Using invariants to change detection in dynamical system with chaos. (English) Zbl 1141.37332
Summary: Change detection is the crucial subject in dynamical systems. There are suitable methods for detecting changes for linear systems and some methods for nonlinear systems, but there is a lack of methods concerning chaotic systems. This paper presents change detection techniques for dynamical systems with chaos. We consider the dynamical system described by the time series which originated from ordinary differential equation and real-world phenomena. We assume that the change parameters are unknown and the change could be either slight or drastic. The process of change detection is based on characteristic dynamical system invariants. Changes in the invariants’ values of the dynamical systems are the indicators of change. We propose a method of change detection based on the fractal dimension and recurrence plot. The automatic detection is provided by control charts. Methods were checked by using small data sets and stream data.
MSC:
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37M10 | Time series analysis of dynamical systems |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 93C05 | Linear systems in control theory |
| 93C10 | Nonlinear systems in control theory |
Keywords:
systems dynamics; quality control; time series; invariant measures; chaotic systems; fractal dimension; recurrence plotReferences:
| [1] | Montgomery, D., Introduction to Statistical Quality Control (2001), Wiley |
| [2] | Ilin, A.; Valpola, H.; Oja, E., Nonlinear dynamical factor analysis for state change detection, IEEE Trans. Neural Netw., 15, 3 (2004) |
| [3] | Basseville, M.; Nikiforov, I., Detection of Abrupt Changes: Theory and Application (1993), Prentice Hall · Zbl 1407.62012 |
| [4] | Gustafsson, F., Adaptive Filtering and Change Detection (2000), John Wiley Sons, Ltd |
| [5] | Schuster, H., Deterministic Chaos (1988), VGH Verlagsgesellschaft: VGH Verlagsgesellschaft Weinheim · Zbl 0707.58003 |
| [6] | Eckmann, J.; Kamphorst, S.; Ruelle, D., Recurrence plots of dynamical systems, Europhys. Lett., 5 (1987) |
| [7] | Ott, E., Chaos in Dynamical Systems (1993), Cambridge University Press · Zbl 0792.58014 |
| [8] | Ilin, A.; Valpola, H., Detecting process state changes by nonlinear blind source separation, IEICE Trans. Fund. Electron. Commun. Comput. Sci. (2001) |
| [9] | F. Gonzalez, D. Dasgupta, Anomaly detection using real-valued negative selection, 2003, URL citeseer.ist.psu.edu/gonzalez04anomaly.html; F. Gonzalez, D. Dasgupta, Anomaly detection using real-valued negative selection, 2003, URL citeseer.ist.psu.edu/gonzalez04anomaly.html |
| [10] | Hively, L. M.; Protopopescu, V. A., Timely detection of dynamical change in scalp eeg signals, Chaos, 10 (2000) · Zbl 1016.92019 |
| [11] | Letellier, C.; Maquet, J.; Sceller, L. L.; Gouesbet, G.; Aguirre, L. A., On the non-equivalence of observables in phase space reconstructions from recorded time series, J. Phys. A 31 (1998) · Zbl 0936.81014 |
| [12] | Takens, F., On the numerical determination of the dimension of an attractor, Lecture Notes in Math., 898 (1981) |
| [13] | Sauer, T.; Yorke, J.; Casdagli, M., Embedology, J. Statist. Phys. (1991) · Zbl 0943.37506 |
| [14] | Haykin, S.; Principe, J., Making sense of complex world, IEEE Signal Process. Mag., 15, 66-81 (1998) |
| [15] | Kennel, M. B.; Brown, R., Determining embedding dimension for phase-space reconstruction using a geometrical construction, Phys. Rev. A, 45 (1992) |
| [16] | Fraser, A. M.; Swinney, H. L., Independent coordinates for strange attractors from mutual information, Phys. Rev. A 33 (1986) · Zbl 1184.37027 |
| [17] | Cheng, B.; Tong, H., Orthogonal projection, embedding dimension and the sample size in chaotic time series from statistical perspective, Philos. Trans. R. Soc. Lond. A, 348 (1994) · Zbl 0859.62077 |
| [18] | Wong, A.; Wu, L., Fast estimation of fractal dimension and correlation integral on stream data, Inf. Process. Lett., 93 (2003) |
