lqs {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 = NULL, ...,
                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, 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` | A 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.)
`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`.
`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")

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