  Massive data bring the big challenges of memory and computation for analysis.
These challenges can be tackled by taking subsamples from the full data as a
surrogate. For functional data, it is common to collect multiple measurements
over their domains, which require even more memory and computation time when
the sample size is large. The computation would be much more intensive when
statistical inference is required through bootstrap samples. To the best of our
knowledge, this article is the first attempt to study the subsampling method
for the functional linear model. We propose an optimal subsampling method based
on the functional L-optimality criterion. When the response is a discrete or
categorical variable, we further extend our proposed functional L-optimality
subsampling (FLoS) method to the functional generalized linear model. We
establish the asymptotic properties of the estimators by the FLoS method. The
finite sample performance of our proposed FLoS method is investigated by
extensive simulation studies. The FLoS method is further demonstrated by
analyzing two large-scale datasets: the global climate data and the kidney
transplant data. The analysis results on these data show that the FLoS method
is much better than the uniform subsampling approach and can well approximate
the results based on the full data while dramatically reducing the computation
time and memory.
