摘要
交通流系统是一个有人参与的、时变的、开放的复杂巨系统,具有高度的非线性和不确定性,在一定条件下会出现混沌现象。通过对交通流混沌研究现状的综述,可以看出研究交通流混沌有很深的理论和实用价值。通过对混沌判定方法的综述,了解了目前各种判定方法的局限性,确定了研究交通流混沌实时快速判定的思路和角度。本文主要给出了两种交通流混沌实时判定的方法。第一种是把改进型小数据量法和改进型替代数据法结合起来,既利用了小数据量法计算简单、抗噪性好、所需数据量少等优点,又利用了替代数据法的严密性避免误判。
对原有的小数据量法和替代数据法都进行了改进,首次给出了它们综合应用的详细计算步骤,并用Bierley跟驰模型产生的交通流、交通流微观仿真软件产生的交通流、实测交通流的时间序列对此方法进行了分析和验证。第二种方法是利用“寻找混沌与初始条件之间的对应关系”这一思路,基于数据挖掘的理念和方法提出了一种交通流混沌实时判定的智能系统的结构体系。介绍了该系统每个功能模块的作用和实现方法,给出了相应算法的原理和步骤,论证或说明了一些方法选取的原因,并对Logistic系统、交通流微观仿真软件产生的交通流、实测交通流时间序列进行了分析和验证。通过不同的实验,讨论了在其它实验条件和步骤不变的情况下,不同的时间序列特征量(功率谱、小波包系数、小波包能量)、不同点数的时间序列、不同层数的小波包分解、不同的小波函数、不同的知识发现算法(BP神经网络和一种支持向量机的快速分类法)等对此智能判定系统的判定结果的影响,从结果中分析得出了各影响因素的选用及原因。通过以上实验结果,说明它们可以很好地满足混沌实时判定的实时性、准确性的要求,并具有较强的鲁棒性。
Traffic flow system is a human-joined, changeable, open and complex huge system. It is high nonlinear and uncertain. Under certain condition, chaos appears in it. The summary of studies on chaos in traffic flow suggests that study on chaos is of great theoretical and practical value. In addition, the summary of methods to identify chaos shows the limitation on current methods and the direction and angle of study on real-time identification of chaos in traffic flow. In this thesis, these are two approaches to real-time identification of chaos in traffic flow. The first one is the combination of improved small-data method and improved surrogate-data technique. Small-data method is easy to compute, antinoise and reliable for small data. Surrogate-data technique can avoid false identification. However, the novel approach has not only the advantage of small-data method, but also the rigor of surrogate-data technique. In this approach, small-data method and surrogate-data technique are all improved and its computing steps are first introduced in detail. The case studies of this approach are given for time series of traffic flows generated by Bierley Car-following model and microcosmic simulation software and real vehicles. The second one is based on the thought of finding the relationship of chaos and the initial condition. Based on the conception and methods of data mining techniques, an intelligent system framework for real-time identification of chaos in traffic flow is proposed. Furthermore, the function and implementation of every module in the system is introduced, the principle and steps of the arithmetic is proposed and the selection of methods is discussed and explained. The case studies of this system are given for time series of traffic flows generated by Logistic system and microcosmic simulation software and real vehicles. Moreover, when other experimental condition and steps are same, the influence of different factors on identification results is discussed. These factors include feature of time series, such as power spectrum, wavelet packet coefficients and wavelet packet energy, the length of time series, layers of wavelet packet decomposition, wavelet function, algorithms of knowledge discovery, such as BP neural network and Fast Classification for Support Vector Machines. The results show the selection for factors. All above experimental results indicate that these two approaches to real-time identification of chaos in traffic flow are robust and can be fit for requirements of real-time performance and veracity.
