摘要
随着全球经济一体化进程的深入,影响经济运行的因素以及这些因素之间的关系更加复杂,传统的经济学理论已无法准确描述,非线性经济学(或混沌经济学)已逐渐成为当代经济学研究的前沿领域,并已取得迅速的进展。本文运用现代非线性动力学理论与随机动力学理论,研究了经济系统中复杂非线性现象的运行规律,并对实际经济序列进行了实证研究,主要完成了以下工作:
1、建立了社会经济动力学系统的非线性经济周期模型;分析了具有周期激励的社会经济动力系统的稳定性,并通过边界划分将社会经济发展形态划分为三种不同的类型,具体研究了我国在计划经济时代与西方发达国家在经济表现方面的差异,指出了社会转型时期可能面临的风险及风险源。结果表明投资加速数ν与边际储蓄倾向s之差a的变化,是引发社会经济系统发生分岔行为的主要因素,并通过数值仿真验证了理论结果。
2、首先考虑了具有随机形式的自主函数,建立了随机经济周期模型;由于该系统是弱阻尼弱激励的拟不可积Hamilton系统,因此运用基于乘积遍历性定理的Lyapunov指数及一维扩散过程的边界分析理论,得出该系统的局部稳定性和全局稳定性条件;根据系统响应联合概率密度和平稳概率密度以及不同的参数条件,研究了该模型的随机Hopf分岔行为,并对分岔参数进行了具体分析,还通过数值仿真进行了验证。
3、采用基于虚假最近领域概念以及Cao方法,确定天津市西青区每天最忙时段话务量最佳的嵌入维数和时间延滞。并用随机森林、随机梯度Boosting、支持向量机和人工神经网络四种算法对预留的数据进行了预测,结果表明人工神经网络的预测误差最小。
With the global economy integration, the factors affecting the economy and the relations of the factors become more complex. The traditional economic theory cannot describe it, but nonlinear economics (chaotic economics) has become the research focus, and made rapid progress. This dissertation studies the complex nonlinear phenomenon in the economic system with the aid of modern nonlinear dynamics and stochastic dynamics theory. The main content is as follows:
1 The Hicks’consumption function model was improved, the Puu’s investment function model with cubic nonlinearity was introduced, and the nonlinear economic period model of social economic system was set up when the effect of periodic fluctuation on economic system was considered. The nonlinear dynamic characteristic and stability of that system was studied, three kinds of social economic form was divided, and the risk in transform process between different economic forms was analyzed. Finally the theoretic results were proved by numerical simulation.
2 A stochastic economic period model considering the stochastic independence function is set up. The complete stable condition has been obtained with Lyapunov exponent based on multiplicatibve ergodic theorem and boundary analysis of one-dimension diffuse process to this weak damp and weak stimulation Quanti-non-integrate Hamiltion system. The stochastic Hopf bifurcation behavior has been studied on the basis of joint probability density and marginal probability density under different parameters condition, and validated with the numerical simulation.
3. For hourly telephonometry within the busiest period in one day of Xiqing, Tianjin, the optimal embedding dimension and delay time were determined by false nearest neighbors and Cao’s method. The predicted errors of preserved 300 records were calculated by random forest, stochastic gradient boosting, support vector and artificial neuron network. The results show that artificial neuron network predict the most precisely.
