  Measurements of systems taken along a continuous functional dimension, such
as time or space, are ubiquitous in many fields, from the physical and
biological sciences to economics and engineering.Such measurements can be
viewed as realisations of an underlying smooth process sampled over the
continuum. However, traditional methods for independence testing and causal
learning are not directly applicable to such data, as they do not take into
account the dependence along the functional dimension. By using specifically
designed kernels, we introduce statistical tests for bivariate, joint, and
conditional independence for functional variables. Our method not only extends
the applicability to functional data of the HSIC and its d-variate version
(d-HSIC), but also allows us to introduce a test for conditional independence
by defining a novel statistic for the CPT based on the HSCIC, with optimised
regularisation strength estimated through an evaluation rejection rate. Our
empirical results of the size and power of these tests on synthetic functional
data show good performance, and we then exemplify their application to several
constraint- and regression-based causal structure learning problems, including
both synthetic examples and real socio-economic data.
