## 6 Discussion
### Iris Data Set--A Motivational Example for the Detection of FS-Ordinal Structures  
The Iris database is one of the most frequently used traditional machine learning data sets consisting of three types of Iris flowers. The classes, i.e., Iris types, are _Setosa_, _Versionolour_, and _Virginica_. The data are characterised by four features, i.e., _sepal length_, _sepal width_, _petal length_, and _petal width_.  
In combination with the SVM-Acc mapping, it was not able to detect an (FS-)ordinal structure specific to the Iris data set. Note that the Iris data set constitutes a 3-class classification task. This observation implies that the SVM-Acc measure-based PSM does neither fulfil the conditions of Theorem 1, nor of Corollary 1. In fact, calculating the corresponding SVM-Acc mapping-based PSM leads to  
\[M^{(id)}=\begin{pmatrix}\omega_{1}&\omega_{2}&\omega_{3}\\ 0&1.00&1.00\\ 1.00&0&0.99\\ 1.00&0.99&0\end{pmatrix}, \tag{17}\]  
whereby \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) denote the classes _Setosa_, _Versionolor_, and _Virginica_, respectively. Note that the matrix \(M^{(id)}\) from Equation (17) constitutes exactly the same example which we discussed in Section 3.2 (with \(0.99=\alpha\)), in combination with Figure 4. More precisely, the class orders \(\omega_{1}-\omega_{2}-\omega_{3}\) (with its reversed order \(\omega_{3}-\omega_{2}-\omega_{1}\)) and \(\omega_{1}-\omega_{3}-\omega_{2}\)(with its reversed order \(\omega_{2}-\omega_{3}-\omega_{1}\)) constitute ordinal arrangements by Definition 2. Therefore, the uniqueness property of Equation (8) is violated. Thus, by Definition 3, the Iris data set is not FS-ordinal with respect to the mapping SVM-Acc.  
On the other hand, using the DR mapping, the Iris data set is identified as FS-ordinal specific to the order \(\omega_{1}\prec\omega_{2}\prec\omega_{3}\), i.e., _Setosa_\(\prec\)_Versicolor_\(\prec\)_Virginica_ (with its reversed order _Virginica_\(\prec\)_Versicolor_\(\prec\)_Setosa_). The question that arises here is the following. Does this specific structure _make sense_? Based solely on the label names (i.e., flower types), one would probably not try to find an ordinal structure in combination with the Iris data set. However, as we already discussed in this work, one can benefit from ordinal class structures that are present in the feature space, from a machine learning-based point of view. Note that the feature space of the Iris data set consists of solely four features (_spal length_ (SL), _spal width_ (SW), _petal length_ (PL), and _petal width_ (PW)), which are even _easy to interpret_. The low amount of features allows us to proceed with the following _eye test_.  
From the total amount of four features, we can form six distinct binary feature combinations, i.e., (SL, SW), (SL, PL), (SL, PW), (SW, PL), (SW, PW), and (PL, PW). In Figure 6, we plotted all of the six binary feature combinations. From Figure 6, we can make the following observations. Except for the top left plot (_spal length_ vs. _spal width_), the detected class structure (_Setosa_\(\prec\)_Versicolor_\(\prec\)_Virginica_, with its reversed order _Virginica_\(\prec\)_Versicolor_\(\prec\)_Setosa_) is reflected in each binary subspace. Therefore, it is to expect that the same class order is present in the complete 4-dimensional feature space. To answer the question stated above, the class order _Setosa_\(\prec\)_Versicolor_\(\prec\)_Virginica makes sense_ in combination with the provided feature space. Thus, applying the proposed DR measure helped us to identify the _correct_ class order of the Iris data set.