Nonparametric regression

Nonparametric regression

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Nonparametric regression is a form of regression analysis
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...

 in which the predictor
Predictor
Predictor may refer to:* a predictor variable, also known as an independent variable* the Kerrison Predictor, a military fire-control computer* something which makes a prediction* a branch predictor, a part of many modern processors...

 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
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

 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.

See also

  • Non-parametric statistics
    Non-parametric statistics
    In statistics, the term non-parametric statistics has at least two different meanings:The first meaning of non-parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:...

  • Semiparametric regression
    Semiparametric regression
    In statistics, semiparametric regression includes regression models that combine parametric and nonparametric models. They are often used in situations where the fully nonparametric model may not perform well or when the researcher wants to use a parametric model but the functional form with...

  • Isotonic regression
  • Multivariate adaptive regression splines
    Multivariate adaptive regression splines
    Multivariate adaptive regression splines is a form of regression analysis introduced by Jerome Friedman in 1991. It is a non-parametric regression techniqueand can be seen as an extension of linear models that...


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