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A spline primer

spline is a mathematical function.  They are used wherever flexibility and smoothness are required, from computer-aided design and cartoon graphics, to the graduation of mortality tables (McCutcheon, 1974).  There are numerous different types of spline, but the most common is the spline proposed by Schoenberg (1964).  Figure 1 shows Schoenberg splines of degrees 0–3, all of which start in 2015:

Figure 1. Schoenberg (1964) splines of degree 0–3 with first non-zero value from 2015.

Written by: Stephen RichardsTags: Filter information matrix by tag: splines

Back to the future with Whittaker smoothing

Many actuaries will be familiar with Whittaker smoothing (1923) but few will be aware of the close connection between this early method and the method of P-splines.
Written by: Iain CurrieTags: Filter information matrix by tag: Whittaker smoothing, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines, Filter information matrix by tag: penalty function

Forecasting with penalty functions - Part III

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.

Written by: Iain CurrieTags: Filter information matrix by tag: forecasting, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines, Filter information matrix by tag: penalty function, Filter information matrix by tag: mortality crossover

Forecasting with penalty functions - Part II

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.

Written by: Iain CurrieTags: Filter information matrix by tag: forecasting, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines, Filter information matrix by tag: penalty function, Filter information matrix by tag: mortality crossover

Forecasting with penalty functions - Part I

There is much to say on the topic of penalty forecasting, so this is the first of three blogs. In this blog we will describe penalty forecasting in one dimension; this will establish the basic ideas. In the second blog we will discuss the case of most interest to actuaries: two-dimensional forecasting. In the final blog we will discuss some of the properties of penalty forecasting in two dimensions.

Written by: Iain CurrieTags: Filter information matrix by tag: forecasting, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines, Filter information matrix by tag: penalty function

Effective dimension

Actuaries often need to smooth mortality rates. Gompertz (1825) smoothed mortality rates by age and his famous law was a landmark in this area. Figure 1 shows the Gompertz model fitted to CMI assured lives data for ages 20–90 in the year 2002. The Gompertz Law usually breaks down below about age 40 and a more general smooth curve would be appropriate. However, a more general smooth curve would obviously require more parameters than the two for the simple Gompertz model.

Written by: Iain CurrieTags: Filter information matrix by tag: effective dimension, Filter information matrix by tag: splines, Filter information matrix by tag: P-splines

Graduation

Graduation is the process whereby smooth mortality rates are created from crude mortality rates.  Smoothness is an important part of graduation, but another is the extrapolation of mortality rates to ages at which data may be unreliable or even non-existent.
Written by: Stephen RichardsTags: Filter information matrix by tag: graduation, Filter information matrix by tag: extrapolation by age, Filter information matrix by tag: smoothing, Filter information matrix by tag: splines