Jakob Runge

Julia Kaltenborn, Charlotte E. E. Lange, Venkatesh Ramesh et al. · mila

Climate models have been key for assessing the impact of climate change and simulating future climate scenarios. The machine learning (ML) community has taken an increased interest in supporting climate scientists' efforts on various tasks such as climate model emulation, downscaling, and prediction tasks. Many of those tasks have been addressed on datasets created with single climate models. However, both the climate science and ML communities have suggested that to address those tasks at scale, we need large, consistent, and ML-ready climate model datasets. Here, we introduce ClimateSet, a dataset containing the inputs and outputs of 36 climate models from the Input4MIPs and CMIP6 archives. In addition, we provide a modular dataset pipeline for retrieving and preprocessing additional climate models and scenarios. We showcase the potential of our dataset by using it as a benchmark for ML-based climate model emulation. We gain new insights about the performance and generalization capabilities of the different ML models by analyzing their performance across different climate models. Furthermore, the dataset can be used to train an ML emulator on several climate models instead of just one. Such a "super emulator" can quickly project new climate change scenarios, complementing existing scenarios already provided to policymakers. We believe ClimateSet will create the basis needed for the ML community to tackle climate-related tasks at scale.

Felix Wagner, Florian Nachtigall, Lukas Franken et al.

Climate change mitigation in urban mobility requires policies reconfiguring urban form to increase accessibility and facilitate low-carbon modes of transport. However, current policy research has insufficiently assessed urban form effects on car travel at three levels: (1) Causality -- Can causality be established beyond theoretical and correlation-based analyses? (2) Generalizability -- Do relationships hold across different cities and world regions? (3) Context specificity -- How do relationships vary across neighborhoods of a city? Here, we address all three gaps via causal graph discovery and explainable machine learning to detect urban form effects on intra-city car travel, based on mobility data of six cities across three continents. We find significant causal effects of urban form on trip emissions and inter-feature effects, which had been neglected in previous work. Our results demonstrate that destination accessibility matters most overall, while low density and low connectivity also sharply increase CO$_2$ emissions. These general trends are similar across cities but we find idiosyncratic effects that can lead to substantially different recommendations. In more monocentric cities, we identify spatial corridors -- about 10--50 km from the city center -- where subcenter-oriented development is more relevant than increased access to the main center. Our work demonstrates a novel application of machine learning that enables new research addressing the needs of causality, generalizability, and contextual specificity for scaling evidence-based urban climate solutions.

Saranya Ganesh S., Tom Beucler, Frederick Iat-Hin Tam et al.

Robust feature selection is vital for creating reliable and interpretable Machine Learning (ML) models. When designing statistical prediction models in cases where domain knowledge is limited and underlying interactions are unknown, choosing the optimal set of features is often difficult. To mitigate this issue, we introduce a Multidata (M) causal feature selection approach that simultaneously processes an ensemble of time series datasets and produces a single set of causal drivers. This approach uses the causal discovery algorithms PC1 or PCMCI that are implemented in the Tigramite Python package. These algorithms utilize conditional independence tests to infer parts of the causal graph. Our causal feature selection approach filters out causally-spurious links before passing the remaining causal features as inputs to ML models (Multiple linear regression, Random Forest) that predict the targets. We apply our framework to the statistical intensity prediction of Western Pacific Tropical Cyclones (TC), for which it is often difficult to accurately choose drivers and their dimensionality reduction (time lags, vertical levels, and area-averaging). Using more stringent significance thresholds in the conditional independence tests helps eliminate spurious causal relationships, thus helping the ML model generalize better to unseen TC cases. M-PC1 with a reduced number of features outperforms M-PCMCI, non-causal ML, and other feature selection methods (lagged correlation, random), even slightly outperforming feature selection based on eXplainable Artificial Intelligence. The optimal causal drivers obtained from our causal feature selection help improve our understanding of underlying relationships and suggest new potential drivers of TC intensification.

Simon Bing, Urmi Ninad, Jonas Wahl et al.