引文
[1] 许国志.系统科学[M],上海:上海科技教育出版社,2000,86
[2] 王炜.交通工程学[M].南京:东南大学出版社,2002,1-5
[3] Kinzer J P. Application of the theory of probability to problem of highway traffic[D]. U.S.A.: Polytechnic Institute of Brooklyn, 1933
[4] Greenshields B D. A study of Traffic Capacity[J]. HRB Proc., 1934, 14(1): 467-469
[5] Adams W F. Road Traffic Considered as a Random Series[J]. J. Inst. Civ. Eng., 1936, (4): 121-130
[6] Greenshields B D, Shapiro D, Ericksen E L. Traffic Performance at Urban Street Intersections[R]. New Haven Conn.: Tech. Rep. No. 1, Yale Bureau of Highway Traffic, 1947
[7] Pipes L A. An operational analysis of traffic dynamics[J]. J. Appl. Phys., 1953, 24(3): 274-281
[8] Reusche L.A. Vehicle movements in a platoon[J]. Oesterreidisches-Arch, 1950, 4: 193-215
[9] Pipes L A. A Proposed Dynamic Analogy of Traffic[R]. Berkeley: Institute of Transportation and Traffic Engineering, University of California, 1953
[10] Chandler R, Herman R E, Montroll E W. Traffic Dynamics: Study in Car following. Opns. Res., 1958,6: 165-167
[11] Herman R E, Montroll E W, Ports R B, et al. Traffic Dynamics: analysis of stability in Car Following[J]. Operations Research, 1959, 7: 86-106
[12] Lighthill M H, Whitham G B. On Kinematics Wave(Ⅱ)-a theory of traffic on long crowded roads[J]. Proc Roy Soc London, Ser A, 1955, 22: 317-345
[13] Richards P I. Shock waves on the highway[J]. Oper. Res., 1956, 4: 42-51
[14] Pipes.L.A. Hydrodynamic approaches[A]. P. I. Gerlough, D. G. Capelle. An Introduction to Traffic Flow Theory[C], Washington D C: Special Report 79, Highway Research Board, 1964, 13-17
[15] Pipes L A. Car-following models and the fundamental diagram of road traffic[J]. J. Transpn. Res., 1967, 1: 21-29
[16] Pipes L A. Vehicle accelerations in the hydrodynamic theory of traffic flow[J]. Transpn. Res., 1969, 3(2): 229-234
[17] Cleveland D E, Capelle D G.. Queuing Theory Approaches: An Introduction to Traffic Flow Theory[R]. Washington D C: Special Report 79, Highway Research Board, 1964
[18] D. L. Gerlough, M. J. Huber. Traffic Flow Theory[R]. Washington D C: TRB, Special Report No.165, Transportation Research Board, National Research Council, 1975
[19] 冯蔚东,贺国光,刘豹.交通流理论评述[J].系统工程学报,1999,13 (3 ):71-82
[20] 冯苏苇.交通流的动力学模型与数值模拟[J].上海大学学报(自然科学版),1997,3(4):443-453
[21] 戴世强,冯苏苇,顾国庆.交通流动力学:它的内容、方法和意义[J].自然杂志,1997,19(4):196-201
[22] 俞洁,杨成斌.交通流理论发展分析[J].合肥工业大学学报,2004,27(2):163-167
[23] Fritzche H T. A model for traffic simulation[J]. Traffic Engineering Control, 1994, 35(3): 317-321
[24] 何 民,刘小明,荣建.交通流跟驰模型研究进展[J].人类工效学,2000,6(2):46-50
[25] 荣 建,刘小明.基于可变跟驰时间和随机因素的通行能力理论计算模型[J].中国公路学报,2001,14(4):81-85
[26] 陈建阳.交通流微观模型与宏观模型的统一[J].同济大学学报,1997,25(1):46-49
[27] Gazis D C, Herman P, Ports R B. Car-following theory of steady state traffic flow[J]. Operations Research, 1959, 7: 499-505
[28] Payne H J. Models of freeway traffic and control[A]. Bekey G A. Mathematical Methods of Public Systems[C]. CA: LaJolla, 1971, 1: 51-61
[29] Papageorgiou M A. hierarchical control system for freeway traffic[J]. Transportation .Research, 1983, 17B(3): 251-261
