时间序列的重分形交叉相关分析及其预测方法
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摘要
如何描述和理解复杂系统是十分艰巨的任务,但我们可以通过研究由复杂系统产生的时间序列来间接研究复杂系统。本文首先研究时间序列的重分形交叉相关性。我们提出研究重分形交叉相关性的新方法——多重分形统计矩交叉相关分析(MFSMXA)法。为了对比说明其分析效果,采用多重分形去趋势波动分析(MFXDFA)和多重分形去趋势滑动平均分析(MFXDMA)作为对比方法。随后分析了四类时间序列的重分形交叉相关性。其中两类是人工生成数据：二元自回归分数求和滑动平均模型(二元ARFIMA)和重分形二项方法(MFbs)生成的序列对,两模型都具有分形性的理论表达,三种方法得出的标度指数τxy接近于其对应的理论值,但MFSMXA方法分析效果最好,与理论曲线最为接近,MFXDMA次之,MFXDFA较差。另两类是实际序列,其中一类为金融时间序列,分析结果说明两个股票市场的波动指数序列间具有重分形特性,且MFSMXA和MFXDMA有相似的分析结果；一类为交通流量和速度序列对,同样分析出其具有重分形性,在q≤O时,三种方法分析结果相似；在q≥0时,MFSMXA和MFXDMA分析结果接近,MFXDFA有偏移。
在分析了时间序列间的重分形交叉相关性后,我们主要针对交通时间序列提出双模式K近邻非参数回归(BKNN)模型,它基于K近邻非参数回归(KNN)模型改进而来。我们使用北京北三环附近站点采集的交通速度进行预测,并引进KNN模型、PKNN模型(KNN模型的修改模型)、季节自回归求和滑动平均(SARIMA)模型以及人工神经网络(ANN)模型作为对比模型,发现BKNN模型给出了最好的短时预测结果,最稳健,应用前景广阔。
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.
在分析了时间序列间的重分形交叉相关性后,我们主要针对交通时间序列提出双模式K近邻非参数回归(BKNN)模型,它基于K近邻非参数回归(KNN)模型改进而来。我们使用北京北三环附近站点采集的交通速度进行预测,并引进KNN模型、PKNN模型(KNN模型的修改模型)、季节自回归求和滑动平均(SARIMA)模型以及人工神经网络(ANN)模型作为对比模型,发现BKNN模型给出了最好的短时预测结果,最稳健,应用前景广阔。
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.
引文
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[3]K. Hu, P. C. Ivanov, Z. Chen, P. Carpena, and H. E. Stanley, Effect of trends on detrended fluctuation analysis [J], Phys. Rev. E, 2001,64:1-20.
[4]Q. D. Y. Ma, R. P. Bartsch, P. Bernaola-Galvan, M. Yoneyama, and P. C. Ivanov, Effect of extreme data loss on long-range correlated and anticorrelated signals quantified by detrended fluctuation analysis [J]. Phys. Rev. E, 2010, 81:1-18.
[5]R. O. Weber and P. Talkner, Spectra and correlations of climate data from days to decades [J]. JOURNAL OF GEOPHYSICAL RESEARCH,2001,106: 131-144.
[6]J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Multifractal detrended fluctuation analysis of nonstationary time series [J]. Physica A, 2002, Vol 316: 87-114.
[7]S. Arianos and A. Carbone, Detrending moving average algorithm:A closed-form approximation of the scaling law [J]. Physica A, 2007, Vol 382:9-15.
[8]E. Alessio, A. Carbone, G. Castelli and V. Frappietro, Second-order moving average and scaling of stochastic time series [J]. The European Physical Journal B,2009, Vol 27:197-200.
[9]Gao-Feng Gu, Wei-Xing Zhou, Detrending moving average algorithm for multifractals [J], Phys. Rev. E, 2010, Vol 82.
[10]L. M. Xu, P. C. Ivanov, K. Hu, Z. Chen, A. Carbone,and H. E. Stanley, Quantifying signals with power-law correlations: A comparative study of detrended fluctuation analysis and detrended moving average techniques. Phys. Rev. E, 2005, Vol 71,051101.
[11]A. Bashan, R. Bartsch, J. W. Kantelhardt, and S. Havlin, Comparison of detrending methods for fluctuation analysis [J]. Physica A, 2008, Vol387: 5080-5090.
[12]于建玲,臧保将,商朋见,股市时间序列的多重分型分析[J],北京交通大学学报,2006,V01.30 No.6：69-72.
[13]张宏,董科强,交通时间序列的多重分形特征分析[J],河北工程大学学报(自然科学版),2009,Vol 26:109-112.
[14]Seungjae Lee, Young-Ihn Lee, Bumcheol Cho, Short-term travel speed prediction models in car navigation systems [J], Journal of Advanced Transportation, 2005, Vol. 40, No. 2:122-138.
[15]Danech-Pajouh M., and Aron M., ATHENA: a method for short-term inter-urban motorway traffic forecasting [J], Recherche Transports Securite, 1991,6: 11-16.
[16]Ahmed, M. S., and Cook, A. R., Analysis of freeway traffic time-series data by using Box-Jenkins techniques [J]. Transportation Research Record 722, Transportation Research Board, Washington, D.C.1979: 1-9.
[17]Moorthy C. K., and Ratcliffe B. G, Short term traffic forecasting using time series methods [J], Transp. Plan. Technol.,1988,12(1):45-56.
[18]Williams, B.M., Modeling and forecasting vehicular traffic flow as a seasonal stochastic time series process, Doctoral dissertation, Department of Civil Engineering, University of Virginia, Charlottesville, 1999.