The task of inferring high-level causal variables from low-level observations, commonly referred to as causal representation learning, is fundamentally underconstrained. As such, recent works to address this problem focus on various assumptions that lead to identifiability of the underlying latent causal variables. A large corpus of these preceding approaches consider multi-environment data collected under different interventions on the causal model. What is common to virtually all of these works is the restrictive assumption that in each environment, only a single variable is intervened on. In this work, we relax this assumption and provide the first identifiability result for causal representation learning that allows for multiple variables to be targeted by an intervention within one environment. Our approach hinges on a general assumption on the coverage and diversity of interventions across environments, which also includes the shared assumption of single-node interventions of previous works. The main idea behind our approach is to exploit the trace that interventions leave on the variance of the ground truth causal variables and regularizing for a specific notion of sparsity with respect to this trace. In addition to and inspired by our theoretical contributions, we present a practical algorithm to learn causal representations from multi-node interventional data and provide empirical evidence that validates our identifiability results.

Andreas Gerhardus, Jonas Wahl, Sofia Faltenbacher et al.

In recent years, a growing number of method and application works have adapted and applied the causal-graphical-model framework to time series data. Many of these works employ time-resolved causal graphs that extend infinitely into the past and future and whose edges are repetitive in time, thereby reflecting the assumption of stationary causal relationships. However, most results and algorithms from the causal-graphical-model framework are not designed for infinite graphs. In this work, we develop a method for projecting infinite time series graphs with repetitive edges to marginal graphical models on a finite time window. These finite marginal graphs provide the answers to $m$-separation queries with respect to the infinite graph, a task that was previously unresolved. Moreover, we argue that these marginal graphs are useful for causal discovery and causal effect estimation in time series, effectively enabling to apply results developed for finite graphs to the infinite graphs. The projection procedure relies on finding common ancestors in the to-be-projected graph and is, by itself, not new. However, the projection procedure has not yet been algorithmically implemented for time series graphs since in these infinite graphs there can be infinite sets of paths that might give rise to common ancestors. We solve the search over these possibly infinite sets of paths by an intriguing combination of path-finding techniques for finite directed graphs and solution theory for linear Diophantine equations. By providing an algorithm that carries out the projection, our paper makes an important step towards a theoretically-grounded and method-agnostic generalization of a range of causal inference methods and results to time series.

Oana-Iuliana Popescu, Andreas Gerhardus, Jakob Runge

Conditional independence testing (CIT) is a common task in machine learning, e.g., for variable selection, and a main component of constraint-based causal discovery. While most current CIT approaches assume that all variables are numerical or all variables are categorical, many real-world applications involve mixed-type datasets that include numerical and categorical variables. Non-parametric CIT can be conducted using conditional mutual information (CMI) estimators combined with a local permutation scheme. Recently, two novel CMI estimators for mixed-type datasets based on k-nearest-neighbors (k-NN) have been proposed. As with any k-NN method, these estimators rely on the definition of a distance metric. One approach computes distances by a one-hot encoding of the categorical variables, essentially treating categorical variables as discrete-numerical, while the other expresses CMI by entropy terms where the categorical variables appear as conditions only. In this work, we study these estimators and propose a variation of the former approach that does not treat categorical variables as numeric. Our numerical experiments show that our variant detects dependencies more robustly across different data distributions and preprocessing types.

Juan L. Gamella, Simon Bing, Jakob Runge · eth-zurich

We evaluate methods for causal representation learning (CRL) on a simple, real-world system where these methods are expected to work. The system consists of a controlled optical experiment specifically built for this purpose, which satisfies the core assumptions of CRL and where the underlying causal factors (the inputs to the experiment) are known, providing a ground truth. We select methods representative of different approaches to CRL and find that they all fail to recover the underlying causal factors. To understand the failure modes of the evaluated algorithms, we perform an ablation on the data by substituting the real data-generating process with a simpler synthetic equivalent. The results reveal a reproducibility problem, as most methods already fail on this synthetic ablation despite its simple data-generating process. Additionally, we observe that common assumptions on the mixing function are crucial for the performance of some of the methods but do not hold in the real data. Our efforts highlight the contrast between the theoretical promise of the state of the art and the challenges in its application. We hope the benchmark serves as a simple, real-world sanity check to further develop and validate methodology, bridging the gap towards CRL methods that work in practice. We make all code and datasets publicly available at github.com/simonbing/CRLSanityCheck

Simon Bing, Jonas Wahl, Jakob Runge

We introduce structural causal bottleneck models (SCBMs), a novel class of structural causal models. At the core of SCBMs lies the assumption that causal effects between high-dimensional variables only depend on low-dimensional summary statistics, or bottlenecks, of the causes. SCBMs provide a flexible framework for task-specific dimension reduction while being estimable via standard, simple learning algorithms in practice. We analyse identifiability in SCBMs, connect them to information bottlenecks in the sense of Tishby & Zaslavsky (2015), and illustrate how to estimate them experimentally. We also demonstrate the benefit of bottlenecks for effect estimation in low-sample transfer learning settings. We argue that SCBMs provide an alternative to existing causal dimension reduction frameworks like causal representation learning or causal abstraction learning.