[30] Phillips W F. A new continuum traffic model obtained from kinetic theory[J]. IEEE Trans. Autom. Control, 1978, 23: 1032-1036
[31] Kuhne R D. Macroscopic freeway model for dense traffic-stop-start waves and incident detection[A]. Volmuller J, Hamerslag R. Proceeding of 9th InternationalSymposium on Transportation and Traffic Theory[C]. Netherlands: VNU Science Press, 1984, 21-42
[32] Michalopoulos P G., Yi P, Lyrintzis A S. Continuum modeling of traffic dynamics for congested freeways[J]. Transpn. Res. B, 1993, 27: 315-332
[33] 黄润生, 黄浩. 混沌及其应用(第 2 版)[M]. 武汉: 武汉大学出版社, 2005,15-86
[34] 乔林.交通经济交互作用中的混沌现象[D].北京:北方交通大学硕士研究生学位论文,1996.3
[35] 洛伦兹(著),刘式达,刘式适,严中伟(译).混沌的本质[M].北京:气象出版社,1997,1-60
[36] 唐巍,李殿璞,陈学允.混沌理论及其应用研究[J].电力系统自动化,2000,24(7):67-70
[37] 黄登仕,李后强.混沌、分形及其应用[M].合肥:中国科学技术大学出版社,1995,1-75
[38] 吕金虎,陆君安,陈士化.混沌时间序列分析及其应用[M].武汉:武汉大学出版社,2002,1-10, 27-45, 46-92
[39] 陈奉苏.混沌控制及其应用[M].北京:中国电力出版社,2006,14-73
[40] 张化光,王智良,黄玮.混沌系统的控制理论[M].沈阳:东北大学出版社,2003,1-16
[41] 吴祥兴,陈忠.混沌学导论[M].上海:上海科学技术文献出版社,1996,1-84
[42] 刘式达,梁福明,刘式适,等.自然科学中的混沌和分形[M].北京:北京大学出版社,2003,120-131
[43] 方锦清.驾驭混沌与发展高新技术[M].北京:原子能出版社,2002,30-41
[44] 邹恩,李祥飞,陈建国.混沌控制及其优化应用[M].长沙:国防科技大学出版社,2002,1-4, 23-49
[45] 关新平,范正平,陈彩莲,等.混沌控制及其保密通信中的应用[M].北京:国防工业出版社,2002,1-10
[46] 龙运佳,梁以德.近代工程动力学 随机·混沌[M].北京:科学出版社,1998,86-103
[47] 陈士华,陆君安.混沌动力学初步[M].武汉:武汉水利电力大学出版社,1998,38-43
[48] 刘秉正.非线性动力学与混沌基础[M].长春:东北师范大学出版社,1994,139-173
[49] Eryk Infeld, George Rowlands. Nonlinear waves, solitons, and chaos[M]. Beijing: World Publishing Corporation, 2005, 304-344
[50] Edward Ott. Chaos in dynamical systems[M]. Beijing: World Publishing Corporation, 2005, 1-20, 115-160
[51] Paul Manneville. Instabilities, chaos and turbulence: an introduction to nonlinear dynamics and complex systems[M]. London: Imperial College Press, 2004, 304-330
[52] 王东山,贺国光.交通混沌研究综述与展望[J].土木工程学报,2003,36(1):68-74
[53] 杨瑞,李洋,吕文超.城市干路交通流实施混沌智能控制方法的研究[J].林业机械与木工设备,2004,32(10):22-2
[54] 王正武,黄中祥,况爱武.短期交通流序列混沌识别及预测精度分析[J].长沙交通学院学报,2004,20(2):73-76
[55] 唐志强,王正武,招晓菊,等.基于神经网络和混沌理论的短时交通流预测[J].山西科技,2005,5:117-118,120
[56] 宗春光,宋靖雁,任江涛,等.基于相空间重构的短时交通流预测研究[J].公路交通科技,2003,20(4):71-75
[57] 王进,史其信.基于延迟坐标状态空间重构的短期交通流预测研究[J].ITS通讯,2005,7(1):14-18
[58] 王进,史其信,陆化普.交通流可预测性分析[J].ITS 通讯,2005,7(2):11-14
[59] 唐明,陈宝星,柳伍生. 基于相空间重构的短时交通流分形研究[J].山西交通学院学报,2004,12(1):50-54,67
[60] 冯蔚东,陈剑,贺国光,等.交通流中的分形研究[J].高技术通讯,2003,13(6):59-65
[61] 杜振财,王瑞峰,纪常伟.基于 Logistic 映射的交通流的混沌态研究[J].交通科技,2005,2:78-80
[62] 徐吉谦.交通工程总论,北京:人民交通出版社,1991,26-29
[63] 王辉球,缪立新,吴世江.基于跟驰模型的交通流混沌研究[J].ITS 通讯,2005,7(1):19-22
[64] 张智勇,荣建,任福田.跟驰车队中的混沌现象研究[J].土木工程学报交通工程分册,2001,1(1):58-59
[65] 贺国光,王东山.仿真交通流混沌现象的传播特性研究[J].土木工程学报,2004,37(1):70-73
[66] 贺国光,万兴义,王东山.基于跟驰模型的交通流混沌研究[J].系统工程,2003,21(2):50-55
[67] 贺国光,万兴义.基于混沌判据评价几类跟驰模型合理性的仿真研究[J].系统工程理论与实践,2004,24(4):123-129
[68] 贺国光,王东山.交通流的有序运动与混沌运动相互转化现象的仿真研究[J].土木工程学报,2003,36(7):53-56
[69] 李英,刘豹,马寿峰.交通流时间序列中混沌特性判定的替代数据方法[J].系统工程,2000,18(6):54-58
[70] 贺国光,万兴义.基于最大 Lyapunov 指数的交通流仿真数据混沌状态识别[J].自动化技术与应用,2003,22(4):8-10,13
[71] Tang Y S, Li J, Tian Y G, Chen X. Chaos forecast for traffic conflict flow[J]. Journal of Jilin University (Engineering and Technology Edition), 2005, 35(6): 646-648
