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 };

For and (three explanatory variables including intercept), you obtain a total of 5,985 different subsets of 4 observations out of 21. If you do not specify OPTN[5], the LMS algorithms draw random sample subsets. Since there is a large number of subsets with singular linear systems that you do not want to print, you can choose OPTN[2]=2 for reduced printed output, as in the following:

title2 "***Use 2000 Random Subsets for LMS***";
a = aa[,2:4]; b = aa[,5];
optn = j(9,1,.);
optn[2]= 2;    /* ipri */
optn[3]= 3;    /* ilsq */
optn[8]= 3;    /* icov */

call lms(sc,coef,wgt,optn,b,a);

Summary statistics are shown in Output 12.2.1. Output 12.2.2 displays the results of LS regression. Output 12.2.3 displays the LMS results for the 2000 random subsets.

For LMS, observations 1, 3, 4, and 21 have scaled residuals larger than 2.5 (output not shown), and they are considered outliers. Output 12.2.4 displays the corresponding WLS results.

The V7 LTS algorithm is similar to the LMS algorithm. Here is the code:

title2 "***Use 2000 Random Subsets for LTS***";
a = aa[,2:4]; b = aa[,5];
optn = j(9,1,.);
optn[2]= 2;    /* ipri */
optn[3]= 3;    /* ilsq */
optn[8]= 3;    /* icov */
optn[9]= 1;    /* V7 LTS */

call lts(sc,coef,wgt,optn,b,a);

In addition to observations 1, 3, 4, and 21, which were considered outliers in LMS, observations 2 and 13 for LTS have absolute scaled residuals that are larger (but not as significantly as observations 1, 3, 4, and 21) than 2.5 (output not shown). Therefore, the WLS results based on LTS are different from those based on LMS.

Output 12.2.6 displays the results for the weighted LS regression.

The FAST-LTS algorithm uses only 500 random subsets and gets better optimization results. Here is the code:

title2 "***Use 500 Random Subsets for FAST-LTS***";
a = aa[,2:4]; b = aa[,5];
optn = j(9,1,.);
optn[2]= 2;    /* ipri */
optn[3]= 3;    /* ilsq */
optn[8]= 3;    /* icov */
optn[9]= 0;    /* FAST-LTS */

call lts(sc,coef,wgt,optn,b,a);

For this example, the two LTS algorithms identify the same outliers; however, the FAST-LTS algorithm uses only 500 random subsets and gets a smaller objective function, as seen in Output 12.2.7. For large data sets, the advantages of the FAST-LTS algorithm are more obvious.

You now report the results of LMS for all different subsets. Here is the code:

title2 "*** Use All 5985 Subsets***";
a = aa[,2:4]; b = aa[,5];
optn = j(9,1,.);
optn[2]= 2;    /* ipri */
optn[3]= 3;    /* ilsq */
optn[5]= -1;   /* nrep: all 5985 subsets */
optn[8]= 3;    /* icov */

ods select IterHist0 BestSubset EstCoeff;
call lms(sc,coef,wgt,optn,b,a);

Output 12.2.8 displays the results for LMS.

Next, report the results of LTS for all different subsets, as follows:

title2 "*** Use All 5985 Subsets***";
a = aa[,2:4]; b = aa[,5];
optn = j(9,1,.);
optn[2]= 2;    /* ipri */
optn[3]= 3;    /* ilsq */
optn[5]= -1;   /* nrep: all 5985 subsets */
optn[8]= 3;    /* icov */
optn[9]= 1;    /* V7 LTS */

ods select IterHist0 BestSubset EstCoeff;
call lts(sc,coef,wgt,optn,b,a);

Output 12.2.9 displays the results for LTS.