Decomposing causality into its synergistic, unique, and redundant components | Nature Communications

Nature Communications volume 15, Article number: 9296 (2024) Cite this article

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Causality lies at the heart of scientific inquiry, serving as the fundamental basis for understanding interactions among variables in physical systems. Despite its central role, current methods for causal inference face significant challenges due to nonlinear dependencies, stochastic interactions, self-causation, collider effects, and influences from exogenous factors, among others. While existing methods can effectively address some of these challenges, no single approach has successfully integrated all these aspects. Here, we address these challenges with SURD: Synergistic-Unique-Redundant Decomposition of causality. SURD quantifies causality as the increments of redundant, unique, and synergistic information gained about future events from past observations. The formulation is non-intrusive and applicable to both computational and experimental investigations, even when samples are scarce. We benchmark SURD in scenarios that pose significant challenges for causal inference and demonstrate that it offers a more reliable quantification of causality compared to previous methods.

The quest for understanding causality is the cornerstone of scientific discovery1. It is through the exploration of cause-and-effect relationships that we are able to understand a given phenomenon and shape the course of events through deliberate actions2. This has accelerated the proliferation of methods for causal inference, as they hold the potential to drive progress across multiple scientific and engineering domains, such as climate research3, neuroscience4, economics5, epidemiology6, social sciences7, and fluid dynamics8,9, among others.

A central aspect of causality is the concept of physical influence10: manipulation of the cause manifests as changes in the effects1,11,12,13. For example, prolonged exposure to elevated air pollution levels has a causal connection to a higher incidence of chronic respiratory conditions14. The precise definition of causality remains elusive, yet it must be distinguished from the concepts of association and correlation. Association indicates a statistical relationship between two variables in which they have a tendency to co-occur more often than would be expected by random chance. Yet, association does not automatically imply causation15. Association may arise from shared causes, statistical coincidences, or the influence of confounding factors. An example of association can be observed in the increased rates of chronic respiratory diseases in regions undergoing significant deforestation. Although it may seem that deforestation directly contributes to respiratory health issues, this might primarily be due to the confounding factor of air pollution. Correlation, on the other hand, refers to a particular type of association that measures the monotonic strength and direction of variables15,16,17,18. Correlation implies association but not causation; causation implies association but not correlation15. Discerning between causality, association, and correlation poses a significant challenge in the development of methods for causal discovery. Here, we introduce an approach for causal inference that facilitates the study of complex systems in a manner that surpasses simple correlational and associational analyses.

The first factor to consider is the nature of interaction among variables. Three building blocks serve as the foundations of causal interactions1: mediator, confounder, and collider effects. These interactions can intertwine and manifest concurrently, leading to more complex causal networks. Therefore, accurately capturing these interactions is key to faithfully characterizing more general causal patterns. Consider the three events denoted by A, B, and C:

Mediator variables (A → B → C) emerge in the causal chain between variable A to variable C, with variable B acting as a bridge. In this scenario, B is often viewed as the mechanism or mediator responsible for transmitting the influence of A to C. Mediator variables help explain the underlying mechanisms by which an independent variable influences a dependent variable. A simple example is ↑ education level →  ↑ job skills →  ↑ income.

Confounding variables (A ← B → C) act as the common cause for two variables: B → A and B → C. They have the potential to create a statistical correlation between A and C, even if there is no direct causal link between them. Consequently, confounding variables can obscure or distort the genuine relationship between variables. Following the example above, air pollution →  ↑ deforestation19, and air pollution →  ↑ respiratory health conditions20.

Collider variables (A → B ← C) represent the effect of multiple factors acting on the same variable: A → B and C → B. This scenario is particularly relevant in nonlinear dynamical systems, where most variables are affected by multiple causes due to their coupling. A collider exhibits redundant causes when both A and C contribute to the same effect or outcome of B, creating overlapping or duplicative influences on the outcome. Consequently, redundant causes result in multiple pathways to the same effect. For instance, both hard work and high intelligence can independently contribute to the good grades of a student. Note that A and C may not necessarily be independent. A collider is synergistic if the combined effect of A and C on B surpasses their individual effects on B when considered separately. For example, two drugs, A and C, may be required in tandem to effectively treat a condition B, when each drug alone is ineffective.

The search for mathematical definitions of causality that accurately identify mediator, confounder, and collider effects remains an active area of research21. One of the most intuitive formulations of causality relies on the concept of interventions1,10. The approach offers a pathway for evaluating the causal effect that the process A exerts on another process B by setting A at a modified value \(\tilde{A}\) and observing the consequences of the post-intervention in B. Despite its intuitiveness, interventional studies are not without limitations3,22. Causality with interventions is intrusive (i.e., it requires modification of the system) and costly (the experiments or simulations need to be repeated). When data are gathered from physical experiments, establishing causality through interventions may become highly challenging or impractical (e.g., we cannot use interventions to assess the causality in the stock market in 2008). Additionally, the notion of causality with interventions prompts questions about the type of intervention that should be introduced and whether this intervention could affect the outcome of the exercise as a consequence of forcing the system out of its natural state. Interventional studies can also pose ethical problems in fields such as neuroscience or climate science23. For example, they might involve manipulating neural functions in living organisms or altering natural environmental conditions, potentially leading to irreversible changes or damage.

The alternative approach to interventions involves discovering causal links through observations. Observational methods are predominantly data-driven and do not require alterations to the original system. In the recent years, the steady advancements in computational power coupled with the exponential growth of big data have significantly contributed to the widespread adoption of observational techniques. One of the pioneering approaches is rooted in the use of forecasting models. The concept was initially proposed by Wiener24 and later quantified by Granger25. Granger causality (GC) measures the causality from the process B to A by evaluating how the inclusion of B in an autoregressive model reduces the forecast error for A. Originally developed for linear bivariate relationships, GC has since expanded to encompass nonlinear and multivariate scenarios26,27,28,29, finding applications in diverse fields ranging from econometrics30,31,32 to fluid dynamics33, and biology34,35.

Model-free approaches for causal discovery have also been proposed to overcome the limitations of GC. A leading method in this domain is convergent cross-mapping (CCM)36 and its variants37,38,39,40,41, which apply Takens’ embedding theorem42 to establish connections between variables and the attractor of the system. An alternative approach, known as continuity scaling43, directly assesses causal relationships by examining the scaling laws governing the continuity of the system.

