## 3 Machine Learning Models
### Deep Learning Models
#### 3.3.3 Long-Short Term Memory Networks (LSTM)
\begin{table}
\begin{tabular}{c c c c} \hline
**Article** & **Algorithms** & **Metrics** & **Dataset** \\ \hline [PERSON] et al. [8] & LSTM & RMSE & stock indices \\  & BiLSTM & & \\ \hline [PERSON] et al. [32] & ARIMA\((5,1,0)\) & MAE & daily stock price \\  & LSTM & (R)MSE & \\  & XGBoost & & \\ \hline  & ARIMA\((6,1,5)\) & RMSE & bitcoin price \\  & FFNN & MAPE & \\  & CNN & & \\  & LSTM & & \\  & SVR & & \\ \hline  & ARIMA\((1,1,0)\) & RMSE & bitcoin price \\  & LSTM & MAPE & \\  & GRU & & \\ \hline [PERSON] et al. [45] & ARIMA\((1,1,0)\) & precision rate & bitcoin price \\  & LSTM & time efficiency & \\ \hline  & ARIMA\((3,1,3)\) & RMSE & short-term bitcoin price \\  & LSTM & MAPE & \\  & MAE & & \\  & MAD & & \\ \hline [PERSON] et al. [47] & ARIMA\((0,1,0)\) & (R)MSE & financial budget \\  & LSTM & MAE & \\ \hline  & Prophet & MAE & wholesale food prices \\ [PERSON] et al. [48] & LSTM & MAPE & \\  & CNN & RMSE & \\ \hline  & ARIMA\((0,1,1)\) & RMSE & oil production \\ [PERSON] et al. [49] & LSTM & MAE & \\  & [PERSON] & & \\ \hline  & ARIMA\((2,2,5)\) & MSE & COVID-19 cases \\  & LSTM & MAPE & \\  & XARN & PSNR & \\  & & SMAPE & \\  & & R-value & \\ \hline  & ARIMA & MSE & COVID-19 trends \\ [PERSON] et al. [51] & SARIMA & RMSE & \\  & RNN-GRU & & \\  & RNN-LSTM & & \\ \hline [PERSON] et al. [52] & custom CNN/LSTM & MSE & temperature forecast \\ \hline  & ARIMA\((5,0,6)\) & RMSE & air quality index \\ [PERSON] et al. [53] & Prophet & MAPE & \\  & LSTM & MAE & \\ \hline  & ARIMA\((2,0,3)\times(2,1,3)_{24}\) & MSE & short-term wind speed \\ [PERSON] et al. [54] & LSTM & MAE & \\  & GRU & \(R^{2}\) & \\ \hline [PERSON] et al. [17] & ARIMA\((1,1,0)\) & RMSE & CO\({}_{2}\) levels forecast \\  & LSTM & MAE & \\ \hline [PERSON] et al. [55] & ARIMA\((1,1,2)\) & RMSE & web traffic \\  & LSTM & MAE & \\ \hline [PERSON] et al. [56] & ARIMA\((6,1,0)\) & RMSE & cellular traffic \\  & LSTM & & \\ \hline \end{tabular}
\end{table} 
Table 4: Summary of studies comparing ARIMA and Deep Learning models in time series forecasting. In the area of economic forecasting, the works of [PERSON] et al. [8] and [PERSON] et al. ([13; 32]) focus on stock market index forecasting, comparing the performance of LSTM networks and autoregressive ARIMA models. In the first paper, the comparison includes bi-directional LSTM networks (BiLSTMs) which enhance the training process of deep learning algorithms by dually feeding the input data to the network (1) from input to output and (2) from output to input. This paper performs a behavioral analysis of LSTM, BiLSTM, and ARIMA algorithms and compares their performances. The aim of the paper is to investigate the contribution of additional training layers to the determination of the required network parameters, and according to its results, the enhancement of the training process through the use of bidirectional LSTM networks improves the predictions of financial indicators. In particular, it has been observed that the predictions of the BiLSTM models are better than those of the LSTM and ARIMA algorithms, while the LSTM models equilibrate much faster than the BiLSTM networks. The work of [PERSON] et al., on the other hand, deals with the ARIMA, LSTM, and XGBoost comparison schemes. The XGBoost algorithm is an implementation of the class of gradient boosting algorithms, which builds a machine learning model from simpler models (usually decision trees) with the goal of improving the final performance. According to the results of this work, LSTM and XGBoost algorithms give better forecasting results than ARIMA for most price sets, while ARIMA achieves lower error values in two of the stock price sets, whose data values are significantly lower than the rest, which seems to affect the results of the analysis. In this particular case, ARIMA outperformed both machine learning methods in incorporating feature size while ignoring units. On the other hand, LSTM gives better results than the other two methods, achieving the lowest error scores in most of the considered datasets.