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{{short description|Moving average and polynomial regression method for smoothing data}}
[[Image:Loess curve.svg|thumb|LOESS curve fitted to a population sampled from a [[sine wave]] with uniform noise added. The LOESS curve approximates the original sine wave.]]
{{Regression bar}}
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LOESS and LOWESS thus build on [[classical statistics|"classical" methods]], such as linear and nonlinear [[least squares regression]]. They address situations in which the classical procedures do not perform well or cannot be effectively applied without undue labor. LOESS combines much of the simplicity of linear least squares regression with the flexibility of [[Non-linear regression|nonlinear regression]]. It does this by fitting simple models to localized subsets of the data to build up a function that describes the deterministic part of the variation in the data, point by point. In fact, one of the chief attractions of this method is that the data analyst is not required to specify a global function of any form to fit a model to the data, only to fit segments of the data.
The trade-off for these features is increased computation. Because it is so computationally intensive, LOESS would have been practically impossible to use in the era when least squares regression was being developed. Most other modern methods for process
A smooth curve through a set of data points obtained with this statistical technique is called a
==History==
Local regression and closely related procedures have a long and rich history, having been discovered and rediscovered in different fields on multiple occasions. An early work by [[Robert Henderson (mathematician)|Robert Henderson]]<ref>Henderson, R. Note on Graduation by Adjusted Average. Actuarial Society of America Transactions 17, 43--48. [https://archive.org/details/transactions17actuuoft archive.org]</ref> studying the problem of graduation (a term for smoothing used in Actuarial literature) introduced local regression using cubic polynomials
Specifically, let <math>Y_j</math> denote an ungraduated sequence of observations. Following Henderson, suppose that only the terms from <math>Y_{-h}</math> to <math>Y_h</math> are to be taken into account when computing the graduated value of <math>Y_0</math>, and <math>W_j</math> is the weight to be assigned to <math>Y_j</math>. Henderson then uses a local polynomial approximation <math>a + b j + c j^2 + d j^3</math>, and sets up the following four equations for the coefficients:
:<math>
\begin{align}
\sum_{j=-h}^h ( a + b j + c j^2 + d j^3) W_j &= \sum_{j=-h}^h W_j Y_j \\
\sum_{j=-h}^h ( aj + b j^2 + c j^3 + d j^4) W_j &= \sum_{j=-h}^h j W_j Y_j \\
\sum_{j=-h}^h ( aj^2 + b j^3 + c j^4 + d j^5) W_j &= \sum_{j=-h}^h j^2 W_j Y_j \\
\sum_{j=-h}^h ( aj^3 + b j^4 + c j^5 + d j^6) W_j &= \sum_{j=-h}^h j^3 W_j Y_j
\end{align}
</math>
Solving these equations for the polynomial coefficients yields the graduated value, <math>\hat Y_0 = a</math>.
Henderson went further. In preceding years, many 'summation formula' methods of graduation had been developed, which derived graduation rules based on summation formulae (convolution of the series of obeservations with a chosen set of weights). Two such rules are the 15-point and 21-point rules of [[John Spencer (Actuary)|Spencer]] (1904).<ref>{{citeQ|Q127775139}}</ref> These graduation rules were carefully designed to have a quadratic-reproducing property: If the ungraduated values exactly follow a quadratic formula, then the graduated values equal the ungraduated values. This is an important property: a simple moving average, by contrast, cannot adequately model peaks and troughs in the data. Henderson's insight was to show that ''any'' such graduation rule can be represented as a local cubic (or quadratic) fit for an appropriate choice of weights.