[19]Dochy T. M., and Lechevallier Y., Short term traffic forecasting using neural network, Transportation system: Theory and application of advanced technology, B. Liu and J. M. Blosseville, eds.,1995:Vol. 2:633-638.
[20]Lajos Kisgyorgy and Laurence R. Rilett, Travel time prediction by advanced neural network, Periodica Polytechnica Ser. Civ. Eng., 2002, Vol 46: 15-32.
[21]Satu Innamaa, Short-term prediction of traffic situation using mlp-neural networks, congress on intelligent transport systems, Turin, Italy, 2000.
[22]Smith, B.L., Demetsky, M.J, Short-term Traffic Flow Prediction Models-a Comparison of Neural Network and Nonparametric Regression Approaches, Transportaion Research Record 1453, 1994:98-104.
[23]Smith, B.L., Demetsky, M.J., Traffic flow forecasting: comparison of modeling approaches, ASCE Journal of Transportation Engineering, 1997, Vol. 123, No. 4: 261-266.
[24]Y. F. Tang, William H. K. Lam, M. ASCE, and Pan L. P. Ng, Comparison of four modeling techniques for short-term AADT forecasting in Hong Kong, Journal of Transportation Engineering, 2003, Vol.129, No. 3:271-277.
[25]F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, Multifractal temperature and flux of temperature variance in fully developed turbulence. Europhys. Lett.,1996, Vol 34:195.
[26]G. Xu, R. A. Antonia, and S. Rajagopalan, Scaling of mixed longitudinal-transverse velocity structure functions, Europhys. Lett.,2007, Vol 79:44001.
[27]T. B. Zeleke and B.C. Si, Scaling Relationships between Saturated Hydraulic Conductivity and Soil Physical Properties, Soil Sci. Soc. AM. J., 2004, 69:1691-1702.
[28]D. C. Lin, Common multifractality in the heart rate variability and brain activity of healthy humans, Physica A, 2008, Vol 20: 387-394.
[29]D. C. Lin and A. Sharif, Common multifractality in the heart rate variability and brain activity of healthy humans. Chaos, 2010, 20, 023121.
[30]B. Podobnik, D. Wang, D. Horvatic, I. Grosse, and H. E. Stanley, Time-lag cross-correlations in collective phenomena. Europhys. Lett., 2010, Vol 90, 68001.
[31]S. Arianos and A. Carbone, Cross-correlation of long-range correlated series. J. Stat. Mech., 2009, P03037.
[32]W. C. Jun, G. Oh, and S. Kim, Understanding volatility correlation behavior with a magnitude cross-correlation function. Phys. Rev. E, 2006, 73,066128.
[33]B. Podobnik and H. E. Stanley, Detrended Cross-Correlation Analysis:A New Method for Analyzing Two Nonstationary Time Series, Phys. Rev. Lett.,2008,100, 084102.
[34]B. Podobnik, I. Grosse, D. Horvatic, S. Ilic, P. Ch. Ivanov, and H. E. Stanley, Quantifying cross-correlations using local and global detrending approaches, Eur. Phys. J.2009,71,243-250.
[35]B. Podobnik, D. Horvatic, A. M. Petersen, and H. E. Stanley, Cross-correlations between volume change and price change, Proc. Natl. Acad. Sci.,2009, 106, 22079-22084.
[36]E. L. Siqueira Jr., T. Stosic, L. Bejan, and B. Stosic, Correlations and cross-correlations in the Brazilian agrarian commodities and stocks, Physica A, 2010, Vol 389, 2739-2743.
[37]N. Xu, P. J. Shang, and S. Kamae, Modeling traffic flow correlation using DFA and DCCA, Nonlin. Dyn., 2010, Vol 61,207-216.
[38]G. F. Zebende, P. A. da Silva, and A. M. Filho, Study of cross-correlation in a self-affine time series of taxi accidents, Physica A, 2011, Vol 390,1677-1683.
[39]W. X. Zhou, Multifractal detrended cross-correlation analysis for two nonstationary signals, Phys. Rev. E, 2008, Vol 77, 066211.
[40]L. Kristoufek, Multifractal height cross-correlation analysis, (2010), uTIA AV CR Research Report No. 2281.
[41]S. Shadkhoo and G. R. Jafari, Multifractal detrended cross-correlation analysis of temporal and spatial. Eur. Phys. J. B, 2009, Vol 72:679-683
[42]S. Hajian and M. S. Movahed, Multifractal Detrended Cross-Correlation Analysis of sunspot numbers and river flow fluctuations. Physica A, 2010, Vol 389:4942-4957.
[43]Y. D. Wang, Y. Wei, and C. F. Wu, Cross-correlations between Chinese A-share and B-share markets. Physica A, 2010, Vol 389: 5468-5478.
[44]L. Y. He and S. P. Chen, Nonlinear bivariate dependency of price-volume relationships in agricultural commodity futures markets: A perspective from Multifractal Detrended Cross-Correlation Analysis, Physica A,2011, Vol 390: 297-308.
[45]Y. D. Wang, Y. Wei, and C. F. Wu, Detrended fluctuation analysis on spot and futures markets of West Texas Intermediate crude oil. Physica A,2011, Vol 390: 864-875.
[46]Zhi-Qiang Jiang, Wei-Xing Zhou. Multifractal detrending moving average cross-correlation analysis, Submitted to Statistical Finance, Physics and Society, 2011.
[47]Jan W. Kantelhardt, Fractal and Multifractal Time Series, 2008, arXiv:0804.0747v1.
[48]J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Multifractal detrended fluctuation analysis of nonstationary time series. Physica A, 2002, Vol 316: 87-114.
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