loess

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Local Polynomial Regression Fitting

Fit a polynomial surface determined by one or more numerical predictors, using local fitting.

Keywords
smooth, loess
Usage
loess(formula, data, weights, subset, na.action, model = FALSE,
span = 0.75, enp.target, degree = 2,
parametric = FALSE, drop.square = FALSE, normalize = TRUE,
family = c("gaussian", "symmetric"),
method = c("loess", "model.frame"),
control = loess.control(...), ...)
Arguments
formula
a formula specifying the numeric response and one to four numeric predictors (best specified via an interaction, but can also be specified additively). Will be coerced to a formula if necessary.
data
an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which loess is called.
weights
optional weights for each case.
subset
an optional specification of a subset of the data to be used.
na.action
the action to be taken with missing values in the response or predictors. The default is given by getOption("na.action").
model
should the model frame be returned?
span
the parameter $\alpha$ which controls the degree of smoothing.
enp.target
an alternative way to specify span, as the approximate equivalent number of parameters to be used.
degree
the degree of the polynomials to be used, normally 1 or 2. (Degree 0 is also allowed, but see the Note.)
parametric
should any terms be fitted globally rather than locally? Terms can be specified by name, number or as a logical vector of the same length as the number of predictors.
drop.square
for fits with more than one predictor and degree = 2, should the quadratic term be dropped for particular predictors? Terms are specified in the same way as for parametric.
normalize
should the predictors be normalized to a common scale if there is more than one? The normalization used is to set the 10% trimmed standard deviation to one. Set to false for spatial coordinate predictors and others known to be on a common scale.
family
if "gaussian" fitting is by least-squares, and if "symmetric" a re-descending M estimator is used with Tukey's biweight function. Can be abbreviated.
method
fit the model or just extract the model frame. Can be abbreviated.
control
control parameters: see loess.control.
...
control parameters can also be supplied directly (if control is not specified).
Details

Fitting is done locally. That is, for the fit at point $x$, the fit is made using points in a neighbourhood of $x$, weighted by their distance from $x$ (with differences in parametric variables being ignored when computing the distance). The size of the neighbourhood is controlled by $\alpha$ (set by span or enp.target). For $\alpha < 1$, the neighbourhood includes proportion $\alpha$ of the points, and these have tricubic weighting (proportional to $(1 - \mathrm{(dist/maxdist)}^3)^3$). For $\alpha > 1$, all points are used, with the maximum distance assumed to be $\alpha^{1/p}$ times the actual maximum distance for $p$ explanatory variables.

For the default family, fitting is by (weighted) least squares. For family="symmetric" a few iterations of an M-estimation procedure with Tukey's biweight are used. Be aware that as the initial value is the least-squares fit, this need not be a very resistant fit.

It can be important to tune the control list to achieve acceptable speed. See loess.control for details.

Value

• An object of class "loess".

Note

As this is based on cloess, it is similar to but not identical to the loess function of S. In particular, conditioning is not implemented.

The memory usage of this implementation of loess is roughly quadratic in the number of points, with 1000 points taking about 10Mb.

degree = 0, local constant fitting, is allowed in this implementation but not documented in the reference. It seems very little tested, so use with caution.

source

The 1998 version of cloess package of Cleveland, Grosse and Shyu. A later version is available as dloess at http://www.netlib.org/a.

References

W. S. Cleveland, E. Grosse and W. M. Shyu (1992) Local regression models. Chapter 8 of Statistical Models in S eds J.M. Chambers and T.J. Hastie, Wadsworth & Brooks/Cole.