MCD and MVE Calls
finds the minimum covariance determinant estimator and
the minimum volume ellipsoid estimator
 CALL MCD( sc, coef, dist, opt, x<, s>);
The MDC call is the robust (resistent) estimation of multivariate location
and scatter, defined by minimizing the determinant of
the covariance matrix computed from h points.
 CALL MVE( sc, coef, dist, opt, x<, s>);
The MVE call is the robust (resistent) estimation of multivariate location
and scatter, defined by minimizing the volume of an
ellipsoid containing h points.
The MVE and MCD subroutines compute the minimum volume ellipsoid
estimator and the minimum covariance determinant estimator.
These robust locations and covariance matrices can be used to
detect multivariate outliers and leverage points. For this
purpose, the MVE and MCD subroutines provide a table of robust
distances.
The inputs to the MCD and MVE subroutine are as follows:
 opt
 refers to an options vector with the following components
(missing values are treated as default values):
 opt[1]
 specifies the amount of printed output.
Higher option values request additional
output and include the output of lower values.
 opt[1]=0
 prints no output except error messages.
 opt[1]=1
 prints most of the output.
 opt[1]=2
 additionally prints case numbers of the
observations in the best subset and some
basic history of the optimization process.
 opt[1]=3
 additionally prints how many subsets
result in singular linear systems.
The default is opt[1]=0.
 opt[2]
 specifies whether the classical, initial, and
final robust covariance matrices are printed.
The default is opt[2]=0.
Note that the final robust covariance
matrix is always returned in coef.
 opt[3]
 specifies whether the classical, initial, and final
robust correlation matrices are printed or returned:
 opt[3]=0
 does not return or print.
 opt[3]=1
 prints the robust correlation matrix.
 opt[3]=2
 returns the final robust correlation matrix in coef.
 opt[3]=3
 prints and returns the final robust correlation matrix.
 opt[4]
 specifies the quantile h used in the objective function.
The default is opt[5]= h = [(N+n+1)/2].
If the value of h is specified outside the range ,
it is reset to the closest boundary of this region.
 opt[5]
 specifies the number N_{Rep} of subset generations.
This option is the same as described
previously for the LMS and LTS subroutines.
Due to computer time restrictions, not all subset combinations
can be inspected for larger values of N and n.
If opt[6] is zero or missing, the default
number of subsets is taken from the following table.
n

1

2

3

4

5

6

7

8

9

10

N_{lower}  500  50  22  17  15  14  0  0  0  0 
N_{upper}  1000000  1414  182  71  43  32  27  24  23  22 
N_{Rep}  500  1000  1500  2000  2500  3000  3000  3000  3000  3000 
n

11

12

13

14

15

N_{lower}  0  0  0  0  0 
N_{upper}  22  22  22  23  23 
N_{Rep}  3000  3000  3000  3000  3000 


 If the number of cases (observations) N is smaller
than N_{lower}, then all possible subsets are used;
otherwise, N_{Rep} subsets are drawn randomly.
This means that an exhaustive search
is performed for opt[6]=1.
If N is larger than N_{upper}, a note is printed
in the log file indicating how many subsets exist.
 x
 refers to an N ×n matrix X of regressors.
 s
 refers to an n vector containing n observation
numbers of a subset for which the objective function
should be evaluated. This subset can be the start for
a pairwise exchange algorithm if opt[4] is specified.
Missing values are not permitted in x.
Missing values in opt cause the default value to be used.
The MCD and MVE subroutines return the following values:
 sc
 is a column vector containing the following scalar information:
 sc[1]
 the quantile h used in the objective function
 sc[2]
 number of subsets generated
 sc[3]
 of subsets with singular linear systems
 sc[4]
 number of nonzero weights w_{i}
 sc[5]
 lowest value of the objective function F_{MVE}
attained (volume of smallest ellipsoid found)
 sc[6]
 Mahalanobislike distance used in the computation of
the lowest value of the objective function F_{MVE}
 sc[7]
 the cutoff value used for the outlier decision
 sc[8]
 the correction factor c(N,n) used in
scaling the initial MVE scatter matrix
 sc[9]
 scaling factor for the initial MVE scatter matrix

