3.4 Subgroup Visualization

A novel visualization method can be used to visualize the output of any subgroup discovery algorithm, provided that the output has the form of rules with a target class in their consequent. It can also be used as a method for visualizing standard classification rules.

Subgroup visualization, as described in this section, allows us to compare distributions of different subgroups. The approach assumes the existence of at least one numeric (or ordered discrete) attribute of expert's interest for subgroup analysis. The selected attribute is plotted on the X-axis of the diagram. The Y-axis represents a class, or more precisely, the number of instances of a given class. Both directions of the Y-axis (Y⁺ and Y^-) are used to indicate the number of instances. In Figure 6, for instance, the X-axis represents age, the Y⁺-axis denotes class coronary heart disease (CHD) and Y^- denotes class `healthy' (non-CHD). Out of four graphs at the Y⁺ side, three represent induced subgroups (A1, A2 and C1) of CHD patients, and the fourth shows the age distribution of the entire population of CHD (all CHD) patients. The graphs at the Y^- side show the distribution of non-CHD (all healthy) patients in the training set and the distribution of healthy subjects included into the subgroup A2 (dashed line).

Figure 6: Distributions of the numbers of CHD patients (all CHD) and healthy subjects (all healthy) in terms of age (in years). Graphs A1, A2, and C1 represent the distributions of CHD patients belonging to the corresponding subgroups. The dashed line represents healthy subjects included in subgroup A2.

Figure 7: Distributions of the numbers of CHD patients (all CHD) and healthy subjects (all healthy), as well as the distributions of patients for subgroups B1 and B2 in terms of age (in years). The dashed line represents healthy subjects included in subgroup B1.

Figure 8: Distributions of all CHD patients and those described by patterns A1 and B2, as well as all healthy subjects and those included into pattern B2 (dashed line) in terms of total cholesterol value in mmol L^-1.

On purpose, the graphs of subgroups A1 and C1 in Figure 6 show only the coverage of positive cases (CHD patients), and in Figure 7 the graph of subgroup B2 shows only the coverage of positive cases, whereas the graphs of A2 in Figure 6 and B1 in Figure 7 indicate that the descriptions of subgroups cover positive cases (CHD patients) as well as some negative cases (healthy individuals). Except for the correct visualization of subgroups A2 and B1 and of the entire CHD and non-CHD distribution, Figures 6 and  7 have been simplified in order to enable a better understanding of the visualization method, by showing just the coverage of positive cases.

In medical domains we typically use the Y⁺ side to represent the number of positive cases (CHD patients, in this paper) in order to reveal properties of induced patterns for subgroups of these patients. On the other hand, the Y^- side is reserved to reveal properties of these same patterns (or other patterns) for the negative cases (patients without CHD). One of the advantages of using Y⁺ and Y^- as proposed above is that in binary classification problems the comparison of the area under the graph of a subgroup and the graph of the entire population visualizes the fractions of $\frac{TP}{Pos} = \frac{TP}{TP + FN}$ at the Y⁺ side (sensitivity TPr), and $\frac{FP}{Neg} = \frac{FP}{TN + FP}$ at the Y^- side (false alarm rate FPr), where Pos and Neg stand for the numbers of positive and negative cases in the entire population, respectively. For instance, in the visualization of subgroup B1 in Figure 7 the area under the dashed line on the Y^- side represents the numbers of misclassified training instances of subgroup B1. In this way, the sensitivity and false alarm rate can be estimated for pattern B1 from Figure 7. The same information for pattern B2 can be found in Figure 8, showing subgroups A1 and B2 in terms of attribute `total cholesterol value'.

The proposed visualization method can be adapted to visualize subgroups also in terms of value distributions of discrete/nominal attributes. An approach to such visualization is presented in Figure 9. However, due to bar chart representation, it is more difficult to compare several subgroups in one visualization.

Figure 9: Distribution of CHD patients and healthy subjects with respect to stress values (low, high, and very high) for the entire population and the five induced patterns.

Figure 10: Distribution of CHD patients and healthy subjects with respect to exercise ECG ST segment depression in millimeters (1mm corresponds to 0.1 mV). Large difference between total healthy and ill populations can be noticed, but differences among patterns are very small. Patterns A1 and C1 are selected as extreme cases. The dashed line presents healthy persons incorrectly described by pattern C1

In general, it is not necessary that Y⁺ and Y^- denote two opposite classes. If appropriate, they may denote any two classes, or even any two different attribute values, which the expert would like to compare.