## 5 Model Results
### Daily Prediction
#### 5.2.1 Model Performance  
In both directions, combining the tables and the plots, the performance of a Negative binomial count data time series model outweighs support vector regression and artificial neural networks in predicting daily crash-frequency. Table 3 displays the model performance of daily prediction. To mitigate the detrimental impact of variables, the logarithmic link function was used. The seasonality of the data was also considered as the regression on \(\lambda\)t\(-\)7, the unobserved conditional mean seven time units, which corresponds to a week back in time. The fitting distribution was decided through the synthesis result of scoring rules and diagnostic histograms. Diagnostic plots of ACF, marginal calibration, and PIT were generated to compare the performance of fitting Poisson conditional distribution and a Negative binomial conditional distribution. Figure 6 shows diagnostic plots after model fitting in Nantung direction. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Direction** & **Model** & **Avg\_Error** & **RMSD** \\ \hline \multirow{4}{*}{Nantung} & Count Data Time & \multirow{2}{*}{0.70} & \multirow{2}{*}{0.96} \\  & Series with NB & & \\  & SVR & 1.11 & 1.32 \\  & ANN & 1.77 & 2.42 \\  & Count Data Time & \multirow{2}{*}{0.72} & \multirow{2}{*}{1.00} \\  & Series with NB & & \\  & SVR & 1.01 & 1.29 \\  & ANN & 1.79 & 2.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model performance of daily prediction. From the ACF plot of fitting Negative Binomial distribution, no obvious serial correlation was shown based on autocorrelation function. By comparing the probability integral transform plots, the Negative Binomial one appears more uniform, and thereby is more proper to be used in the fitting model. Besides, various metrics, namely, logarithmic, quadratic, spherical, rankprob, dawseb, normsq, and sqerro, are considered when comparing the suitability of Poisson and NB distribution. The smaller criterion points refer to better fitting of the distribution model. Table 4 shows the comparison between Poisson and NB model in Nantung direction. Scoring metrics shown in the above table indicate that the NB model is preferred. With the concerns of the over-dispersion issue of data distribution and the combined performance of measure metrics along with diagnostic graphs, the Negative binomial for the time series count data model was adopted. The coefficient beta_1 and beta_7 corresponds to regression on the previous observation; alpha_7 corresponds to regression on values of the conditional mean seven units back in time for consideration of weekly periodicity. Table 5 shows the summary statistics for Negative Binomial Model in Nantung direction. Table 6 shows the summary statistics for Negative Binomial Model in Soochow direction. The fitted model for the number of crashes in Nantung direction \(Y_{t}\) in the time period \(t\), is given by \(Y_{t}|\mathcal{F}_{t}-1\ \sim NegBin(\lambda_{t},\,8.96)\) with:  
\[\lambda_{t}=8.03\times 10^{-6}+0.29Y_{t-1}+0.18Y_{t-7}+\left(7.3\times 10^{-6},-0.01\,-1.29\,-0.02,\ 0.12\right)^{T}\!X_{t},t=1,\ldots,271\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Logarithmic** & **Quadratic** & **Spherical** & **Rankprob** & **Dawseb** & **Normsq** & **Sqerror** \\ \hline Poisson & 1.494 & \(-0.308\) & \(-0.541\) & 0.719 & 1.525 & 1.445 & 2.465 \\ \hline Negative Binomial & 1.441 & \(-0.319\) & \(-0.550\) & 0.713 & 1.408 & 0.967 & 2.465 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between Poisson and NB Model in Nantung direction. Figure 6: Diagnostic plots after model fitting in Nantung direction.