引文
[1] M.B.Priestley. State dependent models, A general approach to nonlinear time series analysis[J]. Time series Aal. 1980,(1), 712~716
[2] Ted, T.and Sayers, C.L. Is chaos generic in economic data? International Journal of Bifurcation and chaos, Vol.3,1993, 745~755
[3] Peters, E.E(1991). A chaotic Attractor for the S&P500, March/April,1991
[4] Barnsley M.G, Fractal everywhere. Academic Press Inc 1998
[5] F.Takens. Detecting strange attractors in fluid turbulence. Dynamical Systems and Turbulence, Springer, Belin,1981
[6] H.D.I Abarbemel. Analysis of observe chaotic data in physical system[J]. Reviews of Modern physics, 1993(65), 1331~1392
[7] A.I.Mees, P.E. Rapp and L.S.Jennings. Singular-value decomposition and embedding dimension[J], Phy.Rev A, 1987, 340~346
[8] M.T. R.osensteinJJ, Collins Reconstruction expansion as a geometry-based framework for choosing proper delay times[J], Phy.D, 1994,(73),82~98
[9] T.T.Hartley and F.Mossayebi.A Classical approach to controlling the Lorenz Equation. Int.J. of bifurcation and chaos[J],1993,(2),407~441
[10] D.S.Broomherd and P.Gregory. Extracting pualitative dynamics from experimental data[J], Phys.D,1986,(20), 217~236
[11] Stfano Redaelli, Wise-law M. Macek, Lyapunov exponent and entropu of the solar wind flow, Planetray and Space Science 49(2001) 1211~1218
[12] Fabio Sattin, Lyap: A Fortran 90 program to computer the Lyapunov exponent of a dynamical system from a time series, Computer Physics Communications 107(1997) 253~257
[13] N. N. Oiwa, N. Fiedler-Ferrara, A Fast algorithm for estimating Lyapunov exponents from time series, Physics Letters A 246(1998) 117~121
[14] D. Lai, G. Chen, Statistical Analysis of Lyapunov Exponent from Time siries,Mathl. Comput. Modelling Vol. 27, No. 7, pp. 1~9, 1998
[15] Tonu Puu, Irina Sushko, A business cycle model with cubic nonlinearity, chaos slitons and fractals, 2004,19:597-612
[16] M.solomonovich, L.P.Apedaile, A dynamical economic model of sustainable agriculture and the ecosphere, applied mathematics and computation, 1997,84:221-246.
[17] P. Chen, "Empirical and Theoretical Evidence of Monetary Chaos," System Dynamics Review, 4,81-108 (1988)
[18] W. A. Brock and C. Sayers, "Is the Business Cycles Characterized by Deterministic Chaos?"Journal of Monetary Economics, 22, 71-80 (1988)
[19] Ahin Imoto, An example of nonlinear endogenous business cycle model:build in the trade union, Economics letters, 2003,81:117-124
[20] Abraham C.-L. Chian, Attractor merging crisis in chaotic business cycles, chaos solitons andfractals, 2005,24:869-875
[21] Grebogi C, Ott E, York JA. Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 1983;7:181–200.
[22] Goodwin RM. The nonlinear accelerator and the persistence of business cycles. Econometrica 1951;19:1–17
[23] Hicks JR. A contribution to the theory of the trade cycle. Oxford: Oxford University Press; 1950.
[24] Puu T. Chaos in business cycles. Chaos, Solitons, & Fractals 1991;1:457–73.
[25] Puu T. Attractors, bifurcations, and chaos--nonlinear phenomena in economics. Springer-Verlag; 2000.
[26] Oseledec V.L., A multiplicative ergodic theorem Lyapunov characteristic numbers for dynamical systems, Transaction of the Moscow Mathematical Society,1968, 19,197-231
[27] Wiggins,S (1990) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Vol 2.Springer-Verlag, New York C. V. Haridas and M. B. Rajarshi,A Stochastic Model for Evolution of Sociality in Insects,Theoretical Population Biology 59, 107-117 (2001)
[28] Li Lin, Stability and Hopf Bifurcation of a Differential Delay System, Journal of Biomathematics,2002,17(2), 157-164
[29] Sunita Gakkhar, Ra’id Kamel Naji, Chaos in seasonally perturbed ratio- dependent prey–predator system, Chaos, Solitons and Fractals 15 (2003) 107–118
[30] Shandelle M. Henson1 , Aaron A. King, R. F. Costantino, J. M. Cushing, Explaining and predicting patterns in stochastic population systems, The Royal Society, 2003.6.20,Online
[31] Ryszard Rudnickia, Long-time behaviour of a stochastic prey–predator model, Stochastic Processes and their Applications , 108 (2003) 93– 107
[32] J. Golec S,. Sathananthan, Stability Analysis of a Stochastic Logistic Model, Mathematical and Computer Modelling 38 (2003) 585-593
[33] Juan Pablo Aparicio, Hern_an Gustavo Solari , Sustained oscillations in stochastic systems, Mathematical Biosciences 169 (2001) 15-25
[34] Khasmiskii.R.Z., Stochastic Stability of Differential Equations, Alpen aan den Rijin, the Netherlands, Sijthoff and Noordhoff,1980.
