The Geometry of Gauged Linear Sigma Model Correlation Functions

Applying advances in exact computations of supersymmetric gauge theories, we study the structure of correlation functions in two-dimensional N=(2,2) Abelian and non-Abelian gauge theories. We determine universal relations among correlation functions, which yield differential equations governing the dependence of the gauge theory ground state on the Fayet-Iliopoulos parameters of the gauge theory. For gauge theories with a non-trivial infrared N=(2,2) superconformal fixed point, these differential equations become the Picard-Fuchs operators governing the moduli-dependent vacuum ground state in a Hilbert space interpretation. For gauge theories with geometric target spaces, a quadratic expression in the Givental I-function generates the analyzed correlators. This gives a geometric interpretation for the correlators, their relations, and the differential equations. For classes of Calabi-Yau target spaces, such as threefolds with up to two Kahler moduli and fourfolds with a single Kahler modulus, we give general and universally applicable expressions for Picard-Fuchs operators in terms of correlators. We illustrate our results with representative examples of two-dimensional N=(2,2) gauge theories.Comment: 76 pages, v2: references added and minor improvement

Causal discovery for time series from multiple datasets with latent contexts

Causal discovery from time series data is a typical problem setting across the sciences. Often, multiple datasets of the same system variables are available, for instance, time series of river runoff from different catchments. The local catchment systems then share certain causal parents, such as time-dependent large-scale weather over all catchments, but differ in other catchment-specific drivers, such as the altitude of the catchment. These drivers can be called temporal and spatial contexts, respectively, and are often partially unobserved. Pooling the datasets and considering the joint causal graph among system, context, and certain auxiliary variables enables us to overcome such latent confounding of system variables. In this work, we present a non-parametric time series causal discovery method, J(oint)-PCMCI+, that efficiently learns such joint causal time series graphs when both observed and latent contexts are present, including time lags. We present asymptotic consistency results and numerical experiments demonstrating the utility and limitations of the method

Causal Discovery for time series from multiple datasets with latent contexts

Causal discovery from time series data is a typical problem setting across the sciences. Often, multiple datasets of the same system variables are available, for instance, time series of river runoff from different catchments. The local catchment systems then share certain causal parents, such as time-dependent large-scale weather over all catchments, but differ in other catchment-specific drivers, such as the altitude of the catchment. These drivers can be called temporal and spatial contexts, respectively, and are often partially unobserved. Pooling the datasets and considering the joint causal graph among system, context, and certain auxiliary variables enables us to overcome such latent confounding of system variables. In this work, we present a non-parametric time series causal discovery method, J(oint)-PCMCI+, that efficiently learns such joint causal time series graphs when both observed and latent contexts are present, including time lags. We present asymptotic consistency results and numerical experiments demonstrating the utility and limitations of the method

Clustering of causal graphs to explore drivers of river discharge

This work aims to classify catchments through the lens of causal inference and cluster analysis. In particular, it uses causal effects (CEs) of meteorological variables on river discharge while only relying on easily obtainable observational data. The proposed method combines time series causal discovery with CE estimation to develop features for a subsequent clustering step. Several ways to customize and adapt the features to the problem at hand are discussed. In an application example, the method is evaluated on 358 European river catchments. The found clusters are analyzed using the causal mechanisms that drive them and their environmental attributes

Conditional Independence Testing with Heteroskedastic Data and Applications to Causal Discovery

Conditional independence (CI) testing is frequently used in data analysis and machine learning for various scientific fields and it forms the basis of constraint-based causal discovery. Oftentimes, CI testing relies on strong, rather unrealistic assumptions. One of these assumptions is homoskedasticity, in other words, a constant conditional variance is assumed. We frame heteroskedasticity in a structural causal model framework and present an adaptation of the partial correlation CI test that works well in the presence of heteroskedastic noise, given that expert knowledge about the heteroskedastic relationships is available. Further, we provide theoretical consistency results for the proposed CI test which carry over to causal discovery under certain assumptions. Numerical causal discovery experiments demonstrate that the adapted partial correlation CI test outperforms the standard test in the presence of heteroskedasticity and is on par for the homoskedastic case. Finally, we discuss the general challenges and limits as to how expert knowledge about heteroskedasticity can be accounted for in causal discovery

Discovering Causal Relations and Equations from Data

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.Comment: 137 page

Vector Causal Inference between Two Groups of Variables

Methods to identify cause-effect relationships currently mostly assume the variables to be scalar random variables. However, in many fields the objects of interest are vectors or groups of scalar variables. We present a new constraint-based non-parametric approach for inferring the causal relationship between two vector-valued random variables from observational data. Our method employs sparsity estimates of directed and undirected graphs and is based on two new principles for groupwise causal reasoning that we justify theoretically in Pearl's graphical model-based causality framework. Our theoretical considerations are complemented by two new causal discovery algorithms for causal interactions between two random vectors which find the correct causal direction reliably in simulations even if interactions are nonlinear. We evaluate our methods empirically and compare them to other state-of-the-art techniques

Increasing Effect Sizes of Pairwise Conditional Independence Tests between Random Vectors

A simple approach to test for conditional independence of two random vectors given a third random vector is to simultaneously test for conditional independence of every pair of components of the two random vectors given the third random vector. In this work, we show that conditioning on additional components of the two random vectors that are independent given the third one increases the tests’ effect sizes while leaving the validity of the overall approach unchanged. We leverage this result to derive a practical pairwise testing algorithm that first chooses tests with a relatively large effect size and then does the actual testing. We show both numerically and theoretically that our algorithm outperforms standard pairwise independence testing and other existing methods if the dependence within the two random vectors is sufficiently high