Further discussions of the historical work on graduation and local polynomial fitting can be found in [[Frederick Macaulay|Macaulay]] (1931),<ref name="mac1931">{{citeQ|Q134465853}}</ref> [[William S. Cleveland|Cleveland]] and [[Catherine Loader|Loader]] (1995);<ref name="slrpm">{{citeQ|Q132138257}}</ref> and [[Lori Murray|Murray]] and [[David Bellhouse (statistician)|Bellhouse]] (2019).<ref>{{cite Q|Q127772934}}</ref>
The [[Savitzky-Golay filter]], introduced by [[Abraham Savitzky]] and [[Marcel J. E. Golay]] (1964)<ref>{{cite Q|Q56769732}}</ref> significantly expanded the method. Like the earlier graduation work, their focus was data with an equally-spaced predictor variable, where (excluding boundary effects) local regression can be represented as a [[convolution]]. Savitzky and Golay published extensive sets of convolution coefficients for different orders of polynomial and smoothing window widths.
Local regression methods started to appear extensively in statistics literature in the
An important extension of local regression is Local Likelihood Estimation, formulated by [[Robert Tibshirani]] and [[Trevor Hastie]] (1987).<ref name="tib-hast-lle">{{
Practical implementations of local regression began appearing in statistical software in the
While Local Regression, LOWESS and LOESS are sometimes used
==Model definition==
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For ease of presentation, the development below assumes a single predictor variable; the extension to multiple predictors (when the <math>x_i</math> are vectors) is conceptually straightforward. A functional relationship between the predictor and response variables is assumed:
<math display="block">Y_i = \mu(x_i) + \epsilon_i</math>
where <math>\mu(x)</math> is the unknown ‘smooth’ regression function to be estimated, and represents the conditional expectation of the response, given a value of the predictor variables. In theoretical work, the ‘smoothness’ of this function can be formally characterized by placing bounds on higher order derivatives. The <math>\epsilon_i</math> represents random error; for estimation purposes these are assumed to have [[mean]] zero. Stronger assumptions (
Local regression then estimates the function <math>\mu(x)</math>, for one value of <math>x</math> at a time. Since the function is assumed to be smooth, the most informative data points are those whose <math>x_i</math> values are close to <math>x</math>. This is formalized with a bandwidth <math>h</math> and a [[kernel (statistics)|kernel]] or weight function <math>W(\cdot)</math>, with observations assigned weights
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\sum_{i=1}^n w_i(x) \left ( Y_i - \beta_0 - \beta_1(x_i-x) - \ldots - \beta_p(x_i-x)^p \right )^2.
</math>
The local
<math display="block">\hat\mu(x) = \hat\beta_0</math>
while the remaining coefficients can be interpreted
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It is to be emphasized that the above procedure produces the estimate <math>\hat\mu(x)</math> for one value of <math>x</math>. When considering a new value of <math>x</math>, a new set of weights <math>w_i(x)</math> must be computed, and the regression coefficient estimated afresh.
===Matrix
As with all least squares estimates, the estimated regression coefficients can be expressed in closed form (see [[Weighted least squares]] for details):
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This matrix representation is crucial for studying the theoretical properties of local regression estimates. With appropriate definitions of the design and weight matrices, it immediately generalizes to the multiple-predictor setting.
==Selection
Implementation of local regression requires specification and selection of several components:
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# The degree of local polynomial, or more generally, the form of the local model.
# The choice of weight function <math>W(\cdot)</math>.
# The choice of fitting criterion (least
Each of these components has been the subject of extensive study; a summary is provided below.
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One question not addressed above is, how should the bandwidth depend upon the fitting point <math>x</math>? Often a constant bandwidth is used, while LOWESS and LOESS prefer a nearest-neighbor bandwidth, meaning ''h'' is smaller in regions with many data points. Formally, the smoothing parameter, <math>\alpha</math>, is the fraction of the total number ''n'' of data points that are used in each local fit. The subset of data used in each weighted least squares fit thus comprises the <math>n\alpha</math> points (rounded to the next largest integer) whose explanatory variables' values are closest to the point at which the response is being estimated.<ref name="NIST">NIST, [http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd144.htm "LOESS (aka LOWESS)"], section 4.1.4.4, ''NIST/SEMATECH e-Handbook of Statistical Methods,'' (accessed 14 April 2017)</ref>
More sophisticated methods attempt to choose the bandwidth ''adaptively''; that is, choose a bandwidth at each fitting point <math>x</math> by applying criteria such as cross-validation locally within the smoothing window. An early example of this is [[Jerome H. Friedman]]'s<ref>{{
===Degree of local polynomials===
Most sources, in both theoretical and computational work, use low-order polynomials as the local model, with polynomial degree ranging from 0 to 3.
