We first consider a two-dimensional (2D) stochastic linear model (Eq. 12 in ): 1dx1=(-x1+0.5x2)dt+0.1dw1,dx2=-x2dt+0.1dw2, where x1 and x2 are the two variables, t is time, and w1 and w2 represent standard Wiener processes in x1 and x2, respectively (wk,t+Δt-wk,t∼ΔtN(0,1), with N(0,1) being a normal distribution with zero mean and unit variance). In this simple system, x2 drives x1 but not vice versa (Fig. f).

We solve this system with the Euler–Maruyama method using a time step Δt=0.001 and 1000 unit times, which brings 106 time steps. We initialize the system with x1(0)=1 and x2(0)=2. For our analysis, we discard the first 10 unit times (first 104 time steps), which is considered to be our spin-up period.

Then, we investigate a six-dimensional (6D) stochastic linear vector autoregressive (VAR) model with only one lag (Eq. 21 in ): 2x1,t+1=0.1-0.6x3,t+u1,t+1,x2,t+1=0.7-0.5x1,t+0.8x6,t+u2,t+1,x3,t+1=0.5+0.7x2,t+u3,t+1,x4,t+1=0.2+0.7x4,t+0.4x5,t+u4,t+1,x5,t+1=0.8+0.2x4,t+0.7x6,t+u5,t+1,x6,t+1=0.3-0.5x6,t+u6,t+1, where xk (k=1,…,6) represents the six variables, and uk represents normal random noises in these six variables (uk∼N(0,1)). By construction, we have two directed cycles, i.e., x1→x2→x3→x1 and x4→x5→x4, and these cycles are driven by a common cause, i.e., x6, which drives both x2 and x5 (Fig. d).

We solve this system using 106 time steps (Δt=1). For our analysis, we discard the first 104 time steps.

The next model is a nine-dimensional (9D) stochastic nonlinear VAR system with a maximum of four lags (Eq. 17 in ): 3x1,t=3.4x1,t-1(1-x1,t-12)e-x1,t-12+2.5x2,t-4+1.8x3,t-2+1.5x4,t-2+0.4u1,t,x2,t=3.4x2,t-1(1-x2,t-12)e-x2,t-12+0.4u2,t,x3,t=3.4x3,t-1(1-x3,t-12)e-x3,t-12+0.25x1,t-1+0.4u3,t,x4,t=3.4x4,t-1(1-x4,t-12)e-x4,t-12+1.5x5,t-3+1.2x6,t-1+0.4u4,t,x5,t=3.4x5,t-1(1-x5,t-12)e-x5,t-12+0.4u5,t,x6,t=3.4x6,t-1(1-x6,t-12)e-x6,t-12+1.5x7,t-3+0.4u6,t,x7,t=3.4x7,t-1(1-x7,t-12)e-x7,t-12+0.4u7,t,x8,t=3.4x8,t-1(1-x8,t-12)e-x8,t-12+0.8x7,t-1+0.4u8,t,x9,t=3.4x9,t-1(1-x9,t-12)e-x9,t-12+1.8x7,t-1+0.4u9,t, where xk (k=1,…,9) represents the nine variables, e is the exponential function, and uk represents normal random noises in these nine variables (uk∼N(0,1)). This system contains a directed chain x7→x6→x4→x1→x3 and a fork, i.e., x7 driving x6, x8, and x9. There are also two colliders, with x5 and x6 both affecting x4 on the one hand, and x2, x3, and x4 driving x1 on the other hand (Fig. d). A particularity of this system compared to the 6D model (Eq. ) is the presence of lags larger than one.

We solve this system using 106 time steps (Δt=1). For our analysis, we discard the first 104 time steps.

We also use the three-dimensional (3D) model, which is deterministic, nonlinear, and non-periodic; it is a simplified model representing atmospheric convection: 4dxdt=10(y-x),dydt=28x-y-xz,dzdt=xy-83z, where x, y, and z are the three variables and are proportional to the convection intensity, the horizontal temperature variation and the vertical temperature variation, respectively. We use the standard parameters of the model.

We solve the model using the fourth-order Runge–Kutta scheme, a time step Δt=0.01, and 1000 unit times, which brings 105 time steps. We initialize the system with x(0)=0, y(0)=1, and z(0)=0. For our analysis, we discard the first 100 unit times (first 104 time steps; the spin-up period).

