Explicit Granger causality in kernel Hilbert spaces

Granger causality (GC) is undoubtedly the most widely used method to infer cause-effect relations from observational time series. Several nonlinear alternatives to GC have been proposed based on kernel methods. We generalize kernel Granger causality by considering the variables' cross-relations explicitly in Hilbert spaces. The framework is shown to generalize the linear and kernel GC methods and comes with tighter bounds of performance based on Rademacher complexity. We successfully evaluate its performance in standard dynamical systems, as well as to identify the arrow of time in coupled Rössler systems, and it is exploited to disclose the El Niño–Southern Oscillation phenomenon footprints on soil moisture globally.

Figure 1

Representation of the GC model for the different kernel functions. Each model encodes explicit relations of past states with future ones. Note the difference between the regular map $\psi \left(\right[\mathbf{x},\mathbf{y}\left]\right)$ working with the concatenation of time series in the input domain and the joint map $\tilde{\psi }(\mathbf{x},\mathbf{y})$ working with the concatenation of maps of $\mathbf{x}$ and $\mathbf{y}$ in Hilbert spaces.

Figure 2

Significance of the positive and negative cases detected for the coupled AR system. Histogram of the difference between the estimated causality index $\delta$ and the associated threshold ${\delta }_{\mathrm{threshold}}$ are shown for each method and direction.

Figure 3

Causality index $\delta$ estimated for each method (solid lines) and their associated thresholds (dashed lines) for the coupled logistic maps as a function of the coupling parameter $\alpha$.

Figure 4

Arrow of time and the causal direction for a coupled Rössler system. Causal strength estimated for each method (solid lines) and associated thresholds (dashed lines) for forward and backward propagation directions versus the time delay $\tau$.

Figure 5

Top left: ENSO4 (red) and estimated SM interannual complex components (blue and black) studied. Right panel: GC analysis of ENSO and SM. Left plots show $\delta$ (red line) with (a) GC, (b) KGC, and (c) XKGC estimated with a 2-year moving window, its variability (shaded), and highlight the three phases in the 2016 ENSO event. CP, Central Pacific; EP, East Pacific; Mix, mixture of both cases). Right plots show the distribution (gray) and averages (red) of $\delta$ and thresholds (black) for lag $\tau$ and 100 runs. Bottom left: Spatial distribution of causal footprints of ENSO on SM obtained by XKGC.

Figure 6

Dependence of ${\delta }_{\mathrm{ENSO}4\to \mathrm{SM}}$ on the phase. Note that all methods preserve the noncausal phase dependence. The causality index-phase relation is cyclic with the $\pi$ period.

Figure 7

Global distribution of the ENSO-SM causality index ${\delta }_{\mathrm{ENSO}4\to \mathrm{SM}}$. A greater $\delta$ indicates that a higher percentage of SM interannual variability can be explained by ENSO.