Travel Time Reliability Estimation in Urban Road Networks: Utilization of Statistics Distribution and Tensor Decomposition

Te travel time reliability (TTR) is crucial for evaluating the reliability of road networks, but real trafc data is often incomplete and sparse. Tis study validates that road network TTR conforms to a normal distribution and devises a quantifcation approach for road network TTR. Two reliability estimation methods are tailored for two data sources: section detectors and mobile detectors. Simulation experiments have confrmed the efectiveness of these methods. Te study emphasizes that the TTR estimation method using trafc section data (S-TTR), which is based on the verifed normal distribution assumption, maintains average absolute errors below 10%. On the other hand, the TTR estimation method that utilizes sparse trajectory data (T-TTR), which relies on tensor decomposition, profciently flls in all missing data with an average error of 0.0059.


Introduction
To foster urban progress, it is crucial to improve transportation infrastructure and assess the status of road networks.Travel time reliability (TTR) is a critical metric for gauging the reliability of road networks [1].Accurately estimating TTR for urban road networks and collecting reliable data can help transportation authorities optimize trafc networks [2][3][4].
Trafc data is crucial for TTR estimation on road networks, and the focus is now on data-driven solutions through Intelligent Transportation Systems [5].By using real-time data, it is possible to estimate network conditions in real-time, which makes it easier to control the network dynamically [6].Uneven deployment of data collection sensors can result in incomplete trafc data.Examples of such sensors include section detectors, foating vehicles, and satellite positioning.Tis leads to data gaps and losses, which can create a situation where the data used is substantial but sparse.Te aim of this study is to estimate road network TTR using sparse data for accurate and reliable results.
Tis study proposes new methods for estimating road network TTR to enhance transportation efciency.It utilizes trafc section detectors and trajectory data to ofer customized estimation techniques for TTR.Te contributions of this study are as follows: (i) A method for measuring the reliability of urban road networks based on TTR has been proposed.Real data from the Huangpu District of Shanghai was used to verify that the network TTR follows a normal distribution.A numerical calculation method for TTR based on this pattern has also been proposed.Additionally, the applicability of these methods has been demonstrated with data that has varying degrees of sparsity.
In the upcoming chapters, we have conducted a thorough examination of TTR.In Section 2, we have reviewed the literature on TTR.In Section 3, we have introduced a quantifcation method for TTR and have validated the normal distribution of TTR in the road network.We have proposed two estimation methods: S-TTR and T-TTR.In Section 4, we have comprehensively validated the efectiveness and applicability of both S-TTR and T-TTR estimation methods.Lastly, in Section 5, we have concluded this study by summarizing the aforementioned content and presenting our fndings.

Literature Review
Tere are two methods to measure road network TTR: mathematical analytical methods and statistical measurement methods.Te former uses trafc distribution models to calculate results, while the latter measures reliability by analyzing travel times.Mathematical methods can efectively consider various factors, but modeling and parameter calibration can be complex, limiting their use [7].Trafc data collection devices now provide more accurate statistical measurements by collecting more data [8].
Due to the challenge of obtaining complete real-world trafc data, some studies use trafc simulation data to explore TTR.Khani and Boyles [9] found a solution for fnding the most reliable path for risk-averse individuals by adding the mean and variance of route travel times.Researchers are now considering the diferences between trafc simulation data and real-world data due to advancements in information technology.Taylor [10] utilized a three-parameter Burr distribution to ft travel time distribution and applied the Fosgerau method to estimate TTR.Li et al. [11] used the Lempel-Ziv algorithm from information theory to analyze TTR based on historical data.
Recent research is enhancing TTR estimation through trafc simulation and real-world data by incorporating origin-destination data and simplifying network models for better accuracy and efciency.Te data-driven methods used in this study can provide a more accurate depiction of actual road network conditions.
Trafc data is collected using section detectors to gather information on fow, occupancy, and speed.However, these cannot provide travel time data directly.Te lack of section detectors in some links and the absence of positioning devices in some vehicles leads to data sparsity, making current trafc fow prediction methods inefective.Tese methods require raw data input and training processes such as time-series-based approaches and machine learningrelated methods.
One efective solution for handling data sparsity is constructing models, such as matrix and tensor factorization methods [12].Tensor factorization is suitable for predicting historical missing data [13].Tang et al. [14] constructed a three-dimensional tensor to simulate travel times for diferent links under varying trafc conditions during certain periods.Te study considered the impact of congestion on travel times but lacked a comprehensive model analysis.Zhong et al. [15] employed tensor factorization to identify trafc patterns.Pastor [16] proposed a low-rank tensor model for handling vehicle trafc volume data, utilizing the correlation between local structures present in multiple models and enhancing tensor sequence rank accuracy by optimizing balanced tensors.Additionally, Tan et al. [17] introduced a tensor-based approach to model and complete missing trafc data values.
Tis study proposes a new method to estimate TTR in road networks using the Bureau of Public Roads (BPR) function.A threshold is determined based on trafc section data to introduce a TTR estimation method.Additionally, a tensor-based TTR estimation method using decomposition is proposed to overcome sparsity in trajectory data.

Methodology
3.1.Notation.Te relevant parameters in this study are listed in Table 1.

