Type:	Package
Title:	Symmetrized MCD
Version:	0.6
Date:	2026-03-31
Imports:	Rcpp (≥ 0.11.0)
LinkingTo:	Rcpp, RcppArmadillo
Description:	Provides implementations of origin-based and symmetrized minimum covariance determinant (MCD) estimators, together with supporting utility functions.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation:	yes
Packaged:	2026-03-31 10:52:44 UTC; nordklau
Author:	Klaus Nordhausen [aut, cre], Christophe Croux [aut]
Maintainer:	Klaus Nordhausen <klaus.nordhausenr@gmail.com>
Repository:	CRAN
Date/Publication:	2026-04-03 15:30:07 UTC

Symmetrized MCD

Description

Provides implementations of origin-based and symmetrized minimum covariance determinant (MCD) estimators, together with supporting utility functions.

Details

The minimum covariance determinant (MCD) estimator is one of the most popular robust scatter matrix estimators; however, it does not possess the independence property. The symmetrized MCD is a variant of the MCD that does satisfy this property. It is, however, computationally more demanding, as it is based on all pairwise differences of the observations. On the other hand, the symmetrized MCD does not require a location estimate, since it is inherently centered at the origin.

To reduce the computational burden, an approximate variant of the symmetrized MCD is also implemented. This variant does not use all pairwise differences, but instead relies on a subset of size n m, where n denotes the sample size and m is a user-specified integer.

In addition, several helper functions are provided to support the computation and use of the symmetrized MCD.

Index of help topics:

MCDzero                 MCD with respect to the Origin
Maha                    Fast squared Mahalanobis distances
mdiff                   Computes Successive Differences for a Data Set
symMCD                  Symmetrized Minimum Covariance Determinant
                        Estimator
symMCD-package          Symmetrized MCD

Author(s)

Klaus Nordhausen [aut, cre] (ORCID: <https://orcid.org/0000-0002-3758-8501>), Christophe Croux [aut] (ORCID: <https://orcid.org/0000-0003-2912-3437>)

Maintainer: Klaus Nordhausen <klaus.nordhausenr@gmail.com>

References

Croux, C. and Nordhausen, K. (2026). Symmetrized MCD. Unpublished manuscript.

MCD with respect to the Origin

Description

Computes the minimum covariance determinant (MCD) estimator with the location fixed at the origin, using either a C++ or a plain R implementation.

Usage

MCDzero(X, alpha = 0.5, ns = 500, nc = 10, delta = 0.01)
MCDzeroR(y, alpha = 0.5, ns = 500, nc = 10, delta = 0.01)

Arguments

Details

The minimum covariance determinant (MCD) estimator is computed using an iterative algorithm based on random initial subsets, followed by a fixed number of concentration steps. In contrast to the classical MCD, the location is fixed at the origin and is not estimated from the data.

The parameter alpha controls the size of the subset retained at each step and thus determines the robustness of the estimator. For each of the ns random initial subsets, the algorithm applies nc concentration steps to improve the determinant of the scatter estimate. The best solution over all starts is retained.

An optional reweighting step is applied using the tuning constant delta, which aims to improve efficiency by downweighting observations with large squared Mahalanobis distances.

Value

For the C++ implementation, a list with the following components:

For the R implementation, a list with the following components:

Examples

X <- matrix(rnorm(300), ncol=3)
colnames(X) <- LETTERS[1:3]

# settings seeds for comparison
set.seed(1)
mcd0cpp <- MCDzero(X, 0.5)
set.seed(1)
mcd0r <- MCDzeroR(X, 0.5)

mcd0cpp$SigmaRaw  
mcd0cpp$SigmaRaw - mcd0r$sigma  

mcd0cpp$SigmaRe  
mcd0cpp$SigmaRe - mcd0r$sigmaR 

Fast squared Mahalanobis distances

Description

Computes squared Mahalanobis distances using different choices of location and scatter matrices.

Usage

Maha(X, center, scatter)
MahaClassical(X)
MahaOrigin(X)

Arguments

Details

The function Maha allows the user to supply a custom location and scatter matrix in order to compute (pseudo) Mahalanobis distances. This makes it possible to use alternative location and scatter estimates instead of the sample mean vector and covariance matrix. In contrast, the function MahaClassical always uses the sample mean and the classical covariance matrix. The function MahaOrigin fixes the location at the origin and computes the corresponding covariance matrix with respect to the origin.

The three functions provided here prioritize computational efficiency, but offer fewer options than mahalanobis. For example, mahalanobis allows the user to provide a pre-inverted scatter matrix.

