Nonlinear Estimation of the Negative Exponential Model of Population Density Decline

Abstract

The negative exponential model of the decline of density with distance from the center was estimated using census tract data for New York in the traditional manner, regressing the natural log of density on distance, log linear estimation. The predicted densities were plotted along with the actual densities and showed significant under prediction near the center, readily explained by the log transformation of density. Nonlinear estimation of the model produced more satisfactory results. The model was estimated using nonlinear and log linear estimation for 43 large urban areas in the United States using tract data for 1970 and 2010. The average log linear estimates of the density gradient and central density were lower than the nonlinear estimates, with some urban areas having very large differences. Comparison of the values obtained from regressing the predicted densities on the actual densities showed the nonlinear values to be either higher or similar to the log linear values. Considering the conclusions that might be drawn regarding changes over time, the mean results for percent change were similar for the nonlinear and log linear estimates, but differences for some individual urban areas were substantial. The conclusion is that nonlinear regression is superior for the estimation of the negative exponential model.