# Pavement Distress Initiation Prediction by Time-Lag Analysis and Logistic Regression
## 4 Results and Discussion
### Temporal Cross-Correlation Analysis  
The time lag is the time between two closely related events, such as cause and effect. The phenomenon of time delay in geography and environment was found as a common explanatory variable because interaction effects between various parts of the environment required an extended period. Moreover, some time delay effects were found in the previous research in pavement engineering and transportation. Road frost and anti-skid performance degradation caused by low temperature and high precipitation will not occur on the first day of the arrival of low-temperature weather but will occur after a short time delay [30; 31]. To understand the effects of the environment including temperature, precipitation, and humidity, a time-lag analysis method with both time-lag cross-correlation (TLCC) and windowed time-lag cross-correlation (WTLCC) was utilized. The time-lag analysis results are presented in the following Figure 4.  
Figure 4: _Cont_. The daily cumulative number of distress initiations is presented in the left column of Figure 4; the maximum value was observed on 26 February, and the maximum value is 57. The second-highest and third-highest daily cumulative number of distress initiations were 52 and 43, observed on 27 February and 13 May following 26 February, respectively. Moreover, two ridge types of the cumulative number of distress initiation variables in the broken line graph illustrate the concentrated outbreak of pavement distress occurrence. The first ridge started on 15 February and ended on 17 March. The time period of the second distress occurrence ridge was from 7 May to 1 June. The broken lines from the meteorological variables in the left column of Figure 4a,c,e are the daily change of maximum temperature, minimum temperature, and the daily average temperature, respectively. They all show high correlations and a significant increase during the study period, which offers an increasing temperature trend from winter to summer. In the left column of Figure 4g, daily total precipitation and relative humidity were also observed with a slight increase during the study period. Significantly, the daily total precipitation variable has been observed in two ridges in March and May, respectively. It is illustrated that there were two concentrated rainfall periods in March and May during the study period. Figure 4: Daily cumulative number of distress initiations (No_DIS_INI) and meteorological variables (left column) and TLCC charts between the cumulative daily number of distress initiations and meteorological variables (right column). (**a**) Comparison of No_DIS_INI and daily maximum temperature; (**b**) TLCC between No_DIS_INI and daily maximum temperature; (**c**) Comparison of No_DIS_INI and daily minimum temperature; (**d**) TLCC between No_DIS_INI and daily minimum temperature; (**e**) Comparison of No_DIS_INI and daily average temperature; (**f**) TLCC between No_DIS_INI and daily average temperature; (**g**) Comparison of No_DIS_INI and daily total precipitation; (**h**) TLCC between No_DIS_INI and daily total precipitation; (**i**) Comparison of No_DIS_INI and relative humidity; (**j**) TLCC between No_DIS_INI and relative humidity. The TLCC charts of each selected variable pair during the whole study period are shown in the right column. The horizontal axis of the TLCC chart is represented as the offset between two relevant signals, and the vertical axis is represented as the cross-correlation value, which measures the similarity of two relevant signals. In this study, the cumulative daily number of distress initiations was set as the S1 signal, and the meteorological variables were set as the S2 signal. The time-lag cross-correlation coefficient can be obtained by calculating the correlation between the S1 signal after shifting and the S2 signal. Therefore, the positive offset value means the S2 signal leads the S1 signal. The offset value represents the time interval in the above lead-follow relationship.