Causal inference uses observational data to infer the causal structure of the data generating system. We study a class of restricted Structural Equation Models for time series that we call Time Series Models with Independent Noise (TiMINo). These models require independent residual time series, whereas traditional methods like Granger causality exploit the variance of residuals. This work contains two main contributions: (1) Theoretical: By restricting the model class (e.g. to additive noise) we provide general identifiability results. They cover lagged and instantaneous effects that can be nonlinear and unfaithful, and non-instantaneous feedbacks between the time series.

Time-series data are ubiquitous in the real world, often comprising a series of data points recorded at varying time intervals. Understanding the underlying structures between variables associated with temporal processes is of paramount importance for numerous real-world applications (Spirtes et al., 2000; Berzuini et al., 2012; Peters et al., 2017). Although randomised experiments are considered the gold standard for unveiling such relationships, they are frequently hindered by factors such as cost and ethical concerns. Structure learning seeks to infer hidden structures from purely observational data, offering a powerful approach for a wide array of applications (Bellot et al., 2021; Löwe et al., 2022; Runge, 2018; Tank et al., 2021; Pamfil et al., 2020; Gong et al., 2022). However, many existing structure learning methods for time series are inherently discrete, assuming that the underlying temporal processes are discretized in time and requiring uniform sampling intervals throughout the entire time range. Consequently, these models face two key limitations: (i) they may misrepresent the true underlying process when it is continuous in time, potentially leading to incorrect inferred relationships; and (ii) they struggle with handling irregular sampling intervals, which frequently arise in fields such as biology (Trapnell et al., 2014; Qiu et al., 2017; Qian et al., 2020) and climate science (Bracco et al., 2018; Raia, 2008). Although there exists a previous work (Bellot et al., 2021) that also tries to infer the underlying structure from the continuous-time perspective, its framework based on ordinary differential equations (ODE)

However, due to the limited recording capabilities Learning causal structure among event types from and storage capacities, retaining event's occurred times discrete-time event sequences is a particularly important with high-resolution is expensive or practically impossible in but challenging task. Existing methods, such many real-world applications, and we usually only can access as the multivariate Hawkes processes based methods, the corresponding discrete-time event sequences. For example, mostly boil down to learning the so-called in large wireless networks, the event sequences are usually Granger causality which assumes that the cause logged at a certain frequency by different devices whose event happens strictly prior to its effect event. Such time might not be accurately synchronized. As a result, lowresolution an assumption is often untenable beyond applications, discrete-time event sequences are obtained and the especially when dealing with discrete-time temporal precedence assumption will be frequently violated event sequences in low-resolution; and typical discrete in discrete-time event sequences, which raises a serious identifiability Hawkes processes mainly suffer from identifiability issue of causal discovery. For example, as shown issues raised by the instantaneous effect, in Figure 1, there are three event sequences produced by three i.e., the causal relationship that occurred simultaneously event types v

Causal representation learning is the task of identifying the underlying causal variables and their relations from high-dimensional observations, such as images. Recent work has shown that one can reconstruct the causal variables from temporal sequences of observations under the assumption that there are no instantaneous causal relations between them. In practical applications, however, our measurement or frame rate might be slower than many of the causal effects. This effectively creates "instantaneous" effects and invalidates previous identifiability results. To address this issue, we propose iCITRIS, a causal representation learning method that allows for instantaneous effects in intervened temporal sequences when intervention targets can be observed, e.g., as actions of an agent. iCITRIS identifies the potentially multidimensional causal variables from temporal observations, while simultaneously using a differentiable causal discovery method to learn their causal graph. In experiments on three datasets of interactive systems, iCITRIS accurately identifies the causal variables and their causal graph.

Discovering causal relationships between different variables from time series data has been a long-standing challenge for many domains such as climate science, finance, and healthcare. Given the complexity of real-world relationships and the nature of observations in discrete time, causal discovery methods need to consider non-linear relations between variables, instantaneous effects and history-dependent noise (the change of noise distribution due to past actions). However, previous works do not offer a solution addressing all these problems together. In this paper, we propose a novel causal relationship learning framework for time-series data, called Rhino, which combines vector auto-regression, deep learning and variational inference to model non-linear relationships with instantaneous effects while allowing the noise distribution to be modulated by historical observations. Theoretically, we prove the structural identifiability of Rhino. Our empirical results from extensive synthetic experiments and two real-world benchmarks demonstrate better discovery performance compared to relevant baselines, with ablation studies revealing its robustness under model misspecification.

Causal inference uses observational data to infer the causal structure of the data generating system. We study a class of restricted Structural Equation Models for time series that we call Time Series Models with Independent Noise (TiMINo). These models require independent residual time series, whereas traditional methods like Granger causality exploit the variance of residuals. This work contains two main contributions: (1) Theoretical: By restricting the model class (e.g. to additive noise) we provide more general identifiability results than existing ones. The results cover lagged and instantaneous effects that can be nonlinear and unfaithful, and non-instantaneous feedbacks between the time series. (2) Practical: If there are no feedback loops between time series, we propose an algorithm based on non-linear independence tests of time series. When the data are causally insufficient, or the data generating process does not satisfy the model assumptions, this algorithm may still give partial results, but mostly avoids incorrect answers. The Structural Equation Model point of view allows us to extend both the theoretical and the algorithmic part to situations in which the time series have been measured with different time delays (as may happen for fMRI data, for example). TiMINo outperforms existing methods on artificial and real data. Code is provided.

Causal inference uses observations to infer the causal structure of the data generating system. We study a class of functional models that we call Time Series Models with Independent Noise (TiMINo). These models require independent residual time series, whereas traditional methods like Granger causality exploit the variance of residuals. There are two main contributions: (1) Theoretical: By restricting the model class (e.g. to additive noise) we can provide a more general identifiability result than existing ones. This result incorporates lagged and instantaneous effects that can be nonlinear and do not need to be faithful, and non-instantaneous feedbacks between the time series. (2) Practical: If there are no feedback loops between time series, we propose an algorithm based on non-linear independence tests of time series. When the data are causally insufficient, or the data generating process does not satisfy the model assumptions, this algorithm may still give partial results, but mostly avoids incorrect answers. An extension to (non-instantaneous) feedbacks is possible, but not discussed. It outperforms existing methods on artificial and real data. Code can be provided upon request.