## 3 Results and Discussions
### Geographical Prediction Model  
The developed Artificial Neural Networks (ANNs) for predicting the country of provenance of an unknown (i.e., not included in the training set) honey sample proved to be very successful, taking into account the overall classification accuracy. For the development of this model, 103 honey samples were used, and the entities were approximately uniformly distributed between the two classes (i.e., 50 Romanian samples and 53 French samples, as can be seen in Figure 1). Hence, the models which separate the Romanian honey objects from the French ones were based on an equilibrated data set. The determined isotope (\(\delta^{2}\)H, \(\delta^{18}\)O, \(\delta^{13}\)C\({}_{\text{honey}}\), \(\delta^{13}\)C\({}_{\text{Protein}}\)) and elemental markers (Li, Na, Mg, Al, P, K, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, As, Rb, Sr, Y, Zr, Nb, Mo, Pd, Sn, Sb, Ba, La, Ce, Pr, Ir) were used as input data for the model development (Table S1--Supplementary Materials). Two main approaches were adopted in the development of the artificial network structures. The first one aimed to use the majority of the samples for the training process of the artificial neural networks by applying leave-one-out cross validation. Thus, for a specific network structure (e.g., the number of neurons used in the ANN, the activation functions chosen), 103 models where created, one for every honey sample being the single entity in the testing set. Each configuration led to an overall accuracy which represented the percentage of correctly classified samples from the total number of honey items. With the aim of obtaining more accurate models, different numbers of iterations (i.e., in the range of 10-20,000) and of hidden neurons (i.e., from 5 to 50) and distinct distributions of hidden units on layers (e.g., the number of neurons on the first hidden layer being greater than the ones on the second hidden layer and vice versa) were tested. The created neural  
Figure 3: K-fold cross validation. structures did not contain dropout layers as a method of avoiding over-fitting, so the challenge was to find the proper value for the number of epochs and the number of hidden units such that, in most of the cases, the tested honey samples were correctly assigned to a group. During this process, it was noticed that as the number of hidden neurons increased, more entities were predicted in a right way. The best accuracy of 95% was achieved for the model illustrated in Figure 4, where to the units on the dense_1 and dense_2 layers the ReLU function was applied. The neurons on the dense_3 layer used the sigmoid function in order to compute the output. Binary crossentropy loss was utilized together with the sigmoid activation function, and the learning phase of each model consisted of 10,000 iterations through the training set. Using this approach, 98 samples out of 103 were correctly classified. The wrongly attributed samples were five honeys originated from Romania. Of these, two samples were acacia honeys and the other three had as floral origin: linden, colza and sunflower. Even though this neural structure presented good results for a high percentage of the honey samples, it was observed that the error of classification for the wrongly predicted samples started to increase from a certain epoch. This pointed out the fact that, despite receiving optimistic results, the training data were over-fitted. Therefore, a second approach was developed, with the aim of solving the drawbacks of the artificial neural network structure previously described. First of all, the model validation was conducted under 10-fold cross validation, meaning that the overall accuracy of the model represented the average of performances obtained by testing consecutively the 10 folds. In this way, the variations of the prediction loss of more entities were observed during the training stage, and the models in which over-fitting occurred were better detected.