Natural systems are often complex and dynamic (i.e. nonlinear), and are difficult to understand using linear statistical approaches. Linear approaches are fundamentally based on correlation and are ill posed for dynamical systems, because in dynamical systems, not only can correlation occur without causation, but causation can also occur in the absence of correlation. To study dynamical systems, nonlinear time series analytical methods have been developed in the past decades [1-5]. These nonlinear statistical methods are rooted in State Space Reconstruction (SSR), i.e. lagged coordinate embedding of time series data [6] (http://simplex.ucsd.edu/EDM_101_RMM.mov). These methods do not assume any set of equations governing the system but recover the dynamics from time series data, thus called Empirical Dynamic Modeling (EDM).
EMD bears a variety of utilities to investigating dynamical systems: 1) determining the complexity (dimensionality) of system [1], 2) distinguishing nonlinear dynamic systems from linear stochastic systems [1], 3) quantify the nonlinearity (i.e. state dependence) [7], 4) determining causal variables [3], 5) tracking strength and sign of interaction [8, 9], 5) forecasting [5], 6) scenario exploration of external perturbation [4], and 7) classifying system dynamics [2, 10]. These methods and applications can be used for mechanistic understanding of dynamical systems and providing effective policy and management recommendations on ecosystem, climate, epidemiology, financial regulation, and much else.

[1]Hsieh, C.H., S.M. Glaser, A.J. Lucas, and G. Sugihara (2005) Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean. Nature. 435: 336-340.
[2]Hsieh, C.H., C. Anderson, and G. Sugihara (2008) Extending nonlinear analysis to short ecological time series. American Naturalist. 171: 71-80.
[3]Sugihara, G., R. May, H. Ye, C. H. Hsieh, E. Deyle, M. Fogarty, and S. Munch (2012) Detecting causality in complex ecosystems. Science 338: 496-500.
[4]Deyle, E., M. F. Fogarty, C. H. Hsieh, L. Kaufman, A. MacCall, S. B. Munch, C. Perretti, H. Ye, and G. Sugihara (2013) Predicting climate effects on Pacific sardine. PNAS. 110: 6430-6435.
[5]Ye, H., R. J. Beamish, S. M. Glaser, S. C. H. Grant, C. H. Hsieh, L. J. Richards, J. T. Schnute, and G. Sugihara (2015) Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling. PNAS. 112: E1569–E1576.
[6]Takens, F. 1981. Detecting strange attractors in turbulence. Pages 366-381 in D. A. Rand and L. S. Young, editors. Dynamic systems and turbulence. Springer-Verlag, New York.
[7]Anderson, C. N. K., C. H. Hsieh, S. A. Sandin, R. Hewitt, A. Hollowed, J. Beddington, R. M. May, and G. Sugihara. 2008. Why fishing magnifies fluctuations in fish abundance. Nature 452:835-839.
[8]Deyle, E. R., R. M. May, S. B. Munch, and G. Sugihara. 2016. Tracking and forecasting ecosystem interactions in real time. Proceedings of the Royal Society of London B: Biological Sciences 283.
[9] Ushio, M., C. H. Hsieh, R. Masuda, E. Deyle, H. Ye, C. W. Chang, G. Sugihara, and M. Kondoh. 2018. Fluctuating interaction network and dynamic stability of natural fish community. Nature 554: 360-363.
[10]Liu, H., M. J. Fogarty, S. M. Glaser, I. Altman, C. Hsieh, L. Kaufman, A. A. Rosenberg, and G. Sugihara. 2012. Nonlinear dynamic features and co-predictability of the Georges Bank fish community. Marine Ecology Progress Series 464:195-207.