Fuzzy descriptor systems and spectral analysis for chaotic time series prediction
Tóm tắt
Predicting future behavior of chaotic time series and systems is a challenging area in the literature of nonlinear systems. The prediction accuracy of chaotic time series is extremely dependent on the model and the learning algorithm. In addition, the generalization property of the proposed models trained by limited observations is of great importance. In the past two decades, singular or descriptor systems and related fuzzy descriptor models have been the subjects of interest due to their many practical applications in modeling complex phenomena. In this study fuzzy descriptor models, as a more recent neurofuzzy realization of locally linear descriptor systems, which have led to the introduction of intuitive incremental learning algorithm that is called Generalized Locally Linear Model Tree algorithm, are implemented in their optimal structure to be compared with several other methods. A simple but efficient technique, based on the error indices of multiple validation sets, is used to optimize the number of neurons as well as to prevent over fitting in the incremental learning algorithms. The aim of the paper is to demonstrate the advantages of fuzzy descriptor models and to make a fair comparison between the most successful neural and neurofuzzy approaches in their best structures according to prediction accuracy, generalization, and computational complexity. The Mackey–Glass time series, Lorenz time series (as two well-known classic benchmarks), Darwin sea level pressure time series and long-term prediction of Disturbance Storm Time index, an important index of geomagnetic activity (as two natural chaotic dynamics) are used as practical examples to evaluate the power of the proposed method in long term prediction of chaotic dynamics.
Tài liệu tham khảo
Rezaei Yousefi MM, Mirmomeni M, Lucas C (2007) Input variables selection using mutual information for neurofuzzy modeling with the application to time series forecasting. In: Proceedings of IEEE international joint conference on neural networks: IJCNN, Orlando, Florida, USA, August 12–17
Koskela T, Varsta M, Heikkonen J, Kaski K (1997) Time series prediction using RSOM with local linear models, Research reports B15, Laboratory of Computational Engineering, Helsinki University of Technology, ISBN 951-22-3788-1
Casdagli M (1989) Nonlinear prediction of chaotic time series. Physica D 35:335–356
Chen S, Wu Y, Luk BL (1999) Combined genetic algorithm optimization and regularized othogonal least squares learning for radial basis function networks. IEEE Trans Neural Netw 10(5):1239–1243
Leung H, Lo T, Wang S (2001) Prediction of noisy chaotic time series using an optimal radial basis function neural network. IEEE Trans Neural Netw 12(5):1163–1172
Weigend A, Huberman B, Rumelhart DE (1990) Predicting the future: a connectionist approach. Int J Neural Syst 1:193–209
Weigend A, Huberman B, Rumelhart DE (1992) Predicting sunspots and exchange rates with connectionist networks. In: Casdagli E (ed) Nonlinear modeling and forecasting. Addison-Wesley, Reading, pp 395–432
Mirmomeni M, Lucas C, Moshiri B (2007) Long term prediction of chaotic time series with the aid of neurofuzzy models, spectral analysis and correlation analysis. In: Proceedings of IEEE international joint conference on neural networks. IJCNN, Orlando, FL, USA, August 12–17
Haykin S (1994) Neural networks: a comprehensive foundation. MacMillan, New York
Hornik K, Stinchombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Comput 2:359–366
Park J, Sandberg IW (1993) Approximation and radial basis function networks. Neural Comput 5(2):305–316
Elsner JB (1992) Predicting time series using a neural network as a method of distinguishing chaos from noise. J Phys A: Math Gen A 25:843–850
Nelles O (2001) Nonlinear system identification. Springer, Berlin
Gholipour A, Lucas C, Nadjar Araabi B, Mirmomeni M, Shafiee M (2007) Extracting the main patterns of natural time series for long-term neurofuzzy prediction. Neural Comput Appl 16(4–5):383–393
Abarbanel HDI (1996) Analysis of observed chaotic data. Springer, New York
Ott E (1993) Chaos in dynamical systems. Cambridge University Press, Cambridge
Ott E, Sauer T, Yorke JA (eds) (1994) Coping with chaos: analysis of chaotic data and the exploitation of chaotic systems. Wiley, New York
Petgen H-O, Jürgens H, Saupe D (1992) Chaos and fractals new frontiers of science. Springer, New York
Leung H, Lo T, Wang S (2001) Prediction of noisy chaotic time series using an optimal radial basis function neural network. IEEE Trans Neural Netw 12:1163–1172
Fredrich K (1986) Estimating the dimension of weather and climate attractors. J Atmos Sci 43:419–432
Medio A (1992) Chaotic dynamics: theory and applications to economics. Cambridge University Press, Cambridge
