(-\)4

10

19

0.0067

0.0049

0.6175

0.9975

0.0031

0.0025

0.3889

0.9944

\(-\)4.25

10

19

0.0067

0.0048

0.5901

0.9979

0.0030

0.0024

0.3690

0.9957

\(-\)4.5

10

19

0.0065

0.0047

0.5583

0.9982

0.0029

0.0023

0.3422

0.9967

\(-\)4.75

10

19

0.0064

0.0045

0.5315

0.9984

0.0034

0.0024

0.3544

0.9974

1.5

8

12

0.0093

0.0073

1.0694

0.9817

0.0052

0.0042

0.7194

0.9493

1.75

8

12

0.0084

0.0065

0.9256

0.9856

0.0049

0.0039

0.6454

0.9611

2

8

12

0.0076

0.0059

0.7683

0.9926

0.0036

0.0030

0.4480

0.9805

2.25

8

12

0.0073

0.0057

0.7108

0.9948

0.0031

0.0025

0.3661

0.9870

2.5

8

12

0.0065

0.0050

0.6051

0.9961

0.0032

0.0026

0.3680

0.9911

2.75

8

12

0.0063

0.0048

0.5753

0.9970

0.0028

0.0023

0.3244

0.9936

3

8

12

0.0060

0.0045

0.5320

0.9976

0.0028

0.0023

0.3206

0.9953

3.25

8

12

0.0060

0.0045

0.5209

0.9980

0.0026

0.0022

0.2938

0.9965

3.5

8

12

0.0058

0.0044

0.4962

0.9983

0.0024

0.0020

0.2664

0.9973

3.75

8

12

0.0059

0.0044

0.4806

0.9985

0.0023

0.0019

0.2518

0.9979

4

8

12

0.0058

0.0043

0.4772

0.9987

0.0023

0.0019

0.2542

0.9983

4.25

8

12

0.0058

0.0043

0.4724

0.9987

0.0020

0.0017

0.2255

0.9987

4.5

8

12

0.0058

0.0042

0.4678

0.9989

0.0020

0.0017

0.2229

0.9989

4.75

8

12

0.0058

0.0042

0.4647

0.9990

0.0020

0.0016

0.2148

0.9991

Fig. 9

ad Return scaling cross-correlation prediction by STSNN for \(p= -2, -4, 2, 4\), respectively

Fig. 10

ad Errors of return scaling cross-correlation prediction by STSNN in testing set for \(p= -2, -4, 2, 4\), respectively

In the present work, the max lag k is 600 (in Sect. 3.1), the maximum training iterations number \(K=200\), \(\eta =0.01\) and the predefined minimum training threshold \(\zeta =10^{-5}\). When using the STSNN model to predict the return scaling cross-correlation \(C_k(p)\). Because the recent data have a stronger effect than the data far from present time and the impact of data follows the time exponential decay, then we choose the drift function such that, the drift value decreases as the variable increases. And the standard deviation characterizes the degree of fluctuation of the data. So we define the drift function \(\mu (t)\) and the volatility function \(\sigma (t)\) as follows
$$\begin{aligned} \mu (t)=\frac{1}{(a-t)^3}, \qquad \sigma (t)=\left[ \frac{1}{N-1}\sum \limits _{i=1}^N(x_i-{\overline{x}})^2\right] ^{1\over 2} \end{aligned}$$
(16)
where a is the parameter which equals to the sample number in the data set, \({\overline{x}}\) represents the mean of the sample data. Here, \(\beta =1.25\), then the stochastic time strength function is given by [see (5)]
$$\begin{aligned} \phi (t_n)= & {} \frac{1}{1.25}\exp \left\{ \int _{t_0}^{t_n}\frac{1}{(c-t)^3}dt\right. \nonumber \\&\left. +\int _{t_0}^{t_n} \left[ {\frac{1}{N-1}\sum \limits _{i=1}^N(x_i-{\overline{x}})^2}\right] ^{1\over 2}dB(t)\right\} . \end{aligned}$$
(17)
To evaluate the forecasting performance of the STSNN model, the following performance measures, absolute error (AE), relative error (RE), root-mean-square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and correlation coefficient R are considered. These measures are defined as follows
$$\begin{aligned}&\text{ AE }=d_t-y_t, \qquad \text{ RE }=\frac{d_t-y_t}{d_t},\nonumber \\&\text{ MAE }={1\over N}\sum \limits _{t=1}^N|d_t-y_t| \end{aligned}$$
(18)
$$\begin{aligned}&\text{ RMSE }=\Big [{1\over N}\sum \limits _{t=1}^N(d_t-y_t)^2\Big ]^{1\over 2},\nonumber \\&\text{ MAPE }=100\times {1\over N}\sum \limits _{t=1}^N\left| {{d_t-y_t}\over {d_t}}\right| \end{aligned}$$
(19)
$$\begin{aligned}&R=\frac{\sum \limits _{t=1}^{N}(d_t-{\overline{d}}) (y_t-{\overline{y}})}{\sqrt{\sum \limits _{t=1}^{N} (d_t-{\overline{d}})^2\sum \limits _{t=1}^{N}(y_t-{\overline{y}})^2}} \end{aligned}$$
(20)
\(d_t\) and \(y_t\) are the actual and prediction values at lag t, respectively, N is the sample size, \({{\overline{d}}}\) and \(\overline{y}\) are the average values, respectively. Noting that AE, RE, MAE, RMSE, and MAPE are measures of the deviation between prediction and actual values, the prediction performance is better when the values of these evaluation criteria are smaller. However, if the results are not consistent among these criteria, we choose the MAPE as the benchmark since MAPE is relatively more stable than other criteria. R can be utilized to evaluate the prediction accuracy, and when R is closer to 1, the predictive sequence and actual sequence will have strong dependency.

