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 Language Reference

## 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 NRep 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 Nlower 500 50 22 17 15 14 0 0 0 0 Nupper 1000000 1414 182 71 43 32 27 24 23 22 NRep 500 1000 1500 2000 2500 3000 3000 3000 3000 3000

 n 11 12 13 14 15 Nlower 0 0 0 0 0 Nupper 22 22 22 23 23 NRep 3000 3000 3000 3000 3000

If the number of cases (observations) N is smaller than Nlower, then all possible subsets are used; otherwise, NRep subsets are drawn randomly. This means that an exhaustive search is performed for opt[6]=-1. If N is larger than Nupper, 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 wi

sc[5]
lowest value of the objective function FMVE attained (volume of smallest ellipsoid found)

sc[6]
Mahalanobis-like distance used in the computation of the lowest value of the objective function FMVE

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 = [(c2(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.
• x1 air flow to the plant
• x2 cooling water inlet temperature
• x3 acid concentration
The response variable yi 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 L1-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.

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