## 3 A Sample of the Experimental Work Results  
In this section, we show the exemplary results for our selected techniques, to show its effectiveness in the context of reducing training data size. For the sake of simplicity, we have chosen the k-NN classifier as a base. We carried out experiments on selected data from the UCI repository [37]--see Table 10. In Tables 11-20 and Figure 1, we have the results for Cross Validation 5 method. Let us move on to the discussion of selected detailed results, starting from description of the results for the Australian Credit data set. The result for Standard (SG) and Concept-dependent granulation (CDG) is in Table 11, where, in case of SG for radius 0.5, we have reduction in training size of around 90 percent preserving classification accuracy in the range of 84.7 percent. For the CDG variant, we have reduction in training size of about 99.5 percent for radius 0.071, where the exhaustive rule set is reduced in 99.9 percent and accuracy of classification is around 77 percent. The results are comparable, but the concept-dependent variant shows a more stable classification as the radius increases. In case of Homogeneous granulation, see Table 15, we have accuracy equal to 0.835 with a 48 percent reduction of training size. The sample of results for exemplary epsilon variant--\(\epsilon\) Homogeneous Granulation--is in Table 20, where we have reduction in training size about 50 percent, with accuracy of 0.842. The layered granulation process is visible in Table 16, where the basic method is concept-dependent granulation and the result is similar to a single concept-dependent variant. In the case of Car data set, see Table 12, the concept-dependent variant works best giving accuracy of 0.864, with a reduction in training size of around 73 percent. For a Hepatitis data set, concept-dependent also works best, for radius 0.474, the accuracy is equal to 0.875, with a 90 percent reduction in training size. In addition, finally, the spectacular result is obtained for Heart Disease data set, where with 99 percent reduction in training size, we have obtained for concept-dependent and standard granulation the accuracy 0.8. The results for homogeneous variants are shown in Tables 15 and 20. The best result we have achieved on the tested data are a reduction of 62 percent in the number of objects with full classification efficiency. Allow us to summarize the results obtained in this section. The internal knowledge from the original training decision systems--measured by ability for classification--seems to be preserved in each mentioned case (the accuracy of classification is fully comparable with nil case, without reduction). Both techniques, standard granulation and concept-dependent, prove to be comparable. In the concept-dependent variant, we observe a higher classification stability with an increasing radius. Another advantage of the concept-dependent variant is the creation of granular reflection, which from the smallest radii contain patterns from all decision-making classes. The multiple variant does not produce spectacular results, but, according to our previous research, see [30]--it allows us to look for the optimal granulation radii. Our research shows that the radius for which the reduction of objects between the first and second layer is greatest is close to the optimal one in most tested systems. In this way, the optimum granulation radius can be estimated without classification tests. The last group of tested techniques are recently discovered homogeneous methods, which work dynamically on every data and do not require estimation of optimal parameters. It is obvious that the effectiveness of our methods depends to a large extent on the data under investigation. We do not plan to present an overview of the effectiveness of the whole range of classification techniques because our aim was to present an example of the effectiveness of approximation methods for decision-making systems. Let us move on to presenting additional test results for selected previously used classifiers.