lqs                   package:lqs                   R Documentation

Resistant Regression

Description:

     Fit a regression to the good points in the dataset, thereby
     achieving a regression estimator with a high breakdown point.
     `lmsreg' and `ltsreg' are compatibility wrappers.

Usage:

     lqs(x, ...)
     lqs.formula(formula, data, ...,
                 method = c("lts", "lqs", "lms", "S", "model.frame"),
                 subset, na.action = na.fail, model = TRUE,
                 x = FALSE, y = FALSE, contrasts = NULL)
     lqs.default(x, y, intercept = TRUE, method = c("lts", "lqs", "lms", "S"),
                 quantile, control = lqs.control(...), k0 = 1.548, seed, ...)
     lmsreg(...)
     ltsreg(...)

Arguments:

 formula: a formula of the form `y ~ x1 + x2 + ...'.

    data: data frame from which variables specified in `formula' are
          preferentially to be taken.

  subset: an index vector specifying the cases to be used in fitting.
          (NOTE: If given, this argument must be named exactly.)

na.action: function to specify the action to be taken if `NA's are
          found.  The default action is for the procedure to fail.  An
          alternative is `na.omit', which leads to omission of cases
          with missing values on any required variable.  (NOTE: If
          given, this argument must be named exactly.) 

   model: logical. If `TRUE' the model frame is returned.

contrasts: an optional list.  See the `contrasts.arg' of
          `model.matrix.default'.

       x: a matrix or data frame containing the explanatory variables.

       y: the response: a vector of length the number of rows of `x'.

intercept: should the model include an intercept?

  method: the method to be used. `model.frame' returns the model frame:
          for the others see the `Details' section. Using `lmsreg' or
          `ltsreg' forces `"lms"' and `"lts"' respectively. 

quantile: the quantile to be used: see `Details'. This is over-ridden
          if `method = "lms"'. 

 control: additional control items: see `Details'.

      k0: the cutoff / tuning constant used for chi() and psi()
          functions when `method = "S"', currently corresponding to
          Tukey's ``biweight''.

    seed: the seed to be used for random sampling: see `.Random.seed'.
          The current value of `.Random.seed' will be preserved if it
          is set.. 

     ...: arguments to be passed to `lqs.default' or `lqs.control'.

Details:

     Suppose there are `n' data points and `p' regressors, including
     any intercept.

     The first three methods minimize some function of the sorted
     squared residuals. For methods `"lqs"' and `"lms"' is the
     `quantile' squared residual, and for `"lts"' it is the sum of the
     `quantile' smallest squared residuals. `"lqs"' and `"lms"' differ
     in the defaults for `quantile', which are `floor((n+p+1)/2)' and
     `floor((n+1)/2)' respectively. For `"lts"' the default is
     `floor(n/2) + floor((p+1)/2)'.

     The `"S"' estimation method solves for the scale `s' such that the
     average of a function chi of the residuals divided by `s' is equal
     to a given constant.

     The `control' argument is a list with components

     `psamp': the size of each sample. Defaults to `p'.

     `nsamp': the number of samples or `"best"' or `"exact"' or
          `"sample"'.  If `"sample"' the number chosen is `min(5*p,
          3000)', taken from Rousseeuw and Hubert (1997). If `"best"'
          exhaustive enumeration is done up to 5000 samples: if
          `"exact"' exhaustive enumeration will be attempted however
          many samples are needed.

     `adjust': should the intercept be optimized for each sample?

Value:

     An object of class `"lqs"'.

Note:

     There seems no reason other than historical to use the `lms' and
     `lqs' options.  LMS estimation is of low efficiency (converging at
     rate n^{-1/3}) whereas LTS has the same asymptotic efficiency as
     an M estimator with trimming at the quartiles (Marazzi, 1993,
     p.201). LQS and LTS have the same maximal breakdown value of
     `(floor((n-p)/2) + 1)/n' attained if `floor((n+p)/2) <= quantile
     <= floor((n+p+1)/2)'. The only drawback mentioned of LTS is
     greater computation, as a sort was thought to be required
     (Marazzi, 1993, p.201) but this is not true as a partial sort can
     be used (and is used in this implementation).

     Adjusting the intercept for each trial fit does need the residuals
     to be sorted, and may be significant extra computation if `n' is
     large and `p' small.

     Opinions differ over the choice of `psamp'.  Rousseeuw and Hubert
     (1997) only consider p; Marazzi (1993) recommends p+1 and suggests
     that more samples are better than adjustment for a given
     computational limit.

     The computations are exact for a model with just an intercept and
     adjustment, and for LQS for a model with an intercept plus one
     regressor and exhaustive search with adjustment. For all other
     cases the minimization is only known to be approximate.

Author(s):

     B.D. Ripley

References:

     P. J. Rousseeuw and A. M. Leroy (1987) Robust Regression and
     Outlier Detection. Wiley.

     A. Marazzi (1993) Algorithms, Routines and S Functions for Robust
     Statistics. Wadsworth and Brooks/Cole.

     P. Rousseeuw and M. Hubert (1997) Recent developments in PROGRESS.
     In L1-Statistical Procedures and Related Topics, ed Y. Dodge, IMS
     Lecture Notes volume 31, pp. 201-214.

See Also:

     `predict.lqs'

Examples:

     data(stackloss)
     set.seed(123)
     lqs(stack.loss ~ ., data = stackloss)
     lqs(stack.loss ~ ., data = stackloss, method = "S", nsamp = "exact")