Martin Rabel, Jakob Runge

Real-world problems, for example in climate applications, often require causal reasoning on spatially gridded time series data or data with comparable structure. While the underlying system is often believed to behave similarly at different Points in space and time, those variations that do exist are relevant twofold: They often encode important information in and of themselves. And they may negatively affect the stability and validity of results if not accounted for. We study the information encoded in changes of the causal graph, with stability in mind. Two core challenges arise, related to the complexity of encoding system-states and to statistical convergence properties in the presence of imperfectly recoverable non-stationary structure. We provide a framework realizing principles conceptually suitable to overcome these challenges - an interpretation supported by numerical experiments. Primarily, we modify constraint-based causal discovery approaches on the level of independence testing. This leads to a framework which is additionally highly modular, easily extensible and widely applicable. For example, it allows to leverage existing constraint-based causal discovery methods (demonstrated on PC, PC-stable, FCI, PCMCI, PCMCI+ and LPCMCI), and to systematically divide the problem into simpler subproblems that are easier to analyze and understand and relate more clearly to well-studied problems like change-point-detection, clustering, independence-testing and more. Code is available at https://github.com/martin-rabel/Causal_GLDF.

Jakob Runge

The problem of selecting optimal backdoor adjustment sets to estimate causal effects in graphical models with hidden and conditioned variables is addressed. Previous work has defined optimality as achieving the smallest asymptotic estimation variance and derived an optimal set for the case without hidden variables. For the case with hidden variables there can be settings where no optimal set exists and currently only a sufficient graphical optimality criterion of limited applicability has been derived. In the present work optimality is characterized as maximizing a certain adjustment information which allows to derive a necessary and sufficient graphical criterion for the existence of an optimal adjustment set and a definition and algorithm to construct it. Further, the optimal set is valid if and only if a valid adjustment set exists and has higher (or equal) adjustment information than the Adjust-set proposed in Perkovi{ć} et al. [Journal of Machine Learning Research, 18: 1--62, 2018] for any graph. The results translate to minimal asymptotic estimation variance for a class of estimators whose asymptotic variance follows a certain information-theoretic relation. Numerical experiments indicate that the asymptotic results also hold for relatively small sample sizes and that the optimal adjustment set or minimized variants thereof often yield better variance also beyond that estimator class. Surprisingly, among the randomly created setups more than 90\% fulfill the optimality conditions indicating that also in many real-world scenarios graphical optimality may hold. Code is available as part of the python package \url{https://github.com/jakobrunge/tigramite}.

Andreas Gerhardus, Jakob Runge

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.

Simon Bing, Jonas Wahl, Urmi Ninad et al.

In causal models, a given mechanism is assumed to be invariant to changes of other mechanisms. While this principle has been utilized for inference in settings where the causal variables are observed, theoretical insights when the variables of interest are latent are largely missing. We assay the connection between invariance and causal representation learning by establishing impossibility results which show that invariance alone is insufficient to identify latent causal variables. Together with practical considerations, we use these theoretical findings to highlight the need for additional constraints in order to identify representations by exploiting invariance.

Wiebke Günther, Oana-Iuliana Popescu, Martin Rabel et al.

Causal systems often exhibit variations of the underlying causal mechanisms between the variables of the system. Often, these changes are driven by different environments or internal states in which the system operates, and we refer to context variables as those variables that indicate this change in causal mechanisms. An example are the causal relations in soil moisture-temperature interactions and their dependence on soil moisture regimes: Dry soil triggers a dependence of soil moisture on latent heat, while environments with wet soil do not feature such a feedback, making it a context-specific property. Crucially, a regime or context variable such as soil moisture need not be exogenous and can be influenced by the dynamical system variables - precipitation can make a dry soil wet - leading to joint systems with endogenous context variables. In this work we investigate the assumptions for constraint-based causal discovery of context-specific information in systems with endogenous context variables. We show that naive approaches such as learning different regime graphs on masked data, or pooling all data, can lead to uninformative results. We propose an adaptive constraint-based discovery algorithm and give a detailed discussion on the connection to structural causal models, including sufficiency assumptions, which allow to prove the soundness of our algorithm and to interpret the results causally. Numerical experiments demonstrate the performance of the proposed method over alternative baselines, but they also unveil current limitations of our method.

