## 4 Application  
The data in Figure 2 and Table 1 are used to illustrate how this new structure QCCS works and how the exact tests are used for testing whether the levels of thematic accuracy are achieved or not. Figure 3 summarizes the specifications from Table 1 regarding the percentage of correctness and the limits on the misclassifications for each category. It presents the set of probability vectors, each one with values in the assumed order, and also groups some categories if the specifications do not analyze them separately. Figure 4 shows the same data as Figure 2, but the values in each column are ordered and grouped following Figure 3.  
The required assumptions for these tests are as follows:  
* The positional accuracy is adequate for the scale of the thematic product;
* The RDS is an adequate quality reference for this new approach of thematic accuracy quality control. Assuming the set of specifications presented in Table 1 (Figure 3), we define the corresponding null hypotheses for each column, and we illustrate the application of this new approach to this example. Woodland.According to Figure 3 (column Wo), for the _Woodland_ category, we define a multinomial distribution with three categories: \(X_{1}\) denoting the number of elements correctly classified; \(X_{2}\), the number of elements confused with _Grassland_; and \(X_{3}\), the number of elements classified as _Non-vegetated_ or _Water_. Therefore, \(q=3\). The specifications lead us to the multinomial \(M(51,p_{0})\) with \(p_{0}=(0.95,\ 0.04,\ 0.01)\). In this case, the observed classification is \(Y^{*}=(47,\ 4,\ 0)\) (see Figure 4, column Wo), and the \(p\)-value is obtained by adding up the probabilities of \(M(51,p_{0})\) for classification outcomes that are worse according to the partial order \(\leq\ _{m}\). The value obtained is \(0.1678\). Grassland.Along the same lines, from Figure 3 (column G), we define \(X=(X_{1},X_{2},X_{3})\), with \(X_{1}\) being the number of elements correctly classified, \(X_{2}\) being the number of misclassifications with _Non-vegetated_, and \(X_{3}\) being the number of misclassifications with _Woodland_ or _Water_. \(X\) follows \(M(48,p_{0})\) with a probability vector of \(p_{0}=(0.88,\ 0.10,\ 0.02)\). The observed classification is \(Y^{*}=(40,5,3)\) (see Figure 4, column G), and the \(p\)-value obtained is \(0.2229\). Non-vegetated.Following similar reasoning to that used in the previous categories, the multinomial distribution is \(M(53,p_{0})\) with \(p_{0}=(0.90,\ 0.08,\ 0.02)\), defined from the specifications shown in Figure 3 (column N). The observed classification is \(Y^{*}=(45,\ 6,\ 2)\) (see Figure 4, column N), and the \(p\)-value obtained is \(0.1785\). Figure 4: Quality control column set (QCCS) from Figure 2, prepared for the proposed exact multinomial test. In each column (i.e., vector), the values are reordered and grouped following Figure 3. Wo = Woodland, G = Grassland, N = Non-vegetated, Wa = Water. Water. In this category, as expressed in Table 1 and Figure 3 (column Wa), the maximum percentage of misclassified elements is 1%, so the classification results lead us to define a binomial distribution with the following parameters: the number of elements to be classified, \(m=48\), and the probability of correctness in the classification, \(p=0.99\), in short, \(B(48,0.99)\). The observed classification is \(Y^{*}=48\) (see Figure 4, column Wa), and the \(p\)-value obtained is 1. Overall decision. For this QCCS, four hypothesis tests were carried out, and in order to assure the same significance level, \(\alpha=0.05\), we applied the Bonferroni correction to compensate for the number of tests. As a consequence, we reject the hypothesis that the thematic quality levels specified in Table 1 and Figure 3 are globally achieved if any of the four \(p\)-values obtained are less than \(\alpha/4=0.0125\). It does not occur in this example, and we can conclude that the data shown in Figure 4 globally satisfy the quality conditions stated in Table 1 (Figure 3).