3.3 Forecasting results

In this section, with the same parameter setting such as the number of input and hidden layer units and so on in neural network model, the return scaling cross-correlation \(C_k(p)\) of SSE and SZSE is forecasted by STSNN and BPNN models (The programming software is MATLAB 7.10.0 (R2010a)). The cross-correlations with lag 1 to 400 are the training sets, and the cross-correlations with lag 401 to 600 are the testing sets in STSNN and BPNN models. In the model, the proper number of the hidden layer nodes requires validation techniques to avoid under-fitting (too few neurons) and over-fitting (too many neurons). Generally, too many neurons in the hidden layers, and hence, too many connections, produce a neural network that memorizes the data and lacks the ability to generalize. In the STSNN model forecasting, the number of hidden layer units is based on the number of input layer units and the neural network training results. In details, we first choose a number range through a few empirical formula, and determine the number of hidden layer units with the least error which is selected by neural network training. Table 1 exhibits the corresponding predicting metrics of return scaling cross-correlations between SSE and SZSE by the STSNN model with different values of p. The RMSE, MAE, and MAPE of the prediction become smaller gradually with the increasing of |p| both in training set and testing set. And the R value becomes larger gradually and closer to 1 with the increasing of |p|. The experiment results exhibit that the cross-correlations forecasting by the STSNN model performs better when the values of |p| are large. With the same values of |p|, the forecasting results with positive p are better than those of negative p, because all the values of RMSE, MAE, MAPE are smaller and the values of R are larger when the values of p are positive. Figures 9 and  10 show the prediction results of return scaling cross-correlations between SSE and SZSE by the STSNN model for \(p= -2, -4, 2, 4\), respectively. The cross-correlations of Fig. 9a (\(p=-2\)) and Fig. 9c (\(p=2\)) have larger fluctuations than those of Fig. 9b (\(p=-4\)) and Fig. 9d (\(p=4\)). Considering the testing set, the actual value curves and the predictive value curves in four figures are more closer than those in training set, this shows that the forecasting effects in testing set are better than those in training set. It illustrates that the STSNN model could catch the interrelatedness of cross-correlation data sets, and the number of the hidden layer nodes is chosen properly because the data are well-fitting. Fig. 10 shows the forecasting absolute error (AE), relative error (RE) in testing set corresponding to Fig. 9. The RE and AE values in Fig. 10a (\(p=-2\)) are more larger than others, and Fig. 10d (\(p=4\)) has the smallest RE and AE values in four figures of Fig. 10. Table 1 and Figs. 9 and  10 show that the STSNN model is feasible in forecasting return scaling cross-correlations between SSE and SZSE.
Table 2

