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LTS Call

CALL LTS( sc, coef, wgt, opt, y<, <x><, sorb>>);

A new algorithm, FAST-LTS, was added to the LTS subroutine in SAS/IML Release 8.1. The FAST-LTS algorithm is set as the default algorithm. The original algorithm is kept for convenience, and can be used temporarily by specifying optn[9]=1. Eventually the original algorithm will be replaced with the new FAST-LTS algorithm.

The original algorithm for the LTS subroutine and the algorithm used in the LMS subroutine are based on the PROGRESS program by Rousseeuw and Leroy (1987). Rousseeuw and Hubert (1996) prepared a new version of PROGRESS to facilitate its inclusion in SAS software, and they have incorporated several recent developments. Among other things, the new version of PROGRESS now yields the exact LMS for simple regression, and the program uses a new definition of the robust coefficient of determination (R²). Therefore, the outputs may differ slightly from those given in Rousseeuw and Leroy (1987) or those obtained from software based on the older version of PROGRESS.

FAST-LTS

1. The default h is (n+p+1)/2, where p is the number of the independent variables. You can choose any integer h with . The LTS's breakdown point (n-h+1)/n is reported. If you are sure that the data contains less than 25% of contamination, you can obtain a good compromise between breakdown value and statistical efficiency by putting h=[.75n].
2. If p=1 (univariate data), then compute the LTS estimator by the exact algorithm of Rousseeuw and Leroy (1987, pp. 171-172) and stop.
3. From here on, . If n<600, draw a random p-subset and compute the regression coefficients using these p points (if the regression is degenerate, draw another p-subset). Compute the absolute residuals for all points in the data set and select the first h points with smallest absolute residuals. From this selected h-subset, carry out C-steps (Concentration step; refer to Rousseeuw and Van Driessen [1998] for details) until convergence. Repeat this procedure 500 or times (as determined by opt[5] of LTS Call) and find the ten (at most) solutions with the lowest sums of h squared residuals. For each of these ten best solutions, take C-steps until convergence and find the best final solution.
4. If n > 600, construct up to five disjoint random subsets with sizes as equal as possible, but not to exceed 300. Inside each subset, repeat the procedure in step 3 500/5 = 100 times and keep the ten best solutions. Pool the subsets, yielding the merged set of size n_merged. In the merged set, for each of the 5×10 = 50 best solutions, carry out two C-steps using n_merged and h_merged = [n_merged (h/n)] and keep the ten best solutions. In the full data set, for each of these ten best solutions, take C-steps using n and h until convergence and find the best final solution.

OPTION Changes

OUTPUT Changes

1. The "Complete Enumeration for LTS" table and the "Resistant Diagnostic" table do not apply for the FAST-LTS algorithm and are not displayed.
2. The analysis based on "Observations of Best Subset" of size p, where p is the number of the independent variables, is replaced by the analysis based on observations of the best h of the entire data set obtained after full iteration, which gives the exact LTS estimator and covariances for small data set (). LTS Objective Function, Preliminary LTS Scale, Robust R Squared, and Final LTS Scale are also reported for the LTS estimator.
3. The LTS residuals are changed, because of the change of the LTS estimator.
4. The "Coef" vector does not include the best subset.

Illustrative Example

title1 'Compare two algorithms for LTS';

proc iml;
  reset noname;

  x = {42, 37, 37, 28, 18, 18, 19, 20, 15};
  y = {80, 80, 75, 62, 62, 62, 62, 62, 58};
  optn = j(9, 1, .);
  optn[1]= 0;    /* --- with intercept      --- */
  optn[2]= 4;    /* --- print all output    --- */
  optn[3]= 3;    /* --- compute LS and WLS  --- */
  optn[8]= 3;    /* --- covariance matrices --- */
  optn[9]= 1;    /* --- Version 7 LTS       --- */

  call lts(sc, coef, wgt, optn, y, x);

  print "sc is " sc, "coef is " coef;

  optn[9]= 0;    /* --- FAST-LTS algorithm   --- */

  call lts(sc, coef, wgt, optn, y, x);

  print "sc is " sc, "coef is " coef;

quit;

Comparison of the outputs is summarized as follows:

1. The summary statistics for dependent and independent variables and the results of the classical (unweighted) least-squares estimation (outputs in Page 1 and first half of Page 2) do not change.
2. The "Complete Enumeration for LTS" table on page 1 and the "Resistant Diagnostic" table on page 3 of the first output is eliminated. The analysis based on "Observations of Best Subset" of size p, where p is the number of the independent variables, is replaced by the analysis based on observations of the best h of the entire data set obtained after full iteration.
3. The best ten (at most) estimates before the final search are available by setting a proper option (see pages 2 and 3 of the second output). The first one is the best with the smallest objective value.
4. The results of the weighted least squared estimation (page 4 for the first output, page 5 for the second output) do not change because the two algorithms detect the same outliers in this simple example. Results will be changed if they detect different outliers.
5. The "Coef" vector does not include the best subset (page 6 of the second output).

Output from Previous Algorithm for LTS

Output from FAST-LTS Algorithm

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