Spatially multi-scale dynamic factor modeling via sparse estimation

In many spatio-temporal data, their spatial variations have inherent global and local structures. The spatially continuous dynamic factor model (SCDFM) decomposes the spatio-temporal data into a small number of spatial and temporal variations, where the spatial variations are represented by the factor loading (FL) functions. However, the FL functions estimated by the maximum likelihood or maximum [Formula: see text] penalized likelihood capture global structures but do not capture local structures. We propose a method for estimating the spatially multiscale FL functions using a sparse penalty. To overcome the problems of existing sparse penalties, we propose the adaptive graph lasso (AGL) penalty. The method with the AGL penalty eliminates redundant basis functions contained in the FL functions, and leads to the FL functions having global and localized structures. We derive the EM algorithm with block coordinate descent that enables us to maximize the AGL penalized log-likelihood stably. Applications to synthetic and real data show that the proposed modeling procedure accurately extract not only the spatially global structures but also spatially local structures, which the [Formula: see text] penalized estimation do not extract.