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.

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.

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.