  This paper focuses on the analysis of spatially correlated functional data.
The between-curve correlation is modeled by correlating functional principal
component scores of the functional data. We propose a Spatial Principal
Analysis by Conditional Expectation framework to explicitly estimate spatial
correlations and reconstruct individual curves. This approach works even when
the observed data per curve are sparse. Assuming spatial stationarity,
empirical spatial correlations are calculated as the ratio of eigenvalues of
the smoothed covariance surface $Cov(X_i(s),X_i(t))$ and cross-covariance
surface $Cov(X_i(s), X_j(t))$ at locations indexed by $i$ and $j$. Then a
anisotropy Mat\'ern spatial correlation model is fit to empirical correlations.
Finally, principal component scores are estimated to reconstruct the sparsely
observed curves. This framework can naturally accommodate arbitrary covariance
structures, but there is an enormous reduction in computation if one can assume
the separability of temporal and spatial components. We propose hypothesis
tests to examine the separability as well as the isotropy effect of spatial
correlation. Simulation studies and applications of empirical data show
improvements in the curve reconstruction using our framework over the method
where curves are assumed to be independent. In addition, we show that the
asymptotic properties of estimates in uncorrelated case still hold in our case
if 'mild' spatial correlation is assumed.
