## 2 Materials and Methods
### Methodology
Both GRU layers are accompanied by Dropout layers with a 20% dropout rate to mitigate overfitting;
* After the GRU layers, a fully connected (Dense) layer with 32 neurons and ReLU activation is used to form higher-level representations of the data. This is followed by a Dense output layer with a single neuron to predict the target value;
* This model was compiled using the Adam optimizer and the Mean Squared Error (MSE) loss function. MSE is suitable for regression tasks as it minimizes the mean squared deviation between the predicted and actual values, thereby enhancing forecasting accuracy. To improve model robustness, Dropout is applied after the attention layer and each GRU layer with a probability of 20%. This reduces the likelihood of overfitting, particularly when handling complex temporal data. Another model considered in this study is ARIMA-LSTM, a hybrid framework that combines the linear ARIMA model with the LSTM neural network [29]. This approach enables the capture of both linear and nonlinear temporal dependencies. This model consists of the following stages:  
* In the initial phase, the ARIMA model is applied separately to each feature of the time series. The use of ARIMA facilitates the extraction of trend and seasonal components,highlighting their linear dependencies. The residuals obtained from the ARIMA model for each feature are then calculated, forming a time series that will be further trained in the LSTM neural network. In this study, the parameters of the ARIMA model are empirically selected for each series;
* The typical order of the ARIMA model used for trend analysis was (5, 1, 0), where 5 represents the autoregressive order; 1 indicates the order of differencing, and 0 signifies the order of the moving average;
* In the second phase, the ARIMA residuals are fed into the LSTM neural network to identify remaining nonlinear dependencies. The data were normalized to a range of 0 to 1 using \"MinMaxScaler\" to accelerate training and enhance the model's robustness;
* The first LSTM layer consists of 64 units, returning sequences (return_sequences = True) for the subsequent layer. A Dropout layer with a probability of 0.2 is included to prevent overfitting;
* The second LSTM layer also contains 64 units but returns only the final hidden state (\"return_sequences = False\");
* A fully connected (Dense) layer with 32 neurons and ReLU activation is applied to enhance nonlinear representations of the data. The output Dense layer contains 22 neurons, corresponding to the number of forecasted features;
* This model was compiled using the Adam optimizer and the Mean Squared Error (MSE) loss function, which is a standard approach for regression tasks. Training was conducted for 100 epochs with a batch size of 32. In addition, we considered the Prophet model (developed by the Facebook research team) as a comparative baseline [37]. Prophet decomposes time series into trend, seasonality, and holiday components, allowing for easy interpretation and flexible incorporation of domain-specific knowledge. Compared to the MAGRU, LSTM, and ARIMA-LSTM approaches, Prophet tends to excel in handling irregular data frequencies and requires fewer manual adjustments of parameters, but it may struggle to capture complex nonlinear dependencies and long-term temporal patterns. Structurally, Prophet assumes an additive model consisting of a piecewise linear (or logistic) growth trend and multiple seasonalities that can be independently specified. For this study, we configured Prophet with a learning rate of 0.05, a seasonality mode set to \"additive\", enabled yearly and weekly seasonalities, and a changepoint prior scale of 0.1 to control trend flexibility. While the relative simplicity and interpretability of Prophet offer valuable advantages, its forecast accuracy may be limited when confronted with intricate time-series dynamics that the more specialized MAGRU, LSTM, and ARIMA-LSTM models are designed to capture.