Information theory, the science of message communication44, has also served as a framework for model-free causality quantification. The success of information theory relies on the notion of information as a fundamental property of physical systems, closely tied to the restrictions and possibilities of the laws of physics45,46. The grounds for causality as information are rooted in the intimate connection between information and the arrow of time. Time-asymmetries present in the system at a macroscopic level can be leveraged to measure the causality of events using information-theoretic metrics based on the Shannon entropy44. The initial applications of information theory for causality were formally established through the use of conditional entropies, employing what is known as directed information47,48. Among the most recognized contributions is transfer entropy (TE)49, which measures the reduction in entropy about the future state of a variable by knowing the past states of another. Various improvements have been proposed to address the inherent limitations of TE. Among them, we can cite conditional transfer entropy (CTE)50,51,52,53, which stands as the nonlinear, nonparametric extension of conditional GC27. Subsequent advancements of the method include multivariate formulations of CTE45 and momentary information transfer54, which extends TE by examining the transfer of information at each time step. Other information-theoretic methods, derived from dynamical system theory55,56,57,58, quantify causality as the amount of information that flows from one process to another as dictated by the governing equations.

Another family of methods for causal inference relies on conducting conditional independence tests. This approach was popularized by the Peter-Clark algorithm (PC)59, with subsequent extensions incorporating tests for momentary conditional independence (PCMCI)23,60. PCMCI aims to optimally identify a reduced conditioning set that includes the parents of the target variable61. This method has been shown to be effective in accurately detecting causal relationships while controlling for false positives23. Recently, new PCMCI variants have been developed for identifying contemporaneous links62, latent confounders63, and regime-dependent relationships64.

The methods for causal inference discussed above have significantly advanced our understanding of cause-effect interactions in complex systems. Despite the progress, current approaches face limitations in the presence of nonlinear dependencies, stochastic interactions (i.e., noise), self-causation, mediator, confounder, and collider effects, to name a few. Moreover, they are not capable of classifying causal interactions as redundant, unique, and synergistic, which is crucial to identify the fundamental relationships within the system. Another gap in existing methodologies is their inability to quantify causality that remains unaccounted for due to unobserved variables. To address these shortcomings, we propose SURD: Synergistic-Unique-Redundant Decomposition of causality. SURD offers causal quantification in terms of redundant, unique, and synergistic contributions and provides a measure of the causality from hidden variables. The approach can be used to detect causal relationships in systems with multiple variables, dependencies at different time lags, and instantaneous links. We demonstrate the performance of SURD across a large collection of scenarios that have proven challenging for causal inference and compare the results to previous approaches.

Consider the collection of N time-dependent variables given by the vector Q = [Q1(t), Q2(t), …, QN(t)]. For example, Qi may represent the regional average of climatological variables (e.g., temperature, pressure,...) or the evolution of human heart rate. The components of Q are the observables and are treated as random variables. Our objective is to quantify the causality from the components of Q to the future of the target variable Qj, denoted by \({Q}_{j}^{+}={Q}_{j}(t+\varDelta T)\), where ΔT > 0 is an arbitrary time increment. The vector Q can include variables at times less or equal to t + ΔT, which allows us to identify both lagged and instantaneous dependencies.

SURD quantifies causality as the increase in information (ΔI) about \({Q}_{j}^{+}\) obtained from observing individual components or groups of components from Q. The information in \({Q}_{j}^{+}\) is measured by the Shannon entropy44, denoted by \(H({Q}_{j}^{+})\), which represents the average number of bits required to unambiguously determine \({Q}_{j}^{+}\). It is also useful to interpret Shannon entropy as a measure of uncertainty. Processes that are highly uncertain (high entropy) are also the ones from which we gain the most information when their states are determined. Conversely, uncertainty is zero when the process is completely deterministic, indicating no information is gained when the outcome is revealed. Using the principle of forward-in-time propagation of information (i.e., information only flows toward the future)45, \(H({Q}_{j}^{+})\) can be decomposed as the sum of all causal contributions from the past and present:

where \(\varDelta {I}_{{{{\boldsymbol{i}}}}\to j}^{R}\), \(\varDelta {I}_{i\to j}^{U}\), and \(\varDelta {I}_{{{{\boldsymbol{i}}}}\to j}^{S}\) are the redundant, unique, and synergistic causalities, respectively, from the observed variables to \({Q}_{j}^{+}\), and ΔIleak→j is the causality from unobserved variables, referred to as the causality leak. Unique causalities are associated with individual components of Q, whereas redundant and synergistic causalities arise from groups of variables from Q. Consequently, the set \({{{\mathcal{C}}}}\) contains all combinations involving more than one variable. For instance, for N = 2, Equation (1) reduces to \(H({Q}_{j}^{+})=\varDelta {I}_{12\to j}^{R}+\varDelta {I}_{1\to j}^{U}+\varDelta {I}_{2\to j}^{U}+\varDelta {I}_{12\to j}^{S}+\varDelta {I}_{{{{\rm{leak}}}}\to j}\). Figure 1 shows the diagram of the redundant, unique, and synergistic causalities for N = 2. The formal definitions of causality can be found in the Supplementary Materials. Here, we offer an interpretation of each term:

Redundant causality from \({{{{\boldsymbol{Q}}}}}_{{{{\boldsymbol{i}}}}}=[{Q}_{{i}_{1}},{Q}_{{i}_{2}},\ldots ]\) to \({Q}_{j}^{+}\) (denoted by \(\varDelta {I}_{{{{\boldsymbol{i}}}}\to j}^{R}\)) is the common causality shared among all the components of Qi, where Qi is a subset of Q. Redundant causality occurs when all the variables in Qi contain the same amount of information about \({Q}_{j}^{+}\). Therefore, any component of Qi offers identical insight into the outcome of \({Q}_{j}^{+}\).

Unique causality from Qi to \({Q}_{j}^{+}\) (denoted by \(\varDelta {I}_{i\to j}^{U}\)) is the causality from Qi that cannot be obtained from any other individual variable Qk ≠ Qi. This causality occurs when observing Qi yields more information about some outcomes of \({Q}_{j}^{+}\) than observing any other isolated variable.

Synergistic causality from \({{{{\boldsymbol{Q}}}}}_{{{{\boldsymbol{i}}}}}=[{Q}_{{i}_{1}},{Q}_{{i}_{2}},\ldots ]\) to \({Q}_{j}^{+}\) (denoted by \(\varDelta {I}_{{{{\boldsymbol{i}}}}\to j}^{S}\)) is the causality arising from the joint effect of the variables in Qi. This causality occurs when more information about \({Q}_{j}^{+}\) is gained by observing a collection of variables simultaneously than by observing each variable individually.

