Simplified Smoothing Splines for APC Models

Smoothing splines are splines fit including a roughness penalty. They can be used across groups of variables in regression models to produce more parsimonious models with improved accuracy. For APC (age-period-cohort) models, the variables in each direction can be numbered sequentially 1:N, which simplifies spline fitting. Further simplification is proposed using a different roughness penalty. Some key calculations then become closed-form, and numeric optimization for the degree of smoothing is simpler. Further, this allows the entire estimation to be done simply in MCMC for Bayesian and random-effects models, improving the estimation of the smoothing parameter and providing distributions of the parameters (or random effects) and the selection of the spline knots.