Sofia Faltenbacher, Jonas Wahl, Rebecca Herman et al.

Causal discovery aims to infer causal graphs from observational or experimental data. Methods such as the popular PC algorithm are based on conditional independence testing and utilize enabling assumptions, such as the faithfulness assumption, for their inferences. In practice, these assumptions, as well as the functional assumptions inherited from the chosen conditional independence test, are typically taken as a given and not further tested for their validity on the data. In this work, we propose internal coherency scores that allow testing for assumption violations and finite sample errors, whenever detectable without requiring ground truth or further statistical tests. We provide a complete classification of erroneous results, including a distinction between detectable and undetectable errors, and prove that the detectable erroneous results can be measured by our scores. We illustrate our coherency scores on the PC algorithm with simulated and real-world datasets, and envision that testing for internal coherency can become a standard tool in applying constraint-based methods, much like a suite of tests is used to validate the assumptions of classical regression analysis.

Rebecca J. Herman, Jonas Wahl, Urmi Ninad et al.

Causal discovery aims to extract qualitative causal knowledge in the form of causal graphs from data. Because causal ground truth is rarely known in the real world, simulated data plays a vital role in evaluating the performance of the various causal discovery algorithms proposed in the literature. But recent work highlighted certain artifacts of commonly used data generation techniques for a standard class of structural causal models (SCM) that may be nonphysical, including var- and R2-sortability, where the variables' variance and coefficients of determination (R2) after regressing on all other variables, respectively, increase along the causal order. Some causal methods exploit such artifacts, leading to unrealistic expectations for their performance on real-world data. Some modifications have been proposed to remove these artifacts; notably, the internally-standardized structural causal model (iSCM) avoids varsortability and largely alleviates R2-sortability on sparse causal graphs, but exhibits a reversed R2-sortability pattern for denser graphs not featured in their work. We analyze which sortability patterns we expect to see in real data, and propose a method for drawing coefficients that we argue more effectively samples the space of SCMs. Finally, we propose a novel extension of our SCM generation method to the time series setting.

Martin Rabel, Wiebke Günther, Jakob Runge et al.

Causal structure learning with data from multiple contexts carries both opportunities and challenges. Opportunities arise from considering shared and context-specific causal graphs enabling to generalize and transfer causal knowledge across contexts. However, a challenge that is currently understudied in the literature is the impact of differing observational support between contexts on the identifiability of causal graphs. Here we study in detail recently introduced [6] causal graph objects that capture both causal mechanisms and data support, allowing for the analysis of a larger class of context-specific changes, characterizing distribution shifts more precisely. We thereby extend results on the identifiability of context-specific causal structures and propose a framework to model context-specific independence (CSI) within structural causal models (SCMs) in a refined way that allows to explore scenarios where these graph objects differ. We demonstrate how this framework can help explaining phenomena like anomalies or extreme events, where causal mechanisms change or appear to change under different conditions. Our results contribute to the theoretical foundations for understanding causal relations in multi-context systems, with implications for generalization, transfer learning, and anomaly detection. Future work may extend this approach to more complex data types, such as time-series.

Saranya Ganesh S., Frederick Iat-Hin Tam, Milton S. Gomez et al.

Improving statistical forecasts of Atlantic hurricane intensity is limited by complex nonlinear interactions and difficulty in identifying relevant predictors. Conventional methods prioritize correlation or fit, often overlooking confounding variables and limiting generalizability to unseen tropical storms. To address this, we leverage a multidata causal discovery framework with a replicated dataset based on Statistical Hurricane Intensity Prediction Scheme (SHIPS) using ERA5 meteorological reanalysis. We conduct multiple experiments to identify and select predictors causally linked to hurricane intensity changes. We train multiple linear regression models to compare causal feature selection with no selection, correlation, and random forest feature importance across five forecast lead times from 1 to 5 days (24 to 120 hours). Causal feature selection consistently outperforms on unseen test cases, especially for lead times shorter than 3 days. The causal features primarily include vertical shear, mid-tropospheric potential vorticity and surface moisture conditions, which are physically significant yet often underutilized in hurricane intensity predictions. Further, we build an extended predictor set (SHIPS+) by adding selected features to the standard SHIPS predictors. SHIPS+ yields increased short-term predictive skill at lead times of 24, 48, and 72 hours. Adding nonlinearity using multilayer perceptron further extends skill to longer lead times, despite our framework being purely regional and not requiring global forecast data. Operational SHIPS tests confirm that three of the six added causally discovered predictors improve forecasts, with the largest gains at longer lead times. Our results demonstrate that causal discovery improves hurricane intensity prediction and pave the way toward more empirical forecasts.