Causality leak represents the effect from unobserved variables that influence \({Q}_{j}^{+}\) but are not contained in Q. This is the amount of information missing that would be required to unambiguously determine the future of Qj after considering all observable variables collectively.

a Diagram of the decomposition of causal dependencies between a vector of observed variables Q = [Q1, Q2] (past) and a target variable \({Q}_{j}^{+}\) (future) into their synergistic (S), unique (U) and redundant (R) components (in yellow, red, and blue, respectively) and contributions to the total, \(I({Q}_{j}^{+};{Q}_{1},{Q}_{2})\), and individual, \(I({Q}_{j}^{+};{Q}_{i})\), mutual information. The causality leak is represented in gray. A version of this diagram for three variables is shown in the Supplementary Materials. b Redundant, unique, and synergistic causalities for the simple examples of (c) a duplicated input (top panel), an output equal to the first input (middle panel), and an exclusive-OR output (bottom panel). The notation used is such that \([R12,U1,U2,S12]\equiv [\Delta {I}_{12\to 3}^{R},\Delta {I}_{1\to 3}^{U},\Delta {I}_{2\to 3}^{U},\Delta {I}_{12\to 3}^{S}]\). The target variable \({Q}_{3}^{+}\) is affected by external stochastic forcing W, which is independent of the observed variables in Q. The effect of W is measured by the causality leak, represented by the gray bar.

SURD exhibits several key properties that facilitate the precise identification of interactions by preventing the duplication of causality. This is illustrated in Fig. 1. First, the terms in Equation (1) are non-negative and such that the sum of redundant, unique, and synergistic causalities equals the information shared between \({Q}_{j}^{+}\) and Q, referred to as the mutual information \(I({Q}_{j}^{+};{{{\bf{Q}}}})\)44,65,66. SURD also satisfies that the mutual information between individual variables Qi and \({Q}_{j}^{+}\), denoted as \(I({Q}_{j}^{+};{Q}_{i})\), is represented by the sum of unique and redundant causalities involving Qi. This condition is consistent with the notion that causality from an individual variable to \({Q}_{j}^{+}\) is composed solely of unique and redundant causalities, while synergistic causalities emerge from the combined effects of two or more variables67. The information-theoretic formulation of SURD is also well-suited for capturing nonlinear dependencies, as well as deterministic and stochastic interactions, and self-causation.

The forward propagation of information from Equation (1) also lays the foundation for normalizing causality within SURD. Unique, redundant, and synergistic causalities to \({Q}_{j}^{+}\) are normalized by \(I({Q}_{j}^{+};{{{\bf{Q}}}})\), such that their sum equals 1. Similarly, the causality leak is normalized by \(H({Q}_{j}^{+})\), which bounds its values between 0 (indicating that all causalities to \({Q}_{j}^{+}\) are accounted for by Q) and 1 (indicating that none of the causalities are accounted for by Q). Figure 1 includes the results of SURD for three simple examples. Each case represents a system characterized exclusively by redundant, unique, or synergistic causality, respectively. These examples also allow us to introduce the notation used in the following figures: \([R12,U1,U2,S12]\equiv [\varDelta {I}_{12\to 3}^{R},\varDelta {I}_{1\to 3}^{U},\varDelta {I}_{2\to 3}^{U},\varDelta {I}_{12\to 3}^{S}]\), where the index of the target variable Q3 is omitted but it will be unambiguous from the context.

We validate SURD across multiple scenarios that pose significant challenges in causal inference. These include systems with mediator, confounder and colliders effects, Lotka–Volterra prey-predator model68,69, three-interacting species system40, the Moran effect model70, turbulent energy cascade71,72,73, experimental data for a turbulent boundary layer74,75,76, deterministic and stochastic systems with time-lagged dependencies proposed by Sugihara et al.36, Ding et al.77, and Bueso et al. 35, logic gates, synchronization of logistic maps78, and the coupled Rössler–Lorenz system79,80,81,82. A summary of the results is shown in Table 1, where the metric for success is based on whether the results are consistent with the functional dependencies of the system, rather than on the concrete value of the causal strength provided by each method.

The ability of SURD to identify causal relationships is compared with other methods for causal inference, which are also included in Table 1. The approaches considered are conditional Granger causality (CGC)26, conditional transfer entropy (CTE)53, convergent cross-mapping (CCM)36,83, and Peter and Clark momentary conditional independence (PCMCI)23. Each method brings distinct capabilities to address specific challenges in the field of causal inference. To facilitate comparison, we have summarized the key properties associated with each approach in Table 2. This classification outlines the ability of each method to handle multivariate relationships, nonlinear dependencies, stochastic (nondeterministic) processes, contemporaneous links (i.e., those occurring at a smaller time-scale than the time resolution of the data), estimation of the causality leak (i.e., causality from unobserved variables), time-delayed dependencies, and self-causation. The table also highlights the ability of SURD to account for all the scenarios described above.

The findings in Table 1 show that, although there are methods capable of effectively tackling certain situations, SURD is consistently successful across all the cases considered. In particular, SURD offers a distinct advantage in the presence of redundant variables and synchronization phenomena. In the following, we focus our discussion on the cases from Table 1 involving fundamental causal interactions: mediators, confounders, and colliders, as these are key to understand the success of SURD. We also discuss the application of SURD to unveil the causality in two turbulent flow scenarios. Readers are referred to the Supplementary Materials for a comprehensive discussion of the results presented in Table 1, as well as a detailed overview of the causal inference methods utilized and their implementation. Furthermore, the Supplementary Materials also contain a section on the application of SURD to select the most effective input variables for temporal forecasting, as well as a section that discusses the non-separability problem for nonlinear dynamical systems and demonstrates the robustness of SURD under such conditions.

The first case investigated corresponds to the system Q3 → Q2 → Q1, where Q3 influences Q1 through the mediator variable Q2. Figure 2 displays a diagram illustrating the relationships among the variables, along with the results derived from SURD and other causal analysis methods. Note that while the exact value of the causal strength may be subject to debate, the predicted causal links by each method should maintain consistency with the arrows depicted in the diagram.

(Left panel) Schematic of the functional dependence among variables and system equations, where Wi represents unobserved, stochastic forcing on the variable Qi. We use the notation \({Q}_{i}^{n}={Q}_{i}({t}_{n})\), where n indicates the time step. (Center and right panels) Results from SURD with redundant (R), unique (U) and synergistic (S) causalities in blue, red and yellow, respectively. The notation employed is such that R123 denotes \(\Delta {I}_{123\to j}^{R}\) and so on. The gray bar is the causality leak. The results from CGC, CTE, PCMCI, and CCM are depicted on the right. In all methods but CCM, the value of the bar represent the strength of the causal link. In CCM, a causal link is detected only when the value converges to 1 as the length of the time series increases, but not otherwise. CGC and CTE use the same normalization as SURD. The values for SURD, CTE and CCM are upper bounded by 1. The values for PCMCI represent conditional mutual information and are unbounded. The equations to quantify causality by each of the methods are in the Supplementary Materials.