| [19] | N. Alon, Y. Matias, M. Szegedy, The space complexity of approximating the frequency moments, 1996, pp. 20-29. URL citeseer.ist.psu.edu/alon96space.html; N. Alon, Y. Matias, M. Szegedy, The space complexity of approximating the frequency moments, 1996, pp. 20-29. URL citeseer.ist.psu.edu/alon96space.html · Zbl 0922.68057 |
| [20] | Schreiber, T., Efficient neighbor searching in nonlinear time series analysis, Internat. J. Bifur. Chaos, 5 (1995) · Zbl 0885.58052 |
| [21] | Hegger, R.; Kantz, H.; Schreiber, T., Practical implementation of nonlinear time series methods: The tisean package, Chaos, 9 (1999) · Zbl 0990.37522 |
| [22] | Grassberger, P., Phys. Lett. A, 63, 148 (1988) |
| [23] | Molteno, T. C., Fast O(N) box-counting algorithm for estimating dimension, Phys. Rev. E, 48, 5 (1993) |
| [24] | Skubalska-Rafajowicz, E., A new method of estimation of the box-counting dimension of multivariate objects using space-filling curves, Nonlinear Anal., 63 (2005) · Zbl 1221.62088 |
| [25] | Tricot, C., Curves and Fractal Dimension (1995), Springer-Verlag · Zbl 0847.28004 |
| [26] | Falconer, K., Fractal Geometry (1990), Wiley: Wiley New York |
| [27] | Miyata, T.; Watanabe, T., Bi-Lipschitz maps and the category of approximate resolutions, Glas. Mat., 58, 38, 129-155 (2003) · Zbl 1050.54013 |
| [28] | N. Marwan, Encounters with neighbours — current developments of concepts based on recurrence plots and their applications, Ph.D. Thesis 2003; N. Marwan, Encounters with neighbours — current developments of concepts based on recurrence plots and their applications, Ph.D. Thesis 2003 |
| [29] | C.W. Jr., J. Zbilut, Dynamical assessment of physiological systems and states using recurrence plot strategies, J. Appl. Physiol. 78; C.W. Jr., J. Zbilut, Dynamical assessment of physiological systems and states using recurrence plot strategies, J. Appl. Physiol. 78 |
| [30] | Marwan, N.; Wessel, N.; Meyerfeldt, U.; Schirdewan, A.; Kurths, J., Recurrence plot based measures of complexity and its application to heart rate variability data, Phys. Rev. E, 66 (2002) |
| [31] | Trulla, L.; Giuliani, A.; Zbilut, J., Recurrence quantification analysis of the logistic equation with transients, Phys. Lett. A, 223 (1996) · Zbl 1037.37507 |
| [32] | Faure, P.; Korn, H., A new method to estimate the kolmogorov entropy from recurrence plots: Its application to neuronal signals, Phyisca D, 122 (1998) |
| [33] | Thiel, M.; Romano, M. C., Estimation of dynamical invariants without embedding by recurrence plots, Chaos, 14 (2004) · Zbl 1080.37091 |
| [34] | Hawkins, D. M.; Olwell, D. H., Cumulative Sum Charts and Charting for Quality Improvement (1998), Springer-Verlag · Zbl 0990.62537 |
| [35] | Borror, C. M.; Montgomery, D. C.; Runger, G. C., Robustness of the ewma control chart to non-normality, J. Quality Technol., 31, 3, 309-316 (1999) |
| [36] | J.M. Lucas, M. Saccucci, Exponentially weighted moving average control schemes: Properties and enhancements, Technometrics 32; J.M. Lucas, M. Saccucci, Exponentially weighted moving average control schemes: Properties and enhancements, Technometrics 32 |
| [37] | L. Scrucca, Quality control package for R-project. Available at: http://www.r-project.org/; L. Scrucca, Quality control package for R-project. Available at: http://www.r-project.org/ |
| [38] | L. Wu, C. Faloutsos, Fracdim, perl package available at: http://www.andrew.cmu.edu/ lw2j/downloads.html; L. Wu, C. Faloutsos, Fracdim, perl package available at: http://www.andrew.cmu.edu/ lw2j/downloads.html |
| [39] | G. Cormode, Massdal public code bank, source code available at: http://www.cs.rutgers.edu/ |
| [40] | E. Kononov, Visual recurrence analysis. Freeware version available: http://www.myjavaserver.com/nonlinear/vra/download.html; E. Kononov, Visual recurrence analysis. Freeware version available: http://www.myjavaserver.com/nonlinear/vra/download.html |
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