Directed dynamical influence is more detectable with noise

Successful identification of directed dynamical influence in complex systems is relevant to significant problems of current interest. Traditional methods based on Granger causality and transfer entropy have issues such as difficulty with nonlinearity and large data requirement. Recently a framework based on nonlinear dynamical analysis was proposed to overcome these difficulties. We find, surprisingly, that noise can counterintuitively enhance the detectability of directed dynamical influence. In fact, intentionally injecting a proper amount of asymmetric noise into the available time series has the unexpected benefit of dramatically increasing confidence in ascertaining the directed dynamical influence in the underlying system. This result is established based on both real data and model time series from nonlinear ecosystems. We develop a physical understanding of the beneficial role of noise in enhancing detection of directed dynamical influence.

The experimental data set studied in the main text came from records of the Didinium-Paramecium protozoan prey-predator system, which is the same data set studied in Ref. [1]. The detailed experimental conditions and related historical studies [2,3,4] are described in the Supplemental Materials of Ref. [1]. The data can be found in Ref. [5] and downloaded from the web site: http://robjhyndman.com/ tsdldata/data/veilleux.dat.
The length of the data set is L = 71, and we remove the first 10 data points to eliminate transient behavior. For this top-down control system, the variables for the predator (Didinium) and the prey (Paramecium) are denoted as x and y, respectively. The stronger driving variable is x. The time series is normalized to having unit mean and variance.

Supplementary Note 2: Effect of measurement noise on detecting directed dynamical influence -additional examples
To better understand the behavior of the measure R in the CCM framework, we show in Figs. S1 and S2 R and the corresponding correlation coefficients, ρ X|M Y and ρ Y |M X , versus the noise amplitude σ. Note that, Fig. 2(a) in the main text shows R versus σ but under a different parameter setting (r x = 3.5 and r y = 3.8). Here, the results are for r x = r y = 3.8 in Fig. S1, where panel (a) plots R versus σ and panel (b) shows how the corresponding correlation coefficients decay with σ. Similar to the phenomenon described in the main text, for a given value of σ, a smaller value of β x,y as compared with that of β y,x results in a larger value of R, due to the faster decay of the correlation coefficientρ Y |M X . From Fig. 3 in the main text, representative structures of the attractor manifold of the system for r x = r y = 3.8, we see that the faster decay in ρ Y |M X can be attributed to the narrower structure of M X , as it can be relatively readily disturbed by noise. For the symmetric case of β x,y = β y,x , the value of R fluctuates about zero as σ is increased.
Figures S2(a) and S2(b), respectively, plot R and the corresponding correlation coefficients ρ X|M Y and ρ Y |M X versus σ for different values of the asymmetric noise ratio η. The parameters are the same as Fig. 2(b) in the main text (r x = 3.8, r y = 3.5, β x,y = 0.05, and β y,x = 0.1). Since β y,x > β x,y , x 0 affects y 0 more than the other way around, and a successful detection of directed dynamical influence would yield positive values for R. As shown in Fig. S2(a) [or Fig. 2(b) in the main text], in absence of noise (σ = 0), the value of R is about 0.0778, reflecting the directed dynamical relationship correctly. As measurement noise is introduced, the value of R increases with σ first, reaches a maximum, and then decreases, and this type of non-monotonic behavior occurs for η > η c = β x,y /β y,x . As shown in Fig. S2(b), both ρ X|M Y and ρ Y |M X decay with σ, where ρ X|M Y is approximately independent of the value of η but ρ Y |M X depends sensitively on η. As η is increased, the decay rate of the correlation coefficient ρ Y |M X in the small σ region increases. Additionally, for a given value of σ, a larger value of η corresponds to larger effective noise on x(t), resulting in a larger value for the ratio δy/δx. As a result, ρ Y |M X decays more rapidly, as shown in Fig. S2(b). In fact, for large values of η (beyond η c ), both the non-monotonic behavior of R and the overall increase of R with η observed in panel (a) can be attributed to the faster decay of the correlation coefficient ρ Y |M X . Figure S3 shows the reconstructed attractor manifolds of x and y for the two cases where there is no intentional noise (a,b) and there is noise of amplitude σ = 0.005 (c,d). The parameters of the system are set to be r x = 3.8 and r y = 3.5 so that, if the x and y variables are decoupled, there is a chaotic attractor for x and a two-piece attractor for y. corresponding to X(t ) will be used to estimate the position of the point Y (t) (indicated by a black thick dashed arrow). We see that Y (t ) is in fact far away from the black cluster of M Y where the actual point Y (t) is located. As the number of incorrectly estimated neighboring points is increased due to noise-induced merging of the parallel clusters in M X , the accuracy of the estimated Y (t) will decay dramatically. (The two correct neighboring points of X(t) and the two corresponding cross mapping points in M X are indicated by thin black arrows.) Supplementary Figure S3 | For the model predator-prey system treated in the main text, reconstructed phase space forŶ (t)|M X : attractor manifolds of x and y for σ = 0 (a,b) and σ = 0.005 (c,d). The two separated clusters of points in the phase space in panels (b,d) are labeled with red (gray) and black colors, respectively, and the corresponding points (i.e, at the same time instant t) in (a,c) are similarly colored in (b,d). Other parameters are r x = 3.8 and r y = 3.5.
Supplementary Figure S4 | For the model predator-prey system subject to dynamical noise of amplitude σ and asymmetry ratio η, measure R in the parameter plane (σ, η). We fix β y,x = 0.1 and choose a number of values of β x,y : (a) 0.01, (b) 0.02, (c) 0.05, and (d) 0.07. Other parameters are r x = 3.8, r y = 3.5, N = 1000, E x = E y = 2, and τ = 1. The shaded region in each panel indicates divergence of system dynamics due to large noise.