# Prediction of Traffic Volume Based on Deep Learning Model for AADT Correction
## 4 Statistical Learning Verification and Machine Learning Model Analysis and Evaluation
### Machine Learning Model Analysis and Evaluation
Figure 5 presents the results of predicting the hourly traffic volume at point 10004 using the Prophet model. The blue line represents the actual traffic values, and the yellow line represents the predicted values. For point 10001, the regularity in the past time series patterns (seasonality, periodicity) was confirmed, leading to a high predictive performance, with an MAE of 0.117 and an RMSE of 0.147. However, for point 10004, irregular past time series patterns presented lower predictive performance, with an MAE of 111.687, an RMSE of 158.050, and an R\({}^{2}\) of 0.847, when compared to point 10001. This indicates the presence of uncertainties that are not recognized by the data. To address these limitations, analyses utilizing deep-learning-based models were conducted. (3) LSTM Model  
The LSTM model was utilized to evaluate the predictive power of the proposed LSTM model, which incorporates a power function. The predictive accuracy of the LSTM model varies based on the time point, as shown in Figure 6. Owing to the structural limitations of the algorithm, a linear transformation layer is incorporated between the input values (inputs) and the predicted outputs (predictions). This architecture lacks densely nonlinear hidden layers between the input and output layers, which, while enhancing the computational speed, considerably diminishes the predictive performance. Figure 5: Prophet model’s prediction results (point 10004). Furthermore, the model predicts the subsequent time step based solely on the values and characteristics of the immediately preceding time step (t = 0). This approach does not adequately capture the long-term trends and characteristics inherent in the data. (4) Proposed LSTM Model Based on Power Function  
Previous efforts to modify the LSTM architectures have aimed at enhancing the inference capabilities for data characterized by long-term trends, typically yielding similar performance levels. However, our analysis identified inconsistencies in the LSTM performance. In particular, within the forget gate, the memory cell, \(c_{t}\), which stores past data, is described by Equation (8):  
\[c_{t}=f_{t}\odot c_{t-1}+i_{t}\odot\overline{c_{t}}. \tag{8}\]  
Figure 6: LSTM model’s prediction results (24 h). Despite the superior performance of the LSTM models with forget gates over the standard LSTM or RNN models, they frequently suffer significant information loss during the decay process, where non-essential information is discarded using a Sigmoid function. [PERSON] et al. [23] suggested an improved decay function using exponential functions to address this issue. However, exponential functions still led to excessive information loss in datasets with pronounced long-term trends. This effect was notably severe in data with substantial long-term dependencies, such as the hourly traffic volume data. Therefore, this study proposes an LSTM model that employs a power function as the decay function within the forget gate, designed to accommodate long-term patterns and reduce unnecessary information decay in the time series data with significant long-term dependencies. The memory cell in the forget gate is redefined by the power function to reduce the rate of information decay:  
\[\mathrm{c_{t}}=\mathrm{c_{0}}\odot(\mathrm{t-t_{0}+1})^{-\mathrm{P}}=(\frac{ \mathrm{t-t_{0}+1}}{\mathrm{t-t_{0}}})\odot\mathrm{c_{t-1}}. \tag{9}\]  
To evaluate the absolute predictive performance of the proposed LSTM model, it was compared against those of linear combination models, multi-step combination models, and convolution neural networks. Figure 7 depicts the comparative prediction performance of the power-function-based LSTM model against that of the traditional LSTM model, demonstrating enhanced prediction accuracy for the traffic volume. To further substantiate these findings, Table 5 presents the prediction performance metrics, demonstrating that the LSTM model achieves an MAE of 0.1288, whereas the power-function-based LSTM model significantly improves this with an MAE of 0.0733.