[72] Nair A S, Liu J C, Rilett L, Gupta S. Non-Linear analysis of traffic flow[A]. IEEE Conference on Intelligent Transportation Systems, Proceedings[C], ITSC, 2001, 681-685
[73] Lin Y, Wang F Y. Predicting chaotic time series using adaptive wavelet-fuzzy inference system[C]. 2005 IEEE Intelligent Vehicles Symposium Proceedings, 2005, 888-893
[74] Dong C J, Liu Z Y, Qui Z L. Prediction of traffic flow in real-time based on chaos theory[J]. Information and Control, 2004, 33(5): 518-522
[75] Huang K, Chen S F, Zhou Z G. Research on a non-linear chaotic prediction model for urban traffic flow[J]. Journal of Southeast University (English Edition), 2003, 19(4): 410-413
[76] Wang J, Shi Q X, Lu H P. The study of short-term traffic flow forecasting based on theory of chaos[C]. 2005 IEEE Intelligent Vehicles Symposium Proceedings, 2005, 869-874
[77] Chen D E, Zhang J P. Time series prediction based on ensemble ANFIS[C]. 2005 International Conference on Machine Learning and Cybernetics, 2005, 3552-3556
[78] Nagatani T. Chaos and dynamical transition of a single vehicle induced by traffic light and speedup[J]. Physica A: Statistical Mechanics and its Applications, 2005, 348: 561-571
[79] Nagatani T. Chaotic and periodic motions of a cyclic bus induced by speedup[J]. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 2002, 66(4): 046103
[80] Nagatani T. Dynamical transitions to chaotic and periodic motions of two shuttle buses[J]. Physica A: Statistical Mechanics and its Applications, 2003, 319(SUPPL.): 568-578
[81] Nagatani T. Fluctuation of riding passengers induced by chaotic motions of shuttle buses[J]. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 2003, 68(3): 36107-1-8
[82] Toledo B A, Munoz V, Rogan J, et al. MModeling traffic through a sequence of traffic lights[J]. Physical Review E, 2004, 70(1-2): 016107-1-016107-7
[83] Nelson P. A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions[J]. Transport Theory and Statistical Physics, 1995, 24(1-3): 383-409
[84] Waldeer K T. A vehicular traffic flow model based on a stochastic acceleration process[J]. Transport Theory and Statistical Physics, 2004, 33(1): 7-30
[85] Zhang X Y, Jarrett D F. Chaos in a dynamic model of traffic flows in an origin-destination network[J]. Chaos, 1998, 8(2): 503-513
[86] Nagatani T. Chaotic jam and phase transition in traffic flow with passing[J]. Physical Review E (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics), 1999, 60(2): 1535-1541
[87] Safonov L A, Tomer E, Strygin V V, et al. Delay-induced chaos with multifractal attractor in a traffic flow model[J]. Europhysics Letters, 2002, 57(2): 151-157
[88] Nagatani T. Jamming transition in traffic flow on triangular lattice[J]. Physica A: Statistical Mechanics and its Applications, 1999, 271(1-2): 200-221
[89] Safonov L A, Tomer E, Strygin V V, Ashkenazy Y, Havlin S. Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic[J]. Chaos, 2002, 12(4): 1006-1014
[90] Li T. Nonlinear dynamics of traffic jams[J]. Physica D: Nonlinear Phenomena, 2005, 207(1-2): 41-51.