[35] Arnold L., Papanicolaou.G. and Wihstutz V., Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications, SIAM Journal of Applied Mechanics, 1986,46(3),427-450
[36] Namachiwaya N.S., Roessel V. and Talwar S., Maximal Lyapunov exponent and almost-sure stability for coupled two degree of freedom stochastic systems, ASME Journal of Applied Mechanics,1994,61, 446-452
[37] Par doux E. and Wihstutz V., Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion, SIAM Journal of Applied Mechanics, 1988,48(2),442-457
[38] Ariaratnam S.T. and Xie W.C., Lyapunov exponents and stochastic stability of coupled linear systems under real noise excitation, ASME Journal of Applied Mechanics,1992,59, 664-673
[39] Namachchiwaya N Sri., Stochastic bifurcation, Applied Mathematics and Computation, 1990,38:101-159
[40] V.S.Anishchenko, T.E.Vadivasova, G.I.Strelkova, G.A.Okrokvertskhov, Statistical properties of dynamical chaos, Mathematical Biosciences and Engineering, 2004,1(1), 161-184
[41] C.Castillo-Chavez and B.Song, Dynamical Models of tuberculosis and their applications, Mathematical Biosciences and Engineering, 2004,1(2), 361-404
[42] Walter, M., Recknagel, F., Carpenter, C., Bormans, M., Prediction Eutrophication Effects in the Burrinjuck Reservior (Australia) by Means of the Deterministic Model SALMO and the Recurrent Neural Network Model ANNA, Ecol.Modelling, 2000
[43] Y.Kuang, J.Huisman and J.J.Elser, Stoichometric plant-herbivore models and their interpretation, Mathematical Biosciences and Engineering, 2004,1(2), 215-222
[44] R.López,-Ruiz and D.Fournier-Prunaret, Complex behavior in a discrete coupled logistic model for the symbiotic interaction of two species, Mathematical Biosciences and Engineering, 2004,1(2), 307-324
[45] W.Q.Zhu, Z.L.Huang, Stochastic Stability of Quasi-nonintegrable- Hamiltonian Systems,Journal of Sound and Vibration, 1998,218(5), 769-789
[46] W.Q.Zhu,Y.Q.Yang, Stochastic Averaging of Quasi- nonintegrable- Hamiltonian Systems, Journal of Applied Mechanics, Vol.64, 1997, 157-164
[47] W.Q.Zhu,Z.L.Huang,Stochastic Hopf bifurcation of Quasi-nonintegrable- Hamiltonian Systems, Internatonal Journal of Non-Linear Mechanics,34(1999), 437-447
[48] Dong-Wei Huang, Hong-Li Wang, Yu Qiao, Zhi-Wen Zhu, Stochastic Hopf Bifurcation of Four-wheel-steering System, Proceedings of the Fifth International Conference on Stochastic Structural Dynamics, Hangzhou, China, 2003,207-214
[49] Kedai Xu . Stochastic pitchfork bifurcation , numerical simulations and symbolic calculations using MAPLE [J]. Mathematics and Computers in Simulation 38 ,(1995),199-209
[50] L. Arnold, K.D. Xu. Normal-Forms for random diffeomorphism [J]. J.Dynamics Differential Equations, 4 (1992) ,445-483.
[51] K.D.Xu. Bifurcations of random differential equations in dimension one [J]. Random Comput. Dynamics 1 (1992/1993) 277-305.
[52] 黄润生等,混沌及其应用,武汉大学出版社,2000 年
[53] 郝柏林,从抛物线谈起——混沌动力学引论,上海科技教育出版社,1993 年
[54] 陈予恕,唐云等,非线性动力学中的现代分析方法,科学出版社,2000 年
[55] 陈予恕,非线性振动系统的分岔和混沌理论,高等教育出版社,1993 年
[56] 王洪礼,张琪昌,现代非线性动力学理论及应用,天津科技出版社,2002 年
[57] 黄东卫,渤海赤潮生态系统的非线性随机动力学研究,博士论文
[58] 戎海武,徐伟,方同,二自由度耦合非线性随机系统的最大Lyapunov指数和稳定性,应用力学学报,1998,15(1),22-29
[59] 戎海武,孟光,徐伟,方同,二自由度耦合线性随机系统的最大Lyapunov指数和稳定性,应用力学学报,2000,17(3),46-53
[60] 戎海武,方同,二阶线性随机微分方程的渐近稳定性,应用力学学报,1996,13(3),72-78
[61] 刘先斌,陈虬,陈大鹏,非线性随机动力系统的稳定性和分岔研究,力学进展,1996,26(4),437-452
[62] 刘先斌,陈大鹏,陈虬,实噪声参激一类余维 2 分叉系统的最大 Lyapunov指数(Ⅰ),应用数学和力学,1999,20(9),902-912
[63] 刘先斌,陈大鹏,陈虬,实噪声参激一类余维 2 分叉系统的最大 Lyapunov指数(Ⅱ),应用数学和力学,1999,20(10),997-1003
[64] 刘先斌,陈虬,陈大鹏,白噪声参激 Hopf 分叉系统的两次分叉研究,应用数学和力学,1997,18(9),779-788
[65] 刘先斌,陈虬, 实噪声参激 Hopf 分叉系统研究,力学进展,1997,29(2),158-166
[66] 刘先斌,一类随机分叉系统概率 1 分叉研究,固体力学学报,2001,22(3),297-302
[67] 钟万勰 . 结构动力方程的精细时程积分法 [J], 大连理工大学学报,1994,34(2):131-136.
