Abstract
During the last years, the concept of recurrence plots has received considerable interest as a tool for analysing nonlinear and non-stationary time series. However, in the case of discrete-valued observables or variations on very different time scales, problems may occur in direct interpretations of the results of recurrence quantification analysis (RQA). As a potential solution, we suggest combining this approach with ideas from symbolic time series analysis, which allows an arbitrary static or dynamic coarse-graining of the dynamics that goes beyond recent recurrence plot based methods. As an illustrative application, we discuss how the resulting symbolic recurrence plots may be used for a quantitative investigation of the dynamics of discrete-valued inventory levels of cooperating firms in a manufacturing network. Based on discrete-event simulations, measures from traditional RQA are used to evaluate the performance of the individual firms under different production strategies as well as order policies. The results of our investigations are an important step towards an anticipative knowledge about the performance of manufacturing systems under different conditions, which is of major importance for the planning and control of both production and logistics.
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References
H.D.I. Aberbanel, Analysis of Observed Chaotic Data (Springer, New York, 1996)
H. Kantz, T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 1997)
C. Diks, Nonlinear Time Series Analysis – Methods and Applications (World Scientific, Singapore, 1999)
R.V. Donner, S.M. Barbosa (eds.), Nonlinear Time Series Analysis in the Geosciences – Applications in Climatology, Geodynamics, and Solar-Terrestrial Physics (Springer, Berlin, 2008)
C.S. Daw, C.E.A. Finney, E.R. Tracy, Rev. Sci. Instrum. 74, 915 (2003)
J.M. Finn, J.D. Goettee, Z. Toroczkai, M. Anghel, B.P. Wood, Chaos 13, 444 (2003)
J.-P. Eckmann, S.O. Kamphorst, D. Ruelle, Europhys. Lett. 5, 973 (1987)
N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Phys. Rep. 438, 237 (2007)
C. Rieke, K. Sternickel, R.G. Andrzejak, C.E. Elger, P. David, K. Lehnertz, Phys. Rev. Lett. 88, 244 (2002)
C. Rieke, R.G. Andrzejak, F. Mormann, K. Lehnertz, Phys. Rev. E 69, 046111 (2004)
J.P. Zbilut, C.L. Webber Jr., Phys. Lett. A 171, 199 (1992)
C.L. Webber Jr., J.P. Zbilut, J. Appl. Phys. 76, 965 (1994)
L.L. Trulla, A. Giuliani, J.P. Zbilut, C.L. Webber Jr., Phys. Lett. A 223, 255 (1996)
N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, J. Kurths, Phys. Rev. E 66, 026702 (2002)
D.B. Vasconcelos, S.R. Lopes, R.L. Viana, J. Kurths, Phys. Rev. E 73, 056207 (2006)
N. Marwan, J. Kurths, P. Saparin, Phys. Lett. A 360, 545 (2007)
A. Groth, Phys. Rev. E 72, 046220 (2005)
A. Groth, Analyse der Wiederkehr in dynamischen Systemen auf einer Ordinalskala, Ph.D. thesis, University of Greifswald, 2006
N. Marwan, A. Groth, J. Kurths, Chaos Complex. Lett. 2, 301 (2007)
S. Schinkel, N. Marwan, J. Kurths, Cognit. Neurodyn. 1, 317 (2007)
P. Grassberger, H. Kantz, Phys. Lett. A 113, 235 (1985)
F. Christiansen, A. Politi, Nonlinearity 9, 1623 (1996)
F. Christiansen, A. Politi, Physica D 109, 32 (1997)
M.B. Kennel, M. Buhl, Phys. Rev. Lett. 91, 084102 (2003)