The causal contributions detected by SURD are the unique causalities \({\varDelta I}_{3\to 3}^{U}\), \({\varDelta I}_{3\to 2}^{U}\), and \({\varDelta I}_{2\to 1}^{U}\), which are consistent with the dependency of variables in the system. PCMCI and CTE yield similar results; however, CGC cannot identify the link Q3 → Q2. CCM also fails to unambiguously capture the causal links, as the only causality converging to one as the length of the time series increases is Q1 → Q2, which is inconsistent with the equations of the system.

SURD also offers an estimate of the causality leak, which exceeds 95% for Q2 and Q3. This is attributed to the influence of the stochastic forcing terms W2 and W3, respectively, which are assumed to be unknown. The causality leak for Q1 is the lowest (below 50%), as the noise from W1 is the smallest among the three Wi. Note that none of the methods offers any insight into the causality leak, and this is also the case for all the subsequent benchmarks.

The second case (Fig. 3) corresponds to a system where Q3 acts as a confounding variable for Q1 and Q2, i.e., Q1 ← Q3 → Q2. The presence of confounding effects is captured in SURD by the synergistic causalities \(\varDelta {I}_{13\to 1}^{S}\) and \(\varDelta {I}_{23\to 2}^{S}\), while the self-induced causality from Q3 is detected by \(\varDelta {I}_{3\to 3}^{U}\). Other causalities manifest as \(\varDelta {I}_{1\to 1}^{U}\) and \(\varDelta {I}_{3\to 2}^{U}\), which are also consistent with the causal structure of the system. Regarding other methods, all of them correctly identified the confounding effects, although the link Q3 → Q1 and the self-induced causalities identified by PCMCI are barely significant. This highlights another advantage of SURD: the relative importance of the causalities is easier to interpret, since the sum of their normalized values must always equal one. Also note that CCM cannot detect self-induced causalities by construction. The largest causality leak occurs for Q3, which is consistent with the fact that the (unobserved) stochastic term acting on Q3 is ten times larger than the (unobserved) stochastic terms acting on Q1 and Q2.

Same as Fig. 2.

Next, we consider the system [Q2, Q3] → Q1, where Q2 and Q3 act together to influence Q1. In reality, Q2 and Q3 behave as a single random variable that drives Q1. The results, presented in Fig. 4, demonstrate that SURD is able to detect the dominant synergistic effect of Q2 and Q3 on Q1 through \(\varDelta {I}_{23\to 1}^{S}\) along with the self-induced causalities \(\varDelta {I}_{2\to 2}^{U}\) and \(\varDelta {I}_{3\to 3}^{U}\). The smallest causality leak is associated with Q1, as it is affected by the lowest stochastic forcing while being strongly influenced by the (observed) variables Q2 and Q3.

Same as Fig. 2.

PCMCI and CTE also identify the self-influence of Q2 and Q3 and the effect of Q2 and Q3 on Q1. However, neither of the two methods can label the interaction as synergistic and, hence, cannot show that both variables are required in combination, rather than individually, to affect Q1. On the other hand, CGC and CCM are unsuccessful in identifying the interactions of Q2 and Q3 with Q1.

The fourth case under examination explores the fundamental interaction Q2 ≡ Q3 → Q1, where Q3 is identical to Q2. In this scenario, Q2 and Q3 equally influence the future outcomes of Q1. SURD identifies \(\varDelta {I}_{23\to 2}^{R}=\varDelta {I}_{23\to 3}^{R}\) as the most significant causalities associated with Q2 and Q3, respectively, as shown in Fig. 5. Moreover, the identical causalities (and causality leaks) for both Q2 and Q3 suggest that they represent the same variable. SURD also identifies the influence of Q2 and Q3 on Q1 mostly via \(\varDelta {I}_{23\to 1}^{R}\) and \(\varDelta {I}_{12\to 1}^{S}\). Given that Q2 and Q3 are identical, SURD assigns a nonzero value only to \(\varDelta {I}_{12\to 1}^{S}\), but not to \(\varDelta {I}_{13\to 1}^{S}\), to prevent the duplication of causality in compliance with Equation (1).

Same as Fig. 2. The symbol ≡ indicates that variables Q2 and Q3 are identical.

For the rest of the methods, CGC and CTE are unable to identify any causal links between different variables. CCM identifies a bidirectional causal connection between Q2 and Q3, but cannot identify their influence to Q1. Finally, PCMCI exhibits consistent results for Q1; however, it does not offer a mechanism to identify the redundancy between Q2 and Q3.

We apply SURD to investigate the causality of the energy cascade in turbulence, which serves as a primary example of a chaotic, multi-scale, high-dimensional system. The energy cascade is the transfer of kinetic energy from large to small scales in the flow (forward cascade), or vice versa (backward cascade), and has been the cornerstone of most theories and models of turbulence since the 1940s71,72,73,84,85,86,87. However, understanding the dynamics of the energy transfer across scales remains an outstanding challenge. Given the ubiquity of turbulence, a deeper understanding of the energy transfer among flow scales could enable significant progress across various fields, ranging from plasma physics88, combustion89, climate90, and astrophysics91 to engineering applications in aero/hydrodynamics92,93,94,95. Despite the progress made in recent decades, the causal interactions of energy among scales in turbulent flows have received less attention. Here, we investigate the redundant, unique, and synergistic causalities involved in the energy transfer.