Predicting metrics of return scaling cross-correlations between SSE and SZSE by STSNN and BPNN in testing set with different values of p

\(C_k(p)\)

BPNN

STSNN

p

RMSE

MAE

MAPE

R

RMSE

MAE

MAPE

R

\(-\)1.5

0.0165

0.0128

13.5968

0.2392

0.0164

0.0126

13.4479

0.2370

\(-\)1.75

0.0132

0.0104

5.9040

0.5859

0.0127

0.0095

5.3873

0.5845

\(-\)2

0.0103

0.0077

2.8504

0.7607

0.0102

0.0074

2.7445

0.7593

\(-\)2.25

0.0080

0.0060

1.7001

0.8815

0.0080

0.0060

1.6850

0.8812

\(-\)2.5

0.0065

0.0050

1.1599

0.9351

0.0065

0.0050

1.1571

0.9350

\(-\)2.75

0.0056

0.0043

0.8892

0.9620

0.0055

0.0042

0.8701

0.9620

\(-\)3

0.0050

0.0039

0.7381

0.9764

0.0044

0.0034

0.6365

0.9763

\(-\)3.25

0.0047

0.0037

0.6534

0.9845

0.0039

0.0030

0.5276

0.9844

\(-\)3.5

0.0045

0.0036

0.6021

0.9893

0.0039

0.0031

0.5379

0.9844

\(-\)3.75

0.0044

0.0036

0.5751

0.9924

0.0033

0.0026

0.4241

0.9923

\(-\)4

0.0044

0.0036

0.5599

0.9944

0.0031

0.0025

0.3889

0.9944

\(-\)4.25

0.0044

0.0036

0.5498

0.9957

0.0030

0.0024

0.3690

0.9957

\(-\)4.5

0.0044

0.0036

0.5418

0.9967

0.0029

0.0023

0.3422

0.9967

\(-\)4.75

0.0044

0.0036

0.5359

0.9974

0.0034

0.0024

0.3544

0.9974

1.5

0.0053

0.0042

0.7217

0.9494

0.0052

0.0042

0.7194

0.9493

1.75

0.0053

0.0042

0.7243

0.9494

0.0049

0.0039

0.6454

0.9611

2

0.0043

0.0035

0.5325

0.9806

0.0036

0.0030

0.4480

0.9805

2.25

0.0042

0.0035

0.5101

0.9871

0.0031

0.0025

0.3661

0.9870

2.5

0.0042

0.0035

0.5043

0.9911

0.0032

0.0026

0.3680

0.9911

2.75

0.0043

0.0036

0.5055

0.9936

0.0028

0.0023

0.3244

0.9936

3

0.0043

0.0037

0.5095

0.9953

0.0028

0.0023

0.3206

0.9953

3.25

0.0044

0.0038

0.5143

0.9965

0.0026

0.0022

0.2938

0.9965

3.5

0.0045

0.0038

0.5173

0.9973

0.0024

0.0020

0.2664

0.9973

3.75

0.0026

0.0022

0.2950

0.9979

0.0023

0.0019

0.2518

0.9979

4

0.0046

0.0039

0.5245

0.9983

0.0023

0.0019

0.2542

0.9983

4.25

0.0046

0.0040

0.5284

0.9987

0.0020

0.0017

0.2255

0.9987

4.5

0.0046

0.0040

0.5320

0.9989

0.0020

0.0017

0.2229

0.9989

4.75

0.0047

0.0041

0.5352

0.9991

0.0020

0.0016

0.2148

0.9991

Table 2 exhibits the forecasting results of return scaling cross-correlations between SSE and SZSE by BPNN and STSNN models with different parameter values of p. The RMSE, MAE, and MAPE of the prediction become smaller gradually with the increasing value of |p| in testing set by the STSNN model and the BPNN model, and the prediction results have the same regular pattern when p is negative. Besides, when p is positive, the prediction metrics RMSE, MAE and MAPE of the BPNN model are larger than those of prediction by the STSNN model. The value R becomes larger gradually and closer to 1 with the increasing of |p|, and the values of R have not much differences when the values of parameter p are same in two models’ forecasting results. Observing an arbitrary fixed value p, the STSNN model performs better than the BPNN because RMSE, MAE, and MAPE values are smaller which illustrates that the STSNN model could catch the fluctuations of the data, and modeling complex relationships between inputs and outputs and finding patterns in data are better than those of the BPNN model. Table 2 indicates that some forecasting effects have been improved by the STSNN model comparing with the BPNN model.

