## 2 Integrated Predictive Traffic Resilience Framework
### Traffic Resilience Quantification
At the start of the day, traffic volume is usually low, which increases to the peak volume during peak hours, typically around 8:00 a.m. or 5:30 p.m. when people commute to and from work, and then decreases slowly towards the end of the day [7]. For a road segment, the daily traffic resilience cycle \(R_{Dn}\) can be quantified based on the general under-the-curve resilience metric as  
\[R_{D1}=R_{D2}=\cdots R_{Dn}=\frac{\int_{t_{0}}^{T}P_{TR}\left(t\right)\,dt}{ \int_{t_{0}}^{T}P_{TT}(t)\,dt} \tag{11}\]  
\[P_{TR}\left(t\right)=1-\frac{V(t)}{C(t)} \tag{12}\]  
where \(R_{Dn}\) is resilience at day cycle \(n\). \(R_{Dn}\) is the ratio of the performance difference between the current traffic performance \(P_{TR}(t)\) and the regular traffic trend \(P_{TT}(t)\). \(t_{o}\) is the starting time of the study, typically at 12:00 a.m., and \(T\) is the end time considered in the study period, typically at 11:59 p.m. It should be noted that in Equation (11), the equal sign signifies that the equation is applicable to the calculation of daily traffic resilience for \(R_{D1}\), \(R_{D2}\), and \(R_{Dn}\). However, it is important to acknowledge that the daily traffic performance PTR may vary, resulting in different levels of daily traffic resilience for \(R_{D1}\), \(R_{D2}\), and \(R_{Dn}\). This implies that while the equation is the same, the specific values and outcomes of daily traffic resilience may differ based on the corresponding daily traffic performance. \(P_{TT}(t)\) represents the anticipated or average traffic demand pattern during normal or non-disrupted conditions. It captures the inherent regularity or periodicity observed in the daily traffic data from time period \(t_{o}\) to \(T\), typically spanning from 12:00 a.m. to 11:59 p.m. The value of \(P_{TT}(t)\) is not necessarily fixed at one; rather, it is contingent upon the specific characteristics of the traffic data and the chosen normalization approach. In this research, historical data spanning two years is utilized to identify the recurring patterns and trends in traffic volume resulting from the disaster. By analyzing the traffic data over an extended duration, we can extract the regular traffic pattern, which serves as a benchmark for quantifying \(P_{TT}(t)\). The traffic resilience follows an upside-down general resilience curve in Figure 2(a). Thus, the current traffic performance \(P_{TR}(t)\) can be calculated as the maximum volume-to-capacity ratio minus the loss of service of a roadway, which is defined as the \(V(t)/C(t)\) ratio, as shown in Equation (11). The value of 1 refers to the maximum volume-to-capacity ratio. The resilience of a given travel segment \(Rsm\) over a given timespan of a route can be calculated by combining all the daily resilience values of that particular segment, as illustrated in Figure 4. Thus, the travel segment resilience can be calculated as  
\[R_{Sm}=Avg(R_{D1},R_{D2},\ldots,R_{Dn}) \tag{13}\]A travel segment is defined as traveling from one stop station to the next, whereas a travel route comprises multiple segments throughout the day. To determine a route's traffic resilience over a timespan, all the resilience values from the associated segments need to be considered, as shown in Figure 4. The overall route traffic resilience \(R_{r}\) can be calculated by averaging all the segments' resilience,  
\[R_{r}=Avg(R_{S1},R_{S2},\ldots R_{Sm}) \tag{14}\]  
Monitoring the \(V(t)/C(t)\) ratio under different conditions can assess an infrastructure's ability to cope with varying demand and capacity levels, and identify critical points that require attention to enhance its resilience. The \(V(t)/C(t)\) ratio measures the performance and level of service of transportation infrastructure, but during extreme weather conditions, abrupt changes to this ratio can occur, and the ability to recover and restore pre-disaster performance becomes crucial.