Value

A numeric vector with the squared distances

See Also

mahalanobis

Examples

X <- matrix(rnorm(300), ncol = 3)

COV <- cov(X)
MEAN <- colMeans(X)

Maha(X, MEAN, COV)
mahalanobis(X, MEAN, COV)
MahaClassical(X)


MEAN0 <- rep(0, 3)
COV0 <- crossprod(X)/99
MahaOrigin(X)
mahalanobis(X, MEAN0, COV0)

Computes Successive Differences for a Data Set

Description

Computes a subset of pairwise differences between observations by forming successive differences up to a fixed lag.

Usage

mdiff(X, m = 20)

Arguments

Details

Let X be a data matrix with n rows, denoted by x_1, \dots, x_n. The function computes a matrix of successive differences consisting of the vectors x_i - x_{i+j}, for j = 1, \dots, m.

To ensure that each observation contributes exactly m differences, the first m observations are appended to the end of the data matrix before computing the differences. As a result, the output contains n m difference vectors.

The resulting set of differences forms a subset of all possible pairwise differences and provides a computationally cheaper alternative to using all n(n-1)/2 differences. Such constructions are useful, for example, in approximations of symmetrized scatter estimators.

Value

A numeric matrix with n m rows and the same number of columns as X, containing the successive difference vectors.

References

Dumbgen, L. and Nordhausen, K. (2024). Approximating symmetrized estimators of scatter via balanced incomplete U-statistics. Annals of the Institute of Statistical Mathematics, 76, 185–207. <doi:10.1007/s10463-023-00879-1>.

Examples

X <- matrix(1:12, ncol=2)
mdiff(X, m=3)

Symmetrized Minimum Covariance Determinant Estimator

Description

Computes the symmetrized minimum covariance determinant (MCD) estimator based on pairwise differences of the observations.

Usage

symMCD(X, gamma = 0.25, m = NULL, ns = 500, nc = 10, delta = 0.01)

Arguments

Details

The symmetrized MCD computes a minimum covariance determinant (MCD) estimator based on pairwise differences x_i - x_j with i > j. Since these differences are centered by construction, the location is fixed at the origin and no location parameter is estimated or returned. In practice, this corresponds to applying an MCD estimator with respect to the origin to the matrix of pairwise differences; see MCDzero.

In contrast to the classical MCD, the symmetrized MCD possesses the independence property: at the population level, the resulting scatter matrix is diagonal for a random vector with independent components.

The argument gamma controls the fraction of pairwise differences used for the scatter computation. Reasonable values are in the interval (0.25, 1). For a given value of gamma, the exact number of differences to be retained is determined analogously to the implementation in the robustbase package, using the function h.alpha.n.

Computing all pairwise differences is computationally expensive. As an approximation, the function therefore allows the use of only m successive differences per observation. In this case, the order of the observations should be random; hence, applying a random permutation to the observations prior to calling symMCD is advisable. While this approximation substantially reduces computational cost, it may be less appropriate when robustness is the primary concern rather than the independence property.

When only a subset of pairwise differences is used, the number of differences employed in the computation is given by \lfloor n m \gamma \rfloor + 1, where n denotes the sample size and m the number of successive differences per observation.

Value

A list with the following components:

References

Croux, C. and Nordhausen, K. (2026). Symmetrized MCD. Unpublished manuscript.

Examples

# data is small to keep the example fast
X <- matrix(rnorm(150), ncol=3)
colnames(X) <- LETTERS[1:3]

# transformed data to check affine equivariance:
A <- matrix(runif(9),3,3)
Y <- X %*% t(A)

# seed needs to be set for comparison
set.seed(1)
mcdall <- symMCD(X, 0.25)
set.seed(1)
mcdallY <- symMCD(Y, 0.25)

mcdall$SigmaRaw  
mcdall$SigmaRe  

A %*% mcdall$SigmaRaw %*% t(A) - mcdallY$SigmaRaw 
A %*% mcdall$SigmaRe %*% t(A) - mcdallY$SigmaRe



# could do same for incomplete variant

set.seed(123)
mcdm20 <- symMCD(X, 0.25, m = 20)
set.seed(123)
mcdm20Y <- symMCD(Y, 0.25, m = 20)

mcdm20$SigmaRaw
mcdm20$SigmaRe 

A %*% mcdm20$SigmaRaw %*% t(A) - mcdm20Y$SigmaRaw 
A %*% mcdm20$SigmaRe %*% t(A) - mcdm20Y$SigmaRe