混沌时序分析中的若干问题及其应用研究
|
摘要
随着非线性混沌动力学的发展,人们对时间序列的复杂性有了更深刻的认识,尤其是混沌时间序列的分析已经成为一个非常重要的研究方向。混沌时间序列分析包括混沌特征参数判别和混沌时序预测等方面,由于混沌现象普遍地存在于自然界中,因此混沌时序的分析在各种动力系统的混沌检验及预测等方面显得尤为重要。
本文对混沌时序的判别及预测进行了研究,主要成果如下:
1.应用Takens’估计方法研究了经典混沌系统所生成时间序列的分形特征。并运用Takens’估计法研究了上海证券数据A股指数开盘价、收盘价、最高价、最低价的分形特征,发现其分形维数随着时间推移有进一步增加的趋势。结合最大Lyapunov指数的计算结果,表明上海证券数据总体表现为低自由度的弱混沌系统。
2.基于K熵理论首先研究了混沌时间序列的最大可预测度问题。在此研究工作的基础上,提出了一种运用RBF神经网络在K熵限定的最大可预测度内对混沌系统进行直接多步预测的方法,并以Lorenz系统为例对其进行了相应的预测研究。研究结果表明:本文提出的在K熵限定的最大可预测度内进行直接多步预测具有比递推多步预测运行速度快,且预测精度更高的特点,便于对系统的实时预测和控制。
3.利用遗传算法强全局随机搜索特点,结合DRNN神经网络对非线性数据具有鲁棒性和自学习能力的优点,通过将历年的农机总动力数据作为时间序列进行分析,建立DRNN神经网络预测模型对农机总量进行了预测。通过与校验用数据的比较证明本文建立的预测模型具有较高的精度。
4.在加权一阶局域预测模型的理论基础上,提出运用径向基神经网络代替加权一阶局域预测模型构成了基于径向基神经网络的局域预测模型。通过对Logistic映射以及Lorenz系统的三个分量的混沌时间序列的预测仿真,表明该预测方法对混沌时间序列的预测具有较好的效果。
With the development of nonlinear chaotic dynamic, there are more profound recognitions to the complexity of time series. Especially the analysis of time series is becoming an important research aspect. The analyses of chaotic time series include identification of chaotic characteristic parameter and prediction of chaotic time series, and so on. Because chaotic phenomenon widely exists in nature, the analysis of chaotic time series is more important in the field of chaotic identification and prediction on many dynamic systems.
The identification and prediction of chaotic time series are researched in the paper. The main contents are as follows:
1. The fractal dimension of time series of classical chaotic systems is firstly confirmed by Takens’estimator method. Then Takens’estimator method is compared with G-P method, it shows that Takens’estimator method inherits all virtues of G-P method and has better characteristics than G-P method such as low computation complexity and fast calculation speed. The fractal characteristics of opening quotation, closing quotation, maximum price and minimum price in Shanghai stock market are calculated by Takens’estimator method. The results demonstrate that the fractal dimension of Shanghai stock data has increasing trends with time. To further prove the chaotic behaviors of Shanghai stock data, the maximum Lyapunov exponent is calculated by the approach of small data sets. The results also indicate that Shanghai stock data are a feeble chaotic system that is in low degree of freedom.
2. The maximum predictability time of chaotic time series is studied on the basis of Kolmogorov entropy theory. Then a method of direct multi-step prediction of chaotic time series is proposed, which is based on Kolmogorov entropy and radial basis functions neural networks. And the Lorenz system is predicted by the method. Simulation results for direct multi-step prediction method are compared with recurrence multi-step prediction method. The results indicate that the direct multi-step prediction is more accurate and rapid than the recurrence multi-step prediction within the maximum predictability time of chaotic time series. So, it is convenient to forecast and control with real time using the method of direct multi-step prediction.
3.Agricultural machine power is forecast by diagonal recurrent neural network (DRNN) forecasting model, which is combined the characteristic of global random search of genetic algorithm with the virtue of robust and self-study for nonlinear data of DRNN and agricultural machine power data in history is analyzed by time series. The forecasting results is compared with verify data, that prove the forecasting model in this paper has higher precision.
4. A local-region linear prediction method based on radial basis function neural net is presented for chaotic time series prediction, which theory foundation is add-weighted one-rank local-region single-step method. The prediction method is built by using RBFNN substitute for add-weighted one-rank model. The Logistic map and the three axes of Lorenz system are applied to verify the method. Simulation results indicate that the method is effective for prediction of chaotic time series.
本文对混沌时序的判别及预测进行了研究,主要成果如下:
1.应用Takens’估计方法研究了经典混沌系统所生成时间序列的分形特征。并运用Takens’估计法研究了上海证券数据A股指数开盘价、收盘价、最高价、最低价的分形特征,发现其分形维数随着时间推移有进一步增加的趋势。结合最大Lyapunov指数的计算结果,表明上海证券数据总体表现为低自由度的弱混沌系统。
2.基于K熵理论首先研究了混沌时间序列的最大可预测度问题。在此研究工作的基础上,提出了一种运用RBF神经网络在K熵限定的最大可预测度内对混沌系统进行直接多步预测的方法,并以Lorenz系统为例对其进行了相应的预测研究。研究结果表明:本文提出的在K熵限定的最大可预测度内进行直接多步预测具有比递推多步预测运行速度快,且预测精度更高的特点,便于对系统的实时预测和控制。
3.利用遗传算法强全局随机搜索特点,结合DRNN神经网络对非线性数据具有鲁棒性和自学习能力的优点,通过将历年的农机总动力数据作为时间序列进行分析,建立DRNN神经网络预测模型对农机总量进行了预测。通过与校验用数据的比较证明本文建立的预测模型具有较高的精度。
4.在加权一阶局域预测模型的理论基础上,提出运用径向基神经网络代替加权一阶局域预测模型构成了基于径向基神经网络的局域预测模型。通过对Logistic映射以及Lorenz系统的三个分量的混沌时间序列的预测仿真,表明该预测方法对混沌时间序列的预测具有较好的效果。
With the development of nonlinear chaotic dynamic, there are more profound recognitions to the complexity of time series. Especially the analysis of time series is becoming an important research aspect. The analyses of chaotic time series include identification of chaotic characteristic parameter and prediction of chaotic time series, and so on. Because chaotic phenomenon widely exists in nature, the analysis of chaotic time series is more important in the field of chaotic identification and prediction on many dynamic systems.
