Anomaly detection in multivariate timeseries is used in various domains, such as finance, IT, or aerospace, to identify irregular behavior in the used applications. Prior research in anomaly detection has focused on estimating the joint probability of all variables. Then, anomalies are scored based on the probability they receive. Thereby, the variables' dependencies are only considered implicitly. This work follows recent work in anomaly detection that integrates information about the causal relations between the variables in the timeseries into the detection mechanism. The causal mechanisms of the variables are then used to identify anomalies. An observation is identified as anomalous if at least one of the variables it contains deviates from its regular causal mechanism. These regular causal mechanisms are estimated via the conditional distribution of a variable given its causal parent variables, i.e., the variables having a causal influence on a variable. We further develop previous work by gathering information about the causal parents of the variables by applying causal discovery algorithms adapted to the timeseries setting. We apply Conditional Kernel Density Estimation and Conditional Variational Autoencoders to estimate the conditional probabilities. With this causal approach, we outperform methods that rely on the joint probability of the variables in our synthetically generated datasets and the C-MAPPS dataset, which provides simulation data of turbofan engines. Moreover, we investigate the causal approach’s inferred scores on the C-MAPPS dataset to gather insights into the measurements responsible for the prediction of anomalies. Furthermore, we investigate the influence of deviations from the true causal graph on the anomaly detection performance using synthetic data.

Systems with variations of the underlying generating mechanism between different contexts, i.e., different environments or internal states in which the system operates, are common in the real world, such as soil moisture regimes in Earth science. Besides understanding the shared properties of the system, in practice, the question of context-specific properties, i.e., the change in causal relationships between contexts, arises. For real-world data, contexts are often driven by system variables, e.g., precipitation highly influences soil moisture. Nevertheless, this setup needs to be studied more. To account for such endogenous contexts in causal discovery, our work proposes a constraint-based method that can efficiently discover context-specific causal graphs using an adaptive testing approach. Our approach tests conditional independence on the pooled datasets to infer the dependence between system variables, including the context, to avoid introducing selection bias. To yield context-specific insights, conditional independence is tested on context-specific data. We work out the theoretical framework for this adaptive testing approach and give a detailed discussion of the connection to structural causal models, including sufficiency assumptions, which allow to prove the soundness of our algorithm and to interpret the results causally. A simulation study to evaluate numerical properties shows that our approach behaves as expected, but also leads to a further understanding of current limitations and viable extensions.

Structural Causal Models (SCMs) are an important tool in causal inference. They induce a graph and if the graph is acyclic, a unique observational distribution. A standard result states that in this acyclic case, the induced observational distribution satisfies a d-separation global Markov property relative to the induced graph. Time series can also be modelled like SCMs: One just interprets the stochastic difference equations that a time series solves as structural equations. However, technical problems arise when time series "start" at minus infinity. In particular, a d-separation global Markov property for time series and the corresponding infinite graphs, the so-called full time graphs, has thus far only been shown for stable vector autoregressive processes with independent finite-second-moment noise. In this paper, we prove a much more general version of this Markov property. We discuss our assumptions and study violations of them. Doing so hints at several pitfalls at the intersection of time series analysis and causal inference. Moreover, we introduce a new projection procedure for these infinite graphs which might be of independent interest.

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.

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 github.com/jakobrunge/tigramite we provide Python code for all methods involved in the simulation studies.