The degree 0 (local constant) model is equivalent to a [[kernel smoother]]; usually credited to [[Èlizbar Nadaraya]] (1964)<ref>{{
Local linear (degree 1) fitting can substantially reduce the boundary bias.
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Local quadratic (degree 2) and local cubic (degree 3) can result in improved fits, particularly when the underlying mean function <math>\mu(x)</math> has substantial curvature, or equivalently a large second derivative.
In theory, higher orders of polynomial can lead to faster convergence of the estimate <math>\hat\mu(x)</math> to the true mean <math>\mu(x)</math>, ''provided that <math>\mu(x)</math> has a sufficient number of derivatives''. See C. J. Stone (1980).<ref>{{
As with bandwidth selection, methods such as cross-validation can be used to compare the fits obtained with different degrees of polynomial.
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As mentioned above, the weight function gives the most weight to the data points nearest the point of estimation and the least weight to the data points that are furthest away. The use of the weights is based on the idea that points near each other in the explanatory variable space are more likely to be related to each other in a simple way than points that are further apart. Following this logic, points that are likely to follow the local model best influence the local model parameter estimates the most. Points that are less likely to actually conform to the local model have less influence on the local model [[Parameter#Statistics|parameter]] [[Statistical estimation|estimates]].
Cleveland (1979)<ref name="cleve79" /> sets out four requirements for the weight function:
# Non-negative: <math>
# Symmetry: <math>W(-x) = W(x)</math>.
# Monotone: <math>W(x)</math> is a nonincreasing function for <math>x \ge 0</math>.
# Bounded support: <math>W(x)=0</math> for <math>|x| \ge 1</math>.
Asymptotic efficiency of weight functions has been considered by [[V. A. Epanechnikov]] (1969)<ref>{{citeQ|Q57308723}}</ref> in the context of kernel density estimation; J. Fan (1993)<ref>{{citeQ|Q132691957}}</ref> has derived similar results for local regression. They conclude that the quadratic kernel, <math>W(x) = 1-x^2</math> for <math>|x|\le1</math> has greatest efficiency under a mean-squared-error loss function. See [[Kernel (statistics)#Kernel functions in common use|"kernel functions in common use"]] for more discussion of different kernels and their efficiencies.
Considerations other than MSE are also relevant to the choice of weight function. Smoothness properties of <math>W(x)</math> directly affect smoothness of the estimate <math>\hat\mu(x)</math>. In particular, the quadaratic kernel is not differentiable at <math>x=\pm 1</math>, and <math>\hat\mu(x)</math> is not differentiable as a result.
The [[Kernel (statistics)#Kernel functions in common use|tri-cube weight function]],
<math display="block">W(x) = (1 - |x|^3)^3; |x|<1</math>
has been used in LOWESS and other local regression software; this combines higher-order differentiability with a high MSE efficiency.
One criticism of weight functions with bounded support is that they can lead to numerical problems (i.e. an unstable or singular design matrix) when fitting in regions with sparse data. For this reason, some authors{{who|date=April 2025}} choose to use the Gaussian kernel, or others with unbounded support.
===Choice of fitting criterion===
As described above, local regression uses a locally weighted least squares criterion to estimate the regression parameters. This inherits many of the advantages (ease of implementation and interpretation; good properties when errors are normally distributed) and disadvantages (sensitivity to extreme values and outliers; inefficiency when errors have unequal variance or are not normally distributed) usually associated with least squares regression.
