Nonparametric regression

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

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

Example of a curve (red line) fit to a small data set (black points) with nonparametric regression using a Gaussian kernel smoother. The pink shaded area illustrates the kernel function applied to obtain an estimate of y for a given value of x. The kernel function defines the weight given to each data point in producing the estimate for a target point.

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.

Decision tree learning algorithms can be applied to learn to predict a dependent variable from data.[1] 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.[2]

• 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 to Nonparametric regression.

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