Quantifying information transfer and mediation along causal pathways in complex systems

Measures of information transfer have become a popular approach to analyze interactions in complex systems such as the Earth or the human brain from measured time series. Recent work has focused on causal definitions of information transfer excluding effects of common drivers and indirect influences. While the former clearly constitutes a spurious causality, the aim of the present article is to develop measures quantifying different notions of the strength of information transfer along indirect causal paths, based on first reconstructing the multivariate causal network (\emph{Tigramite} approach). Another class of novel measures quantifies to what extent different intermediate processes on causal paths contribute to an interaction mechanism to determine pathways of causal information transfer. A rigorous mathematical framework allows for a clear information-theoretic interpretation that can also be related to the underlying dynamics as proven for certain classes of processes. Generally, however, estimates of information transfer remain hard to interpret for nonlinearly intertwined complex systems. But, if experiments or mathematical models are not available, measuring pathways of information transfer within the causal dependency structure allows at least for an abstraction of the dynamics. The measures are illustrated on a climatological example to disentangle pathways of atmospheric flow over Europe.Comment: 20 pages, 6 figure

Conditional independence testing based on a nearest-neighbor estimator of conditional mutual information

Conditional independence testing is a fundamental problem underlying causal discovery and a particularly challenging task in the presence of nonlinear and high-dimensional dependencies. Here a fully non-parametric test for continuous data based on conditional mutual information combined with a local permutation scheme is presented. Through a nearest neighbor approach, the test efficiently adapts also to non-smooth distributions due to strongly nonlinear dependencies. Numerical experiments demonstrate that the test reliably simulates the null distribution even for small sample sizes and with high-dimensional conditioning sets. The test is better calibrated than kernel-based tests utilizing an analytical approximation of the null distribution, especially for non-smooth densities, and reaches the same or higher power levels. Combining the local permutation scheme with the kernel tests leads to better calibration, but suffers in power. For smaller sample sizes and lower dimensions, the test is faster than random fourier feature-based kernel tests if the permutation scheme is (embarrassingly) parallelized, but the runtime increases more sharply with sample size and dimensionality. Thus, more theoretical research to analytically approximate the null distribution and speed up the estimation for larger sample sizes is desirable.Comment: 17 pages, 12 figures, 1 tabl

Necessary and sufficient conditions for optimal adjustment sets in causal graphical models with hidden variables

The problem of selecting optimal valid backdoor adjustment sets to estimate total causal effects in graphical models with hidden and conditioned variables is addressed. Previous work has defined optimality as achieving the smallest asymptotic variance compared to other adjustment sets and identified a graphical criterion for an optimal set for the case without hidden variables. For the case with hidden variables currently a sufficient graphical criterion and a corresponding construction algorithm exists. Here optimality is characterized by an information-theoretic approach based on the mutual informations among cause, effect, adjustment set, and conditioned variables. This characterization allows to derive the main contributions of this paper: A necessary and sufficient graphical criterion for the existence of an optimal adjustment set and an algorithm to construct it. The results are valid for a class of estimators whose variance admits a certain information-theoretic decomposition.Comment: 11 pages, 2 figures; submitted to UAI202

Optimal model-free prediction from multivariate time series

Forecasting a time series from multivariate predictors constitutes a challenging problem, especially using model-free approaches. Most techniques, such as nearest-neighbor prediction, quickly suffer from the curse of dimensionality and overfitting for more than a few predictors which has limited their application mostly to the univariate case. Therefore, selection strategies are needed that harness the available information as efficiently as possible. Since often the right combination of predictors matters, ideally all subsets of possible predictors should be tested for their predictive power, but the exponentially growing number of combinations makes such an approach computationally prohibitive. Here a prediction scheme that overcomes this strong limitation is introduced utilizing a causal pre-selection step which drastically reduces the number of possible predictors to the most predictive set of causal drivers making a globally optimal search scheme tractable. The information-theoretic optimality is derived and practical selection criteria are discussed. As demonstrated for multivariate nonlinear stochastic delay processes, the optimal scheme can even be less computationally expensive than commonly used sub-optimal schemes like forward selection. The method suggests a general framework to apply the optimal model-free approach to select variables and subsequently fit a model to further improve a prediction or learn statistical dependencies. The performance of this framework is illustrated on a climatological index of El Ni\~no Southern Oscillation.Comment: 14 pages, 9 figure

Causal networks for climate model evaluation and constrained projections

Global climate models are central tools for understanding past and future climate change. The assessment of model skill, in turn, can benefit from modern data science approaches. Here we apply causal discovery algorithms to sea level pressure data from a large set of climate model simulations and, as a proxy for observations, meteorological reanalyses. We demonstrate how the resulting causal networks (fingerprints) offer an objective pathway for process-oriented model evaluation. Models with fingerprints closer to observations better reproduce important precipitation patterns over highly populated areas such as the Indian subcontinent, Africa, East Asia, Europe and North America. We further identify expected model interdependencies due to shared development backgrounds. Finally, our network metrics provide stronger relationships for constraining precipitation projections under climate change as compared to traditional evaluation metrics for storm tracks or precipitation itself. Such emergent relationships highlight the potential of causal networks to constrain longstanding uncertainties in climate change projections. Algorithms to assess causal relationships in data sets have seen increasing applications in climate science in recent years. Here, the authors show that these techniques can help to systematically evaluate the performance of climate models and, as a result, to constrain uncertainties in future climate change projections

High-recall causal discovery for autocorrelated time series with latent confounders

We present a new method for linear and nonlinear, lagged and contemporaneous constraint-based causal discovery from observational time series in the presence of latent confounders. We show that existing causal discovery methods such as FCI and variants suffer from low recall in the autocorrelated time series case and identify low effect size of conditional independence tests as the main reason. Information-theoretical arguments show that effect size can often be increased if causal parents are included in the conditioning sets. To identify parents early on, we suggest an iterative procedure that utilizes novel orientation rules to determine ancestral relationships already during the edge removal phase. We prove that the method is order-independent, and sound and complete in the oracle case. Extensive simulation studies for different numbers of variables, time lags, sample sizes, and further cases demonstrate that our method indeed achieves much higher recall than existing methods for the case of autocorrelated continuous variables while keeping false positives at the desired level. This performance gain grows with stronger autocorrelation. At https://github.com/jakobrunge/tigramite we provide Python code for all methods involved in the simulation studies.Comment: 55 pages, 26 figures; added reference to related work plus accompanying dicussion in section 3.