[91] He G G, Wan X Y. Simulation research on evaluating of reasonability for the car-following models based on chaos criterion[C]. Proceedings of the 2003 IEEE International Conference on Intelligent Transportation Systems, 2003, 1: 364-367
[92] Liu L J, He G G, Wang D S. Simulation study of the transition between chaos and order motion in traffic flow[C]. Proceedings of the Conference on Traffic and Transportation Studies, 2004, 4: 461-467
[93] Wang L, Du Z C, Rong J. Study on expected headway model in urbanexpressway[J]. Beijing Gongye Daxue Xuebao / Journal of Beijing University of Technology, 2005, 31(6): 598-603
[94] D.J.Low, P.S.Addison. A Novel Nonliner Car-following Model[J]. Chaos, 1998, 8(4): 791-799
[95] Addison P S, Low D J. Order and chaos in the dynamics of vehicle platoons[J]. Traffic Engineering and Control, 1996, 37(7-8): 456-459
[96] Addison P S, Low D J. The existence of chaotic behavior in a separation distance centred nonlinear car-following model[C]. Proceedings of 2nd International Conference on Road Vehicle Automation, 1997, 171-180
[97] Low D J, Addison P S, A Nonlinear Temporal Headway Model of Traffic Dynamics[J]. Nonlinear Dynamics, 1998, 16(2): 127-151
[98] Frazier C, Kockelman K M. Chaos theory and transportation systems - Instructive example[J]. 2004, (1897): 9-17
[99] Li K P, Gao Z Y. Nonlinear dynamics analysis of traffic time series[J]. Modern Physics Letters B, 2004, 18(26-27): 1395-1402
[100] Kuehne R. Traffic patterns in unstable traffic flow on freeways[C]. Proceedings of the International Symposium on Highway Capacity and Level of Service, 1991, 211-223
[101] Ddendrinos D S. Traffic-flow dynamics - a search for chaos[J]. Chaos, Solitons & Fractals, 1994, 4(4): 605-617
[102] Lo S.-C.S.-C., Cho H.-J.H-J.. Chaos and control of discrete dynamic traffic model[J]. Journal of the Franklin Institute, 2005, 342(7): 839-851
[103] Shahverdiev E M, Tadaki S. Instability control in two dimensional traffic flow model[J]. Physics Letters A, 1999, 256(1): 55-58
[104] Das S, Bowles B A, Simulations of highway chaos using fuzzy logic[C]. Proceedings of NAFIPS-99: 18th International Conference of the North American Fuzzy Information Processing Society, 1999, 130-133
[105] Banz G, Newkirk R T. Traffic chaos in emergency planning[C]. Simulation in Emergency Management and Technology. Proceedings of the SCS Western Multiconference, 1989, 45-47
[106] Takens F. Detecting strange attractors in fluid turbulence[J]. Lecture notes in Math, 1981, 898(2): 361-381
[107] Packard N H, Crutchfield J P, Farmer J D, et al. Geometry from a time series[J]. Phys. Rev. Lett. , 1980, 45(3): 712-716
[108] Albano A M. SVD and Grassberger-Procaccia algorithm[J]. Phys. Rev. A 1988, 38(20): 3017-3026
[109] Fraser A M. Information and entropy in strange attractors[J]. IEEE Trans on IT Mar, 1989, 35(2): 245-262
[110] Albano A.M. Using high-order correlations to define an embedding window[J]. Physica D, 1991, 54(1): 85-97
[111] P. T. Buzug. Optimal delay time and embedding dimension for delay-time coordinates by analysis of the global static and local dynamical behavior of strange attractors[J]. Phys. Rev. A, 1992, 45(10): 7073-7084
[112] P. T. Buzug. Comparison of algorithms calculating optimal embedding parameters for delay time coordinates[J]. Physica D: Nonlinear Phenomena, 1992, 58(2): 127-137
[113] M. T. Rossenstein, J. J. Colins, C. J. Deluca. Reconstruction expansion as a geometry-based framework for choosing proper delay times[J]. Physica D: Nonlinear Phenomena, 1994, 73(1): 82-98
[114] G. Kember, A. C. Fowler. A correlation function for choosing time delays in phase portrait reconstructions[J]. Phys. Lett. A, 1993, 179(2): 72-80