The identification and prediction of chaotic time series are researched in the paper. The main contents are as follows:
1. The fractal dimension of time series of classical chaotic systems is firstly confirmed by Takens’estimator method. Then Takens’estimator method is compared with G-P method, it shows that Takens’estimator method inherits all virtues of G-P method and has better characteristics than G-P method such as low computation complexity and fast calculation speed. The fractal characteristics of opening quotation, closing quotation, maximum price and minimum price in Shanghai stock market are calculated by Takens’estimator method. The results demonstrate that the fractal dimension of Shanghai stock data has increasing trends with time. To further prove the chaotic behaviors of Shanghai stock data, the maximum Lyapunov exponent is calculated by the approach of small data sets. The results also indicate that Shanghai stock data are a feeble chaotic system that is in low degree of freedom.
2. The maximum predictability time of chaotic time series is studied on the basis of Kolmogorov entropy theory. Then a method of direct multi-step prediction of chaotic time series is proposed, which is based on Kolmogorov entropy and radial basis functions neural networks. And the Lorenz system is predicted by the method. Simulation results for direct multi-step prediction method are compared with recurrence multi-step prediction method. The results indicate that the direct multi-step prediction is more accurate and rapid than the recurrence multi-step prediction within the maximum predictability time of chaotic time series. So, it is convenient to forecast and control with real time using the method of direct multi-step prediction.
3.Agricultural machine power is forecast by diagonal recurrent neural network (DRNN) forecasting model, which is combined the characteristic of global random search of genetic algorithm with the virtue of robust and self-study for nonlinear data of DRNN and agricultural machine power data in history is analyzed by time series. The forecasting results is compared with verify data, that prove the forecasting model in this paper has higher precision.
4. A local-region linear prediction method based on radial basis function neural net is presented for chaotic time series prediction, which theory foundation is add-weighted one-rank local-region single-step method. The prediction method is built by using RBFNN substitute for add-weighted one-rank model. The Logistic map and the three axes of Lorenz system are applied to verify the method. Simulation results indicate that the method is effective for prediction of chaotic time series.
引文
[1] Jeong Jaeseung. Kim Dai-Jin, Chae Jeong-Ho, et a1. Nonlinear analysis of the EEG of schizophrenics with optimal embedding dimension. Medical Engineering Physics, 1998, 20: 669～676.
[2]洪时中.非线性时间序列分析的最新进展及其在地球科学中的应用前景.北京:地球科学进展, 1999, 14(6):559～565
[3] Diks Cees, M udelsee M anfred. Redundancies in the Earth’s climatologically time Series. Phys. Lett.A, 2000, 275(5-6):407～414
[4] Agnon Yehuda, Golan Amos, Shearer Matthew. Nonparametric, nonlinear, short term forecasting: theory and evidence for nonlinearities in the commodity markets. Economics Lett. , 1999, 65:293～299
[5] Fernanda Strozzi. Application of Nonlinear Time Series analysis Technology to High Frequency Currency Exchange Data. Physica A: Statistical Mechanics and its Applications, 2002, 312(3-4):520～538
[6]王兴元,顾树生.心电动态生理及病理信息的非线性动力学研究.中国生物医学工程学报, 2000, 19(4):397～403
[7] Chiang T C. Time Series Dynamics of Short Time Interest: Evidence from Eurocurrency markets. Journal of Int1. Financial Markets, Institution and Money, 1997(7):201～220
[8]刘东林,帅典勋.网络流量模型的非线性特征量的提取及分析[J].电子学报, 2003. 31(12):1866～1869
[9] Cao Liangyue, Soofi A S. Nonlinear deterministic forecasting of daily dollar exchange rates. International Journal of Forecasting, 1999, 15:421～430
[10] Vaidya P G, Angadi S. Decoding chaotic cryptography without access to the super key. Chaos, Solitons and Fractals, 2003, 17: 379～386
[11] Tong H. Nonlinear time series analysis since 1990:some personal reflections. Acta Mathematicae Applicatae Sinica (English Series), 2002, 18(2):177～184.
[12] Devaney R L. An introduction to chaotic dynamical systems. enjamin/Cummings: Menlo Park, CA, 1985.
[13] Li T Y and Yorke J A. Periods three implies chaos. Amer Math Monthly. 1975, 82: 985～992.
[14]孙建成,张太镒,刘枫.基于重建相空间的衰落信道非线性预测算法[J].电波科学学报, 2005, 20(2):154～159.
[15]王定成,宋良图. LM滚动优化的SVMR预测控制研究.模式识别与人工智能, 2005, 18(3):281～285.
[16]李广,邹德忠,谈顺涛.基于混沌神经网络理论的小电网短期电力负荷预测.电力自动化设备, 2006, 26(2): 50～52.
[17]杨晓华,杨志峰,沈珍瑶等.水环境非线性时序预测的高精度RBF网络模型.水科学进展, 2005, 16(6): 788～791.
[18]黄志全,樊敬亮,王思敬.混沌时间序列预测的局域法在边坡变形分析中的应用.工程地质学报, 2005, 13(02):252～256.
[19] Grassberger P, Procaccia I. Characterization of strange attractors. Physical Review Letters, 1983, 50(5):346～349.
[20] Grassberger P, Procaccia I. Measuring the strangeness of strange attractors. Physica D. 1983, 9:189～208.
[21] A.Guerrero, L.A.Smith. Towards coherent estimation of correlation dimension. Physics Letters A, 2003, 318 (4-5):373～379.
