## 3 Methodology
### The MLP Neural Network Model  
The first step towards effectively engaging neural networks for predictive modelling is to know the exact type you want to use and determine network architecture. This paper considers the most robust and special type of neural network: the multi-layer perceptrons (MLP). A single perceptron (LP) has limitations in terms of input-desired output mapping capability. This limitation is because it only contains a single neuron per adaptable synaptic weights and bias; thus, it is only proficient in catering to ridge-like function, notwithstanding the type activation function explored [42]. The above limitation can be catered to using an MLP neural network with more source nodes with data input and output layers sandwiched with hidden layers nodes. Multiple layers of neurons in the MLP network provide enhanced input-desired output mapping capability.  
Figure 2 displays a structure of a classic feedforward MLP network model composed of \(g_{1}g_{2},\cdots g_{l}\), inputs, and predicted output, \((y_{1},y_{2},\ldots,y_{N})\) with \(k_{h}\) hidden nodes, h number. The respective weights connecting the input and hidden layer, as well as the weights connecting the hidden layer and the output layer, are designated by \(w_{ij}^{1}\) and \(w_{jn}^{2}\), while \(C_{j}\) indicates the hidden nodes thresholds. The network learns the correlation between input datasets and predicted output feedback by varying weight and bias values. Accordingly,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Site 1** & **Site 2** & **Site 3** \\ \hline BS Operation Transmitting & & & \\ Frequency (MHz) & 2600 & 2600 & 2600 \\ BS Antenna Height (m) & 28 & 30 & 45 \\ MS Antenna Height (m) & 1.5 & 1.5 & 1.5 \\ BS antenna gain (dBi) & 17.5 & 17.5 & 17.5 \\ MS antenna gain (dBi) & 0 & 0 & 0 \\ MS Transmit power (dB) & 30 & 30 & 30 \\ BS Transmit power (dB) & 43 & 43 & 43 \\ Transmitter cable loss (dB) & 0.5 & 0.5 & 0.5 \\ Feeder Loss (dB) & 3 & 3 & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Measurement Path Loss Computation Parameters.  
Figure 1: Illustration of the TEMS Drive Test Measurement System.  
the MLP network predicted output in correspondence to jth neurons with the \(k\)th node could be articulated as (3):  
\[\hat{y}_{n}(t)=\sum_{j=1}^{k_{h}}w_{j}^{2}F\bigg{(}\sum w_{ij}^{1}\delta_{i}(t)+c _{j}\bigg{)} \tag{3}\]  
for \(1\leq n\prec m\), \(1\leq j\prec k_{h}\), \((w_{j},j=0,1,\ldots,k_{h})\), \(\big{(}w_{ij},i=0,1,\ldots,m;j=0,1,\ldots,k_{h}\big{)}\)  
where:  
\(m\), \(h\) and \(k_{h}\) indicate the input node number, hidden node and hidden node number, respectively; \(i\) designate input to \(j\) hidden layer neuron.  
The \(F\left(\cdot\right)\) in Equation (1) denotes the sigmoid activation function, an import function usually utilized in the MLP network. It can be defined by Equation (4):  
\[F(a)=\frac{1}{1+e^{-a}} \tag{4}\]  
where \(F(x)\) is at all times in the range [\(-1\), \(1\)], with \(F(a)\) being a set of real numbers.  
The weights \(w_{ij}^{1}\) and \(w_{jn}^{2}\), including the threshold \(C_{j}\), are unknown and thus can be chosen to update and reduce the error during prediction. The prediction error can be expressed by employing the expression (5):  
\[\varepsilon_{n}=\frac{1}{2}{\sum_{n=1}}(y_{n}-\hat{y}_{n})^{2} \tag{5}\]  
where,  
\(y_{n}\) and \(\hat{y}_{n}\) represent the target (i.e., actual) data and their predicted output; and \(n=1\)..., \(N\), with \(N\) indicating the actual data sample number.  
In MLP training, the error curve for assessing the network learning improvement related to convergence speed is the generalized aggregate error values. It is often computed using mean square error (MSE). The MSE can be obtained from the least square formation of Equation (6).  
\[MSE=\frac{1}{N}{\sum_{n=1}^{N}}(y_{n}-\hat{y}_{n})^{2} \tag{6}\]  
In this work, the feedforward MLP network model explored for path loss predictive modelling is displayed in Figure 3.  
Figure 2: Scheme of a three-layered ANN multi-layer perceptron.