3.4 Forecasting results in USA stock markets

In this section, the return scaling cross-correlation \(C_k(p)\) of Dow Jones Industrial Average Index (DJIA) and Nasdaq Composite index (IXIC) is forecasted by the STSNN model, the corresponding selected data are from the web www.yahoo.com. Similar as the work of Sect. 3.3, the data cover the time period from August 01, 2002, up to September 27, 2012, and the cross-correlations with lag 1 to 400 are the training sets, and the cross-correlations with lag 401 to 600 are the testing sets in the STSNN model. Table 3 exhibits the corresponding predicting metrics of return scaling cross-correlations between DJIA and IXIC by the STSNN model with different values of p. The RMSE, MAE and MAPE of the prediction become smaller gradually with the increasing of |p| both in training set and testing set. And the R value becomes larger gradually and closer to 1 with the increasing of |p|. The experiment results exhibit that the cross-correlations forecasting by the STSNN model performs better when the values of |p| are large. It shows that the STSNN model works for other stock market indexes.
Table 3

Predicting metrics of return scaling cross-correlations between DJIA and IXIC by STSNN with different values of p

\(C_k(p)\)

Numbers of nodes

Training set

Testing set

p

Input

Hidden

RMSE

MAE

MAPE

R

RMSE

MAE

MAPE

R

\(-\)1.5

6

9

0.0063

0.0048

0.7730

0.9945

0.0056

0.0044

0.8570

0.9209

\(-\)2

6

9

0.0062

0.0049

0.7728

0.9912

0.0051

0.0041

0.7910

0.9845

\(-\)2.5

7

11

0.0060

0.0048

0.6756

0.9917

0.0049

0.0039

0.5690

0.9918

\(-\)3

7

11

0.0056

0.0044

0.5562

0.9988

0.0039

0.0036

0.4134

0.9961

\(-\)3.5

7

11

0.0054

0.0040

0.5198

0.9993

0.0037

0.0035

0.3912

0.9971

\(-\)4

7

11

0.0054

0.0042

0.5175

0.9975

0.0035

0.0031

0.3874

0.9974

\(-\)4.5

7

11

0.0051

0.0040

0.4574

0.9982

0.0032

0.0028

0.3184

0.9985

1.5

6

9

0.0075

0.0051

0.7745

0.9937

0.0054

0.0043

0.7013

0.9331

2

6

9

0.0068

0.0049

0.7738

0.9955

0.0050

0.0041

0.7520

0.9855

2.5

7

11

0.0059

0.0046

0.6745

0.9956

0.0040

0.0034

0.5024

0.9967

3

7

11

0.0054

0.0040

0.5120

0.9982

0.0035

0.0031

0.3874

0.9974

3.5

7

11

0.0052

0.0040

0.4570

0.9986

0.0035

0.0030

0.3811

0.9976

4

7

11

0.0049

0.0039

0.4162

0.9991

0.0032

0.0028

0.3184

0.9985

4.5

7

11

0.0045

0.0037

0.3221

0.9997

0.0028

0.0023

0.2878

0.9990

4 Conclusion

In the present paper, the return scaling cross-correlation function of exponential parameter is introduced, and the corresponding return scaling cross-correlations between two stock market indexes are investigated and forecasted by a developed stochastic time strength neural network model. The STSNN model is utilized to forecast the long-term cross-correlations between SSE and SZSE by training the short-term cross-correlations. The empirical research displays the predicting performance of the STSNN model and the BPNN model, the forecasting accuracy has been improved by the STSNN model comparing the BPNN model, and indicates that the STSNN model could better catch the fluctuation characteristic and interrelatedness of financial data than the BPNN model. We also test the STSNN model’s forecasting effect between DJIA and IXIC of USA. The experimental predicting metrics show that the STSNN model works for other stock market indexes. We hope that this new approach in the present work can make some contributions to the analysis and forecast of financial market cross-correlation behaviors.

