Robust Regression Examples

Example 9.3: LMS and LTS Univariate (Location) Problem: Barnett and Lewis Data

If you do not specify matrix of the last input argument, the regression problem is reduced to the estimation problem of the location parameter . The following example is described in Rousseeuw and Leroy (1987, p. 175):

  
    print "*** Barnett and Lewis (1978) ***"; 
       b = { 3, 4, 7, 8, 10, 949, 951 }; 
  
       optn = j(9,1,.); 
       optn[2]= 3;    /* ipri */ 
       optn[3]= 3;    /* ilsq */ 
       optn[8]= 3;    /* icov */ 
  
    call lms(sc,coef,wgt,optn,b);
 

Output 9.3.1 shows the results of the unweighted LS regression.

Output 9.3.1: Table of Unweighted LS Regression

Robust Estimation of Location and Scale LMS: The 4th ordered squared residual will be minimized. Unweighted Least-Squares Estimation Median = 8 MAD ( * 1.4826) = 5.930408874 Mean = 276 Standard Deviation = 460.43602523 LS Residuals N Observed Residual Res / S 1 3.000000 -273.000000 -0.592916 2 4.000000 -272.000000 -0.590744 3 7.000000 -269.000000 -0.584229 4 8.000000 -268.000000 -0.582057 5 10.000000 -266.000000 -0.577713 6 949.000000 673.000000 1.461658 7 951.000000 675.000000 1.466002 Distribution of Residuals MinRes 1st Qu. Median Mean 3rd Qu. MaxRes -273 -272 -268 0 -266 675

Output 9.3.2 shows the results for LMS regression.

Output 9.3.2: Table of LMS Results

Least Median of Squares (LMS) Method Minimizing 4th Ordered Squared Residual. Highest Possible Breakdown Value = 57.14 % LMS Objective Function = 2.5 LMS Location = 5.5 Preliminary LMS Scale = 5.4137257125 Final LMS Scale = 3.0516389039 LMS Residuals N Observed Residual Res / S 1 3.000000 -2.500000 -0.819232 2 4.000000 -1.500000 -0.491539 3 7.000000 1.500000 0.491539 4 8.000000 2.500000 0.819232 5 10.000000 4.500000 1.474617 6 949.000000 943.500000 309.178127 7 951.000000 945.500000 309.833512 Distribution of Residuals MinRes 1st Qu. Median Mean 3rd Qu. MaxRes -2.5 -1.5 2.5 270.5 4.5 945.5

You obtain the LMS location estimate 6.5 compared with the mean 276 (which is the LS estimate of the location parameter) and the median 8. The scale estimate in the univariate problem is a resistant (high breakdown) estimator for the dispersion of the data (see Rousseeuw and Leroy 1987, p. 178).

For weighted LS regression, the last two observations are ignored (that is, given zero weights), as shown in Output 9.3.3.

Output 9.3.3: Table of Weighted LS Regression

Weighted Least-Squares Estimation Weighted Mean = 6.4 Weighted Standard Deviation = 2.8809720582 There are 5 points with nonzero weight. Average Weight = 0.7142857143 Weighted LS Residuals N Observed Residual Res / S Weight 1 3.000000 -3.400000 -1.180157 1.000000 2 4.000000 -2.400000 -0.833052 1.000000 3 7.000000 0.600000 0.208263 1.000000 4 8.000000 1.600000 0.555368 1.000000 5 10.000000 3.600000 1.249578 1.000000 6 949.000000 942.600000 327.181236 0 7 951.000000 944.600000 327.875447 0 Distribution of Residuals MinRes 1st Qu. Median Mean 3rd Qu. MaxRes -3.4 -2.4 1.6 269.6 3.6 944.6

Use the following code to obtain results from LTS:

  
       optn = j(9,1,.); 
       optn[2]= 3;    /* ipri */ 
       optn[3]= 3;    /* ilsq */ 
       optn[8]= 3;    /* icov */ 
  
    call lts(sc,coef,wgt,optn,b);
 

The results for LTS are similar to those reported for LMS in Rousseeuw and Leroy (1987), as shown in Output 9.3.4.

Output 9.3.4: Table of LTS Results

Least Trimmed Squares (LTS) Method Minimizing Sum of 4 Smallest Squared Residuals. Highest Possible Breakdown Value = 57.14 % LTS Objective Function = 2.0615528128 LTS Location = 5.5 Preliminary LTS Scale = 4.7050421234 Final LTS Scale = 3.0516389039 LTS Residuals N Observed Residual Res / S 1 3.000000 -2.500000 -0.819232 2 4.000000 -1.500000 -0.491539 3 7.000000 1.500000 0.491539 4 8.000000 2.500000 0.819232 5 10.000000 4.500000 1.474617 6 949.000000 943.500000 309.178127 7 951.000000 945.500000 309.833512 Distribution of Residuals MinRes 1st Qu. Median Mean 3rd Qu. MaxRes -2.5 -1.5 2.5 270.5 4.5 945.5

Since nonzero weights are chosen for the same observations as with LMS, the WLS results based on LTS agree with those based on LMS (shown previously in Output 9.3.3).

In summary, you obtain the following estimates for the location parameter:

• LS estimate (unweighted mean) = 276
• Median = 8
• LMS estimate = 5.5
• LTS estimate = 5.5
• WLS estimate (weighted mean based on LMS or LTS) = 6.4