PMC9217125_01
COMPAS dataset ISA projections with and without race as an input attribute, colored according to the IH value for each instance
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Fig8
Figure 8 presents the instance spaces of the COMPAS dataset with (in Fig. 8a) and without (in Fig. 8b) race as an attribute, where the points are colored according to their IH values. IS projection of Fig. 8a is similar to that of Fig. 8b, offering the first evidence that the hardness profile of the dataset is maintained despite the usage or not of the race attribute. It took 20 minutes and 35 seconds to generate the IS of the original COMPAS dataset and 19 minutes and 26 seconds to obtain the IS of the dataset counterpart which eliminates the race attribute. We now further refine our analysis by focusing the Lasso selection tool on two specific regions of the ISA of the COMPAS dataset without race as an input attribute (projection shown in Fig. 8b).
Distributions of attributes race_African-American and race_Caucasian for the entire dataset (left) and in different data selections on ISA built with (center) and without (right) race as an attribute
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Fig9
First, we show in Fig. 9 the distribution of race values considering the entire dataset and the selections with low IH and high IH values, respectively, with and without race as an attribute. Interestingly, the proportions of low IH instances per class and per race in Fig. 9 are very similar despite the usage or not of race as an input attribute.
Boxplots of the raw feature values referring to prior and juvenile offenses for three sets of instances of the COMPAS dataset without race as an input attribute: the easiest to classify (in blue), the hardest to classify (in red) and all the other instances (in green)
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Fig10
Figure 10 contrasts the raw feature values referring to prior and juvenile offenses of the easiest (in blue), hardest (in red) and all the other instances (in green) of the COMPAS (without ) dataset. The plot from Fig. 10b sums up the counts of the ,  and  attributes as a total count of juvenile offenses registered for each individual. As shown in Fig. 10a, the easiest data excerpt contains individuals with high numbers of prior crimes (all recidivists) and the others data excerpt has in general minor amounts of prior offenses (although there are many outliers, since this set contains individuals from both recidivist and non-recidivist classes). Taking the total number of juvenile offenses (Fig. 10b), the contrast is more evident.
Boxplots of some of the meta-features values for three sets of instances of the COMPAS dataset without race as an input attribute: the easiest to classify (in blue), the hardest to classify (in red) and all the other instances (in green)
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Fig11
The values of some of the meta-features (boxplots from Fig. 11) in the easiest to classify (in blue), hardest to classify (in red) and others (in green) set of instances also evidence the contradicting patterns of the hardest to classify instances. The hardest set shows: high CL values (Fig. 11a), indicating the instances in this set have a low likelihood of pertaining to their class; high DCP values (Fig. 11b), indicating that such observations tend to be placed in disjunctures with a majority of examples from the other class; and very high kDN values (median at the maximum of 1.0, Fig. 11c), indicating that most of these observations are close and surrounded by instances from the other class.
ISA for the base 2normals dataset, with points colored by class (left) or IH (right)
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Fig12
Figure 12a shows the base dataset used in these experiments. Figure 12b presents the same dataset with data items colored by IH values. Taking the dataset from Fig. 12a as base, we randomly flip the class labels at the following rates: 5%, 10%, 20%, 30%, 40%, 50%, 70%, 90% and 100%.
Instance easiness and classifiers' footprint areas for different noise levels
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Fig13
Figure 13a shows the average easiness footprint area registered in the instance spaces produced for the different noise levels. The same effect can be verified in the footprint areas of the classification techniques (Fig. 13b).