  Spatiotemporal gene expression data of the human brain offer insights on the
spa- tial and temporal patterns of gene regulation during brain development.
Most existing methods for analyzing these data consider spatial and temporal
profiles separately with the implicit assumption that different brain regions
develop in similar trajectories, and that the spatial patterns of gene
expression remain similar at different time points. Al- though these analyses
may help delineate gene regulation either spatially or temporally, they are not
able to characterize heterogeneity in temporal dynamics across different brain
regions, or the evolution of spatial patterns of gene regulation over time. In
this article, we develop a statistical method based on low rank tensor
decomposition to more effectively analyze spatiotemporal gene expression data.
We generalize the clas- sical principal component analysis (PCA) which is
applicable only to data matrices, to tensor PCA that can simultaneously capture
spatial and temporal effects. We also propose an efficient algorithm that
combines tensor unfolding and power iteration to estimate the tensor principal
components, and provide guarantees on their statistical performances. Numerical
experiments are presented to further demonstrate the mer- its of the proposed
method. An application of our method to a spatiotemporal brain expression data
provides insights on gene regulation patterns in the brain.
