MVE Call
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MVE Call

Example

CALL MVE (sc, coef, dist, opt, x <, s> ) ;

The MVE subroutine computes the robust estimation of multivariate location and scatter, defined by minimizing the volume of an ellipsoid that contains $h$ points.

The MVE subroutine computes the minimum volume ellipsoid estimator. These robust locations and covariance matrices can be used to detect multivariate outliers and leverage points. For this purpose, the MVE subroutine provides a table of robust distances.

In the following discussion, $N$ is the number of observations and $n$ is the number of regressors. The input arguments to the 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.

0

prints no output except error messages.

1

prints most of the output.

2

additionally prints case numbers of the observations in the best subset and some basic history of the optimization process.

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. 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. The default is opt[3]=0.

0

does not return or print.

1

prints the robust correlation matrix.

2

returns the final robust correlation matrix in coef.

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]= $h = \left[\frac{N+n+1}{2}\right]$. If the value of $h$ is specified outside the range $\frac{N}{2} + 1 ~ \leq ~ h ~ \leq ~ \frac{3N}{4} + \frac{n+1}{4}$, it is reset to the closest boundary of this region.

opt[5]

specifies the number $N_\mr {Rep}$ 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[5] 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

$N_\mr {lower}$

500

50

22

17

15

14

0

0

0

0

$N_\mr {upper}$

$10^6$

1414

182

71

43

32

27

24

23

22

$N_\mr {Rep}$

500

1000

1500

2000

2500

3000

3000

3000

3000

3000

n

11

12

13

14

15

$N_\mr {lower}$

0

0

0

0

0

$N_\mr {upper}$

22

22

22

23

23

$N_\mr {Rep}$

3000

3000

3000

3000

3000

If the number of cases (observations) $N$ is smaller than $N_\mr {lower}$, as given in the table, then all possible subsets are used; otherwise, $N_\mr {Rep}$ subsets are chosen randomly. This means that an exhaustive search is performed for opt[5]$= -1$. If $N$ is larger than $N_\mr {upper}$, a note is printed in the log file that indicates how many subsets exist.

x

refers to an $N \times n$ matrix $\mb {X}$ of regressors. Missing values are not permitted in $x$.

s

refers to an $n+1$ vector that contains $n+1$ observation numbers of a subset for which the objective function should be evaluated, where $n$ is the number of parameters. In other words, the MVE algorithm computes the minimum volume of the ellipsoid that contains the observation numbers contained in $s$.

The MVE subroutine returns the following values:

sc

is a column vector that contains the following scalar information:

sc[1]

the quantile $h$ used in the objective function

sc[2]

number of subsets generated

sc[3]

number of subsets with singular linear systems

sc[4]

number of nonzero weights $w_ i$

sc[5]

lowest value of the objective function $F_\mr {MVE}$ attained (volume of smallest ellipsoid found)

sc[6]

Mahalanobis-like distance used in the computation of the lowest value of the objective function $F_\mr {MVE}$

sc[7]

the cutoff value used for the outlier decision

coef

is a matrix with $n$ columns that contains 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

coef[2+n+1:2+2n,]

the final robust correlation matrix for opt[3]=1 or opt[3]=3

dist

is a matrix with $N$ columns that contains 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 robust distances; 0 for large robust distances)

Example

Consider results for Brownlee (1965) stackloss data. The three explanatory variables correspond to measurements for a plant that oxidizes 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 $y_ i$ contains the permillage of ammonia lost (stackloss). These data are also given by Rousseeuw and Leroy (1987). 

        /* X1  X2  X3   Y    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 and Leroy (1987) cite a large number of papers where this data set was analyzed 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 chooses only 2,000 randomly selected subsets in its search. There are in total 5,985 subsets of 4 cases out of 21 cases, as shown in Figure 23.193, which is produced by the following statements: 

a = aa[, 2:4];
opt = j(8, 1, .);
opt[1] = 2;              /* ipri */
opt[2] = 1;              /* pcov: print COV */
opt[3] = 1;              /* pcor: print CORR */
opt[5] = -1;             /* nrep: use all subsets */

call mve(sc, xmve, dist, opt, a);

The first part of the output (Figure 23.193) shows the classical scatter and correlation matrix, along with the means of each variable.

Figure 23.193: Classical Estimates of Scatter and Location

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


The second part of the output (Figure 23.194) shows the results of the optimization (complete subset sampling):

Figure 23.194: Subset Sampling and Optimal Subset

Subset Singular Best
Criterion 		Percent
1497 22 253.312431 25
2993 46 224.084073 50
4489 77 165.830053 75
5985 156 165.634363 100

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 (Figure 23.195) shows the optimization results after local improvement:

Figure 23.195: Robust Estimates of Scatter and Location

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 (Figure 23.196) presents a table that contains the classical Mahalanobis distances, the robust distances, and the weights that identify the outlying observations (that is leverage points when explaining $y$ with these three regressor variables):

Figure 23.196: Distances and Weights

Classical Distances and Robust (Rousseeuw) Distances
Unsquared Mahalanobis Distance and
Unsquared Rousseeuw Distance of Each Observation
N Mahalanobis Distances Robust 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

MinRes 1st Qu. Median Mean 3rd Qu. MaxRes
0.6751244996 1.5084120761 1.7159242054 2.2282960174 2.0831826658 5.6373573538


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