# Prediction of Traffic Volume Based on Deep Learning Model for AADT Correction
## 3 Analysis and Procedure of the Development Model  
### Analysis of the Development Model  
This study evaluates the traditional statistical models (ARIMA, SARIMA) alongside deep learning models (recurrent neural network (RNN), LSTM, power-function-based LSTM) to select suitable models and propose effective algorithms by addressing their limitations. (1) ARIMA  
The ARIMA model is a well-established time series analysis method that incorporates trends and seasonality by performing seasonal and first differencing, rendering the series stationary, as expressed in Equation (1):  
\[\gamma_{t}=T_{t}+S_{t}+E_{t}. \tag{1}\]  
Here, \(T\) represents the trend, \(S\) represents the seasonality, and \(E\) represents the randomness. The ARIMA model extends a typical ARMA model where, if the first difference is zero, it represents a stationary series. The formula for estimating the ARIMA model, which combines first-differenced ARMA, is expressed in Equation (2),  
\[\gamma^{\prime}_{t}=I+\alpha_{1}y^{\prime}_{t-1}+\cdots+\alpha_{p}y^{\prime}_{ t-p}+e_{t}+\theta_{1}e_{t-1}+\cdots+\theta_{q}e_{t-q}, \tag{2}\]  
where \(e\) denotes the error term. However, the ARIMA model is best suited to contexts with short seasonal cycles owing to its structure, which limits its effectiveness in analyzing long seasonal cycles and handling multiple seasonalities. (2) SARIMA  
The SARIMA model integrates the seasonal autoregressive model and seasonal moving average model within the ARIMA framework. The model, as formulated in Equation (3), accommodates the probabilistic characteristics of seasonality (m),  
\[SARIMA(p,\ d,\ q)(P,\ D,\ Q)_{m}, \tag{3}\]  
where \(p\) denotes the non-seasonal autoregressive order, \(d\) denotes the degree of differencing, \(q\) denotes the non-seasonal moving average order, \(P\) denotes the seasonal autoregressive order, \(D\) denotes the degree of seasonal differencing, \(Q\) denotes the seasonal moving average order, and \(m\) represents the number of observations per year (indicating annual seasonality). The distinction between ARIMA and SARIMA lies in SARIMA's capacity to encapsulate the seasonal cycle (m), as expressed in Equation (4),  
\[\Phi_{P}(B^{s})\Phi_{P}(B)\Big{(}1-B)^{d}\Big{(}1-B^{s})^{D}Y_{t}=1+\theta_{Q} (B^{s})\theta_{q}(B)\varepsilon_{t}, \tag{4}\]  
where \(\varepsilon_{t}\) denotes the error term (also known as white noise) and \(B\) represents the backward shift operator. If the traffic volume data cycle is recorded monthly, this equation can be modified to Equation (5):  
\[\big{(}1-\Phi_{P}\big{)}\Big{(}1-\Phi_{1}B^{12}\Big{)}(1-B)\Big{(}1-B^{12} \Big{)}y_{t}=(1+\theta_{1}B)\Big{(}1+\theta_{1}B^{12}\Big{)}\varepsilon_{t}. \tag{5}\]  
(3) RNNRNN is predominantly utilized for time series analysis and is particularly adept at processing sequential data. A defining feature of the RNN is its state vector (also known as the hidden unit), which forms a singular neural structure for the sequence and retains the memory of all preceding elements. The schematic recursive structure of an RNN incorporating the backpropagation through a time algorithm is presented in Figure 1[21]. A major assumption for different traditional statistical learning models is the independence among data samples. However, this assumption does not hold for sequential data, where dependencies exist among individual elements over time. This characteristic is particularly relevant to the traffic volume data analyzed in this study, negating the typical neural network advantage of processing each data sample independently. Within an RNN, the transfer of information through the network's weight matrices is described by the following equations:  
\[\mathrm{s_{t}}=\sigma(\mathrm{Ux_{t}}+\mathrm{Ws_{t-1}}+\mathrm{b_{s}}), \tag{6}\]  
\[\mathrm{h_{t}}=\mathrm{softmax}(\mathrm{Vs_{t}}+\mathrm{b_{h}}). \tag{7}\]  
In RNNs, the weights are designed to handle variable-length sequences. Each timestep receives a new input value, thus updating the hidden state, \(\mathrm{s_{t}}\).