  Gaussian processes are a fundamental statistical tool used in a wide range of
applications. In the spatio-temporal setting, several families of covariance
functions exist to accommodate a wide variety of dependence structures arising
in different applications. These parametric families can be restrictive and are
insufficient in some situations. In contrast, process convolutions represent a
flexible, interpretable approach to defining the covariance of a Gaussian
process and have modest requirements to ensure validity. We introduce a
generalization of the process convolution approach that employs multiple
convolutions sequentially to form a "process convolution chain." In our
proposed multi-stage framework, complex dependencies that arise from a
combination of different interacting mechanisms are decomposed into a series of
interpretable kernel smoothers. We demonstrate an application of process
convolution chains to model killer whale movement, in which the paths taken by
multiple individuals are not independent, but reflect dynamic social
interactions within the population. Our proposed model for dependent movement
provides inference for the latent dynamic social structure in the study
population. Additionally, by leveraging the positive dependence among
individual paths, we achieve a reduction in uncertainty for the estimated
locations of the whales, compared to a model that treats paths as independent.