f = [(c^{2}(N,n))/ CINV(0.5,n)]
 coef
 is a matrix with n columns containing
the following results in its rows:
 coef[1]
 location of ellipsoid center
 coef[2]
 eigenvalues of final robust scatter matrix
 coef[3:2+n]
 the final robust scatter matrix for
opt[2]=1 or opt[2]=3
 dist
 is a matrix with N columns containing
the following results in its rows:
 dist[1]
 Mahalanobis distances
 dist[2]
 robust distances based on the final estimates
 dist[3]
 weights (=1 for small, =0 for large robust distances)
Example
Consider results for Brownlee's (1965) stackloss
data.
The three explanatory variables correspond to
measurements for a plant oxidizing ammonia to
nitric acid on 21 consecutive days.
 x_{1} air flow to the plant
 x_{2} cooling water inlet temperature
 x_{3} acid concentration
The response variable y_{i} gives
the permillage of ammonia lost (stackloss).
These data are also
given by Rousseeuw & Leroy (1987, p.76).
print "Stackloss Data";
aa = { 1 80 27 89 42,
1 80 27 88 37,
1 75 25 90 37,
1 62 24 87 28,
1 62 22 87 18,
1 62 23 87 18,
1 62 24 93 19,
1 62 24 93 20,
1 58 23 87 15,
1 58 18 80 14,
1 58 18 89 14,
1 58 17 88 13,
1 58 18 82 11,
1 58 19 93 12,
1 50 18 89 8,
1 50 18 86 7,
1 50 19 72 8,
1 50 19 79 8,
1 50 20 80 9,
1 56 20 82 15,
1 70 20 91 15 };
Rousseeuw & Leroy (1987, p.76) cite a large number of
papers where this data set was analyzed before and state
that most researchers "concluded that observations 1, 3,
4, and 21 were outliers"; some people also reported
observation 2 as an outlier.
By default, subroutine MVE
tries only 2000 randomly selected subsets in its search.
There are in total 5985 subsets of 4 cases out of 21 cases.
a = aa[,2:4];
optn = j(8,1,.);
optn[1]= 2; /* ipri */
optn[2]= 1; /* pcov: print COV */
optn[3]= 1; /* pcor: print CORR */
optn[5]= 1; /* nrep: use all subsets */
CALL MVE(sc,xmve,dist,optn,a);
The first part of the output shows the classical scatter
and correlation matrix:
Minimum Volume Ellipsoid (MVE) Estimation
Consider Ellipsoids Containing 12 Cases.
Classical Covariance Matrix
VAR1 VAR2 VAR3
VAR1 84.057142857 22.657142857 24.571428571
VAR2 22.657142857 9.9904761905 6.6214285714
VAR3 24.571428571 6.6214285714 28.714285714
Classical Correlation Matrix
VAR1 VAR2 VAR3
VAR1 1 0.781852333 0.5001428749
VAR2 0.781852333 1 0.3909395378
VAR3 0.5001428749 0.3909395378 1
Classical Mean
VAR1 60.428571429
VAR2 21.095238095
VAR3 86.285714286
There are 5985 subsets of 4 cases out of 21 cases.
All 5985 subsets will be considered.
The second part of the output shows the results of the
optimization (complete subset sampling):
Complete Enumeration for MVE
Best
Subset Singular Criterion Percent
1497 22 253.312431 25
2993 46 224.084073 50
4489 77 165.830053 75
5985 156 165.634363 100
Minimum Criterion= 165.63436284
Among 5985 subsets 156 are singular.
Observations of Best Subset
7 10 14 20
Initial MVE Location
Estimates
VAR1 58.5
VAR2 20.25
VAR3 87
Initial MVE Scatter Matrix
VAR1 VAR2 VAR3
VAR1 34.829014749 28.413143611 62.32560534
VAR2 28.413143611 38.036950318 58.659393261
VAR3 62.32560534 58.659393261 267.63348175
The third part of the output shows the optimization results after
local improvement:
Final MVE Estimates (Using Local Improvement)
Number of Points with Nonzero Weight=17
Robust MVE Location
Estimates
VAR1 56.705882353
VAR2 20.235294118
VAR3 85.529411765
Robust MVE Scatter Matrix
VAR1 VAR2 VAR3
VAR1 23.470588235 7.5735294118 16.102941176
VAR2 7.5735294118 6.3161764706 5.3676470588
VAR3 16.102941176 5.3676470588 32.389705882
Eigenvalues of Robust
Scatter Matrix
VAR1 46.597431018
VAR2 12.155938483
VAR3 3.423101087
Robust Correlation Matrix
VAR1 VAR2 VAR3
VAR1 1 0.6220269501 0.5840361335
VAR2 0.6220269501 1 0.375278187
VAR3 0.5840361335 0.375278187 1
The final output presents a table containing the classical
Mahalanobis distances, the robust distances, and the weights
identifying the outlying observations (that is leverage points
when explaining y with these three regressor variables):
Classical Distances and Robust (Rousseeuw) Distances
Unsquared Mahalanobis Distance and
Unsquared Rousseeuw Distance of Each Observation
Mahalanobis Robust
N Distances Distances Weight
1 2.253603 5.528395 0
2 2.324745 5.637357 0
3 1.593712 4.197235 0
4 1.271898 1.588734 1.000000
5 0.303357 1.189335 1.000000
6 0.772895 1.308038 1.000000
7 1.852661 1.715924 1.000000
8 1.852661 1.715924 1.000000
9 1.360622 1.226680 1.000000
10 1.745997 1.936256 1.000000
11 1.465702 1.493509 1.000000
12 1.841504 1.913079 1.000000
13 1.482649 1.659943 1.000000
14 1.778785 1.689210 1.000000
15 1.690241 2.230109 1.000000
16 1.291934 1.767582 1.000000
17 2.700016 2.431021 1.000000
18 1.503155 1.523316 1.000000
19 1.593221 1.710165 1.000000
20 0.807054 0.675124 1.000000
21 2.176761 3.657281 0
Distribution of Robust Distances
MinRes 1st Qu. Median
0.6751244996 1.5084120761 1.7159242054
Mean 3rd Qu. MaxRes
2.2282960174 2.0831826658 5.6373573538
Cutoff Value = 3.0575159206
The cutoff value is the square root of
the 0.975 quantile of the chi square
distribution with 3 degrees of freedom.
There are 4 points with large robust distances receiving
zero weights. These may include boundary cases. Only points
whose robust distances are substantially larger than the cutoff
value should be considered outliers.
References

