Mirage correlations result from a fundamental property of nonlinear dynamical systems known as state dependency (Sugihara et al. Such a “mirage correlation” is a hallmark of nonlinear dynamical systems (Sugihara et al. If one uses correlation to infer causality, one may erroneously conclude that the two species have no causal interaction.
1d): positive correlation for a period of time, negative correlation for another period of time, and then no correlation in yet another period of time.
The two species exhibit a mirage correlation (Fig. This can be demonstrated using a two-species competition model (Fig. These methods and applications can be used to provide a mechanistic understanding of dynamical systems.Įven more counter-intuitively, a lack of correlation does not imply lack of causation. state dependence), (3) determining causal variables, (4) forecasting, (5) tracking the strength and sign of interaction, and (6) exploring the scenario of external perturbation. Using model examples, we aim to guide users through several basic applications of EDM, including (1) determining the complexity (dimensionality) of a system, (2) distinguishing nonlinear dynamical systems from linear stochastic systems, and quantifying the nonlinearity (i.e.
HOW TO DECREASE LAG WITH X MIRAGE SOFTWARE
Here, we provide a step-by-step tutorial for EDM applications with rEDM, a free software package written in the R language. EDM bears a variety of utilities to investigating dynamical systems.
HOW TO DECREASE LAG WITH X MIRAGE SERIES
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). lagged coordinate embedding of time series data. These nonlinear statistical methods are rooted in state space reconstruction, i.e. In recent decades, nonlinear methods that acknowledge state dependence have been developed. State dependency means that the relationships among interacting variables change with different states of the system. “Mirage correlation” (i.e., the sign and magnitude of the correlation change with time) is a hallmark of nonlinear systems that results from state dependency. Thus, they are ill-posed for dynamical systems, where correlation can occur without causation, and causation may also occur in the absence of correlation.
Linear approaches are fundamentally based on correlation. nonlinear), making them difficult to understand using linear statistical approaches. Natural systems are often complex and dynamic (i.e.
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