## 3 Case Study
### Real-World Case Study  
To further investigate the performance of the proposed method in real cases, the lithology types in the well-known Swiss Jura data set [24] are used, in which five rock types are sampled at 359 locations in a 14.5 km\({}^{2}\) area (Figure 5). The class proportions of these five categories are (0.2046, 0.3282, 0.2432, 0.0116, 0.2124), respectively. We focus on MANN with sigmoid activation functions and consider the 10 nearest samples as the neighbors of the target point. For the proposed method, we choose the one that has the maximum correlation with output probabilities as the input and we have trained nine compositional ANNs. We furthermore assume that the random interaction noise \(\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{9}\right)\) is a Gaussian random field. By trial and error, we find that the model which has the highest classification accuracy can be specified by mean \(\mu=0\) and standard deviation \(\sigma=0.1\); that is,  
\[P\left(\varepsilon_{i}\right)=\mathrm{N}\left(0,0.01\right),\ \mathrm{cov}\left( \varepsilon_{i},\varepsilon_{j}\right)=0\ \mathrm{for}\ i\
eq j\,. \tag{6}\]  
Correlation coefficients between various transition probabilities are compared with each other and the maximum correlation pairs are shown in Figure 6, where the red pictures (Figure 6d,h) are worthy of discussion. Note that the correlation between \(P\left(D_{4}|A\right)\) and \(P\left(D_{5}|A\right)\) is quite weak with \(r=0.2364\), and the relevance from \(P\left(D_{1}|A\right)\) to \(P\left(D_{3}|A\right)\) with \(P\left(D_{5}|A\right)\) is even lower. In such a case, we choose \(P\left(D_{5}|A\right)\) itself as the input node of \(P\left(D_{5}|AD_{1}\cdots D_{4}\right)\), since no significant interaction information found from \(D_{1}\) to \(D_{5}\). In the same way, \(P\left(D_{9}|A\right)\) is considered as the input node of \(P\left(D_{9}|AD_{1}\cdots D_{8}\right)\). It is rather remarkable that the maximum correlation pairs may not always be \(P\left(D_{i}|A\right)\) and \(P\left(D_{i-1}|A\right)\), just as Figure 6g displays. \(P\left(D_{6}|A\right)\) rather than \(P\left(D_{7}|A\right)\) is chosen as the input node for output \(P\left(D_{8}|AD_{1}\cdots D_{7}\right)\), which means that the main interaction information among \(D_{1},D_{2},\cdots D_{8}\) is encapsulated in \(D_{6}\) and \(D_{8}\). In general, the input nodes of \(P\left(D_{i}|AD_{1}\cdots D_{i-1}\right)\), \(i=2,\,3,\,\cdots,\,10\), should be \(P\left(D_{i-1}|A\right)\) if the correlation coefficients between \(P\left(D_{i-1}|A\right)\) and \(P\left(D_{i}|A\right)\) exceed 0.5. The R package \"nnet\" designed under the framework of feed forward and back propagation algorithms is used during the training process. We consider three units in the hidden layer. The training weights of MANN with interaction noise are given in Table 2.  
Figure 5: Jura lithology data set with 359 samples. As can be seen from Figure 7, the nine ANNs converge after a certain number of iterations. The training error of MANN with interaction noise can be viewed as the summation of the nine ANNs, which is illustrated in Figure 7 by different colors and line styles. Our original image (Figure 8a) is obtained from the \"jura.grid\" data set kindly provided by R software 3.2.5, and the known 5957 grid values are used as the validation set. We find that MANN with interaction noise has a classification accuracy of 65.12% (3879 out of 5957) and can successfully preserve small-scale Quaternary patch features (Figure 8b). We have also used the MCRF method for comparison. Not surprisingly, the latter yields a relatively lower classification accuracy not only in overall prediction, but also in most of the single categories (Table 3). This result is comprehensible, since the general solution of MCRF is based on the conditional independence assumption, which fails to measure the interaction or redundancy information among the neighboring sources.