The relaxed intersection of m sets corresponds to the classical
intersection between sets except that it is allowed to relax few sets in order to avoid an empty intersection.
This notion can be used to solve constraints satisfaction problems
that are inconsistent by relaxing a small number of constraints.
When a bounded-error approach is considered for parameter estimation,
the relaxed intersection makes it possible to be robust with respect
to some outliers.
Figure 1. q-intersection of 6 sets for q=2 (red), q=3 (green), q= 4 (blue), q= 5 (yellow).
Figure 2. Set-membership function associated to the 6 intervals.
Figure 3. The red box corresponds to the 4-relaxed intersection of the 6 boxes
Figure 4. Set of all parameter vectors consistent with exactly 6-q data bars (painted red), for q=1,2,3,4,5.
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the data set. An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses.
Figure 5. q-relaxed intersection of 6 sets for q=2 (red), q=3 (green), q= 4 (blue), q= 5 (yellow).