[22]赵贵兵,石炎福,段文锋等.从混沌时间序列同时计算关联维和Kolmogorov熵.计算物理,1999,16(3):309～315
[23]向小东,郭耀煌.混沌时间序列最大Lyapunov指数的计算.预测, 2001, 20(5):76～78
[24] Ma Junhai, Chen Yushu. The matrix algorithm of lyapunov exponent for the experimental data obtained in dynamic analysis. Applied Mathematics and Mechanics, 1999, 20(9):985～993
[25]李国辉,周世平,徐得名.时间序列最大Lyapunov指数的计算.应用科学学报,2003,21(2):127～131
[26]单华宁,王平立,王执铨.一种利用小波包分析计算混沌时间序列Kolmogorov熵的方法.信息与控制, 2005, 34(1):12～15
[27] H.Tong, KS Lim. Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society B, 1980, (42):245～292.
[28] H.Tong,ZM.Wu. Multi-step-ahead Forecasting of cyclical data by threshold autoregression, in OD.Anderson, JG.Gooijer, KDC.Stoodley, Time Series Analysis:Theory and Practice. North-Holland Pub1.Co.,Amsterdam, 1982.
[29]侯振挺,吴今培. TAR模型建模的新策略.科学探索学报, 1987, (7):13～22.
[30] Lapedes A,Farber R.A self-optimizing,nonsymmetrical neural net for content addressable memory and pattern recognition.Physica 22D, 1986, 247～259.
[31] Vaidyanathan R, Venkatasubramanian V. Representing and diagnosing dynamic process data using neural works. Eng Appl Artif Intell, 1992, 5(1):11～21.
[32] Bhat N, Mcavoy T J. Use of neural nets for dynamic modeling and control of chemical process systems. Compute Chem Eng, 1990, 14(4/5):573～578.
[33]吴建锋,何小荣,陈丙珍.动态系统前馈网络模型及其应用.化工学报,2000, 51(3):378～382
[34]蒋伟进.非线性混沌时序的神经网络预测与控制算法研究.计算机应用与软件, 2004, 21(4):81～83
[35]周文献,孙立军.基于遗传神经网络的水泥路面使用性能预测.计算机应用, 2005, 25(12): 280～281
[36]李冬梅,王正欧.基于RBF神经网络的混沌时间序列预测.模式识别与人工智能, 2001, 14(2):231～234
[37]向小东.基于混沌理论与径向基函数神经网络的混沌时间序列预测.福州大学学报(自然科学版), 2003, 31(4):401～407
[38]王宏伟,马广富.基于模糊模型的混沌时间序列预测.物理学报, 2004, 53(10): 3293～3297
[39]马军海,陈予恕,辛宝贵.基于非线性混沌时序动力系统的预测方法研究.应用数学和力学, 2004, 25(6): 551～557
[40]李春鑫,李天伟,方建平等.基于小波神经网络非线性预测方法的研究.光电技术应用, 2005, 20(2): 44～46
[41]张学工.关于统计学习理论与支持向量机.自动化学报, 2000, 26(1):32～42.
[42] Vapnik V N. The Nature of Statistical Learning Theory. New York: Springer-Verlag, 1995
[43] Mukherjee S,Osuma E,Girosi F. Nonlinear prediction of chaotic time series using support vector machines. Neural Networks for Signal ProcessingⅦ, 1997, 511～520.
[44] Mckay D, Fyfe C. Probability prediction using Support Vector Machine. Proceedings of Fourth International Conference in Knowledge-BasedIntelligent Engineering Systems & Allied Technologies [C]. 2000, 189～192.
[45] Suykens J A K,Vandewalle J.Least squares support vector machine classifiers. Neural Processing Letters, 1999, 9(3):293～300.
[46]崔万照,朱长纯,保文星等.混沌时间序列的支持向量机预测.物理学报, 2004, 53(10): 3303～3310.
[47]叶美盈,汪晓东,张浩然基.线最小二乘支持向量机回归的混沌时间序列预测.物理学报,2005, 54(6):2568～2573
[48] J.D. Farmer, J.J. Sidorowich. Predicting chaotic time series, Phys. Rev. Lett. 1987, 59: 845～848.
[49] Kugiumtzis D,Lingjarde O C,Christophersen N. Regularized local prediction of chaotic time series. Physica D, 1998, 112(3):344～360
[50]甘建超,肖先赐.混沌时间序列基于邻域点的非线性多步自适应预测.物理学报, 2003, 52(12):2995～3001.
[51]李元诚,李波,方廷健.基于小波支持向量机的非线性组合预测方法研究.信息与控制, 2004, 33(3): 303～306.
[52] B.B.Mandelbrot. The fractal geometry of nature. New York: W H. Freeman, 1982, 1～22.
[53]周凌云,王瑞丽,吴光敏等.非线性物理理论及应用.北京:科学出版社,2000, 3～18
[54]林鸿溢,李映雪.分形论-奇异性探索.北京:北京理工大学出版社,1993, 52～72
[55]汪富泉,李后强.分形几何与动力学系统.黑龙江教育出版社,1993.
[56]曾文曲,王向阳.分形理论与分形的计算机模拟.沈阳:东北大学出版社,1993, 1～31.
[57] S. N. Rasband. Chaotic dynamics of nonlinear systems. New York:John Wiley & Sons, Inc, 1989, 71～85
[58]齐东旭.分形及其计算机生成.北京:科学出版社,1994.
[59]宿成建,缪晓波,刘星.中国证券市场的非线性特征与分形维分析.系统工程理论与实践, 2005(5): 68～73
[60] Theiler J. Spurious dimension from correlation algorithms applied to limited time series data. Physical Review A. 1986, 34(3): 2427～2432.
[61] Pecore L. Synchronization in chaotic system. Physica Letter, 1990, 64(8):821～824.
[62]吕金虎,张锁春. Lyapunov指数的数值计算方法.非线性动力学学报, 2001,8(1):84～92.
