R: Robust Location and Scatter Estimation via MCD

covMcd {robustbase} | R Documentation

## Robust Location and Scatter Estimation via MCD

### Description

Compute the Minimum Covariance Determinant (MCD) estimator, a robust multivariate location and scale estimate with a high breakdown point, via the ‘Fast MCD’ or ‘Deterministic MCD’ (“DetMcd”) algorithm.

### Usage

    covMcd(x, cor = FALSE, raw.only = FALSE,
           alpha =, nsamp =, nmini =, kmini =,
           scalefn =, maxcsteps =,
           initHsets = NULL, save.hsets = FALSE, names = TRUE,
           seed =, tolSolve =, trace =,
           use.correction =, wgtFUN =, control = rrcov.control())

### Arguments

`x` | a matrix or data frame.
`cor` | should the returned result include a correlation matrix? Default is `cor = FALSE`.
`raw.only` | should only the “raw” estimate be returned, i.e., no (re)weighting step be performed; default is false.
`alpha` | numeric parameter controlling the size of the subsets over which the determinant is minimized; roughly `alpha*n`, (see ‘Details’ below) observations are used for computing the determinant. Allowed values are between 0.5 and 1 and the default is 0.5.
`nsamp` | number of subsets used for initial estimates or `"best"`, `"exact"`, or `"deterministic"`. Default is `nsamp = 500`. For `nsamp = "best"` exhaustive enumeration is done, as long as the number of trials does not exceed 100'000 (`= nLarge`). For `"exact"`, exhaustive enumeration will be attempted however many samples are needed. In this case a warning message may be displayed saying that the computation can take a very long time. For `"deterministic"`, the deterministic MCD is computed; as proposed by Hubert et al. (2012) it starts from the h most central observations of six (deterministic) estimators.
`nmini, kmini` | for n >= 2 n_0, n_0 := \code{nmini}, the algorithm splits the data into maximally `kmini` (by default 5) subsets, of size approximately, but at least `nmini`. When `nmini*kmini < n`, the initial search uses only a subsample of size `nmini*kmini`. The original algorithm had `nmini = 300` and `kmini = 5` hard coded.
`scalefn` | for the deterministic MCD: `function` to compute a robust scale estimate or character string specifying a rule determining such a function. The default, currently `"hrv2012"`, uses the recommendation of Hubert, Rousseeuw and Verdonck (2012) who recommend `Qn` for n < 1000 and `scaleTau2` for larger n. Alternatively, `scalefn = "v2014"`, uses that rule with cutoff n = 5000.
`maxcsteps` | maximal number of concentration steps in the deterministic MCD; should not be reached.
`initHsets` | NULL or a K x h integer matrix of initial subsets of observations of size h (specified by the indices in `1:n`).
`save.hsets` | (for deterministic MCD) logical indicating if the initial subsets should be returned as `initHsets`.
`names` | logical; if true (as by default), several parts of the result have a `names` or `dimnames` respectively, derived from data matrix `x`.
`seed` | initial seed for random generator, like `.Random.seed`, see `rrcov.control`.
`tolSolve` | numeric tolerance to be used for inversion (`solve`) of the covariance matrix in `mahalanobis`.
`trace` | logical (or integer) indicating if intermediate results should be printed; defaults to `FALSE`; values >= 2 also produce print from the internal (Fortran) code.
`use.correction` | whether to use finite sample correction factors; defaults to `TRUE`.
`wgtFUN` | a character string or `function`, specifying how the weights for the reweighting step should be computed. Up to April 2013, the only option has been the original proposal in (1999), now specified by `wgtFUN = "01.original"` (or via `control`). Since robustbase version 0.92-3, Dec.2014, other predefined string options are available, though experimental, see the experimental `.wgtFUN.covMcd` object.
`control` | a list with estimation options - this includes those above provided in the function specification, see `rrcov.control` for the defaults. If `control` is supplied, the parameters from it will be used. If parameters are passed also in the invocation statement, they will override the corresponding elements of the control object.

### Details

The minimum covariance determinant estimator of location and scatter implemented in `covMcd()` is similar to R function `cov.mcd()` in MASS. The MCD method looks for the h (> n/2) (h = h(α,n,p) = `h.alpha.n(alpha,n,p)`) observations (out of n) whose classical covariance matrix has the lowest possible determinant.

The raw MCD estimate of location is then the average of these h points, whereas the raw MCD estimate of scatter is their covariance matrix, multiplied by a consistency factor (`.MCDcons(p, h/n)`) and (if `use.correction` is true) a finite sample correction factor (`.MCDcnp2(p, n, alpha)`), to make it consistent at the normal model and unbiased at small samples. Both rescaling factors (consistency and finite sample) are returned in the length-2 vector `raw.cnp2`.

The implementation of `covMcd` uses the Fast MCD algorithm of Rousseeuw and Van Driessen (1999) to approximate the minimum covariance determinant estimator.

