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This paper introduces a generalized diffusion entropy analysis method to analyze long-range correlation, then applies this method to stock volatility series, traffic congestion index and daily average temperature series of Beijing. The method uses the techniques of diffusion process and Rényi entropy to focus on the scaling behaviors of regular and extreme volatility of time series. However, crossovers arising from the extrinsic periodic trend make the scaling behavior difficult to analyze. Periodic trend is found to deeply affect correlation, as series present obscure scaling behavior while exhibiting obvious scaling behavior after the trend is removed. We introduce Fourier filtering method to eliminate the trend effects and systematically investigate the multifractal long-range correlation of traffic congestion index and daily average temperature series. For daily average temperature series of Beijing, regular volatility and extreme volatility both exhibit long-range persistence. For traffic congestion index series of Beijing, regular volatility exhibits un-correlation or short correlation, while extreme volatility reveals anti-persistence.

Accurate forecasting of stock market volatility is an important issue in portfolio risk management. In this paper, an ensemble system for stock market volatility is presented. It is composed of three different models that hybridize the exponential generalized autoregressive conditional heteroscedasticity (GARCH) process and the artificial neural network trained with the backpropagation algorithm (BPNN) to forecast stock market volatility under normal, t-Student, and generalized error distribution (GED) assumption separately. The goal is to design an ensemble system where each single hybrid model is capable to capture normality, excess skewness, or excess kurtosis in the data to achieve complementarity. The performance of each EGARCH-BPNN and the ensemble system is evaluated by the closeness of the volatility forecasts to realized volatility. Based on mean absolute error and mean of squared errors, the experimental results show that proposed ensemble model used to capture normality, skewness, and kurtosis in data is more accurate than the individual EGARCH-BPNN models in forecasting the S&P 500 intra-day volatility based on one and five-minute time horizons data.