## 2 Theoretical Background
### Fault Detection Using an Artificial Neural Network for Regression  
When an incident signal is applied through one of the lines of a cable that has \(N\) cores (generator line), \(N-1\) reflected signals are obtained from the other lines (victim lines) in the TFDR. Consequently, the total of \(N(N-1)\) generator-victim reflected signals are obtained from the \(N\)-core cable. Since the features of the reflected signals depend on the connected generation-victim line pairs, the fault location and the faulty line can be identified by analyzing the total big data measured from the cable. The ANN, a statistical learning algorithm inspired from neural networks in a biological nervous system is widely utilized in machine learning and cognitive sciences. Specifically, the ANNs are applied in function approximation, recognition, classification and pattern recognition. We used a multi-layer perceptron (MLP) which is a class of the feed-forward ANN to detect the fault location and to distinguish the cable connection points in a multi-core cable. The topological structure of the MLP used in this study is illustrated in Figure 2. The MLP consists of one input layer with three neurons, one hidden layer with 25 neurons, and one output layer with two neurons [23]. Since the number of hidden layers determines the performance of the data fitting process, an appropriate number of hidden layers is requisite. Figure 1: The example of TFDR and TDR; (**a**) measured TFDR signal in time domain, (**b**) RID-Rihaczek distribution of the incident signal and the measured signal, (**c**) time-frequency cross-correlation results, (**d**) TDR results. In this study, the optimal hidden layer of the MLP is determined by the split-sample method. For the split-sample method, 50% of the data samples have been used for training and the remaining half has been used for testing. The activation function for the hidden layer was the hyperbolic tangent sigmoid function as follows:  
\[f(x)=\tanh(x)=\frac{2}{1+e^{-2x}}-1 \tag{3}\]  
where \(x\) and \(f(x)\) are the input and output of the hidden neuron, respectively. First, we set the thresholds of the average TFCC (\(\mu_{1}\)) and the normalized variance (\(\mu_{2}\)), respectively and classified the TFDR reflected signal cases in a multi-core cable as follows: (1) a normal cable junction, (2) a balanced fault, or faults in all lines, and (3) the unbalanced fault, or a partial line fault resulting in reflected signals due to both the fault and cross-talk. The threshold value of each input parameter was set with preliminary test results considering noise, signal interference, and cross-terms. Next, the input features of the TFDR results for the MLP were set as time domains: average of the TFCC, the normalized energy variance of the reflected signals, and the time delay between the reference signal and the reflected signal. The time delay and the average of the TFCC are strongly relevant with the fault existence and location of the reflected data, respectively. In addition, the normalized energy variance of the reflected signals contain information on the presence of an incipient partial line fault. Partial line faults require the detection of faulty lines through classification before all lines become defective leading to devastating errors in the multi-core cable. The input features can be expressed as follows:  
\[C_{m}(t) =\frac{1}{N(N-1)}\sum_{i=1}^{N}\sum_{j=1}^{N}C_{ij}(t)\] for \[i\
eq j \tag{4}\] \[\sigma^{2}(t) =E[(r_{ij}(t)-r_{m}(t))^{2}],\] for \[i\
eq j\] (5) \[\delta t =t_{r_{ij}}-t_{s} \tag{6}\]  
where \(i\) and \(j\) refer to the connected generation and victim line number, respectively. \(r_{ij}(t)\) is the reflected signal with i-th generation/j-th victim line.