[63] Martin N F G,England J W. Mathematical theory of entropy. London: Addison-Wesley Publishing Company, 1981
[64] C. E. Shannon. A mathematical theory of communication. The bell system technical, 1948, 27:379～423
[65]邢修三.物理熵,信息熵及其演化方程.中国科学(A辑),2001, 37(1):77～84
[66] E.T.Jaynes. Information theory and statistical mechanics. Physical Review. 1957, 106:361～630
[67]何大义.熵在数据分析中的应用研究.统计与决策, 2005, (4):27～29
[68] Adler R, Konheim A., McAndrew M. Topological entropy, Trans. Amer Math. Soc, 1965, 114:309～319
[69] Bowen R., Entropy for group endomorphisms and homogeneous spaces, Trans Amer. Math. Soc., 1971, 153:401～414
[70] Pincus S M. Approximate entropy as a measure of system complexity. Proc Natl Acad Sci USA, 1991, (88):2297～2301
[71] Pincus S. Approximate entropy (ApEn) as a complexity measure. Chaos, 1995, 5(1): 110～117
[72] Steven M Pincus. Quantifying complexity and regularity of neurobiological systems. Methods in Neurosciences, 1995, 28:336～363.
[73]邱红梅,高思莉,刘春霞,肖青.噪声对混沌系统近似熵的影响.东北师大学报自然科学版, 2004, 36(1): 46～49
[74]杨福生,廖旺才.近似熵:一种适用于短数据的复杂性度量.中国医疗器械杂志, 1997, 21(5):283～286
[75] Kolmogorov A. N. A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl.Akad.Sci. SSSR., 1958, 119:861～864
[76] P.Grassberger, I.Procaccia. Characterization of strange attractors, Phys. Rev.Lets., 1983, 50(5): 346～349
[77]马军海,盛昭翰.经济系统混沌时序重构的分析和应用.管理科学学报. 2002, 5(3):73～78.
[78]肖方红,阎桂荣,韩宇航.混沌时序相空间重构参数确定的信息论方法.物理学报,第2005, 54(2):550～556
[79] Takens F. Detecting strange attactors in turbulence. Lecture Notes in Mathematics. Springer Verlag, 1981, 898:366～381.
[80] Packard N H,Crutchfield J P, Farmer J D, et a1. Geometry from a timeseries. Phys. Rev. Lett., 1980,43(9):712～715
[81] Gibson J F,Farmer J,Casdagli M ,et a1. An analytic approach to practical state space reconstruction. Physica D, 1992, 57: 1～30
[82]邓兰松,沈菲.非线性时序的混沌特性分析与预测.天津大学学报,2004, 37(11):1022～1025
[83] Farmer J D,Sidorowich J J. Predicting chaotic time series. Phys Rev Lett,1987, 59(8):845～848.
[84] Linsay P S.An efficient method of forecasting chaotic time series using linear interpolation. Phys Lett A, 1991, 153(6/7):353～356.
[85] Navone H D, Ceceatto H A. Forecasting chaos from small data sets: a comparison of different nonlinear algorithms. J Phys A: Math Gen, 1995, 28(12):3381～3388.
[86]任晓林,胡光锐,徐雄.混沌时间序列局域线性预测方法.上海交通大学学报, 1999, 33(1):19～21.
[87] Wolf A, Swift J B, Swinney H L, et al. Determining Lyapunou exponents from time series. Physica D, 1985,16(2):285～371
[88]温权,张勇传,程时杰.混沌时间序列预测技术.水电能源科学, 2001, 10(3):76～78
[89]吕金虎,陆君安,陈士华.混沌时间序列分析及其应用.武汉:武汉大学出版社,2002, 113
[90] Stutzer.M. Chaotic dynamics & bifurcation in a macro model. Journal of Economic Dynamics & Control November, 1980, 2: 353～376
[91] Day.R. The emergence of chaos from classical economic growth. Quarterly Journal of Economics, 1983, 54:201～213
[92] Chen Ping. Nonlinear dynamics theory of economic chaos and fluctuations. Working paper, National center for Economic research at Peking University, 2000, 10 No.C2000015.
[93]张兵.中国股票市场有效性研究.南京:南京大学出版社,2004
[94] MaJunhai, Chen Yushu. The Non-Linear Chaotic model reconstruction for the experimental data obtained from different dynamic system. Applied Mathematics and Mechanics, 1999, 20(11):1214～1221.
[95] Ma Junhai, Chen Yushu. Study on the prediction method of low-dimension time series that arise from the intrinsic non-linear dynamics. Applied Mathematics and Mechanics, 2001, 22 (5):501～509.
[96] Ma Junhai, Chen Yushu. Study on prediction methods for dynamic systems of nonlinear chaotic time series. Applied Mathematics andMechanics, 2004, 25 (6):605～611.
[97]马军海,盛昭瀚.混沌经济时序非线性动力系统的预测方法研究.系统工程学报,2002, 17(2):97～102.
[98]马军海,齐二石,莫馨.混沌时序重构及上海股票指数预测的应用研究.系统工程理论与实践, 2003, 23(12):86～94.
[99]马军海.复杂非线性系统的重构技术.天津:天津大学出版社,2005
[100] Hongchun Wang, Guanrong Chen , Jinhu Lü. Complex dynamical behaviors of daily data series in stock exchange. Physics Letters A,2004,333(3-4):246～255.
[101]熊正丰.金融时间序列分形维估计的小波方法.系统工程理论与实践,2002, 22(12):48～53.
[102]徐绪松,熊保平等.用RBF神经网络确定上海股市的分形维数.武汉大学学报(理学版),2003, 49(3):309～312.
[103]刘孝贤,郭举修,张庆华.上证综合指数的非线性动力学特点.山东工业大学学报, 2001, 31(5):416～423.
[104]邹志勇.金融数据的分形实证分析.电脑与信息技术,2001(6):62～65.
