## 2 Materials and Methods
### CUI Rate Prediction Modeling  
In developing the final CUI rate prediction model, only the most influential input factors from the field and experiment were included because the process and predictive ability of the artificial neural network (ANN) greatly depends on the best input set. In fact, if the model includes irrelevant variables, the model will be difficult to train, and a less accurate model will be produced. Before this CUI model development, the most influential factors for both field and experimental data were determined [16; 17]. Thus, in the final CUI rate model, the data input consisted of the most influential input parameter, which was the operating temperature (\"C), with an addition of the age of pipe to ease of time-based measurement. The output was the CUI rate in millimeters per year. The final CUI rate prediction was modeled by three (3) neural network layers' structure consisting of two (2) input nodes in the first layer, eight (8) hidden nodes in the second layer, and in the third layer, one (1) output node. The two (2) input units involved elapsed time (years) and operating temperature (\"C), while the output was the CUI rate (mm/year). The input was then weighted and associated with the connection to the neuron in the hidden layer; then, the activation function was balanced to yield the output. The activation function at each node spread the value of the weighted input to the weighted output and produced non-linear values with a sigmoidal function with backpropagation [18; 19]. This expression is shown as in Equation (3). \[Z_{k}=\frac{1}{1+e^{-[\sum\limits_{i}^{w_{jk}y_{j}+\xi_{i}}]}}. \tag{3}\]  
The output node, \(Z_{k}\) is the output value of current layer \(k\)-th neuron, \(\xi_{i}\) is the threshold or bias from the current layer \(k\)-th neuron, while \(w_{jk}\) is the joiner weight between the maiden layer and forward layer neuron. \(y_{j}\) is the input layer value based on the \(j\)-th neuron, whereas the effect of \(\xi_{k}\) is an adjustment to the sigmoid function shape. In this study, a sigmoidal function of a hyperbolic tangent (TanH) function as in Equation (4) was adopted to ensure the value of the CUI rate was in a specified range during prediction [19; 20]. \[f(TanH)=\frac{e^{2y}-1}{e^{2y}+1}. \tag{4}\]  
In this equation, \(y\) is a linear combination of the Y parameters. Only positive values of this hyperbolic tangent function were gathered as the CUI rate. There are a total of 9344 data input, 4424 from field data, and 4918 were from experimental data. The finalized model featuring the age of pipe with its CUI rate was visualized in a graph, and the fitting line was calculated. In determining the best model for data fitness, a range of neurons in the hidden layers of the ANN was tested. Neurons with the highest R\({}^{2}\) value were selected for model development as it indicates the best fit. Although neural networks are very flexible with high capability and applied in various fields, they may overfit data. It can fit the data model very well but poorly perform the forecast observations. This phenomenon can be defined by observing the sum of squared error (SSE) and the R\({}^{2}\) values. If the R\({}^{2}\) shows a high value with high SSE, this means the model might be overfitted. A decent model should have a high R\({}^{2}\) value with low SSE. Thus, to avoid the overfitting problem, the bias value was applied to the model variables and used an independent data set to evaluate the predictive power of the model. Further, during the model development process using the prediction profiler simulator function in the Statistical Discovery JMP 10.0 2012 SAS Integration 9.1.3 Server software package, the original data were separated into parts. A part of a dataset was specified to estimate model parameters, while the other part for checking the predictive capability of the model. The neural network was divided into three sets, which were training data, validation data, and testing data. Training data was used for estimations of model parameters.