  It is commonplace to encounter heterogeneous or nonstationary data, of which
the underlying generating process changes across domains or over time. Such a
distribution shift feature presents both challenges and opportunities for
causal discovery. In this paper, we develop a framework for causal discovery
from such data, called Constraint-based causal Discovery from
heterogeneous/NOnstationary Data (CD-NOD), to find causal skeleton and
directions and estimate the properties of mechanism changes. First, we propose
an enhanced constraint-based procedure to detect variables whose local
mechanisms change and recover the skeleton of the causal structure over
observed variables. Second, we present a method to determine causal
orientations by making use of independent changes in the data distribution
implied by the underlying causal model, benefiting from information carried by
changing distributions. After learning the causal structure, next, we
investigate how to efficiently estimate the "driving force" of the
nonstationarity of a causal mechanism. That is, we aim to extract from data a
low-dimensional representation of changes. The proposed methods are
nonparametric, with no hard restrictions on data distributions and causal
mechanisms, and do not rely on window segmentation. Furthermore, we find that
data heterogeneity benefits causal structure identification even with
particular types of confounders. Finally, we show the connection between
heterogeneity/nonstationarity and soft intervention in causal discovery.
Experimental results on various synthetic and real-world data sets (task-fMRI
and stock market data) are presented to demonstrate the efficacy of the
proposed methods.
