"Variable X is said to Granger cause Y if the predictability of Y (in some idealized model) declines when X is removed from the universe of all possible causative variables ..."

A dynamical system is one that evolves with time either stochastically or deterministically. The evolution is described by a set of differential equations. The instantaeneous state of a dynamical system is represented as a point in some phase space. As the system evolves the point moves describing a trajectory in the phase space. In 1981, Floris Takens established that (unde certain conditions) the shape and structure of the trajectory could be deduced from the time series of any one of its co-ordinates.

Movies from sugihara-2012

b)

a) Lagged vectors

i) Consider two time series of length L, {X} = {X(1), X(2), ..., X(L)} and {Y} = {Y(1), Y(2), ..., Y(L)}.

ii) Construct the lagged-coordinate vectors x(t) = {X(t), X(t-τ), X(t-2τ), ..., X(t-(E-1)τ)} for t = 1+(E-1)τ to t = L. This set of vectors is the "reconstructed manifold" or "shadow manifold" M_{X}

iii) To generate a cross-mapped estimate of Y(t), denoted by Y_{pred}(t) | M_{X} , we begin by locating the contemporaneous lagged-coordinate vector on M_{X} ,x(t), and find its E+1 nearest neighbors. Note that E+1 is the minimum number of points needed for a bounding simplex in an E-dimensional space.

iv) Call the time indices (from closest to farthest) of
the E+1 nearest neighbors of x(t) by t_{1} , ...t_{E+1} . These time indices corresponding to nearest
neighbors to x(t) on M_{X} are used to identify points (neighbors) in Y (a putative neighborhood) to estimate Y(t) from a locally weighted mean of the E+1 Y(t_{i}) values.

Y_{pred}(t) | M_{X} = Σ w_{i} Y(t_{i} ) and i runs from 1 to E+1

where w_{i} = u_{i}/ Σ u_{j} and j runs from 1 to E+1

and u_{i} = exp{-d[x(t),x(t_{i})] / d[x(t),x(t_{1})]}

Panel (A) shows cross mapping for the generic bidirectional case, where the reconstructed manifold M_{X} is diffeomorphic to the original manifold and M_{Y} , because the variable X contains information about the dynamics of Y. When making predictions from the predictor X(t) at time t(represented as a black circle), the nearest points in M_{X} (orange triangles) are mapped to M_{Y}. On M_{Y}, the centroid of these points is the target prediction (green circle) which gives values for Y_{pred}(t)|M_{X}. The range of values for the gray circles corresponds to the uncertainty in prediction. Because these points remain close in the manifold M_{Y} (M_{X} is diffeomorphic), the uncertainty is low for predicting Y_{pred}(t)| M_{X} , as well as X_{pred}(t) | M_{Y} .

In panel (B), predictions are done as in panel (A), but for the special non-generic case where X is insensitive to the state of Y (i.e. Y does not have any influence on X). Since M_{X} is not diffeomorphic to M in this situation, the nearest neighbors on M_{X} (orange triangles) do not map accurately to nearest neighbors on M_{Y} and are far apart. More specifically, on M_{Y}, the corresponding time-indexed neighbors fail to specify the state (are randomly spread apart on M_{Y}) because X contains incomplete information about the dynamics of Y. Thus, estimating the state Y_{pred}(t) | M_{X} has high uncertainty.

a) Sugihara 2012, fig 5

We see strong evidence that B. Eriopoda and Aristida do belong to a dynamical system, and that the CCM method reveals the mutual causality between the time series for these two species(fig 2a,b). The prediction from the Sugihara techniq ue using either the percentage or mean frequency time series improves with the length of the series. The prediction for BOER is consistently better than that for ARSP, indicating that BOER has more influence on ARSP than vice versa.

However, the other pairs of time series within the upland and lowland species exhibit little improvement in CCM prediction as the length of the series increases, making it unlikely that they do belong to the same dynamical system. Other than BOER and ARSP, no other pairs of upland or lowland species seem to belong to a dynamical system.

a) Liang(2005-2014)

In a series of papers, (Liang 2005-2014) formulated a measure of the rate of information flow between two time series, formulated as a discussion in terms of marginal entropy.

In the figure below, Liang displays information flow between the Indian Ocean Dipole index and Pacific tropical Sea Surface Temperature

Applying the Liang technique to the New Mexico plant data we find shows significant flows from the time series for the percentage of upland species B. eriopoda and Aristida spp. Specifically the flow from B. eriopoda (BOER) to Aristidia spp. (ARSP) is 0.14 nat/yr and the reverse flow is approximately the same (0.13 nat/yr.) The time series for mean frequency exhibit information flow of 0.24 nat/yr from ARSP to BOER but not clearly in the other direction.

The lowland species P. mutica (PLMU) and S. brevifolius (SCBR) also exhibit flows of 0.19 nat/yr from PLMU to SCBR and 0.13 nat/yr from SCBR to PLMU. Other combinations show no information flow between them by the Liang measure.

F. Takens, in Dynamical Systems and Turbulence, D. A. Rand, L. S. Young, Eds. (Springer-Verlag, New York, 1981), pp. 366-381.

"Detecting Causality in Complex Ecosystems," George Sugihara et al. Science 338 , 496 (2012), DOI: 10.1126/science.1227079

"Causal feedbacks in climate change," van Nes et al.,Nature Climate Change, 2015, DOI: 10.1038/NCLIMATE2568

"Local predictability and information flow in complex dynamical systems," X. San Liang, Physica D 2013, DOI: 10.1016/j.physd.2012.12.011

"Unraveling the cause-effect relation between time series," X. San Liang, Phys. Rev. E 90, 052150 (2014), DOI: 10.1103/PhysRevE.90.052150