Inductive regression: overcoming OLS limitations with the general regression neural network

Predicting and modeling relationships between spatially varying phenomena is central to understanding the geography of disease, politics, economics, and land use, for example. The purpose of this article is to introduce the General Regression Neural Network (GRNN) to a general geography audience as a powerful tool to use in place of OLS for function approximation with geographic data. Specifically, GRNN is a simple to use, robust predictor that is capable of modeling non-linear relationships without a pre-determined functional form and avoids four OLS assumptions that are frequently violated when using spatially referenced datasets. In OLS, assumptions about data characteristics are encoded into the predictive model and when the assumptions are violated the regression coefficients are biased. Separating model parameters from assumptions of data structure and distribution allows the GRNN to be partially molded by the characteristics of the data themselves, an inductive prediction. The four frequently violated OLS assumptions that do not affect GRNN are (1) linear functional relationship, (2) data distribution, (3) resilience to outliers, and (4) independence of observations. The reasons why these assumptions cause problems for OLS are reviewed and then reexamined in GRNN context. GRNN operation is described in three parts: probability density function estimation, network structure, and interpretation of results. Finally, the case of maize production in the US Great Plains is presented as an example of the practical use of GRNN in function approximation.