Finally, we use eight different regional climate indices affecting the Atlantic and Pacific regions of especially the Northern Hemisphere, following a similar approach as and . Four of these indices are based on atmospheric variables and four of them are based on oceanic ones. Time series of these indices were retrieved from the Physical Sciences Laboratory (PSL) of the National Oceanic and Atmospheric Administration (NOAA; https://psl.noaa.gov/data/climateindices/list/, last access: 20 January 2023). We use monthly values from January 1950 to December 2021 (864 months), and we remove the linear trend in order to get approximately stationary time series, which is a requirement for applying our causal methods.

The four atmospheric indices are computed from the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis:

The Pacific–North American (PNA) index is obtained by projecting the daily 500 hPa geopotential height anomalies over the Northern Hemisphere (0–90° N) onto the PNA loading pattern (second leading mode of rotated empirical orthogonal function (EOF) analysis of monthly mean 500 hPa height anomalies during the 1950–2000 period). A positive PNA features above-average heights in the vicinity of Hawaii and over the intermountain region of North America and below-average heights south of the Aleutian Islands and over the southeastern United States. A negative PNA reflects an opposite pattern of height anomalies over these regions.

Below are the four indices based on ocean conditions:

The Atlantic Multidecadal Oscillation (AMO) index is computed based on version 2 of the extended SST gridded dataset (which uses UK Met Office SST data) averaged over the North Atlantic (0–70° N; unsmoothed time series) and following the procedure described in . Cool and warm phases of the AMO may alternate every 20–40 years.

The LKIF method has been developed by . It has been first applied in bivariate cases and has subsequently been extended to multivariate cases . In our study, we use the multivariate formulation of LKIF. In this framework, causal inference is based on information flow, which has been recognized as a real physical notion, i.e., formulated from first principles of information theory .

Under the assumption of a linear model with additive noise, the maximum likelihood estimate of the information flow reads as follows : 5Tj→i=1detC⋅∑k=1dΔjkCk,di⋅CijCii, where Tj→i is the absolute rate of information transfer from variable xj to variable xi, C is the covariance matrix, d is the number of variables, Δjk represents the cofactors of C (Δjk=(-1)j+kMjk, where Mjk represents the minors), Ck,di is the sample covariance between all xk and the Euler forward difference approximation of dxi/dt, Cij is the sample covariance between xi and xj, and Cii is the sample variance of xi. Note that a nonlinear version of LKIF has recently been developed but will not be used in this study .

To assess the importance of the different cause–effect relationships, we compute the relative rate of information transfer τj→i from variable xj to variable xi following the normalization procedure of : 6τj→i=Tj→iZi, where Zi is the normalizer, computed as follows: 7Zi=∑k=1dTk→i+dHinoisedt, where the first term on the right-hand side represents the information flowing from all the xk to xi (including the influence of xi on itself), and the last term is the effect of noise (taking stochastic effects into account), computed following .

In the following, we will only use the relative rate of information transfer τ (expressed in %). When τj→i is significantly different from 0, xj has an influence on xi; when τj→i = 0, there is no influence. The absolute value of τ indicates the strength of the causal influence. A positive (negative) value is indicative of an increase (decrease) in variability of the target variable xi due to the causal influence of the source xj. However, we will mainly use the absolute value of τ in this study and will only briefly discuss the sign in the case of the model (Fig. ). Statistical significance of τj→i is computed via bootstrap resampling with replacement of all terms included in Eqs. ()–() and using a significance level α=5%. The number of bootstrap realizations varies depending on the case study: 100 for the 2D and models, 300 for the 6D and 9D models, and 1000 for the real-world case study. This number is chosen sufficiently large to achieve convergence of results. The relative rate of information transfer τ is computed for each bootstrap realization, and the error in τ, which we refer to as ϵτ, is calculated as the standard deviation across all τ bootstrapped values. If the confidence interval τ±1.96ϵτ does not contain the zero value, then τ is significant at the 5 % level; otherwise, it is not significant.

The PCMCI method is a causal discovery method based on the Peter and Clark (PC) algorithm , combined with the momentary conditional independence (MCI) approach . Given a set of univariate time series (called “actors”), PCMCI estimates their causal graph representing the conditional dependencies among the time-lagged actors. In its linear application, PCMCI uses partial correlations to iteratively test conditional dependencies in a set of actors, distinguishing between true causal links and spurious links arising from autocorrelation effects, indirect links, or common drivers.

