Nonparametric multiplicative regression

Nonparametric multiplicative regression (NPMR) is a form of nonparametric regression based on multiplicative kernel estimation. This is a smoothing technique that can be cross-validated and applied in a predictive way. Many other smoothing techniques are well known, for example smoothing splines and wavelets. Optimum choice of a smoothing method depends on the specific application. NPMR is useful for habitat modeling. The multidimensionality is provided multiplicatively – this automatically and parsimoniously models the complex interactions among predictors in much the same way that organisms integrate the numerous factors affecting their performance (McCune 2006). Optimizing the selection of predictors and their smoothing parameters in a multiplicative model is computationally intensive. NPMR can be applied to either presence-absence or quantitative response data, with either categorical or quantitative predictors.

NPMR can be applied with a local mean estimator, a local linear estimator, or a local logistic estimator. In each case the weights can be extended multiplicatively to m dimensions. In words, the estimate of the response is a local estimate (for example a local mean) of the observed values, each value weighted by its proximity to the target point in the predictor space, the weights being the product of weights for individual predictors. The model allows interactions, because weights for individual predictors are combined by multiplication rather than addition. A key biological feature of the model is that failure of a population with respect to any single dimension of the predictor space results in failure at that point, because the product of the weights for the point is zero or near zero if any of the individual weights are zero or near zero.

Derivation and application of NPMR are in McCune (2006) and McCune and Mefford (2004).