## 4 Analysis and Experimental Results
### Results  
Following the definition of the optimization procedure using a QPSO algorithm and the technique for calculating the localization error using a standard MLP with an adaptative backpropagation process, the next step in this research was to select the most appropriate values for both the QPSO and MLP algorithms. Several parameters can affect the speed of both processes and strongly alter the total computing time. For the QPSO part, it was decided to choose the standard parameters found in the literature, namely those reported in Table 1. The QPSO procedure was repeated for a total of 100 generations, each producing a new population of 20 vectors of 4 components, recording the size of the tiles as well as the offset position with respect to the bottom left experimental point. Due to the QPSO's nature, the global best individual, i.e., the vector of signal weights producing the lowest positioning error, was recorded at every stage of the algorithm. Each vector was first tested under the preset conditions, which limited the tile size to the interval (0.3, 1.2) m for both horizontal and vertical dimensions and the offset to the tile size. Each non-compliant configuration was discarded and the generation of a new individual was repeated until the preset population size was reached. The positioning errors of the new QPSO generation of individuals were then evaluated by repeating 1000 times the MLP procedure for each individual and calculating the average of the 50 best test positioning errors. Multilayer Perceptron neural networks are subject to several parameters that greatly affect the training accuracy and the computing time. For this research, it was decided to refer to the investigation of the validation runs in our previously published research on the same dataset [32] and to keep the same MLP architecture. In particular, three hidden layers of 18 hidden units each were retained, as well as the standard sigmoid activation function. The values of the learning rates and momentum coefficients for the Adam technique were determined by visual inspection during a small number of dry runs, as well as by measuring the computing time of individual cost calculations. Moreover, the intensities of the input signals were weighted, in agreement with our previous results [32]. The set of parameters for the MLP process are reported in Table 2. Because of the sheer amount of QPSO cost evaluations and the large number of MLP runs for each individual, the overall computing time reached about 28 h despite the fact that each MLP training process of 5000 iterations took only 50 ms to complete. The output of the MLP process was designed to return the integer values of the tile positions in the regular array defined by the vectors \(\mathbf{x}_{i}=(w_{i},\ h_{i},\ u_{i},\ v_{i})\), both for the train and the test data points. Equations (14)-(16) were then applied to calculate the actual position error for both data, while only the test position error was retained as the result of this research. Figure 3: The profiles of the position error relative to the training and testing data in a typical MLP run. The minimum value in the test profile is highlighted. A meaningful comparison of configurations with different tile sizes cannot be done on the basis of the number of tiles mismatch in either direction alone, as larger tiles might be correctly identified despite a relevant actual position error. For this reason, the value of the intrinsic error was taken into account as the average distance between a generic point and the center of the tile, according to Equation (16). Thus, the actual positioning error \(err_{pos}\), i.e., the QPSO cost value for an individual becomes:  
\[err_{pos}=err_{real}+err_{[h,w]} \tag{17}\]  
in which the two parts are defined by Equations (15) and (16). The value of the resulting positioning error as a function of the QPSO iteration number is shown in Figure 4 (right).