## 5 Model Results
### Daily Prediction
#### 5.2.1 Model Performance
The fitted model for the number of crashes in Nantung direction \(Y_{t}\) in the time period \(t\) is given by \(Y_{t}|\mathcal{F}_{t}-1\ \sim NegBin(\lambda_{t},\ 1.98)\) with:  
\[\lambda_{t}=1.04\times 10^{-5}+0.52Y_{t-1}+0.24Y_{t-7}+\left(1.35\times 10^{-6}, -0.01\,-0.54,-0.06,-0.07\right)^{T}X_{t},t=1,\ldots,271\]  
From the tables above, it indicates that the previous crashes indeed influence the number of car accidents occurring in the following days. In the Nantung direction, it shows that vehicle speed and holiday covariates have negative effects on the number of crashes. To be specific, the crash-frequency on a non-holiday day is higher than the frequency on a holiday. The possible reason may be vehicles operating on the studied bridge are usually for daily use or business, and people who travel between Nantung and Soochow on holidays have a predilection for other transportation. Lower vehicle speed relates to more accidents. The possible explanation may be the vehicle speed is the past information. With the common sense that more severe car accidents may cause more severe traffic congestion and much lower vehicle speed, the model has learned the pattern of association between crashes and low vehicle speed, and thereby it indicates a negative relationship between the number of crashes and vehicle speed. However, in the Soochow direction, the covariates are not significant, as observed from the confidence intervals in the above tables. One possible explanation is that the covariates are the records in the past, while only the real-time information may highly affect real crashes. Another explanation may be that the influencing factors of crashes are various [7], and there may exist other factors influencing the number of car accidents. The covariates selected in this study may be limited, and these factors may only interpret part of the reason. Weather is negatively correlated with number of crashes. A possible explanation is that the study zone is part of \begin{table}
\begin{tabular}{c c c c c} \hline  & **Estimate** & **Std.Error** & **CI(Lower)** & **CI(Upper)** \\ \hline Intercept & \(1.04\times 10^{-5}\) & \(0.137\) & \(-0.268\) & \(0.268\) \\ \hline beta\_1 & \(0.521\) & \(0.127\) & \(0.272\) & \(0.771\) \\ \hline beta\_7 & \(0.239\) & \(0.139\) & \(-0.034\) & \(0.512\) \\ \hline alpha\_7 & \(0.240\) & \(0.166\) & \(-0.086\) & \(0.565\) \\ \hline traffic flow & \(1.35\times 10^{-6}\) & \(1.01\times 10^{-5}\) & \(-6.27\times 10^{-6}\) & \(3.32\times 10^{-5}\) \\ \hline vehicle speed & \(-0.010\) & \(0.007\) & \(-0.024\) & \(0.003\) \\ \hline holiday & \(-0.540\) & \(0.413\) & \(-1.35\) & \(0.269\) \\ \hline weather & \(-0.060\) & \(0.077\) & \(-0.211\) & \(0.091\) \\ \hline truck to car & \(-0.069\) & \(0.160\) & \(-0.383\) & \(0.245\) \\ \hline \end{tabular}
\end{table} 
Table 6: Summary statistics for Negative Binomial Model in Soochow direction. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Estimate** & **Std.Error** & **CI(Lower)** & **CI(Upper)** \\ \hline Intercept & \(8.03\times 10^{-6}\) & \(0.089\) & \(-0.174\) & \(0.174\) \\ \hline beta\_1 & \(0.285\) & \(0.094\) & \(0.101\) & \(0.469\) \\ \hline beta\_7 & \(0.177\) & \(0.097\) & \(-0.013\) & \(0.367\) \\ \hline alpha\_7 & \(0.538\) & \(0.099\) & \(0.344\) & \(0.731\) \\ \hline traffic flow & \(7.30\times 10^{-6}\) & \(5.89\times 10^{-6}\) & \(-4.25\times 10^{-6}\) & \(1.89\times 10^{-5}\) \\ \hline vehicle speed & \(-0.010\) & \(0.004\) & \(-0.018\) & \(-0.002\) \\ \hline holiday & \(-1.290\) & \(0.404\) & \(-2.090\) & \(-0.503\) \\ \hline weather & \(-0.016\) & \(0.056\) & \(-0.125\) & \(0.094\) \\ \hline truck to car & \(0.123\) & \(0.071\) & \(-0.016\) & \(0.262\) \\ \hline \end{tabular}
\end{table} 
Table 5: Summary statistics for Negative Binomial Model in Nantung direction.