Navone HD, Ceccatto HA (1995) Forecasting chaos from small data sets: a comparison of different nonlinear algorithms. J Phys A 28(12):3381–3388
Jones AJ (2004) New tools in non-linear modeling and prediction. CMS 1:109–149
Yonchev A, Findeisen R, Ebenbauer C, Allgöower F (2004) Model predictive control of linear continuous time singular systems subject to input constraints. In: 43rd IEEE conference on decision and control, December 14–17, Atlantis, Paradise Island, Bahamas
Raouf J, Boukas EK (2004) Observer-based controller design for linear singular systems with markovian switching. In: 43rd IEEE conference on decision and control, December 14–17, Atlantis, Paradise Island, Bahamas
Mirmomeni M, Lucas C, Shafiee M, Nadjar Araabi B (2008) Introducing an incremental learning method for fuzzy descriptor models to identify nonlinear singular systems. In: 16th Mediterranean conference on control and automation, Congress Centre, Ajaccio, France, June 25–2, pp 7753–758
Taniguchi T, Tanaka K, Wang HO (1999) Fuzzy descriptor systems and fuzzy controller designs. In: Proceedings of the 8th International Fuzzy System Association World Congress, pp 655–659
Taniguchi T, Tanaka K, Yamafuji K, Wang HO (1999) Fuzzy descriptor systems: stability analysis and design via LMIs. In: Proceedings American control conference, pp 1827–1831
Brown M, Harris C (1995) Neurofuzzy adaptive modeling and control. Prentice Hall, UK
Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern 15:116–132
Jang JR (1993) ANFIS: adaptive network-based fuzzy inference system. IEEE Trans Syst Man Cybern 23(3):665–685
Kavli T (1993) ASMOD: an algorithm for adaptive spline modeling of observation data. Int J Control 58(4):947–967
Mirmomeni M, Nadjar Araabi B, Lucas C(2007) Development of LoLiMoT learning method for training neurofuzzy models on-line. In: European symposium on time series prediction (ESTSP07), Otaniemi, Espoo, Finland, February 7–9, pp 113–122
Vautard R, Ghil M (1989) Singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series. Physica D 35:395–424
Vautard R, Yiou P, Ghil M (1992) Singular spectrum analysis: a toolkit for short noisy chaotic signals. Physica D 58:95–126
Kugiumtzis D, Lillekjendlie B, Christophersen N (1994) Chaotic time series, part I: estimation of some invariant properties in state space. Model Identif Control 15(4):205–224
Verghese GC, Levy BC, Kailath T (1981) A generalized state-space for singular systems. IEEE Trans Autom Control AC-26(4)
Ailon A (1991) Decoupling of square singular systems via proportional state feedback. IEEE Trans Autom Control 36(1)
Dai L (1989) Singular control systems. Springer, New York
Wang J, Zhang Q, Liu W, Xin X, Sreeram V (2004) H∞ model reduction for singular systems. In: Proceedings of 2004 American Control Conference, June 30–July 2, Boston, Massachusetts, pp 119–124
Halfmann C, Nelles O, Holzmann H (1999) Modeling and identification of the vehicle suspension characteristics using local linear model trees. In: Proceedings of IEEE international conference on control applications, Khala-Coast Island, Hawaii, USA, pp 1484–1489
Jalili-Kharaajoo M, Ranji R, Bagherzadeh H (2003) Predictive control of a fossil power plant based on locally linear model tree (LOLIMOT). Proceedings of 2003 IEEE conference on control applications, CCA 2003, Istanbul, Turkey, 23–25 June, pp 633–638
Fink A, Nelles O, Isermann R (2002) Nonlinear internal model control for MISO systems based on local linear neuro-fuzzy models. 15th IFAC World Congress, Part I, vol 15, Barcelona, Spain
Gholipour A, Abbaspour A, Araabi BN, Lucas C (2003) Enhancements in the prediction of solar activity by locally linear model tree. In: Proceedings of MIC2003: 22nd international conference on modeling, identification and control, Innsbruck, Austria, pp 158–161
Gholipour A, Lucas C, Araabi BN, Shafiee M (2005) Solar activity forecast: spectral analysis and neurofuzzy prediction. J Atmos Solar Terrestrial Phys 67:595–603
Mackey M, Glass L (1977) Oscillation and chaos in physiological control systems. Science 197:281–287
Lorenz EN (1963) Deterministic non-periodic flow. J Atmos Sci 20:130–141
Fröyland J (1992) Introduction to chaos and coherence. IOP, London
Kaplan A, Kushnir Y, Kane MA (2000) Reduced space optimal interpolation of historical of marine sea level pressure: 1854–1992. J Clim 13:2987–3002
Mendelssohn R, Bograd SJ, Schwing FB, Palacios DM (2005) Teaching old indices new tricks: a state-space analysis of El Niño related climate indices. Geophys Res Lett 32:L07709. doi:10.1029/2005GL022350
Mirmomeni M, Shafiee M, Lucas C, Araabi BN (2006) Introducing a new learning method for fuzzy descriptor systems with the aid of spectral analysis to forecast solar activity. J Atmos Solar Terrestrial Phys 68:2061–2074
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