[115] 林嘉宇,王跃科,黄芝平.语音信号相空间重构中时间延迟的选择——复自相关法[J].1999,15(3):220-225
[116] 王军,石炎福,余华瑞.相空间重构中最优滞时的确定[J].四川大学学报(工程科学版),2001,33(2):47-51
[117] M. B. Kennel, R. Brown, H. D. I. Abarbanel. Determining embedding dimension for phase-space reconstruction using a geometrical construction[J]. Phys. Rev. A, 1992, 45(6): 3403-3411
[118] Cao Liangyue. Practical method for determining the minimum embedding dimension of a scalar time series[J]. Physica D: Nonlinear Phenomena, 1997, 110(1): 43-50
[119] D. Kugiumtzis. State space reconstruction parameters in the analysis of chaotic time series - the role of the time window length[J]. Physica D: Nonlinear Phenomena, 1996, 95(1): 13-28
[120] H. S. Kim, R. J. Eykholt, D. Salas. Nonlinear dynamics, delay times, and embedding windows[J]. Physica D: Nonlinear Phenomena, 1999, 127(1): 48-60
[121] M. Otani, A. J. Jones. Automated embedding and creep phenomenon in chaotic time series[EB/OL]. http://users.cs.cf.ac.uk/Antonia.J.Jones/, 2000-10-14
[122] Gérard Boudjema, Bernard Cazelles. Extraction of nonlinear dynamics from short and noisy time series[J]. Chaos, Solitons & Fractals, 2001, 12(11): 2051-2069
[123] 陈琢,蔡绍洪.基于小波域主分量分解的相空间重构[J].贵州大学学报:自然科学版,2000,17(1):18-22
[124] T. Schreiber. Extremely simple nonlinear noise reduction method[J]. Phys. Rev. E, 1993, 47(4): 2401-2404
[125] T. Schreiber, P. Grassberger. A simple noise-reduction method for real data[J]. Phys. Lett. A, 1991, 160(5): 411-418
[126] R. Hegger, T. Schreiber. A noise reduction method for multivariate time series[J]. Phys. Lett. A, 1992, 170(4): 305-310
[127] T. Schreiber. Efficient neighbor searching in nonlinear time series analysis[J]. International Journal of Bifurcation and Chaos, 1995, 5(2): 349-358
[128] D. S. Broomhead, G. P. King. Extracting qualitative dynamics from experimental data[J]. Physica D: Nonlinear Phenomena, 1986, 20(2-3): 217-236
[129] 姜万录.一种基于正交小波包分析的混沌识别方法[J].燕山大学学报,2004,28(1):7-13
[130] 郑吉兵,高行山.小波变换在分叉与混沌研究中的应用[J].应用数学和力学,1998,19(6):557-563
[131] 吴奇学.经典混沌的理论及应用[J].漳州师范学院,1995,9(4):57-62
[132] M. Johnson, J. Habyarimana. An operational test for distinguishing between complicated and chaotic behavior in deterministic systems[J]. Physica D: Nonlinear Phenomena, 1998, 116(3-4): 289-300
[133] [美]海因斯 著,张建华,卓力,张延华 译.数字信号处理[M].北京:科学出版社,2002,207-211
[134] M. Sano, Y. Sawada. Measurement of the Lyapunov spectrum from a chaotic time series[J]. Phys. Rev. Lett. , 1985, 55: 1082-1085
[135] A. Wolf, J. B. Swift, H. L. Swinney, et al. Determining Lyapunov exponents from a time series[J]. Physica D: Nonlinear Phenomena, 1985, 16(3): 285-317
[136] Gy?rgy Barana and Ichiro Tsuda. A new method for computing Lyapunov exponents[J]. Phys. Lett. A, 1993, 175: 421-427
[137] Keith Briggs. An improved method for estimating Liapunov exponents of chaotic time series[J]. Phys. Lett. A, 1990, 151: 27-32
[138] M. T. Rosenstein, J. J. Collins, C. J. Deluca. A practical method for calculating largest Lyapunov exponents from small data[J]. Physica D: Nonlinear Phenomena,1993, 65: 117-134
[139] 陈琢,梁蓓.基于小波变换的最大 Lyapunov 指数的计算方法[J].贵阳师范大学学报:自然科学版,2000,18(4):58-60
[140] R. Porcher, G. Thomas. Estimating Lyapunov exponents in biomedical time series[J]. Phys. Rev. E, 2001, 64(1): 010902(1-4)
[141] H. Kantz. A robust method to estimate the maximal Lyapunov exponent of a time series[J]. Phys. Lett. A, 1994, 185: 77
[142] 赖建文,周世平,李国辉.非重正交的李雅普诺夫指数谱的计算方法[J].物理学报,2000,49(12):2328-2332
[143] 李尧亭,蔡诗东.混沌和李雅谱诺夫特征指数[J].物理,1996,25(5):282-286
[144] 陈国华,盛昭瀚.基于 Lyapunov 指数的混沌时间序列识别[J].系统工程理论方法应用,2003,12(4):317-320