Note that the term “causal” rests upon a set of assumptions, which are described in . In general, the causal graph should represent a stationary (stable in time) set of causal links, in which causality is determined with a lag l of at least one time step, and it is only true among the specific set of analyzed actors. The PCMCI algorithm is composed of two steps: the PC step and the MCI step. Each step is briefly described in this section.

In the first step, or PC step, for each actor in the (example) set of actors P, the algorithm identifies the initial set of parents P0 based only on the simple correlation between each actor and all other actors up to a maximum lag lmax. Let us assume that with lmax=3, P={A,B,C,D,E} and PA0={Al=-1,Bl=-1,Dl=-2,Cl=-2,El=-1}, where actors A to E in the set of parents of A, PA0, are ordered based on the absolute value of their correlation coefficient with Al=0. Then, in the first iteration of the algorithm, the partial correlation ρ between Al=0 and each actor in PA0 is calculated by conditioning on an additional actor taken from PA0. For example, ρ (Al=0, Al=-1|Bl=-1) = ρ (Res(Al=0), Res(Al=-1)), where Res(Al=0) and Res(Al=-1) are the residuals of Al=0 and Al=-1 after removing the linear influence of Bl=-1. The partial correlation is computed for each actor in PA0 by conditioning (only once) on the strongest available actor. This process is called “iterative conditioning”. At the end of this first iteration, the set of parents of A is updated. Let us assume that in our example PA1={Al=-1,Bl=-1,Cl=-2,El=-1}, then in the second iteration the set of parents PA2 will be identified by conditioning on the first two strongest actors, e.g., ρ (Al=0, Al=-1|Bl=-1, Cl=-2). The PC step ends when the number of actors on which to condition equals the numbers of actors contained in PAn. Then, the same computation is repeated for each actor contained in P, until each actor has its own set of parents Pn.

In the second step, or MCI step, the partial correlation between each possible pair of actors is calculated a second time by regressing once on the combined set of parents. If we assume that PA4={Al=-1,Cl=-2,El=-1} and PB3={Bl=-1,Al=-2,Dl=-1}, then a causal link between Al=0 and Bl=-1 is detected if their partial correlation conditioned on their joint set of parents is significant for a certain threshold α. In this example, ρ (Al=0, Bl=-1|Al=-1, Cl=-2, El=-1, Bl=-2, Al=-3, Dl=-2) is given (note that the lag of PB3 is increased accordingly). At the end of the MCI step, each actor will have its own set of causal parents, and the causal effect of each link can be computed.

The strength of a causal link from variable xj at time t-l to variable xi at time t, noted xj,t-l→xi,t, is expressed in terms of the path coefficient β, which measures the change in the expectation of xi,t following an increase of xj,t-l by 1 standard deviation, keeping all other parents of xi,t constant. The linear coefficients β are calculated as follows: 8xi,t=∑k=1Nβkxj,k+ηxi, where xj,k∈P{xi} (k=1,...,N) is the set of parents of xi,t (N is the number of parents), and ηxi is the residual of xi,t. Note that in order to allow for a meaningful comparison with correlation and LKIF based on a linear model, we use here the PCMCI algorithm along with a linear similarity measure (partial correlation). In principle, PCMCI could also be combined with other statistical association measures that allow for conditioning on the effects of any third variable (like CMI), the study of which is however beyond the scope of the present work. The β coefficients are only calculated for causal links that are significant at the 5 % level, where each p value obtained from the MCI step is corrected using the Benjamini–Hochberg false discovery rate correction method .

Before investigating results from the two causal methods, it is important to highlight the main differences between the two methods, which are summarized in Table . LKIF is directly derived from the propagation of information entropy and quantifies the rate of information transfer from one variable to the other . PCMCI, on the other hand, is a causal network algorithm starting with a fully connected graph from which non-causal links are iteratively removed based on conditioning sets of growing cardinality . The actual underlying PCMCI measure for directional statistical dependence is partial correlations, including the effect of possible causal parents. LKIF does not systematically test the latter but uses a different approach, in which the statistical dependence is measured via the information flowing from one variable to the other.

The metric used by LKIF is the rate of information transfer from variable xj to variable xi and can be expressed either in natural unit of information (nat) per unit time (for T; Eq. ) or in percent (for τ; Eq. ). For PCMCI, the path coefficient β (Eq. ) measures the expected change in xi at time t (in units of standard deviation) if xj is perturbed at time t-l by 1 standard deviation. While time lags must be incorporated with PCMCI, LKIF has not been designed to work with such lags by default, although they can be used in principle . To this end, we can shift in time the time series of the leading variable and recompute LKIF based on the lagged time series.