Tom Hochsprung, Jakob Runge, Andreas Gerhardus

There exist several approaches for estimating causal effects in time series when latent confounding is present. Many of these approaches rely on additional auxiliary observed variables or time series such as instruments, negative controls or time series that satisfy the front- or backdoor criterion in certain graphs. In this paper, we present a novel approach for estimating direct (and via Wright's path rule total) causal effects in a time series setup which does not rely on additional auxiliary observed variables or time series. This approach assumes that the underlying time series is a Structural Vector Autoregressive (SVAR) process and estimates direct causal effects by solving certain linear equation systems made up of different covariances and model parameters. We state sufficient graphical criteria in terms of the so-called full time graph under which these linear equations systems are uniquely solvable and under which their solutions contain the to-be-identified direct causal effects as components. We also state sufficient lag-based criteria under which the previously mentioned graphical conditions are satisfied and, thus, under which direct causal effects are identifiable. Several numerical experiments underline the correctness and applicability of our results.

Gustau Camps-Valls, Andreas Gerhardus, Urmi Ninad et al.

Physics is a field of science that has traditionally used the scientific method to answer questions about why natural phenomena occur and to make testable models that explain the phenomena. Discovering equations, laws and principles that are invariant, robust and causal explanations of the world has been fundamental in physical sciences throughout the centuries. Discoveries emerge from observing the world and, when possible, performing interventional studies in the system under study. With the advent of big data and the use of data-driven methods, causal and equation discovery fields have grown and made progress in computer science, physics, statistics, philosophy, and many applied fields. All these domains are intertwined and can be used to discover causal relations, physical laws, and equations from observational data. This paper reviews the concepts, methods, and relevant works on causal and equation discovery in the broad field of Physics and outlines the most important challenges and promising future lines of research. We also provide a taxonomy for observational causal and equation discovery, point out connections, and showcase a complete set of case studies in Earth and climate sciences, fluid dynamics and mechanics, and the neurosciences. This review demonstrates that discovering fundamental laws and causal relations by observing natural phenomena is being revolutionised with the efficient exploitation of observational data, modern machine learning algorithms and the interaction with domain knowledge. Exciting times are ahead with many challenges and opportunities to improve our understanding of complex systems.

Christian Requena-Mesa, Vitus Benson, Markus Reichstein et al.

Satellite images are snapshots of the Earth surface. We propose to forecast them. We frame Earth surface forecasting as the task of predicting satellite imagery conditioned on future weather. EarthNet2021 is a large dataset suitable for training deep neural networks on the task. It contains Sentinel 2 satellite imagery at 20m resolution, matching topography and mesoscale (1.28km) meteorological variables packaged into 32000 samples. Additionally we frame EarthNet2021 as a challenge allowing for model intercomparison. Resulting forecasts will greatly improve (>x50) over the spatial resolution found in numerical models. This allows localized impacts from extreme weather to be predicted, thus supporting downstream applications such as crop yield prediction, forest health assessments or biodiversity monitoring. Find data, code, and how to participate at www.earthnet.tech

Christian Reimers, Paul Bodesheim, Jakob Runge et al.

Bias in classifiers is a severe issue of modern deep learning methods, especially for their application in safety- and security-critical areas. Often, the bias of a classifier is a direct consequence of a bias in the training dataset, frequently caused by the co-occurrence of relevant features and irrelevant ones. To mitigate this issue, we require learning algorithms that prevent the propagation of bias from the dataset into the classifier. We present a novel adversarial debiasing method, which addresses a feature that is spuriously connected to the labels of training images but statistically independent of the labels for test images. Thus, the automatic identification of relevant features during training is perturbed by irrelevant features. This is the case in a wide range of bias-related problems for many computer vision tasks, such as automatic skin cancer detection or driver assistance. We argue by a mathematical proof that our approach is superior to existing techniques for the abovementioned bias. Our experiments show that our approach performs better than state-of-the-art techniques on a well-known benchmark dataset with real-world images of cats and dogs.

Christian Requena-Mesa, Vitus Benson, Joachim Denzler et al.