[105]李志勇,王春红,陈良生等.股票市场的分形维数分析.东北大学学报(自然科学版), 2003, 24(7):666～669
[106]陈军飞,王嘉松.基于小波分形理论的股价指数信息量测度研究.运筹与管理, 2003, 12(1):65～69
[107]高红兵,潘瑾,陈宏民.深圳成份股指数收益率序列的分形维.预测, 2000, (6):50～51
[108] Fisher Y. Fractal image compression. Fractal, 1994, 3(3):154～156
[109] Barnsley M F. Fractal image compression. New York: Academic Press, 1993, 121～145
[110] Barnsley M F. Fractal function and interpolation. Constructive Approximation, 1986, 8(2):303～329
[111]孙博文.中国股市波动的混沌吸引子的测定与计算.哈尔滨理工大学学报, 2001, 33(5):323～327
[112] Mandelbrot B B. The fractal geometry of nature. Singapore: World Scientific, 1983, 86～88.
[113] Mandelbrot B B. Fractals and scaling in finance. NewYork: Spring-Verlag Technological Education Publishing House, 1997, 35～64.
[114] Grassberger P, Procaccia I. Estimation of the Kolmogorov entropy from a chaotic signa1. Phys Rev A, 1983, 28(4): 2591～2593.
[115]陈增强,袁著祉,张燕.基于神经网络的非线性预测控制综述.控制工程,2002, 9(4): 7～11
[116]阎平凡,张长水编著.人工神经网络与模拟进化计算.北京:清华大学出版社,2000
[117]王洪元,史国栋主编.人工神经网络技术及其应用.北京:中国石化出版社,2002, 41～42.
[118]陈祥光,裴旭东编著.人工神经网络技术及应用.北京:中国电力出版社,2003,71～72
[119]田永青,杨斌,朱仲英.基于RBF神经网络建立税务预测模型的研究.计算机工程, 2002, 28(5):22～27
[120] Holcomb T, Morari M. Local training for radial basis function networks: towards solving the hidden unit problem. American Control Conference, 1991, 2331～2335.
[121]赵贵兵,石炎福,段文锋等.从混沌时间序列同时计算关联维和Kolmogorov熵.计算物理, 1999, 16(3): 309～315.
[122]周佩玲,陶小丽等.基于GA的RBF网络用于旱涝灾害因素预测研究.计算机应用, 2001, 21(1): 7～9.
[123]褚庆全,李立军,张志鹏.基于神经网络的宏观农业生产预测模型的研究.中国农业大学学报,2004,9(5): 1～5.
[124]林丽兰,何勇.改进的BP神经网络方法及其在农业商品总产值预测中的应用.科技通报, 2005, 21(1):6～10.
[125]朱春江,唐德善,马文斌.基于灰色理论和BP神经网络预测观光农业旅游人数的研究.安徽农业科学, 2006, 34(4):612～614.
[126]刘金琨编著.智能控制.北京:电子工业出版社,2005, 100～155.
[127]杨建刚编著.人工神经网络实用教程.杭州:浙江大学出版杜,2001, 148～149
[128]罗四维编著.人工神经网络建造.北京:中国铁道出版社,1998, 134～136
[129] Atiya A F,EI-Shoura S M. A comparison between neural-network forecasting techniques-case study: river flow forecasting. IEEE Trans. on Neural Networks, 1999, 10(2): 402～409.
[130] D.Kugiumtzis, O.C. Lingjaerde, N.Christophersen. Regularized local linear prediction of chaotic time series. Physica D, 1998, 112: 344～360.
[131] Bakirtzis A G, Theocharis J B. Short term load forecasting using fuzzy neural networks. IEEE Tran. on Power System, 1995, 10(3): 1518～1524.
[132]李爱国,覃征.滑动窗口二次自回归模型预测非线性时间序列.计算机学报, 2004, 27(7): 1004～1008
[133]吕金虎,张锁春.加权一阶局域法在电力系统短期负荷预测中的应用.控制理论与应用,2002, 19(5):767～770
[134] Schreiber T. Extremely simple nonlinear noise reduction method. Plays rev E, 1993, 47(4):2401～2404.
[135] Schreiber T, Grassberger P. A simple noise reduction method for real data. Phys Lett A, 1991, 160:411～418
[136] Sivakumar B. Chaos theory in hydrology: important issues and interpretations. J Hydrol, 2000, 227:1～20.
[2]洪时中.非线性时间序列分析的最新进展及其在地球科学中的应用前景.北京:地球科学进展, 1999, 14(6):559～565
[3] Diks Cees, M udelsee M anfred. Redundancies in the Earth’s climatologically time Series. Phys. Lett.A, 2000, 275(5-6):407～414
[4] Agnon Yehuda, Golan Amos, Shearer Matthew. Nonparametric, nonlinear, short term forecasting: theory and evidence for nonlinearities in the commodity markets. Economics Lett. , 1999, 65:293～299
[5] Fernanda Strozzi. Application of Nonlinear Time Series analysis Technology to High Frequency Currency Exchange Data. Physica A: Statistical Mechanics and its Applications, 2002, 312(3-4):520～538
[6]王兴元,顾树生.心电动态生理及病理信息的非线性动力学研究.中国生物医学工程学报, 2000, 19(4):397～403
[7] Chiang T C. Time Series Dynamics of Short Time Interest: Evidence from Eurocurrency markets. Journal of Int1. Financial Markets, Institution and Money, 1997(7):201～220
[8]刘东林,帅典勋.网络流量模型的非线性特征量的提取及分析[J].电子学报, 2003. 31(12):1866～1869
[9] Cao Liangyue, Soofi A S. Nonlinear deterministic forecasting of daily dollar exchange rates. International Journal of Forecasting, 1999, 15:421～430
[10] Vaidya P G, Angadi S. Decoding chaotic cryptography without access to the super key. Chaos, Solitons and Fractals, 2003, 17: 379～386
[11] Tong H. Nonlinear time series analysis since 1990:some personal reflections. Acta Mathematicae Applicatae Sinica (English Series), 2002, 18(2):177～184.
