Research Methods: Visualization in the Numeric Solution of Complex Nonlinear Equations in Mathematical Demography

Abstract

Searching for numeric solutions to complex nonlinear equations is a resource-consuming, risky business. For a given equation, there can be an infinite number of often dramatically quite different solutions meeting a specified goodness-of-fit criterion. Although the leading statistical packages have nonlinear regression procedures, these procedures all require "seeding" with initial estimates of equation parameters. Often one has only the vaguest notion as to what might be realistic estimates of these parameters. Yet one selection might lead to a failure to find any solution, whereas another might lead to a less-than-optimal, or even less-than-close-to-optimal, solution. The source of this problem is, of course, that nonlinear equations can generate a series of local minima. Consequently, for any specific investigation involving nonlinear equations, barring some theoretical, empirical, or mathematical basis for estimating initial parameter values, one is reduced to trial-and-error methods, exceptional good luck, or a combination of both. With the advent of powerful, distributed computing and the diffusion of desktop graphical capability, visualization techniques hold potential for short-circuiting the trial-and-error process. In this paper we report on one such effort using an off-the-shelf Windows-based, curve-fitting program as the visualization engine for an eight-parameter, nonlinear equation derived by Heligman and Pollard to model mortality. First, we review the theory and mathematics of the so-called law of mortality. Second, we develop the working equation for solution and describe the data sources. Third, we attempt unsuccessfully to use standard techniques to solve the equations. Finally, we develop and illustrate a visualization technique designed to generate initial parameter estimates. We then apply the visualization technique and successfully solve the equations for 22 sets of mortality data. Keywords: visualization, law of mortality, nonlinear, mathematical demography.