While for both methods the strength of the metric, in absolute value, indicates how strongly two variables are causally linked (i.e., the larger |τ| and |β|, the larger the causal link), the sign has a different meaning. For LKIF, a positive (negative) value of τj→i means that the variability of the source xj increases (decreases) the variability of the target xi. For PCMCI, the sign of βj→i is closely linked to the correlation between xj and xi (i.e., a positive (negative) value means that an increase in xj leads to an increase (a decrease) in xi in the subsequent time step).

Since correct causal links are known for the three first artificial models (2D, 6D, and 9D models), we can check the performance of the two causal methods, as well as the correlation coefficient, in identifying the ground truth. The diagnostics presented here are not computed for the model and the real-world case study, as no exact solution exists for these two cases. We compute true-positive, true-negative, false-positive, and false-negative rates. The true-positive rate is the percentage of causal links correctly detected by the method among the total number of ground truth causal links. The true-negative rate is the percentage of non-causal links correctly detected by the method among the total number of ground truth non-causal links. The false-positive rate represents the percentage of cases where the method incorrectly detects a causal link among the total number of ground truth non-causal links. The false-negative rate represents the percentage of cases where the method fails to find an existing causal link among the total number of ground truth causal links.

To summarize the results from the confusion matrix, we also compute the ϕ coefficient based on true positives (denoted TP), true negatives (denoted TN), false positives (denoted FP), and false negatives (denoted FN): 9ϕ=TP×TN-FP×FN(TP+FP)(TP+FN)(TN+FP)(TN+FN). The denominator is set to 1 if any of the four sums in the denominator is equal to 0, in which case ϕ=0. A value of 1 represents a perfect prediction of ground truth causal and non-causal links by the method, while a value of 0 means that the result is not better than a random prediction. These diagnostics are presented in Table  and discussed in Sect. .

For the 2D model, the numerical value of the correlation between x1 and x2 is significantly positive (R=0.23; Fig. a) and is similar to the analytical value (Fig. d), but it does not provide any indication on the direction of influence.

LKIF can accurately retrieve the correct causal link, i.e., from x2 to x1, as well as the absence of influence in the reverse direction (Fig. b), as was already demonstrated in . In addition, the numerical estimate of the rate of information transfer (|τ2→1|=5.72%; Fig. b) is very close to the analytical solution (|τ2→1|=5.56%; Fig. e), which provides confidence in the LKIF results found for this simple system.

PCMCI only captures the self-influences of x1 and x2 but is not able to capture any significant causal influence between x1 and x2 with the original time step (i.e., Δt=0.001) (Fig. c). This missed detection is partly due to the fact that PCMCI responds better for discrete maps with finite time steps. Indeed, the time step for discretization is too small, and if we recompute causal links with PCMCI taking every 100 time steps (Δt=0.1), we can recover the influence from x2 to x1, although the value of β is relatively small (Fig. c).

This example shows that LKIF performs well for such a very simple 2D system, while PCMCI struggles with the original time step. In particular, the serial dependency in this particular model might overcast the mutual dependency for a “typical” maximum lag considered by PCMCI, which has not been designed for such conditions.

For the 6D model, the correlations are significant for all 30 pairs of variables (excluding autocorrelations), despite relatively small values for many of them (Fig. a). This shows that a simple correlation analysis fails to only identify the seven causal links that should be identified in this system (Fig. d). The largest correlation of all pairs is between x2 and x5 (R=0.37), but no causal link should exist between the two variables (i.e., this is a false positive). This large correlation probably appears because x6 influences both x2 and x5 by construction (Fig. d) and is thus a confounding variable. Correlations larger than 0.3 in absolute value appear for the two pairs x6–x2 and x6–x5 (Fig. a), which confirms the role of x6 as a confounding variable, but these correlations do not indicate the direction of influences.

Both LKIF (Fig. b; no lag is used) and PCMCI (Fig. c; use of four time lags) can capture the seven correct causal links (Fig. d), i.e., the directed cycle x1→x2→x3→x1, the two-way causal link between x4 and x5, and the influence of x6 on both x2 and x5. Results from PCMCI in terms of self-influences are more accurate based on Eq. (), as it provides two significant self-influences, i.e., x4 and x6, while LKIF identifies all six self-influences as significant. The latter result indicates that the LKIF method may fail at representing the correct self-influences, while PCMCI does not.

