BACON-EEM Algorithm for multivariate outlier detection in incomplete multivariate survey data

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Description

BEM starts from a set of uncontaminated data with possible missing values, applies a version of the EM-algorithm to estimate the center and scatter of the good data, then adds (or deletes) observations to the good data which have a Mahalanobis distance below a threshold. This process iterates until the good data remain stable. Observations not among the good data are outliers.

Usage

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BEM(data, weights, v = 2, c0 = 3, alpha = 0.01, md.type = "m", 
em.steps.start = 10, em.steps.loop = 5, better.estimation = FALSE, monitor = FALSE)

Arguments

data

a matrix or data frame. As usual, rows are observations and columns are variables.

weights

a non-negative and non-zero vector of weights for each observation. Its length must equal the number of rows of the data. Default is rep(1,nrow(data)).

v

an integer indicating the distance for the definition of the starting good subset: v=1 uses the Mahalanobis distance based on the weighted mean and covariance, v=2 uses the Euclidean distance from the componentwise median

c0

the size of initial subset is c0*ncol(data).

alpha

a small probability indicating the level (1-alpha) of the cutoff quantile for good observations

md.type

Type of Mahalanobis distance: "m" marginal, "c" conditional

em.steps.start

Number of iterations of EM-algorithm for starting good subset

em.steps.loop

Number of iterations of EM-algorithm for good subset

better.estimation

If better.estimation=TRUE then the EM-algorithm for the final good subset iterates em.steps.start more.

monitor

If TRUE verbose output.

Details

The BACON algorithm with v=1 is not robust but affine equivariant while v=1 is robust but not affine equivariant. The threshold for the (squared) Mahalanobis distances, beyond which an observation is an outlier, is a standardised chisquare quantile at (1-alpha). For large data sets it may be better to choose alpha/n instead.

The internal function .EM.normal is usually called from BEM. .EM.normal is implementing the EM-algorithm in such a way that part of the calculations can be saved to be reused in the BEM algorithm. .EM.normal does not contain the computation of the observed sufficient statistics, they will be computed in the main program of BEM and passed as parameters as well as the statistics on the missingness patterns.

Value

BEM returns a list whose first component is the sub-list output with the following components:

sample.size

number of observations

discarded.observations

Number of discarded observations

number.of.variables

Number of variables

significance.level

the probability used for the cutpoint, i.e.\ alpha

initial.basic.subset.size

Size of initial good subset

final.basic.subset.size

Size of final good subset

number.of.iterations

Number of iterations of the BACON step

computation.time

Elapsed computation time

center

Final estimate of the center

scatter

Final estimate of the covariance matrix

cutpoint

The threshold MD-value for the cut-off of outliers

The further components returned by BEM are:

outind

Outlier indicator

dist

Final Mahalanobis distances

Note

BEM uses an adapted version of the EM-algorithm in funkction EM-normal.

Author(s)

Beat Hulliger

References

B\'eguin, C. and Hulliger, B. (2008) The BACON-EEM Algorithm for Multivariate Outlier Detection in Incomplete Survey Data, Survey Methodology, Vol. 34, No. 1, pp. 91-103.

Billor, N., Hadi, A.S. and Vellemann, P.F. (2000). BACON: Blocked Adaptative Computationally-efficient Outlier Nominators. Computational Statistics and Data Analysis, 34(3), 279-298.

Schafer J.L. (2000), Analysis of Incomplete Multivariate Data, Monographs on Statistics and Applied Probability 72, Chapman & Hall.

Examples

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# Bushfire data set with 20% MCAR
data(bushfirem,bushfire.weights)
bem.res<-BEM(bushfirem,bushfire.weights,alpha=(1-0.01/nrow(bushfirem))) 
print(bem.res$output)