Notes

Acknowledgements

The authors were supported in part by National Natural Science Foundation of China Grant No. 71271026.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Ao SI (2010) A hybrid neural network cybernetic system for quantifying cross-market dynamics and business forecasting. Soft Comput 15:1041–1053CrossRefGoogle Scholar
  2. Azoff EM (1994) Neural network time series forecasting of financial market. Wiley, New YorkGoogle Scholar
  3. Box GEP, Jenkins GM, Reinsel GC (1994) Time series analysis: forecasting and control, 3rd edn. Prentice Hall, New JerseyMATHGoogle Scholar
  4. Cheng WY, Wang J (2013) Dependence phenomenon analysis of the stock market. Europhys Lett (EPL) 102:18004CrossRefGoogle Scholar
  5. Demuth H, Beale M (1998) Neural network toolbox: for use with MATLAB, 5th edn. The Math Works Inc, NatickGoogle Scholar
  6. Duan WQ, Stanley HE (2011) Cross-correlation and the predictability of financial return series. Phys A 390:290–296CrossRefGoogle Scholar
  7. Fang W, Wang J (2012) Statistical properties and multifractal behaviors of market returns by using dynamic systems. Int J Mod Phys C 23:1250023CrossRefMATHGoogle Scholar
  8. Ghiassi M, Saidane H, Zimbra DK (2005) A dynamic artificial neural network model for forecasting time series events. Int J Forecast 21:341–362CrossRefGoogle Scholar
  9. Haykin S (1999) Neural networks: a comprehensive foundation. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  10. He LY, Chen SP (2011) A new approach to quantify power-law cross-correlation and its application to commodity markets. Phys A 390:3806–3814CrossRefGoogle Scholar
  11. Horvatic D, Stanley HE, Podobnik B (2011) Detrended cross-correlation analysis for non-stationary time series with periodic trends. Europhys Lett (EPL) 94:18007CrossRefGoogle Scholar
  12. Ilinski K (2001) Physics of finance: Gauge modeling in non-equilibrium pricing. Wiley, New YorkGoogle Scholar
  13. Kim KJ, Han I (2000) Genetic algorithms approach to feature discretization in artificial neural networks for the prediction of stock price index. Expert Syst Appl 19:125–132CrossRefGoogle Scholar
  14. Kullmann L, Kertész J, Kaski K (2002) Time-dependent cross- correlations between different stock returns: a directed network of influence. Phys Rev E 66:026125CrossRefGoogle Scholar
  15. Laloux L, Cizeau P, Potters M, Bouchaud JP (2000) Random matrix theory and financial correlations. Int J Theor Appl Financ 3:391–397CrossRefMATHGoogle Scholar
  16. Lamberton D, Lapeyre B (2000) Introduction to stochastic calculus applied to finance. Chapman and Hall/CRC, LondonMATHGoogle Scholar
  17. LeBaron B, Arthur WB, Palmer R (1999) Time series properties of an artificial stock market. J Econ Dyn Control 23:1487–1516CrossRefMATHGoogle Scholar
  18. Lendasse A, Bodt ED, Wertz V, Verleysen M (2000) Non-linear financial time series forecasting—application to the Bel 20 stock market index. Eur J Econ Soc Syst 14:81–91CrossRefMATHGoogle Scholar
  19. Liao Z, Wang J (2010) Forecasting model of global stock index by stochastic time effective neural network. Expert Syst Appl 37:834–841CrossRefGoogle Scholar
  20. Liu FJ, Wang J (2012) Fluctuation prediction of stock market index by Legendre neural network with random time strength function. Neurocomputing 83:12–21CrossRefGoogle Scholar
  21. Liu HF, Wang J (2011) Integrating independent component analysis and principal component analysis with neural network to predict Chinese stock market. Math Problems Eng 382659:15Google Scholar
  22. Mantegna RN (1999) Hierarchical structure in financial markets. Eur Phys J B 11:193–197CrossRefGoogle Scholar
  23. Mantegna RN, Stanley HE (1999) A introduction to econophysics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  24. Mills TC (1999) The econometric modelling of financial time series, 2nd edn. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  25. Mishra AK, Desai VR (2005) Drought forecasting using stochastic models. Stoch Environ Res Risk Assess 19:326–339CrossRefMATHGoogle Scholar
