Bilinear modulation models for seasonal tables of counts

We propose generalized linear models for time or age-time tables of seasonal counts, with the goal of better understanding seasonal patterns in the data. The linear predictor contains a smooth component for the trend and the product of a smooth component (the modulation) and a periodic time series of arbitrary shape (the carrier wave). To model rates, a population offset is added. Two-dimensional trends and modulation are estimated using a tensor product B-spline basis of moderate dimension. Further smoothness is ensured using difference penalties on the rows and columns of the tensor product coefficients. The optimal penalty tuning parameters are chosen based on minimization of a quasi-information criterion. Computationally efficient estimation is achieved using array regression techniques, avoiding excessively large matrices. The model is applied to female death rate in the US due to cerebrovascular diseases and respiratory diseases.