## 2 Preliminaries
### Long Short-Term Memory Network (LSTM)  
In this paper, an integrating model of long short-term memory network based on principal components analysis (PCA-LSTM) is explored to analyze the operational time-series data generated from the dredging process. The proposed model is developed on the basis of long short-term memory network (LSTM), which is a special form of recurrent neural networks (RNN) that can address long-distance dependencies and delay in time-series modeling.  
The LSTM architecture was firstly proposed by [PERSON] and [PERSON] in 1997 [25]. There is a special memory cell unit added to the original hidden layer in classic RNN architecture. The cell state is controlled by three gates: Input gate \(\mathbf{I}_{t}\), Forget gate \(\mathbf{F}_{t}\), and Output gate \(\mathbf{O}_{t}\), as shown in Figure 1.  
The forget gate \(\mathbf{F}_{t}\) decides which information needs to be kept and which can be forgotten. The information consists of the current input \(\mathbf{X}_{t}\) and previous hidden state/short-term memory \(\mathbf{h}_{t-1}\).  
\[\mathbf{F}_{t}=\sigma(\mathbf{W}_{Forget}\cdot[\mathbf{h}_{t-1},\mathbf{X}_{t}])+\mathbf{bias}_{ Forget} \tag{5}\]  
For every time step, the sigmoid function generates values between 0 and 1 that indicate whether the old information is necessary. 0 denotes forget, and 1 means keep. \(\mathbf{W}_{Forget}\) is the weight matrix between forget gate and input gate. \(\mathbf{bias}_{Forget}\) is the connection bias.  
The input gate decides what should be stored in the long-term memory in the new information. It works with the current input \(\mathbf{X}_{t}\) and previous short memory \(\mathbf{h}_{t-1}\) through two layers. In the first layer, the short-term memory and current input is passed through a sigmoid function that values from 0 (not important) to 1 (important):  
\[\mathbf{i}_{t}=\sigma(\mathbf{W}_{Input}\cdot[\mathbf{h}_{t-1},\mathbf{X}_{t}])+\mathbf{bias}_{ Input} \tag{6}\]  
where \(\mathbf{W}_{Input}\) is the weight matrix of sigmoid operator between input gate and output gate. \(\mathbf{bias}_{Input}\) is the bias vector.  
The second layer uses the tanh function to regulate the network. The tanh operator creates a vector \(\widetilde{\mathbf{C}}_{t}\) with all the possible values between \(-1\) and 1:  
\[\widetilde{\mathbf{C}}_{t}=\tanh(\mathbf{W}_{Cell}\cdot[\mathbf{h}_{t-1},\mathbf{X}_{t}])+\bm {bias}_{Cell} \tag{7}\]  
where \(\mathbf{W}_{Cell}\) is the weight matrix of tanh operator between cell state information and network output. \(\mathbf{bias}_{Cell}\) is the bias vector.  
With these two layers input, the cell state updates a new cell state (long-term memory):  
\[\mathbf{C}_{t}=\mathbf{F}_{t}\odot\mathbf{C}_{t-1}+\mathbf{i}_{t}\odot\widetilde{\mathbf{C}}_{t} \tag{8}\]  
where \(\odot\) is the Hadamard product.  
When it comes to the output gate, the current input \(\mathbf{X}_{t}\), the previous short-term memory \(\mathbf{h}_{t-1}\), and the newly obtained cell state \(\mathbf{C}_{t}\) determine the new short-term memory (hidden state) that will be passed on to the cell in the next time step.  
\[\mathbf{O}_{t}=\sigma(\mathbf{W}_{output}\cdot[\mathbf{h}_{t-1},\mathbf{X}_{t}])+\mathbf{bias}_{ output} \tag{9}\]  
Figure 1: The architecture of classical LSTM.  
\[\mathbf{h}_{t}=\mathbf{O}_{t}\odot\tanh(\mathbf{C}_{t}) \tag{10}\]  
\(\mathbf{W}_{output}\) is the weight matrix of the output gate. This hidden state is used for prediction. Both new cell state and hidden state are carried over to the next time step.