Nonparametric regression is a form ofregression analysis where the predictor does not take a predetermined form but is completely constructed using information derived from the data. That is, noparametric equation is assumed for the relationship betweenpredictors and dependent variable. A largersample size is needed to build a nonparametric model having the same level ofuncertainty as aparametric model because the data must supply both the model structure and the parameter estimates.

Nonparametric regression assumes the following relationship, given the random variables${\displaystyle X}$ and${\displaystyle Y}$:

${\displaystyle \mathbb{E}[Y\mid X=x]=m(x),}$

where${\displaystyle m(x)}$ is some deterministic function.Linear regression is a restricted case of nonparametric regression where${\displaystyle m(x)}$ is assumed to be a linear function of the data.Sometimes a slightly stronger assumption of additive noise is used:

${\displaystyle Y=m(X)+U,}$

where the random variable${\displaystyle U}$ is the `noise term', with mean 0.Without the assumption that${\displaystyle m}$ belongs to a specific parametric family of functions it is impossible to get an unbiased estimate for${\displaystyle m}$, however most estimators areconsistent under suitable conditions.

This is a non-exhaustive list of non-parametric models for regression.

• nearest neighbor smoothing (see alsok-nearest neighbors algorithm)
• regression trees
• kernel regression
• local regression
• multivariate adaptive regression splines
• smoothing splines
• neural networks^[1]

In Gaussian process regression, also known as Kriging, a Gaussian prior is assumed for the regression curve. The errors are assumed to have amultivariate normal distribution and the regression curve is estimated by itsposterior mode. The Gaussian prior may depend on unknown hyperparameters, which are usually estimated viaempirical Bayes. The hyperparameters typically specify a prior covariance kernel. In case the kernel should also be inferred nonparametrically from the data, thecritical filter can be used.

Smoothing splines have an interpretation as the posterior mode of a Gaussian process regression.

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(August 2020) (Learn how and when to remove this message)

Kernel regression estimates the continuous dependent variable from a limited set of data points byconvolving the data points' locations with akernel 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.

Decision tree learning algorithms can be applied to learn to predict a dependent variable from data.^[2] Although the original Classification And Regression Tree (CART) formulation applied only to predicting univariate data, the framework can be used to predict multivariate data, including time series.^[3]

• Lasso (statistics)
• Local regression
• Non-parametric statistics
• Semiparametric regression
• Isotonic regression
• Multivariate adaptive regression splines

• Bowman, A. W.; Azzalini, A. (1997).Applied Smoothing Techniques for Data Analysis. Oxford: Clarendon Press.ISBN 0-19-852396-3.
• Fan, J.;Gijbels, I. (1996).Local Polynomial Modelling and its Applications. Boca Raton: Chapman and Hall.ISBN 0-412-98321-4.
• Henderson, D. J.; Parmeter, C. F. (2015).Applied Nonparametric Econometrics. New York: Cambridge University Press.ISBN 978-1-107-01025-3.
• Li, Q.; Racine, J. (2007).Nonparametric Econometrics: Theory and Practice. Princeton: Princeton University Press.ISBN 978-0-691-12161-1.
• Pagan, A.; Ullah, A. (1999).Nonparametric Econometrics. New York: Cambridge University Press.ISBN 0-521-35564-8.

Wikimedia Commons has media related toNonparametric regression.

• HyperNiche, software for nonparametric multiplicative regression.
• Scale-adaptive nonparametric regression (with Matlab software).

• Nonparametric regression

• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from August 2020
• All articles needing additional references
• Commons category link from Wikidata