Language Reference

MVE Call

Subsections:

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 24.223, 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 24.223) shows the classical scatter and correlation matrix, along with the means of each variable.

Figure 24.223: 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 24.224) shows the results of the optimization (complete subset sampling):

Figure 24.224: 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 24.225) shows the optimization results after local improvement:

Figure 24.225: 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 24.226) 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 24.226: 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