[145] J. M. Choi, B. H. Bae, S. Y. Kim. Divergence in perpendicular recurrence plot; quantification of dynamical divergence from short chaotic time series[J]. Physics Letters A, 1999, 263(4-6): 299-306
[146] 赵玉成,肖忠会,许庆余.时间序列分维数用于非线性模型临界参数的识别[J].工程力学,2000,17(1):134-139
[147] P. Grassberger, I. Procaccia. Measuring the strangeness of strange attractors[J]. Physica D: Nonlinear Phenomena, 1983, 9(1-2): 189-208
[148] P. Grassberger, I. Procaccia. Estimation of the Kolmogorov entropy form a chaotic signal[J]. Phys. Rev. A, 1983, 28(4): 2591-2593
[149] A. Cohen, I. Procaccia. Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical system[J]. Phys. Rev. A, 1985, 31(3): 1872-1882
[150] 赵贵兵,石炎福,段文锋.从混沌时间序列同时计算关联维和 Kolmogrorov熵[J].计算物理,1999,16(3):309–315
[151] J. Theiler, S. Eubank, A. Longtin, et al. Testing for nonlinearity in time series - the method of surrogate data[J]. Physica D: Nonlinear Phenomena, 1992, 58(1-4): 77-94
[152] T. Schreiber, A. Schmitz. Improved surrogate data for nonlinearity tests[J]. Phys. Rev. Lett. , 1996, 77(4): 635-638
[153] S. Rombouts, R. Kemen, C. J. Stam. Investigation of nonlinear structure in multichannel EEC[J]. Phys. Lett.A. , 1995, 202(5-6): 352-358
[154] T. Schreiber. Interdisciplinary application of nonlinear time series methods[J].Physics Report, 1999, 308(1): 1-64
[155] C. Nicolis, G. Nicolis. Is there a climatic attractor?[J]. Nature, 311: 529 –532
[156] P. Grassberger. Do climatic attractors exist?[J]. Nature, 323: 609-612
[157] C. Nicolis, G. Nicolis. Weather forecasts still tricky[J]. Nature, 321: 567
[158] P. Grassberger. Evidence for climatic attractors[J]. Nature, 326: 524
[159] William A. Brock, Chera L. Sayers. Is the business cycle characterized by deterministic chaos?[J]. Journal of Monetary Economics, 1988, 20(1): 71-91
[160] J. A. Scheinkman, B. LeBaron. Nonlinear Dynamic and Stock Returns[J]. Journal of Business, 1989, 62: 311-337
[161] 陈颙.分形几何与地球科学(续 2)[J].华南地震,1995,14(4):78-83
[162] B. B. Mandelbrot, J. R. Wallis. Robustness of the rescales rang R/S in the measurement of noncyclic long run statistical dependence[J]. Water Resource research, 1969, 5(5): 967-988
[163] 赵玉成,肖忠会.判定混沌与噪声的局部可变神经网络方法[J].西安交通大学学报,1999,33(10):72–76
[164] 江亚东,陈因颀.一种基于小波网络的混沌时间序列判定[J].北京科技大学学报,2002,24(3):295-298
[165] 向小东.短时含噪时间序列混沌识别方法及其应用[J].工业技术经济,2004,23(2):107-108
[166] J. P. Eckmann, D. Ruelle. Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems[J]. Physica D: Nonlinear Phenomena, 1992, 56(2-3): 185-187
[167] Leonard A. Smith, Intrinsic limits on dimension calculations[J]. Physics Letters A, 1998, 133(6): 283-288
[168] M. A. H. Nerenberg, C. Essex. Correlation dimension and systematic geometric effects[J]. Phys. Rev. A, 1990, 42: 7065-7074
[169] A. Guerrero, L. A. Smith. Towards coherent estimation of correlation dimension[J]. Physics Letters A, 2003, 318: 373-379
[170] A. A. Tsonis, J. B. Elsner. The weather attractor over very short time scales[J]. Nature, 1988, 333: 545-547
[171] Ying-Cheng Lai, David Lerner. Effective scaling regime for computing the correlation dimension from chaotic time series[J]. Physica D: Nonlinear Phenomena, 1998, 115(1-2): 1-18
[172] 盛昭瀚,马军海.管理科学:面时复杂性Ⅲ——经济时序动力系统最佳时序采样间隔分析研究[J].管理科学学报,2000,3(1):100-105
[173] A. R. Osborne, A. D. Kirwan, A. Provenzale, et al. A search for chaotic behavior in large and mesoscale motions in the Pacific Ocean[J]. Physica D: Nonlinear Phenomena, 1986, 23(1-3): 75-83
[174] A. R. Osborne, A. Provenzale. Finite correlation dimension for stochastic systems with power-law spectra[J]. Physica D: Nonlinear Phenomena, 1989, 35(3): 357-381
[175] A. Provenzale, A. R. Osborne, R.soj. Convergence of the K2 entropy for random noises with power law spectra[J]. Physica D: Nonlinear Phenomena, 1991, 47(3): 361-372
