Momentary information transfer as a coupling measure of time series

We propose a method to analyze couplings between two simultaneously measured time series. Our approach is based on conditional mutual sorting information. It is related to other concepts for detecting coupling directions: the old idea of Marko for directed information and the more recent concept of Schreiber’s transfer entropy. By setting suitable conditions we first of all consider momentary information in both time series. This enables the detection not only of coupling directions but also delays. Sorting information refers to ordinal properties of time series, which makes the analysis robust with respect to strictly monotonous distortions and thus very useful in the analysis of proxy data in climatology. Fortunately, ordinal analysis is easy and fast to compute. We consider also the problem of reliable estimation from finite time series.

Figure 1

Setups of three different approaches for coupling analysis. Each box symbolizes the permutation (order pattern) $\pi$ of the corresponding random vectors. (a) Symbolic transfer entropy (STE) according to Ref. [8]. (b) Transfer entropy on rank vectors (TERV) according to Ref. [9]. (c) Momentary sorting information transfer (MSIT, our approach). All three approaches represent special cases of conditional mutual sorting information (CMSI), Eq. (8).

Figure 2

Estimates of mutual sorting information (MSI), Eq. (7), and several conditional MSI (CMSI), Eq. (8), according to the estimator of Eq. (23) for the bidirectionally coupled logistic map (25). The coupling strengths amount to $c_{y\to x}=0.2$ and $c_{x\to y}=0.5$, the coupling delays to ${\tau }_{y\to x}=5$ and ${\tau }_{x\to y}=2$. The sample size is $T=512$. Error bars denote standard deviations of 1000 trials. The gray band denotes the $95\%$ significance level $I^{\mathrm{CMSI}*}$, Eq. (24). Panel (a) shows the cross correlation function $\rho$ in the top plot as well as MSI and CMSI functions for different time combs: (i) MSI for ${\vartheta }_x={\vartheta }_y=\{-1,0\}$, ${\vartheta }_{z_x}={\vartheta }_{z_y}=ø$, (ii) MSI for ${\vartheta }_x={\vartheta }_y=\{-2,-1,0\}$, ${\vartheta }_{z_x}={\vartheta }_{z_y}=ø$, and (iii) CMSI for ${\vartheta }_x={\vartheta }_y=\{-2,-1,0\}$, ${\vartheta }_{z_x}={\vartheta }_{z_y}=\{-2,-1\}$. Note that for these time combs CMSI is equivalent to MSIT [cf. Eqs. (13) and (14)]. (iv) CMSI for ${\vartheta }_x={\vartheta }_y=\{-2,-1,0,1\}$, ${\vartheta }_{z_x}={\vartheta }_{z_y}=\{-2,-1\}$, (v) CMSI for ${\vartheta }_x={\vartheta }_y=\{-2,-1,0\}$ for all lags $\tau$, but ${\vartheta }_{z_x}=ø$, ${\vartheta }_{z_y}=\{-2,-1\}$ for positive $\tau >0$, and ${\vartheta }_{z_x}=\{-2,-1\}$, ${\vartheta }_{z_y}=ø$ for $\tau <0$. The ticks at the ordinate mark the significance threshold $I^{\mathrm{CMSI}*}$, Eq. (24), and the maximum value $I_{max}^{\mathrm{CMSI}}={max}_{\tau }{I}^{\mathrm{CMSI}}$. The figures illustrate that best results for coupling analysis are obtained by the MSIT approach of (a) (iii). (b) shows the maximum MSIT value for positive delays, $I_{max,\tau >0}^{\mathrm{MSIT}}={max}_{\tau >0}{I}^{\mathrm{MSIT}}$ (solid line), and negative delays, $I_{max,\tau <0}^{\mathrm{MSIT}}={max}_{\tau <0}{I}^{\mathrm{MSIT}}$ (dashed line). The coupling strength $c_{y\to x}=0.2$ is constant, and $c_{x\to y}$ is varied. We use here the time combs of (a) (iii), hence CMSI is indeed MSIT, Eqs.(13) and (14). The $95\%$ significance level is separate for each coupling.

Figure 3

Results for the Lorenz system, Eq. (27), for dynamical noise level ${\sigma }_{\mathrm{dyn}}=0$. (a) The signals $x$ and $y$ represent $Z$ components of two unidirectionally delay-coupled Lorenz systems as in Eq. (27), with $T=1500$ samples (only 500 shown), using sampling period $\Delta t=0.02$. (b) MSIT function, gray encoded, for varying time combs ${\vartheta }_x={\vartheta }_y=\{-2d,-d,0\}$, ${\vartheta }_{z_x}={\vartheta }_{z_y}=\{-2d,-d\}$ for different embedding delays $d=1,2,\dots ,30$. Lower plot shows the cumulated MSIT over all time scales (solid), and mutual information (MI, dotted). (c) MSIT for time scales $2d=20$ (top) and $2d=38$ (bottom) where the gray band denotes the $95\%$ significance level ${\mathrm{MSIT}}^{*}=I^{\mathrm{CMSI}*}$, Eq. (24).

Figure 4

Results as in Fig. 3, but with dynamical noise ${\sigma }_{\mathrm{dyn}}=0.3$.

Figure 5

Results as in Fig. 3, but with observational noise. Remember that the $\mathrm{MSIT}$ ($\mathrm{CMSI}$) estimator (23) can be negative as we take Grassbergers entropy estimator.

Figure 6

(a) Path of the North Atlantic Current (NAC) with the locations marked as $\mathbf{x}$ and $\mathbf{y}$ (from Ref. [32]). (b) Anomalized time series (1680 months). (c) MSIT time scales, significant estimates according to the $99.9\%$ significance level (calculated from 10 000 samples) are above the contours. MSIT is shown for varying time combs ${\vartheta }_x={\vartheta }_y=\{-2d,-d,0\}$, ${\vartheta }_{z_x}={\vartheta }_{z_y}=\{-2d,-d\}$ for $d=1,\dots ,60$ covering time scales up to ten years. Lower plot shows cumulated MSIT over all time scales (solid) and MI (dotted, estimated using adaptive binning with eight quantiles). Our MSIT coupling analysis detects two significant coupling delays: The first is about 50 months, it could be related to the surface water current from $\mathbf{x}$ and $\mathbf{y}$ with mean velocity of about $5\hspace{0.5em}\mathrm{cm}/\mathrm{s}$. The second is less than one month, it could be related to the Atlantic surface winds from $\mathbf{x}$ and $\mathbf{y}$ with mean velocity in winter of about $5\hspace{0.5em}\mathrm{m}/\mathrm{s}$.