Brownlee, K.A. (1965),
Statistical Theory and Methodology in Science and
Engineering, New York: John Wiley & Sons, Inc.

Davies, L. (1992),
``The Asymptotics of Rousseeuw's Minimum Volume Ellipsoid
Estimator,'' The Annals of Statistics, 20,
1828 1843.

Rousseeuw, P.J. (1984),
"Least Median of Squares Regression,"
Journal of the American Statistical Association,
79, 871 880.

Rousseeuw, P.J. (1985),
"Multivariate Estimation with High Breakdown Point,"
in Mathematical Statistics and Applications,
ed. by W. Grossmann, G. Pflug, I. Vincze, and W. Wertz,
Dordrecht: Reidel Publishing Company, 283 297.

Rousseeuw, P.J. and Croux, C. (1993),
"Alternatives to the Median Absolute Deviation,"
Journal of the American Statistical Association,
88, 1273 1283.

Rousseeuw, P.J. and Hubert, M. (1997),
"Recent developments in PROGRESS,"
in L_{1}Statistical Procedures and Related Topics,
ed. by Y. Dodge, IMS Lecture Notes  Monograph Series,
No. 31, 201 214.

Rousseeuw, P.J. and Leroy, A.M. (1987),
Robust Regression and Outlier Detection,
New York: John Wiley & Sons, Inc.

Rousseeuw, P.J. and Van Driessen, K. (1997),
``A fast Algorithm for the Minimum Covariance Determinant
Estimator,'' submitted for publication.

Rousseeuw, P.J. and Van Zomeren, B.C. (1990),
"Unmasking Multivariate Outliers and Leverage Points,"
Journal of the American Statistical Association,
85, 633 639.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.