RESEARCH METHODS - VISUALIZATION IN THE NUMERIC SOLUTION OF COMPLEX NONLINEAR EQUATIONS IN MATHEMATICAL DEMOGRAPHY

Searching for numeric solutions to complex nonlinear equations is a re source-consuming, risky business. For a given equation, there can be a n infinite number of often dramatically quite different solutions meet ing a specified goodness-of-fit criterion. Although the leading statis tical 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 e stimates of these parameters. Yet one selection might lead to a failur e to find any solution, whereas another might lead to a less-than-opti mal, 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 minma. Consequently, for any specific investigation involving nonlinear equations, barring some theoretical, empirical, or mathemat ical basis for estimating initial parameter values, one is reduced to trial-and-error methods, exceptional good luck, or a combination of bo th. With the advent of powerful, distributed computing and the diffusi on of desktop graphical capability, visualization techniques hold pote ntial for short-circuiting the trial-and-error process. In this paper we report on one such effort using an off-the-shelf Windows-based, cur ve-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 de scribe the data sources. Third, we attempt unsuccessfully to use stand ard techniques to solve the equations. Finally, we develop and illustr ate a visualization technique designed to generate initial parameter e stimates. We then apply the visualization technique and successfully s olve the equations for 22 sets of mortality data.