# On Predicting Ticket Reopening for Improving Customer Service in 5G Fiber Optic Networks
## 3 Classification for Predicting Ticket Reopening
### Numerical Simulations and Results  
In this subsection, we compare the different classification approaches (described in Section 3.1) in order to predict ticket reopening via supervised learning. According to Section 2, we consider both datasets: the first one (\(X\)) in the original form and the second one in the reduced form \(X_{real}\) (through feature selection analysis). Given a month (i.e., 30 days) of data collected according to the format discussed in Section 2, we reordered the dataset by picking six groups of four days as training sets (\(k\in\{1,6,11,16,21,26\}\)), denoting them, with a slight abuse of notation, with \(X_{training}[k]\) in the case of the original dataset and with \(X_{training}^{real}[k]\) in the case of the reduced dataset, that is,  
\[X_{training}[k]:= \begin{bmatrix}x_{u,(v=k)}^{i}\\ x_{u,(v=k+1)}^{i}\\ x_{u,(v=k+2)}^{i}\\ x_{u,(v=k+3)}^{i}\end{bmatrix} ,\quad\forall u,\quad k\in\{1,6,11,16,21,26\}, \quad i=1,\dots,m, \tag{2}\]  
and analogously for \(X_{training}^{real}[k]\). We then considered the dataset portion relative to each of the remaining six days of the considered month (\(q=k+4\)) as a one-day subset of the ABT providing a suitable test set, denoted with  
\[X_{test}[q]:= \begin{bmatrix}x_{u,(v=q)}^{i}\end{bmatrix} ,\quad\forall u,\quad k \in\{1,6,11,16,21,26\},\quad i=1,\dots,m, \tag{3}\]  
in the case of the original dataset, and with \(X_{test}^{red}[q]\) defined analogously, in the case of the reduced dataset. In order to ensure the statistical robustness of the learned models, we proceeded in the following way. We first trained each classification algorithm on \(X_{training}[k]\) and \(X_{training}^{red}[k]\) alternatively, in order to obtain the learned models for each iteration \(k\) (_training_ phase). Then, we tested each model learned at iteration \(k\) on the one-day test set \(X_{test}[q]\) in the case of the original dataset, and on the one-day test set \(X_{test}^{red}[q]\) in the case of the reduced dataset (_test_ phase). Eventually, we measured the KPIs listed below for each couple \((k,q)\) of training and test sets and we reported in Tables 2 and 3 the average KPI values over all \((k,q)\) couples. For both data preparation and supervised leaning algorithms, all codes are written in **R**. All the simulation runs were performed on a dual-core Intel Core i7-7500U 2.70 GHz (up to 3.50 GHz) processor equipped with 16 GB RAM and running Ubuntu 18.04. Numerical results are provided in terms of _Accuracy_, _Gini coefficient_, _Youden index_ and _AUC_. * _Accuracy_[37]: the accuracy measure tells how well a machine learner, which learned the hypothesis \(h\) as the approximation of the target classification function \(V\), performs in terms of classifying a novel unseen example correctly. The true error of hypothesis \(h\) is the probability that it will misclassify a randomly drawn example \(x\), that is, \[error(h)=Pr[V(x)\
eq h(x)].\] (4)With this in mind, accuracy has the following definition:  
\[Accuracy=\frac{number\ of\ correct\ predictions}{total\ number\ of\ predictions}. \tag{5}\]  
For binary classification, as in the considered case, accuracy can also be calculated in terms of positives and negatives as follows:  
\[Accuracy=\frac{TP+TN}{TP+TN+FP+FN}, \tag{6}\]  
where \(TP\), \(TN\), \(FP\) and \(FN\) are the number of true positives, true negatives, false positives and false negatives, respectively. * _Gini coefficient_[37]: It provides a measure of the degree or probability of the target variable being wrongly classified. It has values between \(-1\) and \(1\). The closer it is to \(1\) the better. * _Youden index_[37]: it is a goodness-of-fit index that represents the maximum separation between the model Receiver Operating Characteristic (ROC) curve and the baseline ROC curve. * _AUC_[37]: an ROC curve shows the probability distribution just for the events, the positive class. It compares the events predicted correctly (true positives) against the events predicted incorrectly (false positives).