[176] J. Theiler. Some comments on the correlation dimension of 1/fα noise[J]. Phys. Lett. A, 1991, 155(8-9): 480-493
[177] J. B. Ramsey, H.-J. Yuan. Bias and error bars in dimension calculations and their evaluation in some simple models[J]. Phys. Lett. A, 1989, 143(5): 287-297
[178] 汪富泉,罗朝盛.G—P 算法的改进及其应用[J].计算物理,1993,10(3):345-351
[179] 李擎,郑德玲.一种新的混沌识别方法(I)[J].北京科技大学学报,1999,21(2):198-201
[180] 王克斌,彭真明.地震数据关联维的快速计算方法[J].物探化探计算技术,2000,22(3):225-228
[181] 杨文强.时间序列关联维的计算方法及程序[J].物探化探计算技术,1996,18(4):339-342
[182] 洪时中,洪时明.一种客观确定分析“无标度区”的方法—自相似比法[J].大自然探索,1993,13(2):53-57
[183] 曾昭才,段虞荣.基于径向基函数网络的混沌时间序列分析[J].重庆大学学报:自然科学版,1999,22(6):113-120
[184] M. T. Rosenstein, J. J. Collins, C. J. Deluca. A practical method for calculating largest Lyapunov exponents from small data[J]. Physica D: Nonlinear Phenomena, 1993, 65: 117-134
[185] Paul V. Mcdonough, Joseph P. Noonan, G. R. Hall. A new chaos detector[J]. Computers and Electrical Engineering, 1995, 21(6): 417-431
[186] Ricard V. Solé, Jordi Bascompte. Measuring chaos from spatial information[J]. Journal of Theoretical Biology, 1995, 175(2): 139-147
[187] Salvador Pueyo. The study of chaotic dynamics by means of very short timeseries[J]. Physica D: Nonlinear Phenomena, 1997, 106(1-2): 57-65
[188] K. H. Chon, K. P. Yip, B. M. Camino, et al. Modeling Nonlinear Determinism in Short Time Series from Noise Driven Discrete and Continuous Systems[J]. International Journal of Bifurcation and Chaos, 2000, 10: 2745-2766
[189] Mototsugu Shintani, Oliver Linton. Nonparametric neural network estimation of Lyapunov exponents and a direct test for chaos[J]. Journal of Econometrics, 2004, 120(1): 1-33
[190] Liebert W, Schuster H G. Proper choice of the time delay for the analysis of chaotic time series[J]. Phys. Lett. A, 1989, 44(3): 107-110
[191] Sato S, Sano M, Sawada Y. Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems[J]. Prog. Theor. Phys, 1987, 77: 1-5
[192] 雷敏,王志中.非线性时间序列的替代数据检验方法研究[J].电子与信息学报,2001,23(3):248-254
[193] 丹尼尔·L. 鸠洛夫,马休·J. 休伯.交通流理论[M].北京,人民交通出版社,1983,32-106
[194] 马寿峰,贺国光,刘豹.一种通用的城市道路交通流微观仿真系统的研究[J].系统工程学报,1998,13(4):8-15,24
[195] [美] Mehmed Kantardzic 著,闪四清,陈茵,程雁等译.数据挖掘——概念、模型、方法和算法[M].北京:清华大学出版社,2003,1-8,171-193
[196] Fangfang Liu, Daolin Xu, Guilin Wen. Tracing initial condition, historical evolutionary path and parameters of chaotic processes from a short segment of scarlar time series[J]. Chaos, Solition & Fractals, 2005, 24: 265-271
[197] Gallant, A.R., White, H. On learning the derivatives of an unknown mapping with multilayer feedforward networks[J]. Neural Networks, 1992, 5: 129-138
[198] 江亚东,丁丽萍,夏克俭等.基于小波神经网络的混沌模式提取[J].北京科技大学学报,2001,23(5):474-477
[199] 赵犁丰,宋洁,姚玉玲等.利用小波包分析和混沌特征提取进行船舶辐射噪声分类[J].中国海洋大学学报,2004,34 (6):1036-1040
[200] 刘树勇,潜伟建,朱石坚.小波分析在混沌信号识别中的应用研究[J].海军工程大学学报,2003,15(1):76-79
[201] 李弼程,罗建书.小波分析及其应用[M].北京:电子工业出版社,2003,26-35,77-85
[202] [美]Albert Boggess,Francis J. Narcowich 著,芮国胜,康健等译.小波与傅里叶分析基础[M].北京:电子工业出版社,2004,170-214
[203] 郑治真,沈萍,杨选辉.小波变换及其 MATLAB 工具的应用[M].北京:地震出版社,2001,62-76
[204] 飞思科技产品研发中心.MATLAB 6.5 辅助小波分析与应用[M].北京:电子工业出版社,2003,6-27
[205] 张青贵.人工神经网络导论[M].北京:中国水利水电出版社,2004,4-15,107-111
[206] 高隽.人工神经网络原理及仿真实例[M].北京:机械工业出版社,2003,45-86
[207] 飞思科技产品研发中心.MATLAB 6.5 辅助神经网络分析与应用[M].北京:电子工业出版社,2003,11-12,64-74
[208] Simon Haykin.神经网络原理(叶世伟,史忠植)[M].北京:机械工业出版社,2004,36-58
[209] 刘向东,陈兆乾.一种快速支持向量机分类算法的研究[J].计算机研究与发展,2004,41(8):1327-1332
[210] 徐进永,张子达,陆爽.基于 K-L 变换和支持向量机的滚动轴承故障模式的识别[J].吉林大学学报:工学版,2005,35(5):500-504
[211] 李岳,陶利民,温熙森.用于滚动轴承故障检测与分类的支持向量机方法[J].中国机械工程,2005,16(6):498-501
[212] 张国云,童兢,向文江.滚动轴承技术故障诊断的支持向量机方法研究[J].计算机工程与应用,2005,41(16):227-229