# A Causal View on

Dynamical Systems

Workshop @ NeurIPS 2022

December 3, 2022, New Orleans, USA

## Overview

Multivariate time series are a common data modality in many scientific fields, such as physics, genomics, neuroscience or economics. A major goal in these disciplines is to answer causal questions, such as gene regulatory network inference in biology or identifying causal drivers of extreme (compound) weather events in climate science, which requires the inference of causal relationships in dynamical systems. Similar to the static setting, accessibility and feasibility restrictions as well as ethical considerations complicate direct experimentation and measurement of such relations in time-dependent settings.

While certain properties of time series may aid practical causal inference, such as the arrow of time, most identifiability results for causal discovery as well as cause effect estimation are restricted to the static setting (as in structural causal models or potential outcomes). These results typically do not extend to temporally evolving dynamics. Consequently, natural sciences mostly resort to continuous-time dynamical systems such as ODEs and PDEs to describe hypothesized mechanisms underlying the data generating processes. While such models also allow for causal interpretations as well as predictions under hypothetical interventions, data-driven discovery of continuous-time dynamical systems remains a challenging and relatively underexplored area of research.

In this workshop, we bring together researchers in dynamical systems, time-series methods, causality, infinite-depth neural networks, and machine learning. We believe a side-by-side discussion of dynamical systems and causal inference (discovery and estimation) will allow to develop novel approaches, transfer expertise across communities, and enable us to overcome current limitations of each perspective individually. Connections to other scientific disciplines as well as practitioners’ perspectives will be highlighted to showcase successful applications of causal inference in dynamical settings.

## Important dates

**Sept. 2****6, 2022 (****11:59 pm AoE)**: Paper contribution deadline**Oct. 20, 2022**: Author notification**Dec. 03, 2022**: Date of Workshop

## Invited Talks

**Atalanti Mastakouri ** (Amazon Web Services)

**Causal Feature Selection in Time-Series Data**Knowing the causal graph that underlies the variables of a system is of major importance in several domains, such as biology, earth science and finance. Nevertheless, in the absence of randomized controlled trials and domain knowledge, causal discovery of the full-time graph in time series settings with latent confounders is a rather challenging problem. In such cases we suggest to focus on a slightly smaller task, that of identification of causal features of a target node, not caring to recover the relationships between the candidate causes themselves, but at the same time not constraining them. In this talk, I will present a conditional independence based method which, by treating the time series as discreet time stamps of a Markov chain, and by assuming that the target cannot cause the potential causes, allows us to identify the set of true unconfounded causes of a target time series. I will explain the fundamentals of the method and provide results from application on two real datasets.

**Biwei Huang** (University of California San Diego)

**Causal discovery from nonstationary time series**It is commonplace to encounter nonstationary time series, of which the underlying generating process changes over time. Such a distribution shift feature presents both challenges and opportunities for causal discovery. In this talk, I will discuss two ways for causal discovery from nonstationary time series. One way is to leverage time index, together with kernel representations, to capture smoothly changing (unmeasured) change factors. The other is by modeling (unmeasured) change factors with the Markov process. We further show that causality and nonstationarity are coupled. Nonstationarity provides additional information for causal structure determination, and causal models describing how the distribution change help forecasting and other downstream tasks.

**Cristopher Salvi** (Imperial College London)

**Signature Kernel Methods**Kernel methods provide a rich and elegant framework for a variety of learning tasks including supervised learning, hypothesis testing, Bayesian inference, generative modelling and scientific computing. Sequentially ordered information often arrives in the form of complex streams taking values in non-trivial ambient spaces (e.g. a video is a sequence of images). In these situations, the design of appropriate kernels is a notably challenging task. In this talk, I will outline how rough path theory, a modern mathematical framework for describing complex evolving systems, allows to construct a family of characteristic kernels on pathspace known as signature kernels. I will then present how signature kernels can be used to develop a variety of algorithms such as two-sample hypothesis and (conditional) independence tests for stochastic processes, generative models for time series and numerical methods for path-dependent PDEs.

**Joris M. Mooij** (University of Amsterdam)

**Towards Markov Properties for Continuous-Time Dynamical Systems**Continuous-time dynamical systems are often modelled by differential-algebraic equations (DAEs), systems of equations in which first and higher-order derivatives may appear. Well-known examples are chemical reactions, electric circuits and constrained mechanical systems. DAEs generalize systems of (ordinary, algebraic) equations as well as ordinary differential equations. Stochasticity can be introduced via exogenous variables and initial states. However, DAEs have no intrinsic notion of causality, unlike for example structural (dynamical) causal models. We show how the structure of the equations nonetheless gives rise to Markov properties by means of Simon's causal ordering algorithm, thereby extending well-known Markov properties for directed graphical models to continuous-time dynamical systems represented by DAEs and their bipartite graphical structure. As an immediate consequence, we obtain possible causal interpretations of the DAE as well as formal causal reasoning methods for domain adaptation that extend Pearl's do-calculus. Our approach also provides the building blocks towards generalizing identification methods (such as Tian's ID algorithm) and constraint-based causal discovery (such as the FCI algorithm) towards a very general class of (possibly) cyclic and (possibly) dynamical systems.

**Karl Friston** (University College London)

**Dynamic Causal Modelling**In recent years, dynamic causal modelling has become established in the analysis of medical timeseries. In this talk, I will review the basic idea behind dynamic causal modelling; namely, to equip a static forward model with a state-space model that embodies interactions within and between populations of molecules, neurons or people. The model is then inverted using straightforward variational techniques, usually under Gaussian assumptions about random effects (i.e., variational Laplace). I hope to showcase the application of dynamic causal modelling to epilepsy and epidemiological modelling. The underlying idea is to test hypotheses about abnormalities in dynamic coupling and thereby identify the causal architecture of the system at hand, which can then be used for simulation or forecasting.

**Vanessa Didelez**

(University of Bremen,

Leibniz Institute for Prevention Research and Epidemiology - BIPS,

SFB 1320: Everyday Activity Science and Engineering (EASE))

**Causal and Graphical Models for Continuous-Time Event Data**In this work, we connect the early work of Robins (1986) on time-varying confounding and g-computation with dynamic models; where we use the equivalent identification approach based on inverse probability of treatment weighting which can also be seen as a change of measure technique. Causal effects in continuous-time event data are formalised as stochastic interventions that change the intensities of certain types of events. Finally, folded graphs (across time) with an asymmetric notion of separation are used to determine identifiability of causal effects in these settings by characterising possible latent processes as eliminable. An application to HPV-tesing serves as illustration.

## Organizers

**Sören Becker**Helmholtz AI

**Alexis Bellot**DeepMind

**Cecilia Casolo**Helmholtz AI

**Niki Kilbertus**

TU Munich

Helmholtz AI

**Sara Magliacane **UvA

## Workshop sponsor

## Contact

In case of questions feel free to reach out to **causaldyn.neurips22@gmail.com**.