We use data from a high-fidelity simulation of isotropic turbulence in a triply periodic domain87. This case is the testbed used by the community to understand the fundamental physical processes of the energy cascade. The total number of degrees of freedom of the system is of the order of 109. The kinetic energy transfer was obtained by filtering the velocity field at four different length scales (denoted by Δi, for i = 1, 2, 3, 4) and calculating the energy flux across those scales. Causality is computed among the time signals of the volume-averaged energy transfer, denoted by Σi for i = 1, 2, 3, 4, where the index signifies energy transfer at scale Δi. The top panels of Fig. 6 depicts a visualization of the filtered velocity from the largest to the smallest flow scale, together with the time evolution of the energy transfer signals at that scale. The causal relationships among energy transfers identified by SURD are shown in the left panels of Fig. 6. The dominant contributions come from redundant and unique causalities, whereas synergistic causalities play a minor role. The unique causalities (depicted in red) vividly capture the forward energy cascade of causality from large to smaller scales, inferred from the non-zero terms \(\varDelta {I}_{1\to 2}^{U}\), \(\varDelta {I}_{2\to 3}^{U}\), and \(\varDelta {I}_{3\to 4}^{U}\). Curiously, no unique causality is observed from smaller to larger scales, and any causality from the backward cascade arises solely through redundant relationships. In the context of SURD, this implies that no new information is conveyed from the smaller scales to the larger scales, which is consistent with recent views of the backward energy cascade in the literature96,97. From the modeling perspective, this justifies the success of subgrid-scale modeling in large-eddy simulation, as the information contained in the smaller scales is redundant and does not constitute a key ingredient in solving the closure model problem. The results obtained from SURD also provide support for classic hypotheses about the energy cascade from a new causal-effect perspective. Among them, we can cite Taylor’s dissipation surrogate assumption98 and the dissipation anomaly99. The former posits that the dissipation rate can be determined by large-scale dynamics, even if dissipation is formally a small-scale feature of the flow. SURD clearly supports this assumption due to the lack of unique and synergistic causality from small to large scales. The results from SURD are also consistent with the dissipation anomaly (i.e., the constant rate of energy dissipation despite decreasing viscosity), which is enabled by the forward directionality of the energy cascade process.

a Visualization of the magnitude of the velocity field for four filter sizes Δi at the same instant. The parameter η is the Kolmogorov length-scale and represents the smallest scale in the flow. b An extract of the time history of Σ1, Σ2, Σ3, and Σ4. The time is non-dimensionalized by Tϵ, which is the integral time-scale of the flow. c Redundant (R), unique (U), and synergistic (S) causal contributions from SURD. Only the top 12 contributions, satisfying the condition \({\Delta I}_{(\cdot )\to j}/I({\Sigma }_{j}^{+};{{{\mathbf{\Sigma }}}})\ge 1{0}^{-3}\), are represented, where Σ = [Σ1, Σ2, Σ3, Σ4]. The gray bar is the causality leak. The results of CGC, CTE, CCM, and PCMCI are depicted on the right.

SURD also provides the causality leak (ΔIleak→j) that measures the amount of causality unaccounted for due to unobserved variables. The largest causality leak occurs for Σ2, where ~47% of the causality is carried by variables not included within [Σ1, Σ2, Σ3, Σ4]. This implies that there are other factors affecting Σ2 that have not been accounted for and that explain the remaining 53% of the causality of the variable. Conversely, the energy transfer at the largest scale Σ1 bears the smallest leak of 14%, which is due to the high value of the unique causality \(\varDelta {I}_{1\to 1}^{U}\). The latter implies that the future of the largest scales is mostly determined by its own past.

Finally, the results from SURD are compared with the other methods. CGC and CCM do not support the hypothesis of a forward energy cascade, which disagrees with the consensus within the fluid dynamics community85,87,100,101,102,103,104. The formulation of CCM used in this study adheres to the original work by Sugihara et al. 36 However, more recent iterations of CCM, such as Extended CCM39, which explicitly account for time delays, have demonstrated efficacy in accurately detecting causality in systems with strongly synchronized variables. Hence, these and other improved versions of CCM might be more suitable for analyzing the turbulent energy cascade, where smaller scales are enslaved to the larger ones. CTE and PCMCI are consistent with the forward propagation of energy, but the strength of the causal links detected is extremely weak. Beyond the failure of some methods to support the forward energy cascade hypothesis, different approaches also yield conflicting outcomes regarding the path followed by the energy across scales and the significance of the backward energy cascade. Additionally, none of the other previous methods offer quantification of missing causality due to unobserved variables, in contrast to the causality leak provided by SURD.

The interaction of turbulent motions of different size within the thin fluid layers immediately adjacent to solid boundaries poses a significant challenge for both physical understanding and prediction. These layers are responsible for nearly 50% of the aerodynamic drag on modern airliners and play a crucial role in the first hundred meters of the atmosphere, influencing broader meteorological phenomena94. Here, we leverage SURD to investigate the interaction between flow velocity motions in the outer layer (far from the wall) and inner layer (close to the wall) of a turbulent boundary layer. Figure 7a illustrates the configuration used to examine the causal interactions between velocity motions. More specifically, the hypotheses under consideration are either (i) a dominant influence of motions far from the wall on those closer, indicating top-down causality (a.k.a. Townsend’s outer-layer similarity hypothesis105), or (ii) the opposite scenario, where influences move from areas closer to the wall outward, suggesting bottom-up causality.

a Schematic of outer-layer and inner-layer streamwise velocity motions in a turbulent boundary and their interactions via unique causality. The velocity signals uI(t) and uO(t) are experimentally measured at the wall-normal locations yI and yO, respectively, and are shown in the panel below. The superscript * denotes the inner scaling with friction velocity, uτ, and kinematic viscosity, ν. b Redundant (R), unique (U), and synergistic (S) causalities among velocity signals in the inner (I) and outer (O) layer of a turbulent boundary layer. The gray bar is the causality leak. The results of CGC, CTE, CCM, and PCMCI are shown on the right. Details about data are provided in “Methods”.

We use experimental data from a zero-pressure gradient turbulent boundary layer from the high Reynolds number wind tunnel at the University of Melbourne74,75,76. The friction Reynolds number is Reτ = uτδ/ν = 14,750, based on the thickness of the boundary layer δ, the kinematic viscosity ν, and the average friction velocity at the wall uτ. The time signals consists of the streamwise velocity at two wall-normal locations within the inner (I) and outer (O) layers, denoted by uI(t) and uO(t), respectively.

Figure 7b shows the redundant, unique, and synergistic causalities from SURD between the inner and outer layers. We use the subindices I and O to refer to causalities from/to uI(t) or uO(t), respectively. The primary observation is that the inner layer motions are predominantly influenced by the unique causality from the outer layer, \(\varDelta {I}_{O\to I}^{U}\). The redundant and synergistic causalities are lower, but they remain significant. Curiously, the unique causality \(\varDelta {I}_{I\to I}^{U}\) is zero, implying that, at the time scale considered, the inner layer motions are independent of their past history. For the outer-layer motions, most of the causality is self-induced \(\varDelta {I}_{O\to O}^{U}\) with no apparent influence from the inner layer. The results distinctly support the prevalence of top-down interactions: causality flows predominantly from the outer-layer large-scale motions to the inner-layer small-scale motions. The outcome is consistent with the modulation of near-wall scales by large-scale motions reported in previous investigations106,107. The lack of bottom-up causality from the inner to the outer layer also aligns with Townsend’s outer-layer similarity hypothesis105 and previous observations in the literature108,109,110,111,112,113.