Y. Hirata, K. Judd, D. Kilminster, Phys. Rev. E 70, 016215 (2004)
M. Buhl, M.B. Kennel, Phys. Rev. E 71, 046213 (2005)
E.M. Bollt, T. Stanford, Y.-C. Lai, K. Życzkowski, Physica D 154, 259 (2001)
W. Lee, L. Luo, Phys. Rev. E 56, 848 (1997)
W. Li, K. Kaneko, Europhys. Lett. 17, 655 (1992)
C. Bandt, B. Pompe, Phys. Rev. Lett. 88, 174102 (2002)
M. Staniek, K. Lehnertz, Phys. Rev. Lett. 100, 158101 (2008)
C.J. Cellucci, A.M. Albano, P.E. Rapp, Phys. Rev. E 71, 066208 (2003)
W. Ebeling, G. Nicolis, Chaos Solit. Frac. 2, 635 (1992)
M. Thiel, M.C. Romano, J Kurths, Phys. Lett. A 330, 343 (2004)
G. Robinson, M. Thiel [arXiv:0706.4032] [math.DS] (2007)
P. Faure, H. Korn, Physica D 122, 265 (1998)
J.S. Iwanski, E. Bradley, Chaos 8, 861 (1998)
M. Thiel, M.C. Romano, P.L. Read, J. Kurths, Chaos 14, 234 (2004)
C. Letellier, Phys. Rev. Lett. 96, 254102 (2006)
M.C. Romano, M. Thiel, J. Kurths, W. von Bloh, Phys. Lett. A 330, 214 (2004)
J.D. Sterman, Business Dynamics: Systems Thinking and Modelling for a Complex World (McGraw-Hill, Boston, 2000)
W.J. Hopp, M.L. Spearman, Factory Physics (McGraw-Hill, Boston, 2000)
G. Radons, R. Neugebauer (eds.), Nonlinear Dynamics of Production Systems (Wiley Europe, Weinheim, 2004)
C. Daganzo, A Theory of Supply Chains (Springer, Berlin, 2005)
J.W. Forrester, Harvard Business Rev. 36, 37 (1958)
H. Lee, V. Padmanabhan, S. Whang, Sloan Management Rev. 38, 93 (1997)
D. Helbing, New J. Phys. 5, 90 (2003)
D. Helbing, S. Lämmer, U. Witt, T. Brenner, Phys. Rev. E 70, 056118 (2004)
D. Helbing, S. Lämmer, Networks of Interacting Machines: Production Organization in Complex Industrial Systems and Biological Cells, edited by D. Armbruster, A.S. Mikhailov, K. Kaneko (World Scientific, Singapore, 2005), p. 33
I. Katzorke, A. Pikovsky, Discr. Dyn. Nature Soc. 5, 179 (2000)
H.-P. Wiendahl, J. Worbs, J. Mater. Process. Technol. 139, 28 (2004)
B. Rem, D. Armbruster, Chaos 13, 128 (2003)
S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.U. Hwang, Phys. Rep. 424, 175 (2006)
B. Scholz-Reiter, U. Hinrichs, R. Donner, A. Witt, Proc. IASTED Conf. Modelling and Simulation (2006), p. 178
R. Donner, U. Hinrichs, B. Scholz-Reiter, A. Witt, Proc. NDES 2006 (2006), p. 22
B. Scholz-Reiter, U. Hinrichs, R. Donner, Proc. CARV 2007 (2007), p. 263
R. Donner, B. Scholz-Reiter, J. Hinrichs, Manufact. Sci. (in press)
R. Donner, U. Hinrichs, B. Scholz-Reiter, Dynamics in Logistics, edited by H.-D. Haasis, H.-J. Kreowski, B. Scholz-Reiter (Springer, Berlin, 2008), p. 161
A. Kraskov, H. Stögbauer, P. Grassberger, Phys. Rev. E 69, 066138 (2004)
T. Schreiber, A. Schmitz, Phys. Rev. Lett. 77, 635 (1996)
E.I. Vlahogianni, M.G. Karlaftis, J.C. Golias, Transp. Res. C 14, 351 (2006)
E.I. Vlahogianni, Transp. Res. Board Ann. Meeting 2007, 07-0814 (2007)
E.I. Vlahogianni, C.L. Webber Jr., N. Geroliminis, A. Skabardonis, Transp. Res. C 15, 392 (2007)
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Donner, R., Hinrichs, U. & Scholz-Reiter, B. Symbolic recurrence plots: A new quantitative framework for performance analysis of manufacturing networks. Eur. Phys. J. Spec. Top. 164, 85–104 (2008). https://doi.org/10.1140/epjst/e2008-00836-2
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DOI: https://doi.org/10.1140/epjst/e2008-00836-2