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Publications and Preprints

J. Bodik and O. Pasche (2024, preprint)
Granger causality in extremes

Abstract: We introduce a rigorous mathematical framework for Granger causality in extremes, designed to identify causal links from extreme events in time series. Granger causality plays a pivotal role in uncovering directional relationships among time-varying variables. While this notion gains heightened importance during extreme and highly volatile periods, state-of-the-art methods primarily focus on causality within the body of the distribution, often overlooking causal mechanisms that manifest only during extreme events. Our framework is designed to infer causality mainly from extreme events by leveraging the causal tail coefficient. We establish equivalences between causality in extremes and other causal concepts, including (classical) Granger causality, Sims causality, and structural causality. We prove other key properties of Granger causality in extremes and show that the framework is especially helpful under the presence of hidden confounders. We also propose a novel inference method for detecting the presence of Granger causality in extremes from data. Our method is model-free, can handle non-linear and high-dimensional time series, outperforms current state-of-the-art methods in all considered setups, both in performance and speed, and was found to uncover coherent effects when applied to financial and extreme weather observations.

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J. Bodik and V. Chavez-Demoulin (2023, preprint)
Structural restrictions in local causal discovery: identifying direct causes of a target variable

Abstract: We consider the problem of learning a set of direct causes of a target variable from an observational joint distribution. Learning directed acyclic graphs (DAGs) that represent the causal structure is a fundamental problem in science. Several results are known when the full DAG is identifiable from the distribution, such as assuming a nonlinear Gaussian data-generating process. Often, we are only interested in identifying the direct causes of one target variable (local causal structure), not the full DAG. In this paper, we discuss different assumptions for the data-generating process of the target variable under which the set of direct causes is identifiable from the distribution. While doing so, we put essentially no assumptions on the variables other than the target variable. In addition to the novel identifiability results, we provide two practical algorithms for estimating the direct causes from a finite random sample and show their effectiveness on several benchmark data-sets. We apply this framework to learn direct causes of the reduction in the fertility rate in different countries.

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J. Bodik and V. Chavez-Demoulin (2023, preprint)
Identifiability of causal graphs under nonadditive conditionally parametric causal models

Abstract: Causal discovery from observational data is a very challenging, often impossible, task. However, estimating the causal structure is possible under certain assumptions on the data-generating process. Many commonly used methods rely on the additivity of the noise in the structural equation models. Additivity implies that the variance or the tail of the effect, given the causes, is invariant; the cause only affects the mean. However, the tail or other characteristics of the random variable can provide different information about the causal structure. Such cases have received only very little attention in the literature. It has been shown that the causal graph is identifiable under different models, such as linear non-Gaussian, post-nonlinear, or quadratic variance functional models. We introduce a new class of models called the Conditional Parametric Causal Models (CPCM), where the cause affects the effect in some of the characteristics of interest. We use sufficient statistics to show the identifiability of the CPCM models in the exponential family of conditional distributions. We also propose an algorithm for estimating the causal structure from a random sample under CPCM. Its empirical properties are studied for various data sets, including an application on the expenditure behavior of residents of the Philippines.

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J. Bodik (2024, Mathematics)
Extreme Treatment Effect: Extrapolating Causal Effects Into Extreme Treatment Domain

Abstract: The potential outcomes framework serves as a fundamental tool for quantifying the causal effects. When the treatment variable (exposure) is continuous, one is typically interested in the estimation of the effect curve (also called the average dose-response function), denoted as mu(t). In this work, we explore the ``extreme causal effect,’’ where our focus lies in determining the impact of an extreme level of treatment, potentially beyond the range of observed values–that is, estimating mu(t) for very large t. Our framework is grounded in the field of statistics known as extreme value theory. We establish the foundation for our approach, outlining key assumptions that enable the estimation of the extremal causal effect. Additionally, we present a novel and consistent estimation procedure that utilizes extreme value theory in order to potentially reduce the dimension of the confounders to at most 3. In practical applications, our framework proves valuable when assessing the effects of scenarios such as drug overdoses, extreme river discharges, or extremely high temperatures on a variable of interest

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J. Bodik, Z. Pawlas, M. Paluš (2023, Extremes)
Causality in extremes of time series.

Abstract: Consider two stationary time series with heavy-tailed marginal distributions. We aim to detect whether they have a causal relation, that is, if a change in one of them causes a change in the other. Usual methods for causality detection are not well suited if the causal mechanisms only manifest themselves in extremes. This paper aims to detect the causal relations in extremes between time series. We define the so-called causal tail coefficient for time series, which, under some assumptions, correctly detects the asymmetrical causal relations between extremes of the time series. The advantage is that this method works even if nonlinear relations and common ancestors are present. Moreover, we mention how our method can help detect a time delay between the two time series. We describe some of its asymptotic properties and show how it performs on some simulations. Finally, we show how this method works on space-weather and hydro-meteorological data sets.

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J. Bodik, L. Mhalla, V. Chavez-Demoulin (2022, preprint)
Detecting causal covariates for extreme dependence structures

Abstract: Determining the causes of extreme events is a fundamental question in many scientific fields. An important aspect when modelling multivariate extremes is the tail dependence. In application, the extreme dependence structure may significantly depend on covariates. As for the general case of modelling including covariates, only some of the covariates are causal. In this paper, we propose a methodology to discover the causal covariates explaining the tail dependence structure between two variables. The proposed methodology for discovering causal variables is based on comparing observations from different environments or perturbations. It is a desired methodology for predicting extremal behaviour in a new, unobserved environment. The methodology is applied to a dataset of NO2 concentration in the UK. Extreme NO2 levels can cause severe health problems, and understanding the behaviour of concurrent severe levels is an important question. We focus on revealing causal predictors for the dependence between extreme NO2 observations at different sites.

