# Pore Water Pressure Prediction Based on Machine Learning Methods--Application to an Earth Dam Case
## 3 Application Case: Montbel Dam, an Existing Earth Dam
### Dataset Preparation  
The PWPs have been monitored every month (or twice a month) from 1990 to 2019, and 450 data sets were available. Different factors are influential for the PWP, such as the hydrostatic level (\(H\)), season (\(S\)), and time or irreversible effects (\(T\)) [4]. Lagged features are the classic way to transform a time series prediction problem into a supervised learning problem. The simplest approach is to predict the \(f\) (\(t+1\)) value given the previous value \(f\) (\(t-1\)), which can represent the time effect. The hydrostatic (\(H\)) factor is defined by the hydraulic head, which measures the reservoir level. The seasonal (\(S\)) effect is intended to represent the day of the year (e.g., 1 to 365) corresponding to the measurement. One typical expression of this factor is given in Equation (4) below, in which the season \(S_{i}\) is an angle equaling to 0 on the 1 st of January and 2\(\pi\) on the 31 st of December [4]. \[S_{i}=\frac{2\pi(t_{i}-t_{0})}{365} \tag{4}\]  
where \(t_{i}\) and \(t_{0}\) are, respectively, the day of the currently analyzed measurement and the initial reference date for the data, with \(t_{i}\) also standing for the factor of irreversible effects \(T\). The dataset is shown in Table 1, and it has been divided randomly into training datasets (352), validation datasets (44), and test datasets (44). The ML model function is defined by  
\[f(t+1)=f(H(t),S(t),T,\ f(t-1)) \tag{5}\]  
Pearson coefficients demonstrate that the four factors, \(T,H,f\) (\(t-1\)), have a relatively large effect on the PWP, as shown in Figure 6. Although \(S\) has a limited influence on PWP, in this case, its effect should be considered for building a more general prediction model in theory [47]. The first histogram, \"T\", shows that the frequency of measurements has \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Min** & **Max** & **Mean** & **Unit** \\ \hline Hydrostatic (H) & 382.42 & 400.27 & 394.8 & m \\ Season (S) & 0 & 6.27 & 3.118 & radian \\ Time (T) & 0 & 10,939 & 6327 & day \\ Pore water pressure (PWP) & 77.3 & 131.6 & 100.8 & kPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of range for input and target output variables (C2). Figure 5: The profile of studied earth dam and position of sensors. doubled in the last several years, which has an impact (correlation coefficient is \(-0.743\)) on the variable \(\mathrm{P(t-1)}\) corresponding to the previous measurement. For most measurements, \(\mathrm{P(t-1)}\) corresponds to the measurement made 1 month before, but for the measurements of the last several years, \(\mathrm{P(t-1)}\) corresponds to the measurement made 2 weeks before. The value and distribution are shown by yellow histograms, and cloud graphs represent areas of concentration, trends, and changes in the data (in terms of two kinds of data), making it easier to discover the structure and patterns of the dataset. For example, in the cloud graph of T and S (noted as \"a1b\"), T concentrates more in the latest years, and S has two peaks, which are approximately symmetrically distributed with a mean value of 3.118. Before inputting the dataset, some preprocessing work should be conducted so that convergence velocity as well as accuracy can be improved drastically. The MinMaxScaler is a normalization method, which scales each feature to a designated range of [0, 1], utilizing two parameters: the lower and the upper bounds [49]. Normalization is a basic procedure  
Figure 6: Pearson coefficients (C2) between the target (PWP) and influential factors (The numbers above the diagonal line indicate the correlation coefficients of the parameters corresponding to the horizontal and vertical coordinates, and the curves on the scatter plot indicate the regression curves of the two parameters. The cloud graphs illustrate the density of each pair of related parameters: denser circles indicate more concentrated data points; the height of the yellow histogram and the curve above it represent the density of data of every single factor).