[68] 钟万勰,欧阳华江,邓子辰. 计算力学与最优控制[M],大连,大连理工大学出版社,1993
[69] 顾元宪,陈飙松,张洪武. 结构动力方程的增维精细时程积分法[J],力学学报,2000,32(4): 447-456.
[70] 朱位秋. 随机振动[M]. 北京: 科学出版社, 1998
[71] 张素英,邓子辰. 非线性动力方程的增维精细积分法[J],计算力学学报,2003,20(4):423-426
[72] 裘春航,吕和祥,蔡志勤,在哈密顿体系下分析非线性动力学问题,计算力学学报,2000,17(2), 127-132
[73] 戎海武,孟光,徐伟,方同,二阶随机参激系统的Lyapunov指数和稳定性,振动工程学报,2002,15(3),295-299
[74] 戎海武,王命宇,方同,二阶随机系统的 Lyapunov指数与稳定性, 振动工程学报,1997,10(2),213-218
[75] 杨槐,朱华,褚亦清,一类非线性系统在随机激励下的分叉,应用力学学报,1993,10(4),69-71
[76] 陆启韶,常微分方程的定性方法和分岔,北京:北京航空航天大学出版社,1989
[77] 陆启韶,分岔与奇异性,上海:上海科技教育出版社,1995
[78] 张锦炎,冯贝叶,常微分方程几何理论与分岔问题,北京:北京大学出版社,1987
[79] 戎海武,徐伟,方同,随机分叉定义小议,广西科学,1997,4(1), 15-19
[80] 徐伟,戎海武,方同, C r随机中心流形定理,1997,14(3), 8-13
[81] 戎海武,孟光,王向东,徐伟,方同,FPK 方程的近似闭合解,应用力学学报,2003,20(3), 95-98
[82] 朱位秋,非线性随机动力学与控制—Hamilton 理论体系框架,科学出版社,2003;
[83] 曹庆杰,非线性系统随机振动与分叉理论研究:[博士学位论文],天津;天津大学,1991
[84] 廖晓昕,动力系统的稳定性和应用,国防工业出版社,2000;
[85] 张志祥,随机扰动间断动力系统的极限性质及其应用,数学的实践与认识,2002,32(4),651-657
[86] 戎海武,王命宇,方同,随机 ARNOLD 系统的稳定性与分叉,应用力学学报,1996,13(4),112-116
[87] Packard, N.H., Grutchfield, J.P., Farmer, J.D. and Shaw, R.S., Geometry from a Time Series [J], Physical. Review Letters, 1980(45): 712~716
[88] Peitgen H O, Jugens H, Saupe D, Chaos and fractals[M], New York: Springer, 1992
[89] Peng C. K., Havilin S., Stanley H. E., et al, Qualitification of scaling exponents and crossover phenomena in nonstationary heartbeat time series[J], Chaos, 1995, 5(1): 82~87
[90] Richard H. Day, Oleg V. Pavlov, Computing economic chaos[J], Computational economics, 2004, 23(4): 289~301
[91] Serletis A., Gogas P., Chaos in east European black-market exchange rates[J],Research in economics, 1997(51):359~385
[92] Steve Gunn. Support Vector Machines for Classification and Regression [R]. ISIS Technical Report, 1998, 5.
