Multifractal Cross-correlation Analysis And Forecasting Methods Of Time Series

The characterization and understanding of complex system is a difficult task, however, complex system can be studied by analyzing time series recording some variables from such system. In this paper, we first discuss the multifractal cross-correlation between time series. We proposed a new method named multifractal cross-correlation analysis based on statistical moments (MFSMXA) to explore the multifractal cross-correlations. The performances of the MFSMXA method are compared with the multifractal detrended fluctuation cross-correlation analysis (MFXDFA) and multifractal detrended moving average cross-correlation analysis (MFXDMA) by extensive numerical experiments on pairs of four types of time series. The first two types are generated from two-component autoregressive fractionally integrated moving average processes (2-ARFIMA) and binomial measures (MFbs), which have theoretical expressions of multifractal nature. In all cases, the scaling exponents τxy extracted from the three methods are very close to the theoretical values. The MFSMXA method outperform the other two, which is closer to the theoretical curves, MFXDMA performs slightly worse and MFXDFA performs worst. The last two types are real data. One is financial time series, the MFSMXA and MFXDFA give similar results and succeed to extract rational multifractality; the other is traffic volume and speed series which also have multifractal nature, the three methods have comparative performance when q≤0, and MFSMXA and MFXDMA have comparative performance while MDXDFA method have slightly deviation when q≥0.After analyzing the multifractal cross-correlation between time series, we then proposed bi-pattern recognition K-nearest neighbor (KNN) nonparametric regression (BKNN) model, which is modified from KNN model, to predict traffic time series. Then the proposed BKNN model is applied to predict one day real traffic speed series from the site locating near the North3rd Ring Road in Beijing. In comparison with KNN model, PKNN model (a modified model based on KNN), seasonal autoregressive integrated moving average (SARIMA) and the artificial neural networks (ANN), the BKNN model appears to be the most promising and robust of the five models to provide better short-term traffic prediction.