wBACON {wbacon}

Weighted BACON Algorithm for Multivariate Outlier Detection

Description

wBACON is an iterative method for the computation of multivariate location and scatter (under the assumption of a Gaussian distribution).

Usage

wBACON(x, weights = NULL, alpha = 0.05, collect = 4, version = c("V2", "V1"),
    na.rm = FALSE, maxiter = 50, verbose = FALSE, n_threads = 2)
distance(x)
## S3 method for class 'wbaconmv'
print(x, digits = max(3L, getOption("digits") - 3L), ...)
## S3 method for class 'wbaconmv'
summary(object, ...)
center(object)
## S3 method for class 'wbaconmv'
vcov(object, ...)

Arguments

x

[matrix] or [data.frame].

weights

[numeric] sampling weight (default weights = NULL).

alpha

[numeric] tuning constant, level of significance, 0 < \alpha < 1; (default: alpha = 0.05).

collect

determines the size m of the initial subset to be m = collect \cdot p, where p is the number of variables, [integer].

version

[character] method of initialization; "V1": weighted Mahalanobis distances (not robust but affine equivariant); "V2" (default): Euclidean norm of the data centered by the coordinate-wise weighted median.

na.rm

[logical] indicating whether NA values should be removed before the computation proceeds (default: FALSE).

maxiter

[integer] maximal number of iterations (default: maxiter = 50).

verbose

[logical] indicating whether additional information is printed to the console (default: TRUE).

n_threads

[integer] number of threads used for OpenMP (default: 2).

digits

[integer] minimal number of significant digits.

...

additional arguments passed to the method.

object

object of class wbaconmv.

Details

The algorithm is initialized from a set of uncontaminated data. Then the subset is iteratively refined; i.e., additional observations are included into the subset if their Mahalanobis distance is below some threshold (likewise, observations are removed from the subset if their distance larger than the threshold). This process iterates until the set of good data remain stable. Observations not among the good data are outliers; see Billor et al. (2000). The weighted Bacon algorithm is due to Béguin and Hulliger (2008).

The threshold for the (squared) Mahalanobis distances is defined as the standardized chi-square 1 - \alpha quantile. All observations whose squared Mahalanobis distances is larger than the threshold are regarded as outliers.

If the sampling weights weights are not explicitly specified (i.e., weights = NULL), they are taken to be 1.0.

Incomplete/missing data

The wBACON cannot deal with missing values. In contrast, function BEM in package modi implements the BACON-EEM algorithm of Béguin and Hulliger (2008), which is tailored to work with outlying and missing values.

If the argument na.rm is set to TRUE the method behaves like na.omit.

Assumptions

The BACON algorithm assumes that the non-outlying data have (roughly) an elliptically contoured distribution (this includes the Gaussian distribution as a special case). "Although the algorithms will often do something reasonable even when these assumptions are violated, it is hard to say what the results mean." (Billor et al., 2000, p. 289)

In line with Billor et al. (2000, p. 290), we use the term outlier "nomination" rather than "detection" to highlight that algorithms should not go beyond nominating observations as potential outliers; see also Béguin and Hulliger (2008). It is left to the analyst to finally label outlying observations as such.

Utility functions and tools

Diagnostic plots are available by the plot method.

The method center and vcov return, respectively, the estimated center/location and covariance matrix.

The distance method returns the robust Mahalanobis distances.

The function is_outlier returns a vector of logicals that flags the nominated outliers.

Value

An object of class wbaconmv with slots

x

see function arguments

weights

see function arguments

center

estimated center of the data

dist

Mahalanobis distances

n

number of observations

p

number of variables

alpha

see function arguments

subset

final subset of outlier-free data

cutoff

see function arguments

maxiter

number of iterations until convergence

version

see functions arguments

collect

see functions arguments

cov

covariance matrix

converged

logical that indicates whether the algorithm converged

call

the matched call

References

Billor N., Hadi A.S. and Vellemann P.F. (2000). BACON: Blocked Adaptive Computationally efficient Outlier Nominators. Computational Statistics and Data Analysis 34, pp. 279–298. doi:10.1016/S0167-9473(99)00101-2

Béguin C. and Hulliger B. (2008). The BACON-EEM Algorithm for Multivariate Outlier Detection in Incomplete Survey Data. Survey Methodology 34, pp. 91–103. https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X200800110616

Schoch, T. (2021). wbacon: Weighted BACON algorithms for multivariate outlier nomination (detection) and robust linear regression, Journal of Open Source Software 6 (62), 3238 doi:10.21105/joss.03238

See Also

plot and is_outlier

Examples

data(swiss)
dt <- swiss[, c("Fertility", "Agriculture", "Examination", "Education",
    "Infant.Mortality")]
m <- wBACON(dt)
m
which(is_outlier(m))