  26. Niu HL, Wang J (2014) Financial time series prediction by a random data-time effective RBF neural network. Soft Comput 18:497–508CrossRefGoogle Scholar
  27. Niu HL, Wang J (2013) Volatility clustering and long memory of financial time series and financial price model. Digit Signal Process 23:489–498MathSciNetCrossRefGoogle Scholar
  28. Faruk DO (2010) A hybrid neural network and ARIMA model for water quality time series prediction. Eng Appl Artif Intell 23:586–594CrossRefGoogle Scholar
  29. Onnela JP, Chakraborti A, Kaski K, Kertész J (2002) Dynamic asset trees and portfolio analysis. Eur Phys J B 30:285–388MathSciNetCrossRefMATHGoogle Scholar
  30. Pei AQ, Wang J (2013) Nonlinear analysis of return time series model by oriented percolation dynamic system. Abstr Appl Anal 2013(612738): 12Google Scholar
  31. Plerou V, Gopikrishnan P, Rosenow B, Amaral LAN, Guhr T, Stanley HE (2002) Random matrix approach to cross correlations in financial data. Phys Rev E 65:066126CrossRefGoogle Scholar
  32. Podobnik B, Stanley HE (2008) Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series. Phys Rev Lett 100:084102CrossRefGoogle Scholar
  33. Refenes AP (1994) Neural networks in the capital markets. Wiley, New YorkGoogle Scholar
  34. Ross SM (1999) An introduction to mathematical finance. Cambridge University Press, CambridgeMATHGoogle Scholar
  35. Ruan GC, Tan Y (2010) A three-layer back-propagation neural network for spam detection using artificial immune concentration. Soft Comput 14:139–150CrossRefGoogle Scholar
  36. Rumelhart DE, McClelland JL (1986) Parallel distributed processing: explorations in the microstructure of cognition. The MIT Press, CambridgeGoogle Scholar
  37. Trippi RR, Turban E (1993) Neural networks in finance and investing: using artificial intelligence to improve real-world performance. Probus, ChicagoGoogle Scholar
  38. Tsay RS (2005) Analysis of financial time series. Wiley, HobokenCrossRefMATHGoogle Scholar
  39. Utsugi A, Ino K, Oshikawa M (2004) Random matrix theory analysis of cross correlations in financial markets. Phys Rev E 70:026110CrossRefGoogle Scholar
  40. Wang F, Wang J (2012) Statistical analysis and forecasting of return interval for SSE and model by lattice percolation system and neural network. Comput Ind Eng 62:198–205CrossRefGoogle Scholar
  41. Wang J, Wang QY, Shao JG (2010) Fluctuations of stock price model by statistical physics systems. Math Comput Model 51:431–440MathSciNetCrossRefMATHGoogle Scholar
  42. Wang TS, Wang J, Zhang JH, Fang W (2011) Voter interacting systems applied to Chinese stock markets. Math Comput Simul 81:2492–2506MathSciNetCrossRefMATHGoogle Scholar
  43. Wang YD, Wei Y, Wu CF (2010) Cross-correlations between Chinese A-share and B-share markets. Phys A 389:5468–5478CrossRefGoogle Scholar
  44. Wichard JD, Merkwirth C, Ogorzalek M (2004) Detecting correlation in stock market. Phys A 344:308–311MathSciNetCrossRefGoogle Scholar
  45. Wilcox D, Gebbie T (2007) An analysis of cross-correlations in an emerging market. Phys A 375:584–598CrossRefGoogle Scholar
  46. Xiao D, Wang J (2012) Modeling stock price dynamics by continuum percolation system and relevant complex systems analysis. Phys A 391:4827–4838CrossRefGoogle Scholar
  47. Yu Y, Wang J (2012) Lattice oriented percolation system applied to volatility behavior of stock market. J Appl Stat 39:785–797MathSciNetCrossRefGoogle Scholar
  48. Zhang JH, Wang J (2010) Modeling and simulation of the market fluctuations by the finite range contact systems. Simul Model Pract Theory 18:910–925CrossRefGoogle Scholar
  49. Zhou WX (2008) Multifractal detrended cross-correlation analysis for two nonstationary signals. Phys Rev E 77:066211CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of ScienceBeijing Jiaotong UniversityBeijingPeople’s Republic of China

Personalised recommendations