# Nonparametric regression

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Nonparametric regression

Nonparametric regression is a form of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. Nonparametric regression requires larger sample sizes than regression based on parametric models because the data must supply the model structure as well as the model estimates.

## Kernel regression

Kernel regression estimates the continuous dependent variable from a limited set of data points by convolving the data points' locations with a kernel function - approximately speaking, the kernel function specifies how to "blur" the influence of the data points so that their values can be used to predict the value for nearby locations.

## 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. 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.

## Regression trees

Decision tree learning algorithms can be applied to learn to predict a dependent variable from data. Although the original CART formulation applied only to predicting univariate data, the framework can be used to predict multivariate data including time series.

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