This example shows the strength of causal methods, which can capture the correct causal influences, while the correlation is not able to provide such information and cannot identify confounding variables and the direction of causality.

For the 9D model, the correlation does a poor job at identifying correct causal influences (Fig. a). In particular, the largest correlation is between x8 and x9, which is not a correct causal link by construction (Eq. ). As for the 6D model, this is due to the fact that x7 should influence both variables (Fig. d). x7 is indeed significantly correlated to both x8 and x9, but the causal direction is not identified by the correlation analysis.

Using LKIF without any lag shows that the method can detect all correct links, except x5→x4, although only four causal influences have a rate of information transfer |τ| larger than 1 % (Fig. b). These four influences are the ones that should appear at lag -1 (Fig. d, i.e., x1→x3, x6→x4, x7→x8, and x7→x9. The method also wrongly identifies 13 causal influences, even if values of information transfer remain small.

The use of time lags up to l=3 time steps with LKIF (we use 9 variables × 4 lags =36 variables in total) allows us to improve results (Fig. b, where the maximum value of all lags is plotted). In particular, all nine correct causal links can now be identified with |τ|>3%, except the influence of x3 on x1, which is significant but has a much smaller value (|τ|=0.68%). Five additional causal influences are wrongly identified by the method with lags up to 3 time steps, but with relatively small values (|τ|<0.4%).

Using PCMCI with lags up to l=4 time steps also allows us to correctly reproduce all causal links, except that it wrongly identifies four additional causal influences but with very small values (Fig. c). All self-influences are also correctly identified by the two methods.

This example also demonstrates the power of causal methods compared to a correlation analysis when using an appropriate number of lags: all expected links are correctly identified. Although some wrong causal links are identified by both methods, the strength of the relationship remains small for these wrong influences.

The only large correlation (excluding autocorrelation) in this system is between x and y, with R=0.88 (Fig. a). The other correlations are very small (R=-0.01) but significant, probably due to the length of the time series.

According to LKIF, a two-way causal link appears between x and y (Fig. b). This causal link is also identified by PCMCI with lags up to l=3 time steps (Fig. c). PCMCI also identifies a significant two-way causal link between y and z, but the value is very close to 0.

Then, we investigate whether there is a lag dependence on the results. For the correlation and LKIF, we repeat the computation by shifting the three variables one by one with a lag from 0 to 1 unit time (100 time steps) with 0.1 unit time increment (i.e., every 10 time steps). For example, we take xt-l at lag l=0.1 unit time and keep yt and zt at lag 0, and we recompute the correlation and relative rate of information transfer. Then, we take xt-l at lag l=0.2 unit time, keeping yt and zt at lag 0 and so on until lag l=1 unit time. We do the same when y leads x and z and when z drives x and y. For PCMCI, all lags from 0 to 1 unit time with 0.01 unit time increment (i.e., every time step) are included in the same computation as the method is designed to work with multiple lags by default. Results are presented in Fig. .

The correlation coefficient between x and y decreases exponentially with increasing lag when x leads y (Fig. a), and it first increases from l=0 to l=0.1 unit time before decreasing exponentially when y leads x (Fig. b). No correlation appears between z and any of the two other variables at any lag (Fig. a–c).

The LKIF rates of information transfer from x to y and from y to x also decrease with increasing lag between 0 and 1 unit time, but starting with a plateau of |τ|∼50% (Fig. d–e). This plateau lasts from l=0 to l=0.2 unit time for τxt-l→yt (Fig. d) and from l=0 to l=0.4 unit time for τyt-l→xt (Fig. e). No information transfer exists between z and the two other variables at any lag (Fig. d–f), in agreement with the absence of correlation (Fig. a–c).

The PCMCI path coefficients between x and y also generally decrease (in the two directions) with increasing lag, although the decrease presents more variability than the correlation and LKIF, with β=0 at lag 0, the largest β value when l=0.1 unit time, and then an oscillatory behavior until l=1 unit time (Fig. g–h). As for the correlation and LKIF, no causal influence is found between z and the two other variables at any lag (Fig. g–i).

