Local regression: Difference between revisions

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Local regression methods started to appear extensively in statistics literature in the 1970s; for example, [[Charles Joel Stone|Charles J. Stone]] (1977),<ref>{{cite Q|Q56533608}}</ref> [[Vladimir Katkovnik]] (1979)<ref>{{citation |first=Vladimir|last=Katkovnik|title=Linear and nonlinear methods of nonparametric regression analysis|journal=Soviet Automatic Control|date=1979|volume=12|issue=5|pages=25–34}}</ref> and [[William S. Cleveland]] (1979).<ref name="cleve79">{{cite Q|Q30052922}}</ref> Katkovnik (1985)<ref name="katbook">{{cite Q|Q132129931}}</ref> is the earliest book devoted primarily to local regression methods.

Extensive theoreticalTheoretical work continued to appear throughout the 1990s. Important contributions include [[Jianqing Fan]] and [[Irène Gijbels]] (1992)<ref>{{cite Q|Q132202273}}</ref> studying efficiency properties, and [[David Ruppert]] and [[Matthew P. Wand]] (1994)<ref>{{cite Q|Q132202598}}</ref> developing an asymptotic distribution theory for multivariate local regression.

An important extension of local regression is Local Likelihood Estimation, formulated by [[Robert Tibshirani]] and [[Trevor Hastie]] (1987).<ref name="tib-hast-lle">{{cite Q|Q132187702}}</ref> This replaces the local least-squares criterion with a likelihood-based criterion, thereby extending the local regression method to the [[Generalized linear model]] setting; for example binary data;, count data; or censored data.

Practical implementations of local regression began appearing in statistical software in the 1980s. Cleveland (1981)<ref>{{cite Q|Q29541549}}</ref> introduces the LOWESS routines, intended for smoothing scatterplots. This implements local linear fitting with a single predictor variable, and also introduces robustness downweighting to make the procedure resistant to outliers. An entirely new implementation, LOESS, is described in Cleveland and [[Susan J. Devlin]] (1988).<ref name="clevedev">{{cite Q|Q29393395}}</ref> LOESS is a multivariate smoother, able to handle spatial data with two (or more) predictor variables, and uses (by default) local quadratic fitting. Both LOWESS and LOESS are implemented in the [[S (programming language)|S]] and [[R (programming language)|R]] programming languages. See also Cleveland's Local Fitting Software.<ref>{{cite web |last=Cleveland|first=William|title=Local Fitting Software|url=http://www.stat.purdue.edu/~wsc/localfitsoft.html|archive-url=https://web.archive.org/web/20050912090738/http://www.stat.purdue.edu/~wsc/localfitsoft.html |archive-date=12 September 2005 }}</ref>