# Contributing towards Representative PM Data Coverage by Utilizing Artificial Neural Networks
## 2 Materials and Methods
### Methodology
Lower MAE values correspond to a better performing network in relation to a lower degree of complexity. Different FFNN configurations were tested for multiple runs (10 repetitions), due to the initial weights of the neurons of the model being randomly established, and thus, the average result of these runs guaranteed the randomness of the process. The number of hidden neurons tested in all cases ranged from one to forty. To avoid pattern exploitation in the training subset (overfitting), the early stopping approach was used [60], which, according to the validation subset error (when it started to increase), stopped the training process. The final networks that were developed were evaluated for their estimation accuracy by applying two difference and correlation statistical measures, the MAE and the  
Figure 3: An example of an artificial neuron and its basic characteristics. coefficient of determination (\(R^{2}\)), [38; 59; 61; 62] on the test subset of the output vector. These criteria are calculated by using the following equations:  
\[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}\left|E_{i}-A_{i}\right| \tag{2}\]  
\[R^{2}=\left(\frac{\sum_{i=1}^{n}\left(A_{i}-\overline{A}\right)\left(E_{i}- \overline{E}\right)}{\sqrt{\sum_{i=1}^{n}\left(A_{i}-\overline{A}\right)^{2} \sqrt{\sum_{i=1}^{n}\left(E_{i}-\overline{E}\right)^{2}}}}\right)^{2} \tag{3}\]  
where \(n\) is the number of data points, \(E\) the estimated and \(A\) the observed concentrations. The best-performing models are associated with lower MAE and higher \(R^{2}\) values and are evaluated based on the results of both statistical parameters. For the MAE metric, the standard deviation (SD) was also calculated to indicate the dispersion of the estimated concentrations from the MAE value. Additionally, the FFNN models' results were compared with the corresponding estimations of a multiple linear regression model (MLR) [38; 63] in order to further establish the superior predictive ability of the FFNNs. Finally, the accuracy of the FFNN models is also examined by plotting scatter diagrams which additionally contribute towards an easier comparison among the models. The scatter diagrams provide information considering the relationship between the observed and estimated values at high, medium and low concentration levels. The last part of the methodology includes an analysis regarding the distinguishing of the significance of every input variable to the output, for all the FFNN models that were developed, utilizing an algorithm proposed by [PERSON] [50]. This methodology is based on recognizing the associations that the synaptic weights reveal considering the inputs and output relationship and was also used in other studies in the field of air quality, to quantify the importance of each station's data (inputs) to the estimated values for the target station [38; 63; 64]. The Relative Importance (RI) percentage is calculated with the use of Equation (4),  
\[RI_{IK}(\%)=\frac{\sum_{j=1}^{h}\left|\frac{w_{ij}w_{ij}}{\sum_{i=1}^{n}\left| \frac{w_{ij}|w_{ij}}{\sum_{i=1}^{n}\left|w_{ji}\right|}\right|}}{\sum_{i=1}^{ n}\sum_{j=1}^{h}\left|\frac{|w_{ij}|w_{ji}|}{\sum_{i=1}^{n}\left|w_{ji}\right|}\right|} \tag{4}\]  
where \(w_{ij}\), \(w_{ij}\) are the connection weights between the \(i\)-th input and \(j\)-th hidden neuron, and between the \(j\)-th hidden and \(k\)-th output neuron, respectively. In general, ANNs provide little explanatory insight into the individual contribution of the input variables in the estimation procedure. The \(RI\) method addresses this issue and can be used as a variable selection technique for similar problems.