## 3 Results
### Estimation of the Capability of Detection and the Capability of Discrimination  
In addition to the stringent official regulations imposed within the agri-food sector (where the maximum permitted limits of some residues are established, as in the case of some agrochemicals in olives in this work), sometimes, some companies internally establish some standards (target values) that ensure the quality of their products, either to fulfill legal requirements or for internal reasons, for example, because the value of the characteristic determines the subsequent stages of processing. In this sense, it is crucial to know the level of certainty with which these limits are not exceeded, whether they are minimum or maximum values, that is, to know the minimum discriminable quantity with a probability of false compliance (\(\beta\)) and false non-compliance (\(\alpha\)) in order to decide whether to accept a sample or not and continue with the process. In the following, some possible real scenarios that may occur in the agri-food industry are presented. The results of applying the global procedure described in Section 2.2 can be seen in Tables 2-4, and graphically, for some of the Operative Characteristic Curves of the applied tests, in Figures 5-8. In these figures, the Operating Characteristics Curves are represented, \(\beta\) on ordinate axis versus the true concentration on the abscissa axis. In each one of them, the dashed line indicates the value of CD\(\beta\) (or CC\(\beta\)) on the \(x\)-axis for a probability \(\beta\) = 0.05 (in the \(y\)-axis), when the probability \(\alpha\) = 0.05 is fixed. In Section 2.2.2., the use of the operative curve of the test to discriminate a concentration has been described. This allows one to evaluate the performance of an NIR-PLS calibration in relation to another different one obtained with another different instrument or measuring in another matrix. Note that by setting the values of \(\alpha\) and \(\beta\) and the number of replicates, r, the interval [CD\(\beta_{\text{i}}\), CD\(\beta_{\text{u}}\)] allows one to immediately obtain the best option in terms of the discrimination capability. This approach is of great interest in scenarios such as food safety and food fraud control, as shown in this section. The curves were plotted for the case of different future replicates (r = 1, r = 3, r = 6 and r = \(\infty\), the theoretical value in the case of infinite samples). It can be observed that as more replicates are performed, the values obtained for the capability of detection and/or the capability of discrimination become closer to the desired PL value. This happens as a consequence of the fact that the expressions CC\(\alpha\), CC\(\beta\), CD\(\alpha\), and CD\(\beta\) contain the factor w\({}_{\text{x0}}\), which also contains the summand 1/r, which decreases as the number of replicates, r, increases. In some cases, increasing the number of replicates may allow a limit to be fulfilled without modifying any other aspect of the measurement system. However, it is not always possible to perform a high number of replicates in certain industrial processes; it will always depend on the conditions imposed by each industry. In this work, r = 3 replicates were considered. It can also be seen in every representation that for the same value of r, the greater the \(\mid\)x \(-\)x\({}_{0}\)\(\mid\) distance, the lower the value of \(\beta\) is. In other words, the lower the risk of committing false compliances (or false negatives) is in an analysis. In particular, Figure 5 shows these Operative Curves when the discriminable amount wants to be detected; for instance, for fat in butter, for the 85% \(w/w\) established value, the two-tailed test applied is the one in Equation (9). As can be seen in Table 4, this test, as it is a two-tailed test, has a lower CD\(\beta\) and an upper one that encompass the range [84.27, 85.73].