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J. Bodik, Z. Pawlas, M. Paluš (2021, Master thesis, won a 10000KC prize in a competition for best master thesis)
Detection of causality in time series using extreme values.

Abstract: In this thesis we deal with the following problem: Let us have two stationary time series with heavy-tailed marginal distributions. We want to detect whether they have a causal relation, i.e. if a change in one of them causes a change in the other. The question of distinguishing between causality and correlation is essential in many different science fields. Usual methods for causality detection are not well suited if the causal mechanisms only manifest themselves in extremes. In this thesis, we propose a new method that can help us in such a non-traditional case distinguish between correlation and causality. We define the so-called causal tail coefficient for time series, which, under some assumptions, correctly detects the asymmetrical causal relations between different time series. We will rigorously prove this claim, and we also propose a method on how to statistically estimate the causal tail coefficient from a finite number of data. The advantage is that this method works even if nonlinear relations and common ancestors are present. Moreover, we will mention how our method can help detect a time delay between the two time series. We will show how our method performs on some simulations. Finally, we will show on a real data set how this method works, discussing a cause of electromagnetic storms.

Selected conferences and talks

UAI2024 (Invited speaker at UAI2024 Workshop on Causal inference in time series in Barcelona, Spain, 15.7.2024-19.7.2024)
EGU24 (Poster at Vienna, Austria, 14.4.2024-19.4.2024)
Causality in extremes workshop (Poster at UNIGE Geneva, Switzerland, 12.2.2024-14.2.2024)
CMStatistics (Presented in HTW Berlin, Germany, 15.12.2023-19.12.2023)
EUROCIM: European causal inference meeting (Presented in University of Oslo, Norway, 18.4.2023-22.4.2023)
CUSO: statistics and applied probability (Participant in Les Diablerets, Switzerland, 5.2.2023-8.2.2023, 4.9.2022-7.9.2022, 6.2.2022-9.2.2022, 12.9.2021-15.9.2021)
BIRS: Combining Causal Inference and Extreme Value Theory in the Study of Climate Extremes and their Causes (Presented via Zoom, Kelowna, Canada, 26.6.2022-1.7.2022)
Robust (Presented in Prague, Czech republic, 12.6.2022-17.6.2022)
CMStatistics (Presented via Zoom, London, UK, 18.12.2021 - 20.12.2021)
Valpred workshop (Presented in Aussois, France, 4.10.2021-7.10.2021)
EVA: Extreme value analysys (Presented via Zoom, Edinburgh, UK, 5.6.2021-9.6.2021)

Keywords: Mathematical statistics; Causal inference; Statistical inference; Probability theory; Hypothesis testing; Estimation theory; Confidence intervals; Statistical models; Causality; Counterfactuals; Treatment effects; Observational studies; Randomized experiments; Potential outcomes; Causal diagrams; Identification; Confounding; Selection bias; Propensity scores; Instrumental variables; Mediation analysis; Sensitivity analysis; Bayesian statistics; Nonparametric methods; Structural equation modeling; Granger causality; Directed acyclic graphs (DAGs); Structural causal models; Regression analysis; Time series analysis; Conditional probability; Markov chains; Estimator bias; Maximum likelihood estimation; Resampling methods; Cross-validation; Model selection; Robust statistics; Multivariate analysis; Experimental design; Statistical power; Sequential analysis; Missing data; Latent variable models; Bayesian networks; Marginalization; Causal effect heterogeneity; Principal component analysis; Survival analysis; Time-to-event data; Nonlinear regression; Longitudinal data analysis; Factor analysis; Structural equation models; Bootstrap methods; Spatial statistics; Cluster analysis; Decision trees; Dimensionality reduction; Bayesian hierarchical models; Statistical learning; Machine learning; Classification methods; Regression analysis; Propensity score matching; Network analysis; Granger causality testing; Observational data analysis; Quasi-experimental designs; Robust causal inference; Counterfactual estimation; Time series modeling; Structural equation modeling; Instrumental variable regression; Causal mediation analysis; Nonignorable missing data; Multiple imputation; Factorial designs; Cluster randomized trials; Survival analysis; Propensity-based weighting; Propensity score calibration; Sensitivity analysis; Nonparametric causal inference; Random forests; Causal discovery algorithms; Generalized linear models; Bayesian model averaging; Longitudinal causal inference; Structural causal models; Neural networks; Deep learning; Artificial neural networks; Feedforward networks; Backpropagation; Activation functions; Convolutional neural networks (CNN); Recurrent neural networks (RNN); Long Short-Term Memory (LSTM); Gated Recurrent Units (GRU); Autoencoders; Generative adversarial networks (GAN); Transfer learning; Fine-tuning; Dropout regularization; Batch normalization; Gradient descent; Stochastic gradient descent (SGD); Mini-batch gradient descent; Learning rate; Momentum; Adaptive learning rate methods (e.g., Adam, RMSprop); Hyperparameters; Overfitting; Underfitting; Model capacity; Weight initialization; Convergence criteria; Loss functions; Optimizers.