If we replace x by x2 to take nonlinearities into account and look at the triplet (x2, y, z), a strong positive correlation now appears between x2 and z (R=0.65; Fig. d). In addition, a strong two-way causal link now appears between x2 and z with both LKIF (|τ|∼50% in the two directions; Fig. e) and PCMCI (|β|=1.7 in the two directions; Fig. f). This shows that the linear versions of LKIF and PCMCI can detect causal links between nonlinear transformed variables in nonlinear models. In this case, the correlation between x and y, combined with the nonlinear forcing product xy in the third equation of the model (z equation; Eq. ), results in a linear correlation between z and the nonlinear non-invertible variable change x2.

The correlation between xt+l2 and zt oscillates between R∼-0.7 (l=0.2 unit time) and R∼0.9 (l=-0.1 unit time) with a period of ∼0.7 unit time (Fig. a). The rates of information transfer from xt+l2 to zt and from zt to xt+l2 also show an oscillatory behavior with a period of ∼0.35 unit time (Fig. b), i.e., half of the correlation oscillation. PCMCI does not exhibit such an oscillatory behavior but rather a quickly decreasing β value for small lags (Fig. c).

The real-world case study with climate indices shows that 54 % of the pairs of variables (excluding autocorrelations) are related by significant correlations when considering no lag (Fig. a). However, it is obvious that a large number of these pairs are correlated but not causally linked. The use of causal methods allows us to remove such spurious links, as demonstrated by the application of LKIF without any lag (Fig. b) and PCMCI with lags up to l=2 months (Fig. c).

Results from the two causal methods present several similarities, including the AO influence on both PDO and TNA (Fig. b–c). Another similarity is the two-way causal link between AMO and TNA, in agreement with and . The AMO–TNA influence is not surprising as both indices are computed from SST anomalies in the North Atlantic, with AMO spanning the majority of the North Atlantic and TNA focusing on the tropical region. Values of the AMO–TNA influence in the two directions are relatively strong for LKIF (|τAMO→TNA|=22% and |τTNA→AMO|=38%) compared to other pairs of influence (Fig. b). In addition, ENSO influences PDO according to both methods (Fig. b–c), and the positive sign of the correlation means that a warm Niño3.4 phase results in a positive PDO (Fig. a). The ENSO influence on PDO was recently reported by , also using LKIF, and , based on the pseudo-transfer entropy. Spatial patterns of ENSO and PDO are very similar, and PDO is often being viewed as an ENSO-like interdecadal climate variability, with PDO occurring at decadal timescales, while ENSO is predominantly an interannual phenomenon .

In terms of differences between the two causal methods, LKIF identifies additional causal influences of AO on PNA, NAO, and AMO, while PCMCI does not identify these causal links (Fig. b–c). It is well known that there is a clear relationship between AO and NAO and that NAO is often referred to as the local manifestation of the AO . Also, according to LKIF, there are two-way causal influences between ENSO and PNA and between ENSO and TNA, which do not appear with PCMCI with lags up to l=2 months (Fig. b–c). It is well known that ENSO has a major influence on the extratropical Northern Hemisphere climate variability, in particular on PNA . However, the influence of ENSO on PNA is complicated by the fact that other mechanisms can affect this relationship, such as the position of the Pacific jet stream . Our results with LKIF suggest that PNA has a stronger influence on ENSO than the reverse, which would go in favor of more complex mechanisms in action. Finally, the influence of ENSO on TNA has also been reported in the literature, and different mechanisms have been proposed . It is interesting to find that the influence of TNA on ENSO is stronger than the reverse influence with LKIF (Fig. b).

The use of 12 time lags (0 to 11 months) with both methods (bringing 8 variables ×12 lags =96 variables in total for LKIF) provides additional insights (Figs. –). PNA influences ENSO with a 1-month lag with LKIF (Fig. a) and with a 4-month lag with PCMCI (Fig. a). Additionally, PNA influences PDO with a 4-month lag and AMO with a 11-month lag using LKIF (Fig. a). However, all PNA influences appear relatively weak in intensity (|τ|<1% with LKIF and |β|<0.1).

NAO influences PDO with both methods but with very different lags depending on the method, i.e., 11 months with LKIF (Fig. b) and 1 month with PCMCI (Fig. b). It also influences TNA with LKIF with a 1-month lag. As for PNA, all significant NAO influences remain limited in intensity (|τ|<1% with LKIF and |β|<0.1).