[12] Devaney R L. An introduction to chaotic dynamical systems. enjamin/Cummings: Menlo Park, CA, 1985.
[13] Li T Y and Yorke J A. Periods three implies chaos. Amer Math Monthly. 1975, 82: 985～992.
[14]孙建成,张太镒,刘枫.基于重建相空间的衰落信道非线性预测算法[J].电波科学学报, 2005, 20(2):154～159.
[15]王定成,宋良图. LM滚动优化的SVMR预测控制研究.模式识别与人工智能, 2005, 18(3):281～285.
[16]李广,邹德忠,谈顺涛.基于混沌神经网络理论的小电网短期电力负荷预测.电力自动化设备, 2006, 26(2): 50～52.
[17]杨晓华,杨志峰,沈珍瑶等.水环境非线性时序预测的高精度RBF网络模型.水科学进展, 2005, 16(6): 788～791.
[18]黄志全,樊敬亮,王思敬.混沌时间序列预测的局域法在边坡变形分析中的应用.工程地质学报, 2005, 13(02):252～256.
[19] Grassberger P, Procaccia I. Characterization of strange attractors. Physical Review Letters, 1983, 50(5):346～349.
[20] Grassberger P, Procaccia I. Measuring the strangeness of strange attractors. Physica D. 1983, 9:189～208.
[21] A.Guerrero, L.A.Smith. Towards coherent estimation of correlation dimension. Physics Letters A, 2003, 318 (4-5):373～379.
[22]赵贵兵,石炎福,段文锋等.从混沌时间序列同时计算关联维和Kolmogorov熵.计算物理,1999,16(3):309～315
[23]向小东,郭耀煌.混沌时间序列最大Lyapunov指数的计算.预测, 2001, 20(5):76～78
[24] Ma Junhai, Chen Yushu. The matrix algorithm of lyapunov exponent for the experimental data obtained in dynamic analysis. Applied Mathematics and Mechanics, 1999, 20(9):985～993
[25]李国辉,周世平,徐得名.时间序列最大Lyapunov指数的计算.应用科学学报,2003,21(2):127～131
[26]单华宁,王平立,王执铨.一种利用小波包分析计算混沌时间序列Kolmogorov熵的方法.信息与控制, 2005, 34(1):12～15
[27] H.Tong, KS Lim. Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society B, 1980, (42):245～292.
[28] H.Tong,ZM.Wu. Multi-step-ahead Forecasting of cyclical data by threshold autoregression, in OD.Anderson, JG.Gooijer, KDC.Stoodley, Time Series Analysis:Theory and Practice. North-Holland Pub1.Co.,Amsterdam, 1982.
[29]侯振挺,吴今培. TAR模型建模的新策略.科学探索学报, 1987, (7):13～22.
[30] Lapedes A,Farber R.A self-optimizing,nonsymmetrical neural net for content addressable memory and pattern recognition.Physica 22D, 1986, 247～259.
[31] Vaidyanathan R, Venkatasubramanian V. Representing and diagnosing dynamic process data using neural works. Eng Appl Artif Intell, 1992, 5(1):11～21.
[32] Bhat N, Mcavoy T J. Use of neural nets for dynamic modeling and control of chemical process systems. Compute Chem Eng, 1990, 14(4/5):573～578.
[33]吴建锋,何小荣,陈丙珍.动态系统前馈网络模型及其应用.化工学报,2000, 51(3):378～382
[34]蒋伟进.非线性混沌时序的神经网络预测与控制算法研究.计算机应用与软件, 2004, 21(4):81～83
[35]周文献,孙立军.基于遗传神经网络的水泥路面使用性能预测.计算机应用, 2005, 25(12): 280～281
[36]李冬梅,王正欧.基于RBF神经网络的混沌时间序列预测.模式识别与人工智能, 2001, 14(2):231～234
[37]向小东.基于混沌理论与径向基函数神经网络的混沌时间序列预测.福州大学学报(自然科学版), 2003, 31(4):401～407
[38]王宏伟,马广富.基于模糊模型的混沌时间序列预测.物理学报, 2004, 53(10): 3293～3297
[39]马军海,陈予恕,辛宝贵.基于非线性混沌时序动力系统的预测方法研究.应用数学和力学, 2004, 25(6): 551～557
[40]李春鑫,李天伟,方建平等.基于小波神经网络非线性预测方法的研究.光电技术应用, 2005, 20(2): 44～46
[41]张学工.关于统计学习理论与支持向量机.自动化学报, 2000, 26(1):32～42.
[42] Vapnik V N. The Nature of Statistical Learning Theory. New York: Springer-Verlag, 1995
[43] Mukherjee S,Osuma E,Girosi F. Nonlinear prediction of chaotic time series using support vector machines. Neural Networks for Signal ProcessingⅦ, 1997, 511～520.
[44] Mckay D, Fyfe C. Probability prediction using Support Vector Machine. Proceedings of Fourth International Conference in Knowledge-BasedIntelligent Engineering Systems & Allied Technologies [C]. 2000, 189～192.
[45] Suykens J A K,Vandewalle J.Least squares support vector machine classifiers. Neural Processing Letters, 1999, 9(3):293～300.
[46]崔万照,朱长纯,保文星等.混沌时间序列的支持向量机预测.物理学报, 2004, 53(10): 3303～3310.
[47]叶美盈,汪晓东,张浩然基.线最小二乘支持向量机回归的混沌时间序列预测.物理学报,2005, 54(6):2568～2573
[48] J.D. Farmer, J.J. Sidorowich. Predicting chaotic time series, Phys. Rev. Lett. 1987, 59: 845～848.
[49] Kugiumtzis D,Lingjarde O C,Christophersen N. Regularized local prediction of chaotic time series. Physica D, 1998, 112(3):344～360
[50]甘建超,肖先赐.混沌时间序列基于邻域点的非线性多步自适应预测.物理学报, 2003, 52(12):2995～3001.
