Information flow within stochastic dynamical systems.

Information flow or information transfer is an important concept in general physics and dynamical systems which has applications in a wide variety of scientific disciplines. In this study, we show that a rigorous formalism can be established in the context of a generic stochastic dynamical system. An explicit formula has been obtained for the resulting transfer measure, which possesses a property of transfer asymmetry and, if the stochastic perturbation to the receiving component does not rely on the giving component, has a form the same as that for the corresponding deterministic system. This formula is further illustrated and validated with a two-dimensional Langevin equation. A remarkable observation is that, for two highly correlated time series, there could be no information transfer from one certain series, say x_{2} , to the other (x_{1}) . That is to say, the evolution of x_{1} may have nothing to do with x_{2} , even though x_{1} and x_{2} are highly correlated. Information flow analysis thus extends the traditional notion of correlation analysis and/or mutual information analysis by providing a quantitative measure of causality between dynamical events.

[1]  Richard Kleeman,et al.  Information Flow in Ensemble Weather Predictions , 2007 .

[2]  D. Ruelle.  Positivity of entropy production in the presence of a random thermostat , 1997 .

[3]  P. Grassberger,et al.  A robust method for detecting interdependences: application to intracranially recorded EEG , 1999, chao-dyn/9907013.

[4]  Martin Ehrendorfer,et al.  Optimal Prediction of Forecast Error Covariances through Singular Vectors , 1997 .

[5]  Andrew J. Majda,et al.  A mathematical framework for quantifying predictability through relative entropy , 2002 .

[6]  E. Epstein,et al.  Stochastic dynamic prediction , 1969 .

[7]  S. Griffies,et al.  A Conceptual Framework for Predictability Studies , 1999 .

[8]  Richard Kleeman.  Statistical predictability in the atmosphere and other dynamical systems , 2007 .

[9]  Andrew J. Majda,et al.  Predictability in a Model of Geophysical Turbulence , 2005 .

[10]  E. Lorenz.  A study of the predictability of a 28-variable atmospheric model , 1965 .

[11]  George F. Carnevale,et al.  Information decay and the predictability of turbulent flows , 1982 .

[12]  L. Kantha,et al.  Predictability, uncertainty, and hyperbolicity in the ocean , 2003 .

[13]  T. Palmer.  Predicting uncertainty in forecasts of weather and climate , 2000 .

[14]  Robert N. Miller,et al.  Ensemble Generation for Models of Multimodal Systems , 2002 .

[15]  David Ruelle,et al.  Positivity of entropy production in nonequilibrium statistical mechanics , 1996 .

[16]  Swinney,et al.  Information transport in spatiotemporal systems. , 1988, Physical review letters.

[17]  Pierre F. J. Lermusiaux,et al.  Uncertainty estimation and prediction for interdisciplinary ocean dynamics , 2006, J. Comput. Phys..

[18]  Andrew J. Majda,et al.  Information flow between subspaces of complex dynamical systems , 2007, Proceedings of the National Academy of Sciences.

[19]  Schreiber,et al.  Measuring information transfer , 2000, Physical review letters.

[20]  Richard Kleeman,et al.  Information transfer between dynamical system components. , 2005, Physical review letters.

[21]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[22]  E. Epstein,et al.  Stochastic dynamic prediction1 , 1969 .

[23]  Andrew J. Majda,et al.  Information theory and predictability for low-frequency variability , 2005 .

[24]  T. Schreiber,et al.  Information transfer in continuous processes , 2002 .

[25]  Brian F. Farrell,et al.  Small Error Dynamics and the Predictability of Atmospheric Flows. , 1990 .

[26]  C. Sims.  MACROECONOMICS AND REALITY , 1977 .

[27]  Michael C. Mackey,et al.  Chaos, Fractals, and Noise , 1994 .

[28]  E. Kalnay,et al.  Ensemble Forecasting at NCEP and the Breeding Method , 1997 .

[29]  A. Majda,et al.  Quantifying predictability in a model with statistical features of the atmosphere , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[30]  A. Moore.  The dynamics of error growth and predictability in a model of the Gulf stream. Part II : Ensemble prediction , 1999 .

[31]  K. Kaneko.  Lyapunov analysis and information flow in coupled map lattices , 1986 .

[32]  C. Leith.  Theoretical Skill of Monte Carlo Forecasts , 1974 .

[33]  K. Hlavácková-Schindler,et al.  Causality detection based on information-theoretic approaches in time series analysis , 2007 .

[34]  Peter C. B. Phillips,et al.  Vector Autoregressions and Causality , 1993 .

[35]  R. Kleeman.  Measuring Dynamical Prediction Utility Using Relative Entropy , 2002 .

[36]  C. Granger.  Investigating Causal Relations by Econometric Models and Cross-Spectral Methods , 1969 .

[37]  Richard Kleeman,et al.  A rigorous formalism of information transfer between dynamical system components. I. Discrete mapping , 2007 .

[38]  X. Liang,et al.  A rigorous formalism of information transfer between dynamical system components. II. Continuous flow , 2007 .