AO is by far the climate index that influences most variables with LKIF (Fig. c), in agreement with . When considering no lag, AO influences all other indices, except QBO and ENSO (Fig. c). The largest value of rate of information transfer is from AO to NAO with |τ|=4%, in agreement with the value considering no lag (Fig. b). AO also influences TNA and AMO at larger lags with LKIF (l=1, 2, and 4 months for TNA and l=2, 5, and 11 months for AMO). With PCMCI, AO only influences TNA at lags l=1 to 4 months, PDO at lag l=1 month, and QBO at lag l=4 months (Fig. c). It is intriguing to notice that no AO influence on NAO appears with PCMCI.

QBO does not have any influence on any other climate indices with any of the methods (Figs. d and d).

The AMO–TNA two-way causal influence already identified in Fig.  also appears in the lagged plots but with contrasting behaviors depending on the causal method. With LKIF, AMO only influences TNA at lag 0 (Fig. e) and TNA influences AMO at lags l=0 and 11 months (Fig. g). With PCMCI, the AMO influence on TNA increases with increasing lag from l=0 to 6 months, then decreases and stays relatively constant until l=11 months (Fig. e), and TNA influences AMO at lags l=2, 4–6, 8, and 10–11 months (Fig. g). TNA has additional influences with PCMCI at lags l≥2 months (on NAO, QBO, PDO, and ENSO; Fig. g) and with LKIF at lags l≥9 months (on PDO and ENSO). The TNA influences on PDO and ENSO, appearing for both causal methods, remain limited to large lags (Figs. g–g).

PDO has an influence on PNA with LKIF at lag l=6 months (Fig. f), which is consistent with using sensitivity experiments with a coupled model. According to PCMCI, PDO influences ENSO at lags l≥3 months (Fig. f).

Finally, ENSO influences PDO at lags l=0,2 and 6 months, and influences TNA at lags l=2 and 6 months with LKIF (Fig. h). With PCMCI, ENSO is the climate index that influences most variables (all but NAO), especially PDO from l=2 to 10 months, TNA from l=3 to 11 months, and AMO from l=4 to 11 months (Fig. h). The large role of ENSO was also reported using pseudo-transfer entropy using lags l=1 to 9 months .

For the simplest (2D) model used here, we show that LKIF can accurately reproduce the correct causal link, with relatively high accuracy compared to the analytical solution, while PCMCI fails to reproduce this link when using the original time step (Sect.  and Fig. ). PCMCI provides the correct influence for the 2D model when taking every 100 time steps (although the β value is small), which shows that PCMCI responds better for discrete maps with finite time steps. For the 6D model, both LKIF and PCMCI can detect the correct causal links (Sect.  and Fig. ). For the 9D nonlinear model, PCMCI allows us to retrieve the correct causal relationships, while some care with the number of lags is needed with LKIF to achieve appropriate results (Sect.  and Figs. –). This shows that LKIF performs better for simpler systems and presents a few more difficulties with more complex models with several lags. On the other hand, PCMCI does not work well in the presence of very strong autocorrelations but may be preferential over LKIF as the number of variables increases. Results from the model are more complicated to interpret as the system is highly nonlinear and chaotic. Both methods detect the same causal links (Sect.  and Fig. ), although some differences appear in the dependence of the causal influence on the time lag (Figs. –). Moreover, the combination of model nonlinearities and nonlinear variable changes can result in linear causal links detectable by LKIF and PCMCI (Fig. ).

The above results are not entirely comparable to findings from from a methodological perspective, as the latter used other causal methods and different coupled systems. However, a similarity is the fact that some methods (Granger causality and its extensions) better perform with the simplest models, while other methods (CCM and predictability improvement) are better suited for more complex systems . This goes in hand with LKIF being better with the specific time-continuous 2D model studied here, while PCMCI is well suited for the time-discrete 9D model of our analysis. Thus, the key finding from , that “it is important to choose the right method for a particular type of data”, is also valid for our study.

The main novelties compared to are that (1) we use two causal methods that have not been compared yet, (2) we compare our causal methods to the classical correlation coefficient, (3) we assess causality between nonlinear variable changes, and (4) we apply the two methods to a real-world case study. Regarding (1), no definite conclusion can be provided as to which method is the best: it depends on the system used. For certain very simple models, LKIF appears to be preferential over PCMCI, although PCMCI has not been designed for the particular 2D model used here (Sect. ). For a more complex model involving more variables and several lags, like the 9D model, PCMCI may be better suited. In any case, we recommend to use as many methods as possible for a specific problem to increase the robustness of results. Regarding (2), we show that both LKIF and PCMCI are superior to correlation, as they allow us to remove spurious links. Regarding (3), we show that the combination of model nonlinearities with nonlinear variable changes can result in linear causal links, detectable by both LKIF or PCMCI. Point (4) is discussed in Sect. .