[51]李元诚,李波,方廷健.基于小波支持向量机的非线性组合预测方法研究.信息与控制, 2004, 33(3): 303～306.
[52] B.B.Mandelbrot. The fractal geometry of nature. New York: W H. Freeman, 1982, 1～22.
[53]周凌云,王瑞丽,吴光敏等.非线性物理理论及应用.北京:科学出版社,2000, 3～18
[54]林鸿溢,李映雪.分形论-奇异性探索.北京:北京理工大学出版社,1993, 52～72
[55]汪富泉,李后强.分形几何与动力学系统.黑龙江教育出版社,1993.
[56]曾文曲,王向阳.分形理论与分形的计算机模拟.沈阳:东北大学出版社,1993, 1～31.
[57] S. N. Rasband. Chaotic dynamics of nonlinear systems. New York:John Wiley & Sons, Inc, 1989, 71～85
[58]齐东旭.分形及其计算机生成.北京:科学出版社,1994.
[59]宿成建,缪晓波,刘星.中国证券市场的非线性特征与分形维分析.系统工程理论与实践, 2005(5): 68～73
[60] Theiler J. Spurious dimension from correlation algorithms applied to limited time series data. Physical Review A. 1986, 34(3): 2427～2432.
[61] Pecore L. Synchronization in chaotic system. Physica Letter, 1990, 64(8):821～824.
[62]吕金虎,张锁春. Lyapunov指数的数值计算方法.非线性动力学学报, 2001,8(1):84～92.
[63] Martin N F G,England J W. Mathematical theory of entropy. London: Addison-Wesley Publishing Company, 1981
[64] C. E. Shannon. A mathematical theory of communication. The bell system technical, 1948, 27:379～423
[65]邢修三.物理熵,信息熵及其演化方程.中国科学(A辑),2001, 37(1):77～84
[66] E.T.Jaynes. Information theory and statistical mechanics. Physical Review. 1957, 106:361～630
[67]何大义.熵在数据分析中的应用研究.统计与决策, 2005, (4):27～29
[68] Adler R, Konheim A., McAndrew M. Topological entropy, Trans. Amer Math. Soc, 1965, 114:309～319
[69] Bowen R., Entropy for group endomorphisms and homogeneous spaces, Trans Amer. Math. Soc., 1971, 153:401～414
[70] Pincus S M. Approximate entropy as a measure of system complexity. Proc Natl Acad Sci USA, 1991, (88):2297～2301
[71] Pincus S. Approximate entropy (ApEn) as a complexity measure. Chaos, 1995, 5(1): 110～117
[72] Steven M Pincus. Quantifying complexity and regularity of neurobiological systems. Methods in Neurosciences, 1995, 28:336～363.
[73]邱红梅,高思莉,刘春霞,肖青.噪声对混沌系统近似熵的影响.东北师大学报自然科学版, 2004, 36(1): 46～49
[74]杨福生,廖旺才.近似熵:一种适用于短数据的复杂性度量.中国医疗器械杂志, 1997, 21(5):283～286
[75] Kolmogorov A. N. A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces, Dokl.Akad.Sci. SSSR., 1958, 119:861～864
[76] P.Grassberger, I.Procaccia. Characterization of strange attractors, Phys. Rev.Lets., 1983, 50(5): 346～349
[77]马军海,盛昭翰.经济系统混沌时序重构的分析和应用.管理科学学报. 2002, 5(3):73～78.
[78]肖方红,阎桂荣,韩宇航.混沌时序相空间重构参数确定的信息论方法.物理学报,第2005, 54(2):550～556
[79] Takens F. Detecting strange attactors in turbulence. Lecture Notes in Mathematics. Springer Verlag, 1981, 898:366～381.
[80] Packard N H,Crutchfield J P, Farmer J D, et a1. Geometry from a timeseries. Phys. Rev. Lett., 1980,43(9):712～715
[81] Gibson J F,Farmer J,Casdagli M ,et a1. An analytic approach to practical state space reconstruction. Physica D, 1992, 57: 1～30
[82]邓兰松,沈菲.非线性时序的混沌特性分析与预测.天津大学学报,2004, 37(11):1022～1025
[83] Farmer J D,Sidorowich J J. Predicting chaotic time series. Phys Rev Lett,1987, 59(8):845～848.
[84] Linsay P S.An efficient method of forecasting chaotic time series using linear interpolation. Phys Lett A, 1991, 153(6/7):353～356.
[85] Navone H D, Ceceatto H A. Forecasting chaos from small data sets: a comparison of different nonlinear algorithms. J Phys A: Math Gen, 1995, 28(12):3381～3388.
[86]任晓林,胡光锐,徐雄.混沌时间序列局域线性预测方法.上海交通大学学报, 1999, 33(1):19～21.
[87] Wolf A, Swift J B, Swinney H L, et al. Determining Lyapunou exponents from time series. Physica D, 1985,16(2):285～371
[88]温权,张勇传,程时杰.混沌时间序列预测技术.水电能源科学, 2001, 10(3):76～78
[89]吕金虎,陆君安,陈士华.混沌时间序列分析及其应用.武汉:武汉大学出版社,2002, 113
[90] Stutzer.M. Chaotic dynamics & bifurcation in a macro model. Journal of Economic Dynamics & Control November, 1980, 2: 353～376
[91] Day.R. The emergence of chaos from classical economic growth. Quarterly Journal of Economics, 1983, 54:201～213
[92] Chen Ping. Nonlinear dynamics theory of economic chaos and fluctuations. Working paper, National center for Economic research at Peking University, 2000, 10 No.C2000015.
[93]张兵.中国股票市场有效性研究.南京:南京大学出版社,2004
[94] MaJunhai, Chen Yushu. The Non-Linear Chaotic model reconstruction for the experimental data obtained from different dynamic system. Applied Mathematics and Mechanics, 1999, 20(11):1214～1221.