These disadvantages can be addressed by replacing the local least-squares estimation by something else. Two such ideas are presented here: local likelihood estimation, which applies local estimation to the [[generalized linear model]], and robust local regression, which localizes methods from [[robust regression]].
====Local likelihood estimation====
In local likelihood estimation, developed in Tibshirani and Hastie (1987),<ref name="tib-hast-lle" /> the observations <math>Y_i</math> are assumed to come from a parametric family of distributions, with a known probability density function (or mass function, for discrete data),
<math display="block">
Y_i \sim f(y,\theta(x_i)),
</math>
where the parameter function <math>\theta(x)</math> is the unknown quantity to be estimated. To estimate <math>\theta(x)</math> at a particular point <math>x</math>, the local likelihood criterion is
<math display="block">
\sum_{i=1}^n w_i(x) \log \left ( f(Y_i,
\beta_0 + \beta_1(x_i-x) + \ldots + \beta_p (x_i-x)^p \right ).
</math>
Estimates of the regression coefficients (in, particular, <math>\hat\beta_0</math>) are obtained by maximizing the local likelihood criterion, and
the local likelihood estimate is
<math display="block">
\hat\theta(x) = \hat\beta_0.
</math>
When <math>f(y,\theta(x))</math> is the normal distribution and <math>\theta(x)</math> is the mean function, the local likelihood method reduces to the standard local least-squares regression. For other likelihood families, there is (usually) no closed-form solution for the local likelihood estimate, and iterative procedures such as [[iteratively reweighted least squares]] must be used to compute the estimate.
''Example'' (local logistic regression). All response observations are 0 or 1, and the mean function is the "success" probability, <math>\mu(x_i) = \Pr (Y_i=1 | x_i)</math>. Since <math>\mu(x_i)</math> must be between 0 and 1, a local polynomial model should not be used for <math>\mu(x)</math> directly. Insead, the logistic transformation
<math display="block">
\theta(x) = \log \left ( \frac{\mu(x)}{1-\mu(x)} \right )
</math>
can be used; equivalently,
<math display="block">
\begin{align}
1-\mu(x) &= \frac{1}{1+e^{\theta(x)}} ;\\
\mu(x) &= \frac{e^{\theta(x)}}{1+e^{\theta(x)}}
\end{align}
</math>
and the mass function is
<math display="block">
f(Y_i,\theta(x_i)) = \frac{ e^{Y_i \theta(x_i)}}{1+e^{\theta(x_i)}}.
</math>
An asymptotic theory for local likelihood estimation is developed in J. Fan, [[Nancy E. Heckman]] and M.P.Wand (1995);<ref>{{cite Q|Q132508409}}</ref> the book Loader (1999)<ref name="loabook">{{citeQ|Q59410587}}</ref> discusses many more applications of local likelihood.
====Robust local regression====
To address the sensitivity to outliers, techniques from [[robust regression]] can be employed. In local [[M-estimator|M-estimation]], the local least-squares criterion is replaced by a criterion of the form
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\right )
</math>
where <math>\rho(\cdot)</math> is a robustness function and <math>s</math> is a scale parameter. Discussion of the merits of different choices of robustness function is best left to the [[robust regression]] literature. The scale
The robustness iterations in LOWESS and LOESS correspond to the robustness function defined by
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and a robust global estimate of the scale parameter.
If <math>\rho(u)=|u|</math>, the local <math>L_1</math> criterion
<math display="block">
\sum_{i=1}^n w_i(x) \left | Y_i - \beta_0 - \ldots - \beta_p(x_i-x)^p \right |
</math>
results; this does not require a scale parameter.
==Advantages==
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Finally, as discussed above, LOESS is a computationally intensive method (with the exception of evenly spaced data, where the regression can then be phrased as a non-causal [[finite impulse response]] filter). LOESS is also prone to the effects of outliers in the data set, like other least squares methods. There is an iterative, [[robust statistics|robust]] version of LOESS [Cleveland (1979)] that can be used to reduce LOESS' sensitivity to [[outliers]], but too many extreme outliers can still overcome even the robust method.