Table  provides true-positive, true-negative, false-positive, and false-negative rates, as well as the ϕ coefficient, for the correlation and the two causal methods used in this study and for the three first artificial models (2D, 6D, and 9D models). For the 9D model, a distinction is made between the case where lags are not considered (PCMCI is not used in this case) and the case where lags are considered. Results show that the correlation has a large chance of detecting false positives (i.e., incorrect detection of causal influences) for all models; thus, the correlation largely overestimates causal links. LKIF and PCMCI allow us to substantially reduce false positives, with 0 % for the 2D and 6D models with both methods, 21 % for the 9D model without lag with LKIF, and <10% for the 9D model with lags with both methods. For the 2D model, LKIF perfectly reproduces the right causal links (ϕ=1), while the correlation coefficient and PCMCI (with the original time step) do not make better than a random prediction (ϕ=0). Only when using a larger sampling time step can PCMCI reproduce the correct causal links. For the 6D model, both LKIF and PCMCI accurately reproduce the ground truth (ϕ=1), while the correlation coefficient again does not make better than a random prediction and identifies all relationships as causal (ϕ=0 and false-positive rate =100%). For the 9D model without lag (PCMCI not included), the correlation does a better job at identifying a certain amount of true negatives (60 %) compared to the 2D and 6D models, but LKIF provides overall better results (ϕ=0.5 for LKIF vs. ϕ=0.4 for correlation), despite the identification of one false negative with LKIF (Fig. b). For the 9D model with lags, the performance of the two causal methods is clearly better than the correlation (ϕ=0.21), with PCMCI (ϕ=0.81) performing slightly better than LKIF (ϕ=0.77).

In our study, we extend previous analyses from and by using monthly time series of climate indices in the Atlantic and Pacific regions. We use the same seven climate indices as ; add QBO to the list to have four indices characterizing both the atmosphere and ocean; and do not use local air temperature, precipitation, or insolation. also computed LKIF based on these indices but focused on the dependence of the rate of information transfer on the timescale (using a time-moving window) and did not compare LKIF to another method. also used NAO, QBO, AMO, PDO, and ENSO (Niño3.4); they used a slightly different index for TNA, and they incorporated seven additional indices. The causal method used by is the pseudo-transfer entropy method .

Due to the small methodological differences in our analysis compared to (see above), some small differences appear, but key results with LKIF remain similar. In particular, we find that AO is the largest driver of all variables as it influences all other indices, except QBO and ENSO (Sect.  and Fig. b). We show that the AO influence mainly occurs at lag l=0 (Fig. c). This is in agreement with , who find that AO plays a key role at short timescale. PCMCI only identifies two AO influences with lags shorter than 2 months, i.e., to PDO and TNA (Fig. c). It is particularly intriguing to see that PCMCI does not detect the AO influence on NAO (Fig. c), while LKIF does (Fig. b), as NAO is often referred to as the local manifestation of AO . This discrepancy might hide seasonal differences, as for example winter and summer NAO have different spatial patterns .

ENSO has a relatively large influence on other climate indices, especially on PDO for both LKIF and PCMCI (Fig. b–c). The pivotal role of ENSO was already identified by and is not surprising due to its importance on the global climate . ENSO has a clear influence on PDO at lags 2 to 10 months for PCMCI (Fig. h), while it only appears at lags 0, 2 and 6 months for LKIF (Fig. h). This ENSO–PDO influence was detected from lags 1 to 7 with pseudo-transfer entropy , thus somewhere in between PCMCI and LKIF. The other clear ENSO influence according to PCMCI, LKIF and pseudo-transfer entropy is on TNA, at lags 2 and 6 months with LKIF (Fig. h), at lags 3–11 months with PCMCI (Fig. h), and at lags 1–9 months with pseudo-transfer entropy . According to PCMCI and pseudo-transfer entropy, ENSO also largely influences other climate indices than PDO and TNA at different lags, which is not the case for LKIF. More research would be needed to further investigate this difference between causal methods.