[95] Ma Junhai, Chen Yushu. Study on the prediction method of low-dimension time series that arise from the intrinsic non-linear dynamics. Applied Mathematics and Mechanics, 2001, 22 (5):501～509.
[96] Ma Junhai, Chen Yushu. Study on prediction methods for dynamic systems of nonlinear chaotic time series. Applied Mathematics andMechanics, 2004, 25 (6):605～611.
[97]马军海,盛昭瀚.混沌经济时序非线性动力系统的预测方法研究.系统工程学报,2002, 17(2):97～102.
[98]马军海,齐二石,莫馨.混沌时序重构及上海股票指数预测的应用研究.系统工程理论与实践, 2003, 23(12):86～94.
[99]马军海.复杂非线性系统的重构技术.天津:天津大学出版社,2005
[100] Hongchun Wang, Guanrong Chen , Jinhu Lü. Complex dynamical behaviors of daily data series in stock exchange. Physics Letters A,2004,333(3-4):246～255.
[101]熊正丰.金融时间序列分形维估计的小波方法.系统工程理论与实践,2002, 22(12):48～53.
[102]徐绪松,熊保平等.用RBF神经网络确定上海股市的分形维数.武汉大学学报(理学版),2003, 49(3):309～312.
[103]刘孝贤,郭举修,张庆华.上证综合指数的非线性动力学特点.山东工业大学学报, 2001, 31(5):416～423.
[104]邹志勇.金融数据的分形实证分析.电脑与信息技术,2001(6):62～65.
[105]李志勇,王春红,陈良生等.股票市场的分形维数分析.东北大学学报(自然科学版), 2003, 24(7):666～669
[106]陈军飞,王嘉松.基于小波分形理论的股价指数信息量测度研究.运筹与管理, 2003, 12(1):65～69
[107]高红兵,潘瑾,陈宏民.深圳成份股指数收益率序列的分形维.预测, 2000, (6):50～51
[108] Fisher Y. Fractal image compression. Fractal, 1994, 3(3):154～156
[109] Barnsley M F. Fractal image compression. New York: Academic Press, 1993, 121～145
[110] Barnsley M F. Fractal function and interpolation. Constructive Approximation, 1986, 8(2):303～329
[111]孙博文.中国股市波动的混沌吸引子的测定与计算.哈尔滨理工大学学报, 2001, 33(5):323～327
[112] Mandelbrot B B. The fractal geometry of nature. Singapore: World Scientific, 1983, 86～88.
[113] Mandelbrot B B. Fractals and scaling in finance. NewYork: Spring-Verlag Technological Education Publishing House, 1997, 35～64.
[114] Grassberger P, Procaccia I. Estimation of the Kolmogorov entropy from a chaotic signa1. Phys Rev A, 1983, 28(4): 2591～2593.
[115]陈增强,袁著祉,张燕.基于神经网络的非线性预测控制综述.控制工程,2002, 9(4): 7～11
[116]阎平凡,张长水编著.人工神经网络与模拟进化计算.北京:清华大学出版社,2000
[117]王洪元,史国栋主编.人工神经网络技术及其应用.北京:中国石化出版社,2002, 41～42.
[118]陈祥光,裴旭东编著.人工神经网络技术及应用.北京:中国电力出版社,2003,71～72
[119]田永青,杨斌,朱仲英.基于RBF神经网络建立税务预测模型的研究.计算机工程, 2002, 28(5):22～27
[120] Holcomb T, Morari M. Local training for radial basis function networks: towards solving the hidden unit problem. American Control Conference, 1991, 2331～2335.
[121]赵贵兵,石炎福,段文锋等.从混沌时间序列同时计算关联维和Kolmogorov熵.计算物理, 1999, 16(3): 309～315.
[122]周佩玲,陶小丽等.基于GA的RBF网络用于旱涝灾害因素预测研究.计算机应用, 2001, 21(1): 7～9.
[123]褚庆全,李立军,张志鹏.基于神经网络的宏观农业生产预测模型的研究.中国农业大学学报,2004,9(5): 1～5.
[124]林丽兰,何勇.改进的BP神经网络方法及其在农业商品总产值预测中的应用.科技通报, 2005, 21(1):6～10.
[125]朱春江,唐德善,马文斌.基于灰色理论和BP神经网络预测观光农业旅游人数的研究.安徽农业科学, 2006, 34(4):612～614.
[126]刘金琨编著.智能控制.北京:电子工业出版社,2005, 100～155.
[127]杨建刚编著.人工神经网络实用教程.杭州:浙江大学出版杜,2001, 148～149
[128]罗四维编著.人工神经网络建造.北京:中国铁道出版社,1998, 134～136
[129] Atiya A F,EI-Shoura S M. A comparison between neural-network forecasting techniques-case study: river flow forecasting. IEEE Trans. on Neural Networks, 1999, 10(2): 402～409.
[130] D.Kugiumtzis, O.C. Lingjaerde, N.Christophersen. Regularized local linear prediction of chaotic time series. Physica D, 1998, 112: 344～360.
[131] Bakirtzis A G, Theocharis J B. Short term load forecasting using fuzzy neural networks. IEEE Tran. on Power System, 1995, 10(3): 1518～1524.
[132]李爱国,覃征.滑动窗口二次自回归模型预测非线性时间序列.计算机学报, 2004, 27(7): 1004～1008
[133]吕金虎,张锁春.加权一阶局域法在电力系统短期负荷预测中的应用.控制理论与应用,2002, 19(5):767～770
[134] Schreiber T. Extremely simple nonlinear noise reduction method. Plays rev E, 1993, 47(4):2401～2404.
[135] Schreiber T, Grassberger P. A simple noise reduction method for real data. Phys Lett A, 1991, 160:411～418
[136] Sivakumar B. Chaos theory in hydrology: important issues and interpretations. J Hydrol, 2000, 227:1～20.