  Functional data analysis is an important area in modern statistics and has
been successfully applied in many fields. Although many scientific studies aim
to find causations, a predominant majority of functional data analysis
approaches can only reveal correlations. In this paper, average treatment
effect estimation is studied for observational data with functional covariates.
This paper generalizes various state-of-art propensity score estimation methods
for multivariate data to functional data. The resulting average treatment
effect estimators via propensity score weighting are numerically evaluated by a
simulation study and applied to a real-world dataset to study the causal effect
of duloxitine on the pain relief of chronic knee osteoarthritis patients.
