MASS (version 7.3-57)

# cov.rob: Resistant Estimation of Multivariate Location and Scatter

## Description

Compute a multivariate location and scale estimate with a high breakdown point -- this can be thought of as estimating the mean and covariance of the `good` part of the data. `cov.mve` and `cov.mcd` are compatibility wrappers.

## Usage

```cov.rob(x, cor = FALSE, quantile.used = floor((n + p + 1)/2),
method = c("mve", "mcd", "classical"),
nsamp = "best", seed)cov.mve(…)
cov.mcd(…)```

## Arguments

x

a matrix or data frame.

cor

should the returned result include a correlation matrix?

quantile.used

the minimum number of the data points regarded as `good` points.

method

the method to be used -- minimum volume ellipsoid, minimum covariance determinant or classical product-moment. Using `cov.mve` or `cov.mcd` forces `mve` or `mcd` respectively.

nsamp

the number of samples or `"best"` or `"exact"` or `"sample"`. The limit 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.

seed

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

arguments to `cov.rob` other than `method`.

## Value

A list with components

center

the final estimate of location.

cov

the final estimate of scatter.

cor

(only is `cor = TRUE`) the estimate of the correlation matrix.

sing

message giving number of singular samples out of total

crit

the value of the criterion on log scale. For MCD this is the determinant, and for MVE it is proportional to the volume.

best

the subset used. For MVE the best sample, for MCD the best set of size `quantile.used`.

n.obs

total number of observations.

## Details

For method `"mve"`, an approximate search is made of a subset of size `quantile.used` with an enclosing ellipsoid of smallest volume; in method `"mcd"` it is the volume of the Gaussian confidence ellipsoid, equivalently the determinant of the classical covariance matrix, that is minimized. The mean of the subset provides a first estimate of the location, and the rescaled covariance matrix a first estimate of scatter. The Mahalanobis distances of all the points from the location estimate for this covariance matrix are calculated, and those points within the 97.5% point under Gaussian assumptions are declared to be `good`. The final estimates are the mean and rescaled covariance of the `good` points.

The rescaling is by the appropriate percentile under Gaussian data; in addition the first covariance matrix has an ad hoc finite-sample correction given by Marazzi.

For method `"mve"` the search is made over ellipsoids determined by the covariance matrix of `p` of the data points. For method `"mcd"` an additional improvement step suggested by Rousseeuw and van Driessen (1999) is used, in which once a subset of size `quantile.used` is selected, an ellipsoid based on its covariance is tested (as this will have no larger a determinant, and may be smaller).

There is a hard limit on the allowed number of samples, \(2^{31} - 1\). However, practical limits are likely to be much lower and one might check the number of samples used for exhaustive enumeration, `combn(NROW(x), NCOL(x) + 1)`, before attempting it.

## 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. J. Rousseeuw and B. C. van Zomeren (1990) Unmasking multivariate outliers and leverage points, Journal of the American Statistical Association, 85, 633--639.

P. J. Rousseeuw and K. van Driessen (1999) A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212--223.

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.

`lqs`

## Examples

Run this code
```# NOT RUN {
set.seed(123)
cov.rob(stackloss)
cov.rob(stack.x, method = "mcd", nsamp = "exact")
# }
```

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