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

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## 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.

## 2 Methodology for STSNN model

*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

*n*is defined as

*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

*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

*k*th training iterations is defined as

*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

*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 \(y_{t+1}=f_O\left( \sum \nolimits _{j=1}^{m}v_jf_H\left( \sum \nolimits _{i=1}^{n}w_{ij}x_{it}\right) \right) \).

## 3 Empirical analysis

### 3.1 Return scaling cross-correlation and data selection

*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)

*L*. Then, the cross-correlation function is defined as

*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*|.

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*|.

### 3.2 Parameter determining of STSNN model

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 | |||||||
---|---|---|---|---|---|---|---|---|---|---|

| Input | Hidden | RMSE | MAE | MAPE | | RMSE | MAE | MAPE | |

\(-\)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 |

*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

*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)]

*R*are considered. These measures are defined as follows

*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

*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.

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 | ||||||
---|---|---|---|---|---|---|---|---|

| RMSE | MAE | MAPE | | RMSE | MAE | MAPE | |

\(-\)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

*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.

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 | |||||||
---|---|---|---|---|---|---|---|---|---|---|

| Input | Hidden | RMSE | MAE | MAPE | | RMSE | MAE | MAPE | |

\(-\)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.

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