The causality leak, also shown in Fig. 7b, is 99% for both uI and uO. Such a high value implies that most of the causality determining the future of uI and uO is contained in other variables not considered in the analysis. This high value is unsurprising since most of the millions of degrees of freedom in the turbulent flow field have been neglected, and only two pointwise signals, uI and uO, are retained to evaluate the causality.

Finally, the results from SURD are contrasted with other methods. In this case, CCM and PCMCI do not support the hypothesis of top-down interactions between velocity motions. The reason behind the failure of these methods is unclear, but it might be related to the high causality leak. CGC and CTE are consistent with the flow of causality from the outer-layer large-scale motions to the inner-layer small-scale motions. However, as already highlighted in previous cases, none of these methods offer a detailed decomposition into redundant, unique, and synergistic causality, nor they account for the effect of unobserved variables as quantified by the causality leak in SURD.

The cases presented in this study show that the faithful quantification of causality remains elusive even in simple causal networks. The difficulties originate from a range of factors, including complexities introduced by mediator, confounding and collider effects; synergistic and duplicated variables; the influence of stochastic noise; and the presence of unobserved variables. SURD addresses these challenges by introducing several unique advances in the field of causal inference that go beyond the insights provided by other methods.

The first distinctive feature of SURD is its suitability for analyzing causal networks involving mediator, confounder, and collider effects—the building blocks of causal interactions between variables. The success of SURD in capturing these fundamental interactions stems from its ability to distinguish between redundant, unique, and synergistic causalities, which is lacking in previous methods. We have shown that the inability to disentangle redundant and synergistic causalities can obscure the relationships among variables within the system, leading to spurious causalities. In the case of PCMCI, incorporating a redundant variable into the set of conditioning variables may lead to the identification of erroneous links23. The challenge posed by redundant causality also extends to CTE27,50,51,52,53, as it evaluates the causality between pairs of variables conditioned on the remaining set of observed variables. CGC encounters difficulties in the same situations as CTE, since the former is a linear, parametric version of the latter. Several attempts have been made in the literature to account for synergistic and redundant causality45,114,115,116,117,118,119, for example, through the calculation of CTE in its multivariate form45. However, these methods may yield negative values of causality, which hinders the interpretability of the results. In contrast, SURD ensures the non-negativity of all the terms. This property significantly enhances the interpretability of SURD terms, allowing for a clear distinction between redundant, unique, and synergistic causality among variables.

SURD also introduces the concept of the causality leak, which quantifies the extent of causality that remains unaccounted for due to unobserved variables. The causality leak serves as a fundamental metric to evaluate the significance of the causal links identified. Low values of the causality leak imply that most of the causality is accounted for by the observed variables. Conversely, high values of the causality leak indicate that most of the causality can be attributed to hidden, exogenous variables. In such situations, SURD highlights the necessity of incorporating additional, currently overlooked variables into the analysis. The capability to detect and quantify missing causality is absent in other methods for causal inference.

The foundational principle of SURD from Equation (1), along with the non-negativity of causalities, also provides a natural normalization for causality that is both intuitive and easily interpretable. Unique, redundant, and synergistic causalities are normalized using the mutual information between all observed variables and the target, ensuring that their sum equals 1. This normalization measures the relative importance of each causality within the group of observed variables. Additionally, the causality leak is naturally normalized by the information content of the target variable, which bounds its value between 0 and 1. For example, in a system with three variables [Q1, Q2, Q3] where we are interested in the causes of Q2, a normalized unique causality from Q1 → Q2 equal to 90% implies that Q2 and Q3 play a minor role (i.e., 10%) in influencing the future of Q2. However, the causality from Q1 → Q2 would still be deemed insignificant if the causality leak of Q2 is 99%, indicating that most of the causality to Q2 resides in other variables not contained in the vector [Q1, Q2, Q3]. Other methods, such as CGC, do not offer bounded values nor a measure of causality leak. Outcomes from PCMCI (when based on correlations) and CCM can be normalized between 0 and 1; however, unlike SURD, the sum of causalities in PCMCI and CCM does not add up to 1 or to any conserved quantity. CTE shares many normalization properties with SURD and allows for the concept of causality leak; however, the possibility for negative values of causality in CTE complicates its interpretation45.

Another essential aspect of SURD is its foundation in transitional probability distributions, which ensures its invariance to transformations such as shifting, rescaling, and other general invertible transformation of the variables. This is applicable to other methods such as CTE and PCMCI. SURD is also robust across scenarios with varying amounts of samples for causal inference, providing consistent causal links with fewer than a thousand samples. This capability is realized through the application of transport maps specifically designed for estimating high-dimensional conditional probability distributions120. Even in situations where the sample size is exceptionally small (e.g., of the order of hundreds) and the number of variables is large, SURD is still capable of identifying causal relationships up to a certain order of synergies [see “Methods”], while higher-order synergistic interactions can be considered part of the causality leak. Finally, methods such as CCM suffer from the presence of increased noise, which complicates the reconstruction of attractor manifolds and reduces the efficacy121,122,123. In contrast, our test cases have shown that SURD is reliable even in the presence of noise. In summary, SURD stands as an effective tool in the field of causal inference with the potential to drive progress across multiple scientific and engineering domains, such as climate research, neuroscience, economics, epidemiology, social sciences, and fluid dynamics, among others.

SURD is an observational non-intrusive method that operates within a probabilistic framework, where causal relationships emerge as a result of transitional probabilities between states. The method adheres to the principle of forward-in-time propagation of information, which states that causation cannot occur backward in time. This formulation is consistent with the identification of contemporaneous links, as these can be interpreted as causal influences acting on a time scale shorter than the measurement interval (e.g., inferring causality on an 8-h scale from daily measured data). Furthermore, the method incorporates the concept of causality leak, which acts as a mechanism of quality control by assessing the impact of unobserved variables. This measure alleviates the need for the assumption of causal sufficiency (i.e., all common causes of the variables must be accurately measured), as it offers a quantifiable measure of the extent to which information from unobserved variables remains unaccounted for. Additionally, the method is model-free, i.e., no prior knowledge about the system dynamics is required. This makes SURD appealing for applications involving deterministic or stochastic multivariate systems with linear and nonlinear dependencies. The method also assumes that the time signals are stationary, which ensures that their statistical properties do not vary over time. Finally, the method can identify cyclic causal relationships, provided that they adhere to the principle of forward-in-time propagation of information.