[93] 埃德加?E?彼得斯著,储海林,殷勤译,分形市场分析——将混沌理论应用到投资与经济理论上[M],北京:经济科学出版社,2002
[94] 陈国华,盛昭翰,基于 Lyapunov 指数的混沌时间序列识别[J],系统工程理论方法与应用,2003,12(4): 317~320
[95] 陈平,文明分岔、经济混沌和演化经济动力学[M],北京:北京大学出版社,2004
[96] 李天云,刘自发. 电力系统负荷的混沌特性及预测. 中国电机工程学报,2000,Vol.20(11):36~40
[97] 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.
[98] 罗雪晖,李霞,支持向量机及其应用研究[J],深圳大学学报(理工版),2003,20(3):40-46
[99] Breiman L, Fredman J, Olshen R, et al. Classification and Regression Trees[M].New York: Chapman and Hall, 1984.
[100] Breiman L. Random Forests[J]. Mach. Learn, 2001, 45(1):5–32. citeseer.ist. psu.edu/breiman01random.html.
[101] Efron B. Bootstrap Methods: Another Look at the Jackknife[J]. The Annals of Statistics, 1979, 7(1):1–26.
[102] Singh K. On the Asymptotic Accuracy of Efron’s Bootstrap[J]. The Annals of Statistics, 1981, 9(6):1187–1195.
[103] Freidman J H. Greedy function approximation: a gradient boosting machine[J]. Annals of Statistics, 2001, 29:1189–1232.
[104] Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning[M].Berlin: Springer-Verlag, 2001.
[105] Lindgren J T, Rousu J. Microscopy Image Analysis of Bread Using Machine Learning Methods[ 缺 文 献 类 型 标 志 代 码 ]. citeseer.ist.psu.edu/lindgren02microscopy.html.
[106] Friedman J. Tutorial: Getting Started with MART in R[缺文献类型标志代码]. citeseer.ist.psu.edu/friedman02tutorial.html.
[107] Jiang W. Some Theoretical Aspects of Boosting in the Presence of Noisy Data[C]//Proc. 18th International Conf. on Machine Learning.[S.l.]: Morgan Kaufmann, San Francisco, CA, 2001:234–241.
[108] Lawrence R, Bunn A, Powell S, et al. Classification of remotely sensed imagery using stochastic gradient boosting as a refinement of classification tree analysis[J]. Remote Sensing of Environment, 2004, 90(3):331–336.
[109] Hancock T, Put R, Coomans D, et al. A performance comparison of modern statistical techniques for molecular descriptor selection and retention prediction in chromatographic QSRR studies[J]. Chemometrics and Intelligent Laboratory Systems, 2005, 76(2):185–196.
[110] Bricklemyer R S, Lawrence R L, Miller P R, et al. Predicting tillage practices and agricultural soil disturbance in north central Montana with Landsat imagery[J]. Agriculture, Ecosystems & Environment, 2006, 114(2-4):210–216.
[111] Moisen G G, Freeman E A, Blackard J A, et al. Predicting tree species presence and basal area in Utah: A comparison of stochastic gradient boosting, generalized additive models, and tree-based methods[J]. Ecological Modelling, 2006, 199(2):176–187.
[112] Bricklemyer R S, Lawrence R L, Miller P R, et al. Monitoring and verifying agricultural practices related to soil carbon sequestration with satellite imagery[J]. Agriculture, Ecosystems & Environment, 2007, 118(1-4):201–210.
[113] Hegger R, Kantz H, Schreiber T. Practical implementation of nonlinear time series methods: The TISEAN package[J]. CHAOS, 1999, 9:413–435.
[114] Kennel M B, Brown R, Abarbanel H D I. Determining embedding dimension for phase-space reconstruction using a geometrical construction[J]. Physical Review A, 1992, 45:3403 – 3411.
[115] Cao L. Practical method for determining the minimum embedding dimension of a scalar time series[J]. Physcai D, 1997, 110:43–50. [18] Liaw A, Wiener M. Classification and regression by randomForest[J]. Rnews, 2002, 2:18–22.
[116] Ridgeway G. A note on out-of-bag estimation for estimating the optimal number of boosting iterations[M]. working paper: http://www.ipensieri. com/gregr/gbm.shtml, 2003.
[117] Rosenstein M T , Collins J J, Luca C J D. A practical method for calculating largest Lyapunov exponents from small data sets[J]. Physica D, 1993,65:117–134.
[118] Nychka D, Ellner S, Gallant A, et al. Finding Chaos in Noisy Systems[J]. J. R. Stat. Soc. B, 1992, 54(2):399–426.