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Nonparametric graduation techniques as a common framework for the description of demographic patterns

therefore serve in order to provide a clear description of the real shape of the various age-specific
patterns, and consequently provide a real basis for population analysis and projections. The classical
way to graduate empirical demographic rates is to fit a model that presents these rates as a paramet-
ric function of age. For the graduation of the age-specific rates of each one of the three demographic
phenomena, specific parametric models have been proposed.
Several parametric models have been proposed for the graduation of the age-specific mortality
rates and many authors have contributed to the problem of estimating their parameters (Heligman
and Pollard, 1980; Keyfitz, 1982; Forfar, McCutcheon, and Wilkie, 1988; Kostaki, 1992; Hannerz,
1999; Karlis and Kostaki, 2000). A variety of models presenting the empirical age-specific fertility
rates as a parametric function of age have also been proposed for the graduation of the age-specific
fertility rates. A thorough description of these models is provided by Kostaki and Peristera (2007).
Finally, for the description of nuptiality patterns alternative parametric models have been proposed
(Coale and McNeil, 1972; Liang, 2000). However, for graduation purposes, a possible way to
smooth demographic rates is the utilization of non-parametric smoothing techniques. Kernels have
already been used for graduating mortality patterns (Copas and Haberman, 1983; Gavin, Haberman,
and Verrall, 1993; Gavin, Haberman, and Verrall, 1994; Felipe, Guillen, and Nielsen, 2000). An
evaluation of kernels as tools for graduating mortality patterns is provided by Peristera and Kostaki
(2005).
An alternative nonparametric way for graduating age-specific demographic rates would be the
utilization of Support Vector Machines (SVM). These techniques appeared in 1995 in the framework
of Vapnik’s Statistical Learning Theory (Vapnik, 1995; Moguerza and Muñoz, 2006) for classifica-
tion and regression purposes. In particular, SVM have been used in a number of applications
(Chongfuangprinya, Kim, Park et al., 2011; Erdogan, 2013). They have been also used successfully
for smoothing noisy data such as neighbourhood curves (Muñoz and Moguerza, 2005) and nonlinear
profiles (Moguerza et al., 2007). Therefore, they can a priori be considered as a promising tool for
demographic graduation tasks. In addition, the use of SVMs is affordable by practitioners with a lack
of advanced statistical or computational skills. The reason is that documentation at all levels is
available through the Internet and new libraries and easy-to-use software are continuously being de-
1
2
veloped (see Weka or the software package known as “R” ).
The focus of this paper is to evaluate and compare the performance of kernels and SVMs for
graduation purposes of demographic rates for each one of the three basic demographic phenomena.
Both kernels and SVMs have been adjusted and applied to empirical data sets of mortality, fertility,
and nuptiality rates of a variety of populations and years. In particular, a cross-validation approach
has been conducted for the SVM models and a plug-in technique has been used for kernel models, in
order to fit their corresponding parameters. For comparison purposes parametric models are also
fitted to the same empirical data sets. In the next section, a short description of existing parametric
models for fitting mortality, fertility, and nuptiality data is provided. Sections 3 and 4 are devoted to
a presentation of kernels and SVMs, respectively. Then, Section 5 provides the results of our calcu-
lations in order to assess the utilization of kernels and SVM techniques as tools for estimating
age-specific mortality, fertility, and nuptiality patterns. Some concluding remarks and some issues
for further research are given in Section 6.
2 Parametric Models

2.1 Mortality Models
A wide variety of mortality laws has been presented in the literature (Brass, 1971; Mode and Busby,
1982) since the first attempt by de Moivre in 1725. Among all of these laws, the most successful
attempt to describe the age-specific mortality pattern for the total life span through a parametric

1 https://weka.wikispaces.com/LibSVMor
2 https://en.wikibooks.org/wiki/Data_Mining_Algorithms_In_R/Classification/SVM

2 International Journal of Population Studies | 2016, Volume 2, Issue 1