==Further reading==
Books substantially covering local regression and extensions:
* Macaulay (1931) "The Smoothing of Time Series",<ref name="mac1931">{{citeQ|Q134465853}}</ref> discusses graduation methods with several chapters related to local polynomial fitting.
* Katkovnik (1985) "Nonparametric Identification and Smoothing of Data"<ref name="katbook">{{citeQ|Q132129931}}</ref> in Russian.
* Fan and Gijbels (1996) "Local Polynomial Modelling and Its Applications".<ref>{{citeQ|Q134377589}}</ref>
* Loader (1999) "Local Regression and Likelihood".<ref name="loabook">{{citeQ|Q59410587}}</ref>
* Fotheringham, Brunsdon and Charlton (2002), "Geographically Weighted Regression"<ref name="gwrbook">{{citeQ|Q133002722}}</ref> (a development of local regression for spatial data).
Book chapters, Reviews:
* "Smoothing by Local Regression: Principles and Methods"<ref name="slrpm">{{citeQ|Q132138257}}</ref>
* "Local Regression and Likelihood", Chapter 13 of ''Observed Brain Dynamics'', Mitra and Bokil (2007)<ref>{{citeQ|Q57575432}}</ref>
* [[Rafael Irizarry (scientist)|Rafael Irizarry]], "Local Regression". Chapter 3 of "Applied Nonparametric and Modern Statistics".<ref>{{cite web|last=Irizarry|first=Rafael|title=Applied Nonparametric and Modern Statistics|url=https://rafalab.dfci.harvard.edu/pages/754/|access-date=2025-05-16}}</ref>
==See also==
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{{refbegin|30em|indent=yes}}
*{{cite book|last1=Fox|first1=John |last2=Weisberg|first2=Sanford |title=An R Companion to Applied Regression|url=https://books.google.com/books?id=SfNrDwAAQBAJ|edition=3rd|date= 2018|publisher=SAGE |isbn=978-1-5443-3645-9|chapter=Appendix: Nonparametric Regression in R|chapter-url=https://socialsciences.mcmaster.ca/jfox/Books/Companion/appendices/Appendix-Nonparametric-Regression.pdf}}
*{{Cite report|title=A Simple Introduction to Moving Least Squares and Local Regression Estimation|first=Rao Veerabhadra|last=Garimella|date=22 June 2017|publisher=Los Alamos National Laboratory|osti = 1367799|doi=10.2172/1367799}}
*{{cite book|last=Harrell |first=Frank E. Jr. |title=Regression Modeling Strategies: With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis|url=https://books.google.com/books?id=94RgCgAAQBAJ&pg=PA29|year=2015|publisher=Springer|isbn=978-3-319-19425-7}}
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==External links==
{{
*[http://www.itl.nist.gov/div898/handbook/pmd/section1/pmd144.htm NIST Engineering Statistics Handbook Section on LOESS]
*[https://stat.ethz.ch/R-manual/R-devel/library/stats/html/loess.html R: Local Polynomial Regression Fitting] The Loess function in [[R (programming language)|R]]
*[https://stat.ethz.ch/R-manual/R-devel/library/stats/html/lowess.html R: Scatter Plot Smoothing] The Lowess function in [[R (programming language)|R]]
*[https://stat.ethz.ch/R-manual/R-devel/library/stats/html/supsmu.html The supsmu function] (Friedman's SuperSmoother) in R
*[http://www.r-statistics.com/2010/04/quantile-loess-combining-a-moving-quantile-window-with-loess-r-function/ Quantile LOESS] – A method to perform Local regression on a '''Quantile
*[http://fivethirtyeight.blogs.nytimes.com/2013/03/26/how-opinion-on-same-sex-marriage-is-changing-and-what-it-means/?hp Nate Silver, How Opinion on Same-Sex Marriage Is Changing, and What It Means] – sample of LOESS versus linear regression
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