### Forecasting with penalty functions - Part III

#### (Jun 2, 2015)

This is the last of my three blogs on forecasting with penalties. I discussed the 1-d case in the first blog and the 2-d case in the second. Here we discuss some of the properties of 2-d forecasting. Some readers may find some of my remarks surprising, even paradoxical.

In our first blog we used the Lee-Carter model as an example where a time series is used to forecast mortality. The method is (a) estimate the parameters in the model by fitting the model to suitable data and (b) forecast a subset of the parameters with a suitable time series. The fit to data, by definition, does not depend on the forecast horizon. This is a familiar and attractive property; we will refer to this as the invariance property. It is easy to overlook…

### Forecasting with penalty functions - Part II

#### (Mar 18, 2015)

Our first blog in this series of three looked at forecasting log mortality with penalties in one dimension, i.e. forecasting with data for a single age. We now look at the same problem, but in two dimensions. Figure 1 shows our data. We see an irregular surface sitting on top of the age-year plane. Just as in the 1-d case, we see an underlying smooth surface, and it is this surface that we wish both to estimate and to forecast.

Figure 1: Crude log mortality rates for Australian males ages 50-90 over 1960-2010.

In the 1-d case we saw (a) how a basis of B-splines sat under the data, and (b) that each B-spline had a coefficient associated with the peak of its B-spline. The same idea applies in 2-d.  Figure 2 shows (for a reduced…