Estimating missing data in hierarchical space-time series with a short temporal extent

A challenging problem exists in the estimation of missing space-time data where the time series are relatively short, and the space series belong to a spatial hierarchy. An example is provided by the population estimates for regions belonging to the NUTS hierarchy which are available from the EUROSTAT data portal. The table demo_r_gind3 provides estimates of the population of NUTS0/1/2/3 regions at the 1st January 2000…2012 inclusive. Inspection of the table reveals that estimates are missing for 2000-2003 for two of the five NUTS3 regions in the NUTS2 region of Liège. There are other instances of missing data at NUTS3 where there are data for the corresponding higher level NUTS regions. The EUROSTAT table demo_r_d2jan provides estimates of the population on the 1st January for a longer time period, 1990…2012 inclusive, but these are only to NUTS2. Again, there is missing data. The question then arises as to whether it is possible to estimate the missing series. The NUTS2 values act as a constraint on the NUTS3 values – the total population of the NUTS3 regions should equal those of the corresponding NUTS2 regions. However, the relative shortness of the available series is a challenge if conventional methods of time series analysis are adopted. Furthermore, the imposition of the spatial constraints is both a check as well as a challenge

The indecomposable symplectic and quadratic modules of the Klein-four group

We determine all indecomposable symplectic modules for the Klein-four group K4 over a perfect field of characteristic 2 and classify the symplectic forms up to isometry. We also determine all K4-invariant quadratic forms which polarize to a given symplectic form and classify such quadratic forms up to isometr