Spline interpolation

A spline is a smooth numeric function that is piecewise defined by polynomial functions connected by points called knots. To interpolate data points with splines, polynomials are fitted piecewise resulting in a continuous curve that passes through each of the known data points. The degree of the polynomial is arbitrary but usually second or third orders polynomials are chosen to ensure smoothness, meaning that the spline has continuous derivatives of the first or second order.

Spline interpolation applied to age-specific data was illustrated by McNeil et al. (1977) [16] and based on Schoenberg (1964) [17]. When spline interpolation is used to estimate single-year age distributions from grouped data, the cumulative number of counts is interpolated since the only known values are those at the boundaries of each age group. If we observe data points (x_i,y_i) with i=1,..,n, where x_i correspond to the sequence of age intervals and y_i to the cumulative numbers of death up to age x_i, the spline function F(x) interpolates all points and consists of polynomials between each consecutive pair of knots x_i and x_i+1. Then, to obtain the death counts for each individual age group, one proceeds by differencing, i.e. f(x)=F^′(x)=F(x+1)−F(x), where f(x) stands for the single-year age-at-death distribution. Spline interpolation is extensively used. It is applied for example by the Human Mortality Database [7] to split aggregated death counts grouped into 5-years age classes, see [8] Appendix B of the protocol. However, the method does not provide reliable estimates for open-ended age groups since the spline function starts declining at old ages leading to erroneous death counts estimates by single year of age that are negative [5]. To overcome the problem Wilmoth et al. use parametric models to fit the late life span, see [8] Appendix C of the protocol. Smith et al. (2004) [5] instead propose and apply a monotonicity constraint, the Hyman filter [18], to cubic spline interpolation for ungrouping deaths of Australian females (see Figure 1 in Smith et al. (2004) [5]). Another method that ensures non-negative values is the piecewise cubic Hermite interpolating polynomial used in the Human Fertility Database [19–21].