ABSTRACT
From microbial communities, human physiology to social and biological/neural networks, complex interdependent systems display multi-scale spatio-temporal patterns that are frequently classified as non-linear, non-Gaussian, non-ergodic, and/or fractal. Distinguishing between the sources of nonlinearity, identifying the nature of fractality (space versus time) and encapsulating the non-Gaussian characteristics into dynamic causal models remains a major challenge for studying complex systems. In this paper, we propose a new mathematical strategy for constructing compact yet accurate models of complex systems dynamics that aim to scrutinize the causal effects and influences by analyzing the statistics of the magnitude increments and the inter-event times of stochastic processes. We derive a framework that enables to incorporate knowledge about the causal dynamics of the magnitude increments and the inter-event times of stochastic processes into a multi-fractional order nonlinear partial differential equation for the probability to find the system in a specific state at one time. Rather than following the current trends in nonlinear system modeling which postulate specific mathematical expressions, this mathematical frame-work enables us to connect the microscopic dependencies between the magnitude increments and the inter-event times of one stochastic process to other processes and justify the degree of nonlinearity. In addition, the newly presented formalism allows to investigate appropriateness of using multi-fractional order dynamical models for various complex system which was overlooked in the literature. We run extensive experiments on several sets of physiological processes and demonstrate that the derived mathematical models offer superior accuracy over state of the art techniques.
- Chiuso A. Regularization and bayesian learning in dynamical systems: Past, present and future. Annual Reviews in Control - submitted, 20XX.Google Scholar
- J. Bassingthwaighte, L. Liebovitch, and B. West. Properties of fractal phenomena in space and time. In Fractal physiology. Springer, 1994. Google ScholarCross Ref
- M. Besserve, N. Logothetis, and B. Schölkopf. Statistical analysis of coupled time series with kernel cross-spectral density operators. In NIPS, 2013.Google Scholar
- P. Bogdan. Mathematical modeling and control of multifractal workloads for data-center-on-a-chip optimization. In Proceedings of the 9th International Symposium on Networks-on-Chip, page 21. ACM, 2015. Google ScholarDigital Library
- R. Bousseljot, D. Kreiseler, and A. Schnabel. Nutzung der ekg-signaldatenbank cardiodat der ptb über das internet. Biomedizinische Technik/Biomedical Engineering, 40(s1):317--318, 1995.Google Scholar
- J. Crutchfield and B. McNamara. Equations of motion from a data series. Complex systems, 1(417--452):121, 1987.Google Scholar
- C. Granger and R. Joyeux. Essays in econometrics. chapter An Introduction to Long-memory Time Series Models and Fractional Differencing, pages 321--337. Harvard University Press, Cambridge, MA, USA, 2001.Google Scholar
- R. Herrmann. Fractional Calculus: An Introduction for Physicists. World Scientific.Google Scholar
- J. R. M. Hosking. Fractional differencing. Biometrika, 68(1):165--176, 1981. Google ScholarCross Ref
- Friston K., Harrison L., and Penny W. Dynamic causal modelling. NeuroImage, 19(4):1273 -- 1302, 2003. Google Scholar
- L. Ljung and T. Glad. On global identifiability for arbitrary model parametrizations. Automatica, 30(2):265--276, 1994. Google ScholarDigital Library
- Lennart Ljung. Perspectives on system identification. Annual Reviews in Control, 34(1):1 -- 12, 2010. Google ScholarCross Ref
- P. Mehta and S. Meyn. Q-learning and pontryagin's minimum principle. In Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proc. of the 48th IEEE Conference on. IEEE, 2009.Google ScholarCross Ref
- E. Montroll and M. Shlesinger. Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: a tale of tails. Journal of Statistical Physics, 1983. Google ScholarCross Ref
- H. Ohlsson, J. Roll, and L. Ljung. Manifold-constrained regressors in system identification. In Decision and Control, 2008. CDC 2008. 47th IEEE Conference on, pages 1364--1369, Dec 2008. Google ScholarCross Ref
- K. Oldham and J. Spanier. The fractional calculus : theory and applications of differentiation and integration to arbitrary order. Mathematics in science and engineering. Academic Press, New York, 1974.Google Scholar
- M. Opper and G. Sanguinetti. Variational inference for markov jump processes. In J.C. Platt, D. Koller, Y. Singer, and S.T. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 1105--1112. Curran Associates, Inc., 2008.Google Scholar
- N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw. Geometry from a time series. Phys. Rev. Lett., 45:712--716, Sep 1980. Google ScholarCross Ref
- D. Rubin. Causal inference using potential outcomes: Design, modeling, decisions. Journal of the American Statistical Association, 100, 2005. Google ScholarCross Ref
- J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer Publishing Company, Incorporated, 1st edition, 2007. Google ScholarCross Ref
- G. Schalk, D.J. McFarland, T. Hinterberger, N. Birbaumer, and J.R. Wolpaw. Bci2000: A general-purpose brain-computer interface (bci) system. IEEE Transactions on Biomedical Engineering, 51(6):1034--1043, 2004. Google ScholarCross Ref
- M. Shlesinger, G. Zaslavsky, and J. Klafter. Strange kinetics. Nature, 363(6424):31--37, 1993. Google ScholarCross Ref
- A. Svenkeson, B. Glaz, S. Stanton, and B. West. Spectral decomposition of non-linear systems with memory. Physical Review E, 93(2), 2016. Google ScholarCross Ref
- Constantino Tsallis. Possible Generalization of Boltzmann-Gibbs Statistics. J. Statist. Phys., 52:479--487, 1988. Google ScholarCross Ref
- B. West. Fractal physiology and chaos in medicine. World Scientific, 2012.Google Scholar
- Y Xue, S. Pequito, J. Coelho, P. Bogdan, and G. Pappas. Minimum number of sensors to ensure observability of physiological systems: A case study. In Communication, Control, and Computing (Allerton), 2016 54th Annual Allerton Conference on, pages 1181--1188. IEEE, 2016. Google ScholarCross Ref
- Y. Xue, S. Rodriguez, and P. Bogdan. A spatio-temporal fractal model for a cps approach to brain-machine-body interfaces. In Design, Automation & Test in Europe Conference & Exhibition (DATE), 2016, pages 642--647. IEEE, 2016. Google ScholarCross Ref
Index Terms
- Constructing compact causal mathematical models for complex dynamics
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