Based on these raw MCD estimates, (unless argument `raw.only` is true), a reweighting step is performed, i.e., `V <- cov.wt(x,w)`, where `w` are weights determined by “outlyingness” with respect to the scaled raw MCD. Again, a consistency factor and (if `use.correction` is true) a finite sample correction factor (`.MCDcnp2.rew(p, n, alpha)`) are applied. The reweighted covariance is typically considerably more efficient than the raw one, see Pison et al. (2002).

The two rescaling factors for the reweighted estimates are returned in `cnp2`. Details for the computation of the finite sample correction factors can be found in Pison et al. (2002).

### Value

An object of class `"mcd"` which is basically a `list` with components

`center` | the final estimate of location.
`cov` | the final estimate of scatter.
`cor` | the (final) estimate of the correlation matrix (only if `cor = TRUE`).
`crit` | the value of the criterion, i.e., the logarithm of the determinant. Previous to Nov.2014, it contained the determinant itself which can under- or overflow relatively easily.
`best` | the best subset found and used for computing the raw estimates, with `length(best) == quan = h.alpha.n(alpha,n,p)`.
`mah` | mahalanobis distances of the observations using the final estimate of the location and scatter.
`mcd.wt` | weights of the observations using the final estimate of the location and scatter.
`cnp2` | a vector of length two containing the consistency correction factor and the finite sample correction factor of the final estimate of the covariance matrix.
`raw.center` | the raw (not reweighted) estimate of location.
`raw.cov` | the raw (not reweighted) estimate of scatter.
`raw.mah` | mahalanobis distances of the observations based on the raw estimate of the location and scatter.
`raw.weights` | weights of the observations based on the raw estimate of the location and scatter.
`raw.cnp2` | a vector of length two containing the consistency correction factor and the finite sample correction factor of the raw estimate of the covariance matrix.
`X` | the input data as numeric matrix, without `NA`s.
`n.obs` | total number of observations.
`alpha` | the size of the subsets over which the determinant is minimized (the default is (n+p+1)/2).
`quan` | the number of observations, h, on which the MCD is based. If `quan` equals `n.obs`, the MCD is the classical covariance matrix.
`method` | character string naming the method (Minimum Covariance Determinant), starting with `"Deterministic"` when `nsamp="deterministic"`.
`iBest` | (for the deterministic MCD) contains indices from 1:6 denoting which of the (six) initial subsets lead to the best set found.
`n.csteps` | (for the deterministic MCD) for each of the initial subsets, the number of C-steps executed till convergence.
`call` | the call used (see `match.call`).

### Author(s)

Valentin Todorov valentin.todorov@chello.at, based on work written for S-plus by Peter Rousseeuw and Katrien van Driessen from University of Antwerp.

Visibility of (formerly internal) tuning parameters, notably `wgtFUN()`: Martin Maechler

### References

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

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

Pison, G., Van Aelst, S., and Willems, G. (2002) Small Sample Corrections for LTS and MCD, Metrika 55, 111–123.

Hubert, M., Rousseeuw, P. J. and Verdonck, T. (2012) A deterministic algorithm for robust location and scatter. Journal of Computational and Graphical Statistics 21, 618–637.

### See Also

`cov.mcd` from package MASS; `covOGK` as cheaper alternative for larger dimensions.

`BACON` and `covNNC`, from package robustX;

### Examples

    data(hbk)
    hbk.x <- data.matrix(hbk[, 1:3])
    set.seed(17)
    (cH <- covMcd(hbk.x))
    cH0 <- covMcd(hbk.x, nsamp = "deterministic")
    with(cH0, stopifnot(quan == 39,
         iBest == c(1:4,6), # 5 out of 6 gave the same
         identical(raw.weights, mcd.wt),
         identical(which(mcd.wt == 0), 1:14), all.equal(crit, -1.045500594135)))

    ## the following three statements are equivalent
    c1 <- covMcd(hbk.x, alpha = 0.75)
    c2 <- covMcd(hbk.x, control = rrcov.control(alpha = 0.75))
    ## direct specification overrides control one:
    c3 <- covMcd(hbk.x, alpha = 0.75,
                 control = rrcov.control(alpha=0.95))
    c1

    ## Martin's smooth reweighting:

    ## List of experimental pre-specified wgtFUN() creators:
    ## Cutoffs may depend on  (n, p, control$beta) :
    str(.wgtFUN.covMcd)

    cMM <- covMcd(hbk.x, wgtFUN = "sm1.adaptive")

    ina <- which(names(cH) == "call")
    all.equal(cMM[-ina], cH[-ina]) # *some* differences, not huge (same 'best'):
    stopifnot(all.equal(cMM[-ina], cH[-ina], tol = 0.2))

[Package robustbase version 0.93-6 Index]