Climate change is global, yet its concrete impacts can strongly vary between different locations in the same region. Seasonal weather forecasts currently operate at the mesoscale (> 1 km). For more targeted mitigation and adaptation, modelling impacts to < 100 m is needed. Yet, the relationship between driving variables and Earth's surface at such local scales remains unresolved by current physical models. Large Earth observation datasets now enable us to create machine learning models capable of translating coarse weather information into high-resolution Earth surface forecasts. Here, we define high-resolution Earth surface forecasting as video prediction of satellite imagery conditional on mesoscale weather forecasts. Video prediction has been tackled with deep learning models. Developing such models requires analysis-ready datasets. We introduce EarthNet2021, a new, curated dataset containing target spatio-temporal Sentinel 2 satellite imagery at 20 m resolution, matched with high-resolution topography and mesoscale (1.28 km) weather variables. With over 32000 samples it is suitable for training deep neural networks. Comparing multiple Earth surface forecasts is not trivial. Hence, we define the EarthNetScore, a novel ranking criterion for models forecasting Earth surface reflectance. For model intercomparison we frame EarthNet2021 as a challenge with four tracks based on different test sets. These allow evaluation of model validity and robustness as well as model applicability to extreme events and the complete annual vegetation cycle. In addition to forecasting directly observable weather impacts through satellite-derived vegetation indices, capable Earth surface models will enable downstream applications such as crop yield prediction, forest health assessments, coastline management, or biodiversity monitoring. Find data, code, and how to participate at www.earthnet.tech .

Gustau Camps-Valls, Dino Sejdinovic, Jakob Runge et al.

Earth observation (EO) by airborne and satellite remote sensing and in-situ observations play a fundamental role in monitoring our planet. In the last decade, machine learning and Gaussian processes (GPs) in particular has attained outstanding results in the estimation of bio-geo-physical variables from the acquired images at local and global scales in a time-resolved manner. GPs provide not only accurate estimates but also principled uncertainty estimates for the predictions, can easily accommodate multimodal data coming from different sensors and from multitemporal acquisitions, allow the introduction of physical knowledge, and a formal treatment of uncertainty quantification and error propagation. Despite great advances in forward and inverse modelling, GP models still have to face important challenges that are revised in this perspective paper. GP models should evolve towards data-driven physics-aware models that respect signal characteristics, be consistent with elementary laws of physics, and move from pure regression to observational causal inference.

Jakob Runge

The paper introduces a novel conditional independence (CI) based method for linear and nonlinear, lagged and contemporaneous causal discovery from observational time series in the causally sufficient case. Existing CI-based methods such as the PC algorithm and also common methods from other frameworks suffer from low recall and partially inflated false positives for strong autocorrelation which is an ubiquitous challenge in time series. The novel method, PCMCI$^+$, extends PCMCI [Runge et al., 2019b] to include discovery of contemporaneous links. PCMCI$^+$ improves the reliability of CI tests by optimizing the choice of conditioning sets and even benefits from autocorrelation. The method is order-independent and consistent in the oracle case. A broad range of numerical experiments demonstrates that PCMCI$^+$ has higher adjacency detection power and especially more contemporaneous orientation recall compared to other methods while better controlling false positives. Optimized conditioning sets also lead to much shorter runtimes than the PC algorithm. PCMCI$^+$ can be of considerable use in many real world application scenarios where often time resolutions are too coarse to resolve time delays and strong autocorrelation is present.

Jakob Runge

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.

Jakob Runge, Reik V. Donner, Jürgen Kurths

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 Niño Southern Oscillation.

Jakob Runge, Jobst Heitzig, Norbert Marwan et al.

While it is an important problem to identify the existence of causal associations between two components of a multivariate time series, a topic addressed in Runge et al. (2012), it is even more important to assess the strength of their association in a meaningful way. In the present article we focus on the problem of defining a meaningful coupling strength using information theoretic measures and demonstrate the short-comings of the well-known mutual information and transfer entropy. Instead, we propose a certain time-delayed conditional mutual information, the momentary information transfer (MIT), as a measure of association that is general, causal and lag-specific, reflects a well interpretable notion of coupling strength and is practically computable. MIT is based on the fundamental concept of source entropy, which we utilize to yield a notion of coupling strength that is, compared to mutual information and transfer entropy, well interpretable, in that for many cases it solely depends on the interaction of the two components at a certain lag. In particular, MIT is thus in many cases able to exclude the misleading influence of autodependency within a process in an information-theoretic way. We formalize and prove this idea analytically and numerically for a general class of nonlinear stochastic processes and illustrate the potential of MIT on climatological data.