# An IoT-Based System for Monitoring the Health of Guyed Towers in Overhead Power Lines
## 5 Evaluation of the Tower Parameters for Machine Learning Estimation Model
### Data and Variables Correlation  
The monitoring system for cable-stayed towers comprises several sensors that monitor the state of the structure. The system model estimates traction across four cables and angle between the cable direction and \(\mathbf{g}\) vector through multiple three-axis accelerometers. These variables have a cause-and-effect relationship to each other. One way to measure and understand the relationship between data is the correlation coefficient. A higher absolute correlation coefficient implies a more significant relationship between the data.  
Before using the data for training, it is interesting to evaluate what correlation features are behind the collected samples. It is essential to inspect the correlation among variables to save computing resources by eliminating the irrelevant ones from the training phase. For the current analysis, we use the Pearson correlation coefficients to measure the linear correlation between two data sets. The Pearson coefficient \(\rho\) is calculated as follows:  
\[\rho=\frac{\sum_{i=1}^{n}{(x_{i}-\overline{x})\times(y_{i}-\overline{y})}}{ \sqrt{\sum_{i=1}^{n}{(x_{i}-\overline{x})^{2}\times\sqrt{\sum_{i=1}^{n}{(y_{i}- \overline{y})^{2}}}}}} \tag{3}\]  
where \(x_{i}\), \(y_{i}\) are the samples, \(\overline{x}\), \(\overline{y}\) are the average value and \(n\) is the total amount of samples. The current results comes from a simulation where we consider \(n=10,000\). Notice that the product of sample standard deviations divides the covariance, and the result is a normalized measurement where \(-1<\rho<1\). Table 1 shows a classification range for the values of \(\rho\), where it allows us to consider if the quantity is irrelevant for the machine learning training phase. Usually, quantities below the weak range are discarded.  
The current analysis of \(\rho\) considers the data generated from the simulation of the analytical model from previous section. It is vital to remind that the considered quantities are the tension, accelerometer axis, and \(\mathbf{g}\) vectors. The technique of drawing out information from multiple quantities and using it for estimating is called sensor fusion [31,32]. Table 2 show the data calculated from the data set using Equation (3). There, X and Y are the displacement coordinates of the tip of the tower; T1, T2, T3, and T4 correspond to the tension of the cables; and A1, A2, A3, and A4 correspond to the angle between the y-axis of the accelerometer and \(\mathbf{g}\) vector.  
The relation between the quantities is also closely investigated. Figure 17 shows a plot of the correlation between (a) A1 and A4; (b) A2 and A3. Notice the inverse relationship at Figure 17a,b that reflects the stretching and loosening of the cables as the model of the tower displace. Figure 18 shows a plot of the correlation between (a) T1 and T3; and (b) T2 and T4. Notice the direct relationship at Figure 18a,b that reflects the stretching and loosening, also validating the model.  
\begin{table}
\begin{tabular}{c c} \hline \hline \(\rho\) & **Correlation** \\ \hline \(\pm\)0.9 & Very strong \\ \hline \(\pm\)0.7 up to \(\pm\)0.9 & Strong \\ \hline \(\pm\)0.5 up to \(\pm\)0.7 & Moderate \\ \hline \(\pm\)0.3 up to \(\pm\)0.5 & Weak \\ \hline \(0\) up to \(\pm\)0.3 & Irrelevant \\ \hline \hline \end{tabular}
\end{table}
Table 1: Classification ranges for the very values of \(\rho\).