Abstract
Spline interpolations and the examples of their application to demographic data were introduced first by H. S. Shryock, J. S. Siegel et al. and then D. R. McNeil et al. Though these are very useful, no further detailed introduction has been done. But there are many other formulas of interpolation and curve fitting with spline functions. From these, the one-and-two-dimensional useful formulas and examples of their applications to demographic data will be introduced in a somewhat orderly form. Formulas introduced here are the following ones. (a) Interpolation and curve fitting with a cubic spline function. (b) Interpolation and curve fitting with B-spline functions. (c) Interpolation with a rational spline function by Spath. (d) Quasi Hermite interpolation with a piecewise cubic polynomial by Akima. (e) Curve fitting with a piecewise cubic polynomial (automatic method) by Yoshimoto et al. The formula (c) is the one for one-dementional data. In (d) and (e), the continuity of interpolation functions and their derivatives are assurned. (b) is comprehensive and most useful, but it is not easy for us to use, because users should be skilled in deciding the number and position of knots. So its computing procedure should be done with big computers. (a) is a special case of (b) with a polynomial of degree three, and is widely used because of its moderate precision and easiness for use. But (a) may provide us unfavourable results for certain specific data. In such cases, formulas (c)-(e) may be used instead of (a). Examples of application are as follows. (1) Interpolating and smoothing values of life table functions, e.g. survivorship function (l_x) and mortality rates (q_x). (2) Interpolating births by five-year age groups of mothers into single ages. Angular transformation and (c) are used for it. (3) Interpolating cause-specific deaths by five-year age groups into single ages. (a) is usually used, but (c) and (d) may be necessary for deaths by certain specific causes of death. (4) Interpolating and smoothing two-demensional demographic data e. g. mortality rates and expectations of life by year and age. Depicting spatial patterns of demographic data using contour map and spline surface. It should be noted that a certain model explaining the data (one-or two-demensional) is taken into consideration whenever these formulas are used. It is interesting that we apply the formulas of interpolation or curve fitting with two-demensional B-spline function of degree k in x and l in x (not necessarily k=l) to two-demensional demographic data. In interpolating population by year and age as a two-demensional demographic data, interpolation by birth cohort may have to be taken into account.