To perform the decomposition proposed in Equation (1), we rely on the concept of mutual information44,65,66 between the target variable \({Q}_{j}^{+}\) and the vector of observed variables Q. This quantity can be mathematically described as:

where \({q}_{j}^{+}\) and q represent all possible values or states of \({Q}_{j}^{+}\) and Q, respectively. Mutual information measures how different the joint probability distribution \(p({q}_{j}^{+},{{{\boldsymbol{q}}}})\) is from the hypothetical distribution \(p({q}_{j}^{+})p({{{\boldsymbol{q}}}})\), where \({q}_{j}^{+}\) and q are assumed to be independent. For instance, if \({Q}_{j}^{+}\) and Q are not independent, then \(p({q}_{j}^{+},{{{\boldsymbol{q}}}})\) will differ significantly from \(p({q}_{j}^{+})p({{{\boldsymbol{q}}}})\). Hence, we assess causality by examining how the probability of \({Q}_{j}^{+}\) changes when accounting for Q.

However, the source of causality might change depending on different states \({q}_{j}^{+}\) of the target variable \({Q}_{j}^{+}\). For example, Q1 can only be causal to positive values of \({Q}_{j}^{+}\), whereas Q2 can only be causal to negative values of \({Q}_{j}^{+}\). Therefore, this decomposition must be performed for all possible values of \({Q}_{j}^{+}\). To do that, we define the specific mutual information124 from Q to a particular event \({Q}_{j}^{+}={q}_{j}^{+}\) as

Note that the specific mutual information is a function of the random variable Q (which encompasses all its states) but only a function of one particular state of the target variable (namely, \({q}_{j}^{+}\)). Similarly to Equation (2), the specific mutual information quantifies the dissimilarity between \(p({q}_{j}^{+},{{\boldsymbol{q}}})\) and \(p({q}_{j}^{+}) p({{\boldsymbol{q}}})\) but in this case for the particular state \({Q}_{j}^{+}={q}_{j}^{+}\). The mutual information between \({Q}_{j}^{+}\) and Q is recovered by \(I({Q}_{j}^{+};{{{\boldsymbol{Q}}}})={\sum }_{{q}_{j}^{+}}p({q}_{j}^{+})\tilde{\imath }({q}_{j}^{+};{{{\boldsymbol{Q}}}})\). For simplicity, we will use \({\tilde{\imath }}_{{{{\boldsymbol{i}}}}}({q}_{j}^{+})=\tilde{\imath }({q}_{j}^{+};{{{{\boldsymbol{Q}}}}}_{{{{\boldsymbol{i}}}}})\).

To perform the decomposition of the specific mutual information in its redundant \(\varDelta {\tilde{\imath }}_{{{{\boldsymbol{i}}}}}^{R}\), unique \(\varDelta {\tilde{\imath }}_{i}^{U}\), and synergistic \(\varDelta {\tilde{\imath }}_{{{{\boldsymbol{i}}}}}^{S}\) components, we quantify the increments in specific information \(\varDelta \tilde{\imath }\) about \({q}_{j}^{+}\) obtained by observing an individual or groups of components from Q. For a given state \({q}_{j}^{+}\) of the target variable \({Q}_{j}^{+}\), the specific causalities \(\tilde{\imath }\) are computed for all the possible combinations of past variables. These components are organized in ascending order, which allows to assign the redundant, unique, and synergistic causalities. Figure 8 shows an example of the decomposition of \(\tilde{\imath }({q}_{j}^{+};{{{\boldsymbol{Q}}}})\) for a particular state of the target variable and for the three simple examples illustrated in Fig. 1. The quantities in Equation (1) are then obtained as the expectation of their corresponding values:

a For a given state \({q}_{j}^{+}\) of the target variable \({Q}_{j}^{+}\), the specific causalities \(\tilde{\imath }\) are computed for all the possible combinations of past variables. These components are organized in ascending order, which allows to assign the redundant (blue), unique (red), and synergistic (yellow) causalities. This process is performed for all possible values \({q}_{j}^{+}\) of \({Q}_{j}^{+}\). b Schematic of simple examples and associated specific mutual information for (top panel) duplicated input, (middle panel) output equal to first input, and (bottom panel) exclusive-OR output.

We compared SURD with other established methods for causal inference in time series: CGC26, CTE27, CCM36, and PCMCI23. The detailed calculation of the previous methods has been documented in the literature, and their corresponding source codes have been employed23,77,83,125. In this paper, an embedding dimension equivalent to the number of variables was used for the application of the CCM method, which was executed with a library size that ensured convergence of the prediction skill for all cases. Therefore, the CCM value should be close to 1 in order to consider a link significantly causal. We used the version of PCMCI based on conditional mutual information (CMI) independence test with the estimator k-nearest neighbor (k-NN)60. All cases were evaluated at a significance level of 1%. Furthermore, PCMCIA was estimated with αPC = 0.05 and CMI-kNN parameters kCMI = 0.1, sperm = 5, and B = 200 permutation surrogates. The same time lag was used for all the methods. The reader is referred to the Supplementary Materials for a more detailed discussion of the packages used, the validation with test cases provided in each of the sources, and the effect of convergence for the CCM method. A summary of the results for PCMCI using different independence tests is also provided in the Supplementary Materials, where the optimal confidence interval αPC during the initial condition selection phase (PC phase) was selected based on the Akaike Information criterion126 from a default list of values, i.e., αPC = [0.05, 0.1, 0.2, 0.3, 0.4, 0.5].

The mediator, confounder, and collider systems considered comprise three variables Q1(tn), Q2(tn), and Q3(tn) at discrete times tn = n. The system is initially set to Q1(1) = Q2(1) = Q3(1) = 0. A stochastic forcing, represented by Wi(tn), acts on Qi(tn) and follows a Gaussian distribution with a mean of zero and a standard deviation of one. The computation of SURD is performed for a time lag of ΔT = 1 using 100 uniform bins per variable. The integration of the system is carried out over 108 time steps, with the first 10,000 steps excluded from the analysis to avoid transient effects. CGC and CTE used the same samples as SURD, while CCM and PCMCI used 5 × 105 samples due to computational constraints. The latter methods were also evaluated using a smaller number of samples, which were one order of magnitude lower, and no significant differences were detected. An analysis of the impact of the number of samples and the sensitivity to partition refinement for SURD is provided in the Supplementary Materials.

