## 5 De-Noising Filter Method and Experimental Results
### De-Noising Filter Model
First, a sigmoid layer is used to determine what parts of the cell state will be output. After that, the cell state is passed through a \"tanh\" function and the ranging of the cell state values is from \(-1\) to \(1\). Finally, the results are multiplied by the output of the sigmoid gate, and the output parts are decided. The output gate's activation vector and hidden state vector are given by:  
\[o_{k}=\sigma(W_{o}x_{k}+R_{o}h_{k-1}+b_{o}) \tag{81}\]  
\[h_{k}=o_{k}*\tanh C_{k} \tag{82}\]  
The above Equations (77)-(82) are the steps to compute the output and the cell state of a LSTM unit. LSTM network can be applied for classification or regression tasks with sequence and time series data. In this paper, we make use of it for the regression task. From Figure 7, we obtain outputs \"h\" with the dimensions (\(H\times S\)), while we want have predicted results \"Y\" with dimesions (\(M\times S\)); therefore we need a weight matrix with dimension (\(M\times H\)) to tranforme \"h\" to \"Y\". As a result, we have the full architecture of the LSTM network with one LSTM layer for regression as the Figure 9.  
Applying this LSTM network, the inputs data (X) are the positions estimated by robust-EKF in Section 3, and the outputs data (Y) are the errors on the estimated positions. For training and testing our model, we should divide our data into three distinct sets: training, validation and prediction. For the \"training part\", Figure 10 presents the rEKF-LSTM training architecture. We determine the estimated position by robust EKF using the combination of GPS, Galileo and GLONASS data. The ENU\({}_{\textit{reKF}}\) is the East, north, and up positions by robust EKF. Before the estimated positions enter the LSTM model, we need to standardize them to have zero mean and unit variance. As a result, the standardized estimated positions are now the training inputs of the LSTM model. The target outputs are the errors of the estimated positions by robust-EKF (\(\delta_{ENU}\)). The errors in the estimated positions are the difference between the true and the estimated (robust-EFK) positions. Similarly, we standardize the target outputs to have zero mean and unit variance and the standardized target outputs are now the training target outputs of the LSTM model. The model is trained on the training dataset using supervised learning with optimization methods such as: stochastic gradient descent, Adam or RMSProp. Each time, the LSTM model is run with the training inputs and returns the training outputs, which are then compared with the targeted ones. Based on the results of the comparison, the parameters of the model (weights and biases) are adjusted. The training process works until the training errors reach the error threshold and the inferred LSTM model is referred to as the \"fitting LSTM model\". The subsequent part is the \"validation part\" for which the architecture is illustrated in Figure 11. Firstly, we standardize the estimated east, north, and up positions. After, we use the \"fitting LSTM model\" from the \"training part\" to predict the standardized ENU errors. Then, we unstandardize them to obtain the predicted ENU errors. Following that, we compare the predicted ENU errors with the true ENU errors. Based on the comparison, we can see how well the LSTM model is generalizing in the training part. If the model is overfitting or underfitting, it is necessary to go back to the training part to tune the model's hyperparameters (the number of layers, the number of hidden units in each layer etc.). We can return to the training part many times until we get the best LSTM model to ensure confidence on our model performance. Figure 10: rEKF-LSTM training architecture. Figure 9: Architecture of LSTM network for regression. After obtaining the best LSTM model, this model is used in the \"prediction part\". Figure 12 presents the prediction architecture of rEKF-LSTM. First, we standardize the estimated east, north, and up positions similarly to the validation part.