Smoothing Continuous Functions

We said that rates and covariate multipliers are continuous functions of age and time. It takes a little work to parametrize an interpolated function of age and time.

• You have to tell it where the control points are. In Cascade, we call this the AgeTimeGrid. It’s a list of ages and a list of times that define a rectangular grid.

• At each of the control points of the age time grid, Dismod-AT will evaluate how close the rate or covariate multiplier is to some reference value. At these points, we define prior distributions. Cascade makes these value priors part of the PriorGrid.

• It’s rare to have data points that are dense across all of age and time. Dismod-AT needs to take a data point at one end, a data point at the other end, and draw a line that connects them. We help it by introducing constraints on how quickly a value can change over age and time. These are a kind of regularization of the problem, called age-time difference priors. They apply to the difference in value between one age-time point and the next greater in age and the next-greater in time. As with value priors, these are specified in the Cascade as part of the PriorGrid.

The random effect for locations is also a continuous quantity.