# Pavement Distress Initiation Prediction by Time-Lag Analysis and Logistic Regression
## 4 Results and Discussion
### Logistic Regression Analysis Results  
We assessed the time delay effect in predicting pavement distress initiation by comparing Bayesian logistic regression models with or without time-lag variables. The first Bayesian logistic regression model considered variables including geometric data (UP_RD_GRADE and DOWN_RD_GRADE, DIS_NEAR_INT), historical distress data (NO_AD_DIS), and meteorological data, which considers the time delay effect. On the contrary, the second Bayesian logistic regression model considered variables including geometric data, historical distress data, and meteorological data without time lag. The pavement initiation dataset was randomly split into train and test datasets with a ratio of 70:30. Then, the Akaike information criterion (AIC), Bayesian information criterion (BIC), and the area under the receiver operating characteristic (AUC) curve were selected to assess the model performance. A lower AIC indicates a better model performance with simple parameters. The AUC range is from 0.5 to 1.0. A higher AUC represents a better potential precision ability. When the AUC range is from 0.7 to 0.8, it indicates the model performance is acceptable, and if the range is from 0.8 to 0.9, this means the model is a good fit of the dataset, and if AUC is higher than 0.9, this indicates perfect prediction. The results of the logistic regression model are shown in Table 3. The values of AIC and BIC for the model with weather parameters with time lags (model 1) are 1335.703 and 1375.964, both of which are smaller than 1371.775 and 1412.036 for the model without considering weather parameters with time lags (model 2). The values of AIC and BIC are determined by considering both the complexity of the model and the likelihood function, and taking the model likelihood function is considered as a penalty term, which means \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **PD\_INI** & **UP\_RD\_GRADE** & **DOWN\_RD\_GRADE** \\ \hline PD\_INI & 1.000 & 0.046 & \(-\)0.039 \\ UP\_RD\_GRADE & 0.046 & 1.000 & \(-\)0.787 \\ DOWN\_RD\_GRADE & \(-\)0.039 & \(-\)0.787 & 1.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation Coefficients Matrix of Dependent Variable and Categorical Variables. Figure 6: Correlation Coefficient Matrix of Dependent variables and Numerical Variables. (**a**) Correlation coefficients of distress initiation and geometric data, meteorological data without time lag. (**b**) Correlation coefficients of distress initiation and geometric data, meteorological data with time lag. that the model complexity and prediction accuracy are most balanced when the AIC and BIC are smallest [45]. Therefore, the introduction of the time-lag relationship makes the model performance of the prediction model better. Similarly, from the AUC values of 0.759 and 0.761 for the training and test sets of model 1, which are both better than 0.748 and 0.749 for model 2, it can be found that introducing the time-lag relationship can improve the prediction accuracy of the prediction model. In addition, the variables of weather factors (average daily temperature, total daily precipitation, and relative humidity) showed significance in the logistic regression model with or without time-lag characteristics, and the conclusions of the correlation analysis can be analyzed together to know that weather factors still play a non-negligible role in the prediction of short-term distress initiation, despite their relatively small influence. Similarly, the distance to the nearest intersection and the pavement distress accumulation in the adjacent area also showed significance in the logistic regression model, which corroborates with the high correlation between the two variables and distress initiation in the previous correlation study.