The case chosen to study the energy cascade is forced isotropic turbulence in a triply periodic box. The data were obtained from a direct numerical simulation127, which is publicly available in https://torroja.dmt.upm.es/turbdata/. In this simulation, the Navier–Stokes equations are numerically integrated by resolving the whole range of spatial and temporal scales of the flow. The conservation of momentum and mass equations for an incompressible flow are given by:

where repeated indices imply summation, x = [x1, x2, x3] are the spatial coordinates, ui for i = 1, 2, 3 are the velocities components, Π is the pressure, ν is the kinematic viscosity, and fi is a linear forcing sustaining the turbulent flow128. The simulation was conducted with 10243 spatial Fourier modes, which is enough to accurately resolve all the relevant length-scales of the flow127. To quantify the transfer of kinetic energy among eddies at different length scales over time, the i-th component of the instantaneous flow velocity in Equation (5), denoted as ui(x, t), is decomposed into contributions from large and small scales according to \({u}_{i}({{{\boldsymbol{x}}}},t)={\bar{u}}_{i}({{{\boldsymbol{x}}}},t)+{u}_{i}^{{\prime} }({{{\boldsymbol{x}}}},t)\). The operator \(\bar{(\cdot )}\) signifies the low-pass Gaussian filter and is given by:

where \(\bar{\varDelta }\) is the filter width and V denotes integration over the whole flow domain. Examples of filtered velocity fields at four different filter widths are included in Fig. 6. The kinetic energy of the large-scale field evolves as

where \({\tau }_{ij}=(\overline{{u}_{i}{u}_{j}}-{\bar{u}}_{i}{\bar{u}}_{j})\) is the subgrid-scale stress tensor, which represents the effect of the (filtered) small-scale eddies on the (resolved) large-scale eddies and \({\bar{S}}_{ij}=(\partial {\bar{u}}_{i}/\partial {x}_{j}+\partial {\bar{u}}_{j}/\partial {x}_{i})/2\) denotes the filtered strain-rate tensor. The interscale energy transfer \({\tilde{\Sigma }}_{i}({{{\boldsymbol{x}}}},t;{\bar{\varDelta }}_{i})\) between the filtered and unfiltered scales is given by

which is the quantity of interest. The velocity field was low-passed filtered at four filter widths: \({\bar{\varDelta }}_{1}=163\eta\), \({\bar{\varDelta }}_{2}=81\eta\), \({\bar{\varDelta }}_{3}=42\eta\), and \({\bar{\varDelta }}_{4}=21\eta\). The filter widths are selected to represent four different flow scales and are located within the inertial range of the simulation: \({L}_{\varepsilon } > {\bar{\varDelta }}_{i} > \eta\), for i = 1, 2, 3 and 4, where Lε represents the size of the largest scales and η is the size of the smallest scales. Finally, we volume-averaged \({\tilde{\Sigma }}_{i}\) over the entire domain, denoted by Σi, which served as a marker for the dynamics of the energy cascade. The generated data are also resolved in time, with flow fields stored at intervals of Δt = 0.0076Tε, where Tε is the characteristic time of the largest flow scales. The simulation was intentionally run for an extended period to ensure the accurate computation of specific mutual information. The total simulated time after transient effects was equal to 165Tε. For a given target variable, Σj, the time delay ΔTj used to evaluate causality was determined as the time required for maximum \(\varDelta {I}_{i\to j}^{U}\) with j ≠ i, where \({\Sigma }_{j}^{+}\) is evaluated at t + ΔTj.

The data used for analyzing inner/outer interactions in a turbulent boundary layer were obtained from a experimental campaign at the high Reynolds number wind tunnel at the University of Melbourne74,75,76, which is publicly available in https://fluids.eng.unimelb.edu.au/. In this campaign, measurements were made at a streamwise distance of x = 21.65 m from the trip at the test section inlet (x = 0), with a free-stream velocity of nominally 20 m/s. The boundary layer at this location has a thickness of δ = 0.361 m, and the friction velocity is uτ = 0.626 m/s. Based on these values, the friction Reynolds number is Reτ = uτδ/ν = 14,750. The data used in this study includes measurements of the streamwise velocity obtained using two synchronous hot-wire anemometry probes with an acquisition rate Δt* = 1.28 at two different wall-normal locations: \({y}_{I} { \! \!*}=4.33\) (for the inner layer) and yO/δ = 0.31 (for the outer layer). The superscript * denotes the inner scaling with friction velocity, uτ, and kinematic viscosity, ν. At each location, the acquisition time consists of three cycles of approximately TU∞/δ = 20,000. Further details about the experimental setup can be found in ref. 74 and Marusic76. The time lag utilized to evaluate causality is ΔT* ≈ 756, which corresponds to the time lag for maximum cross-induced unique causality.

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

The data generated in this study as well as the analysis and simulation code have been deposited in a Zenodo database under identifier https://doi.org/10.5281/zenodo.13750918.

The codes129 developed for this work are available at: https://github.com/Computational-Turbulence-Group/SURD.

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This work was supported by the National Science Foundation under Grant No. 2140775 and MISTI Global Seed Funds and UPM. Á.M.-S. received the support of a fellowship from the ”la Caixa” Foundation (ID 100010434). The fellowship code is LCF/BQ/EU22/11930094. G.A. was partially supported by the Predictive Science Academic Alliance Program (PSAAP; grant DE-NA0003993) managed by the NNSA (National Nuclear Security Administration) Office of Advanced Simulation and Computing and the STTR N68335-21-C-0270 with Cascade Technologies, Inc. and the Naval Air Systems Command. The authors acknowledge the MIT SuperCloud and Lincoln Laboratory Supercomputing Center for providing HPC resources that have contributed to the research results reported within this paper. The authors would like to thank Yuenong Ling for his contributions to this work and Mathieu Le Provost for his assistance with the implementation of the transport map method.

Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, USA

Álvaro Martínez-Sánchez, Gonzalo Arranz & Adrián Lozano-Durán

Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA, USA

Adrián Lozano-Durán

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Á.M.-S.: Methodology, Software, Validation, Investigation, Data Curation, Writing—Original Draft, Writing—Review & Editing, Visualization. G. A.: Methodology, Software, Investigation, Writing—Review & Editing. A.L.-D.: Ideation, Methodology, Writing—Review & Editing, Supervision, Resources, Funding acquisition.

Correspondence to Álvaro Martínez-Sánchez.

The authors declare no competing interests.

Nature Communications thanks Siyang Leng, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

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Martínez-Sánchez, Á., Arranz, G. & Lozano-Durán, A. Decomposing causality into its synergistic, unique, and redundant components. Nat Commun 15, 9296 (2024). https://doi.org/10.1038/s41467-024-53373-4

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Received: 30 April 2024

Accepted: 09 October 2024

Published: 01 November 2024

DOI: https://doi.org/10.1038/s41467-024-53373-4

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