Robust Fitting of Linear Models
Fit a linear model by robust regression using an M estimator.
rlm(x, ...)
## S3 method for class 'formula'
rlm(formula, data, weights, ..., subset, na.action,
method = c("M", "MM", "model.frame"),
wt.method = c("inv.var", "case"),
model = TRUE, x.ret = TRUE, y.ret = FALSE, contrasts = NULL)
## Default S3 method:
rlm(x, y, weights, ..., w = rep(1, nrow(x)),
init = "ls", psi = psi.huber,
scale.est = c("MAD", "Huber", "proposal 2"), k2 = 1.345,
method = c("M", "MM"), wt.method = c("inv.var", "case"),
maxit = 20, acc = 1e-4, test.vec = "resid", lqs.control = NULL)
psi.huber(u, k = 1.345, deriv = 0)
psi.hampel(u, a = 2, b = 4, c = 8, deriv = 0)
psi.bisquare(u, c = 4.685, deriv = 0)formula |
a formula of the form |
data |
an optional data frame, list or environment from which variables
specified in |
weights |
a vector of prior weights for each case. |
subset |
An index vector specifying the cases to be used in fitting. |
na.action |
A function to specify the action to be taken if |
x |
a matrix or data frame containing the explanatory variables. |
y |
the response: a vector of length the number of rows of |
method |
currently either M-estimation or MM-estimation or (for the
|
wt.method |
are the weights case weights (giving the relative importance of case, so a weight of 2 means there are two of these) or the inverse of the variances, so a weight of two means this error is half as variable? |
model |
should the model frame be returned in the object? |
x.ret |
should the model matrix be returned in the object? |
y.ret |
should the response be returned in the object? |
contrasts |
optional contrast specifications: see |
w |
(optional) initial down-weighting for each case. |
init |
(optional) initial values for the coefficients OR a method to find
initial values OR the result of a fit with a |
psi |
the psi function is specified by this argument. It must give
(possibly by name) a function |
scale.est |
method of scale estimation: re-scaled MAD of the residuals (default)
or Huber's proposal 2 (which can be selected by either |
k2 |
tuning constant used for Huber proposal 2 scale estimation. |
maxit |
the limit on the number of IWLS iterations. |
acc |
the accuracy for the stopping criterion. |
test.vec |
the stopping criterion is based on changes in this vector. |
... |
additional arguments to be passed to |
lqs.control |
An optional list of control values for |
u |
numeric vector of evaluation points. |
k, a, b, c |
tuning constants. |
deriv |
|
Fitting is done by iterated re-weighted least squares (IWLS).
Psi functions are supplied for the Huber, Hampel and Tukey bisquare
proposals as psi.huber, psi.hampel and
psi.bisquare. Huber's corresponds to a convex optimization
problem and gives a unique solution (up to collinearity). The other
two will have multiple local minima, and a good starting point is
desirable.
Selecting method = "MM" selects a specific set of options which
ensures that the estimator has a high breakdown point. The initial set
of coefficients and the final scale are selected by an S-estimator
with k0 = 1.548; this gives (for n >> p)
breakdown point 0.5.
The final estimator is an M-estimator with Tukey's biweight and fixed
scale that will inherit this breakdown point provided c > k0;
this is true for the default value of c that corresponds to
95% relative efficiency at the normal. Case weights are not
supported for method = "MM".
An object of class "rlm" inheriting from "lm".
Note that the df.residual component is deliberately set to
NA to avoid inappropriate estimation of the residual scale from
the residual mean square by "lm" methods.
The additional components not in an lm object are
s |
the robust scale estimate used |
w |
the weights used in the IWLS process |
psi |
the psi function with parameters substituted |
conv |
the convergence criteria at each iteration |
converged |
did the IWLS converge? |
wresid |
a working residual, weighted for |
Prior to version 7.3-52, offset terms in formula
were omitted from fitted and predicted values.
P. J. Huber (1981) Robust Statistics. Wiley.
F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw and W. A. Stahel (1986) Robust Statistics: The Approach based on Influence Functions. Wiley.
A. Marazzi (1993) Algorithms, Routines and S Functions for Robust Statistics. Wadsworth & Brooks/Cole.
Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.
summary(rlm(stack.loss ~ ., stackloss)) rlm(stack.loss ~ ., stackloss, psi = psi.hampel, init = "lts") rlm(stack.loss ~ ., stackloss, psi = psi.bisquare)
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