Hypothesis and Validation.
Te delay travel time ratio measures the ratio of the total delay incurred by an individual vehicle during its journey within a road network to the total travel time.We analyzed trafc network and trajectory data collected in Huangpu, Shanghai, China, between May 25 and May 31, 2020, with 29,785 links and 94,658,941 trajectories.Based on our analysis, we determined that the delay travel time ratios of vehicles in the road network follow a normal distribution, as depicted in Figures 1 and 2. Furthermore, a normal distribution ftting was applied to the probability distribution of delay travel time ratios, and the goodness of ft is presented in Table 2. Te ftting results show that the distribution of delay travel time ratios for road network vehicles conforms to the characteristics of normal distribution.
Tus, the mathematical expression for the delayed travel time ratio of vehicles in the road network is We defne reliable travel as when the travel time of a single vehicle falls below or meets a predetermined threshold for its delay and travel time, otherwise it is considered unreliable.Te corresponding reliability formula simplifes as P(pc m ≤ pc 0 ) � P(d/t ≤ pc 0 ).In accordance with defnition and verifed hypothesis, we conclude the TTR of the road network during dt as follows: where μ represents the mean of the delay travel time ratio for vehicles in the road network, and σ 2 denotes the variance of the delay travel time ratio for vehicles in the road network.
According to the defnition of the TTR of the road network, the ratio of the delay to the travel time of each vehicle passing through the road network is calculated according to the obtained vehicle trajectory data, and the 2 Journal of Advanced Transportation probability distribution of the delay travel time ratio of the road network vehicles is shown in Figure 3.For a certain vehicle m, the ratio of delay to travel time is pc m .If pc m ≤ pc 0 , the trip of vehicle m is considered reliable; If pc m > pc 0 , it is considered that the trip of vehicle m is unreliable.Hence, the reliability of the dynamic road network is meant to quantify the likelihood of all vehicles traversing the road network successfully within T time under specifc conditions.Te integral formulation of this reliability is displayed in equation ( 3), whilst its discrete counterpart can be seen in equation (4).
Increased R(T) lead to more reliable travel times for vehicles, enhancing network reliability.Tus, TTR can be expressed as a normal distribution ratio of delay to travel time during a time period dt.Tensor with missing data, Estimate of x -X: Sparse tensor of size Estimate of X w: Binary indicator tensor of the same dimension as x -A (N) : Factor matrices of tensor x on each mode a (n)  r : Column vector of index r of matrix Hadamard product of tensors, also known as element-wise product -∘: Outer product of vectors - Trace norm -Equation (4) calculates real-time reliability for the road network determined within a specifc time span instead of integrating over the entire temporal scale.

Reliability Estimation Method.
Te threshold pc 0 is determined by referencing the percentile of the delay travel time ratio probability distribution.Given that the population For urban road networks, research indicates that travel times increase monotonically with trafc demand and exhibit a convex function, consistent with the BPR function [18].Zhao et al. [19] formulated a BPR correction model for urban road networks based on the BPR model, applicable for networks with signalized intersections and conditions close to trafc capacity saturation.Within the same time interval dt, the delay for link a is given by where t a (x a ) � t 0 a [1 + α(x a /c a ) β ] and α � ω(1 − s l ) + θ • g s .Consequently, the delay travel time ratio for the entire road network U within a specifc time period is given by Te estimation of the variance σ 2 of the road network vehicle delay travel time is the focus and difculty of this study.To assess the variance σ 2 of the delay travel time ratio distribution, we need to analyze the unevenness of travel time per unit distance on the road network.Network density distribution is studied about the macroscopic fundamental diagram discreteness [1,20,21].Knoop et al. [21] proposed the theory of the generalized macroscopic fundamental diagram, emphasizing the correlation between road network performance (i.e., total travel distance (TTD) in this study), cumulative vehicles in the network (i.e., total time spent (TTS) in this study), and the unevenness of link vehicle density.Based on this research, this study employs the diference between the ideal value of TTD(TTS) with completely uniform network density and the TTD of vehicles detected in real-time TTD(t) as input parameters for estimating the variance σ 2 of the delay travel time ratio distribution.
Figure 4 illustrates the variance estimation method.Te ideal value is situated on the envelope of the macroscopic fundamental diagram, as demonstrated by the dotted box in the fgure, demonstrating the correlation between the difference and the evenness of the network density.Te position of the point generated from the sum of the TTD and the number of vehicles can be found below the envelope when the road network encounters unexpected events like trafc accidents.Te study assumes that when the vehicle density and expectations of the road network remain constant, scatter points inclined upward indicate higher reliability with higher TTD and smaller variance, while those inclined downward signify decreased reliability and increased variance.Terefore, a variance estimation method is established accordingly.
Trough section detection, the TTD and number of vehicles on a network can be estimated, including their dynamics curve and related point positioning at diferent moments [22].Notably, the normal curve for the TTD and number of vehicles in an homogeneous network accounts for signal control within the road network [23].Real-time methods to predict TTD and TTS are as follows: Due to network confguration, signaling, organization, and incidents, trafc lane inequality leads to network heterogeneity.Tis could render an equilibrium curve for the TTD and total vehicle count, despite homogeneous network conditions [23].However, under such non-homogeneous network conditions, a defnite relationship still exists between the two parameters.Te actual value of the TTD(TTS) may deviate from the ideal value TTD(TTS).Te extent of deviation signifes the stability of TTR in the road network to some degree; greater deviation suggests increased instability in TTR and a larger variance in the delay travel time ratio.In practice, the TTD will not exceed the idealized TTD.If TTD � TTD, it signifes uniform vehicle density in the road network.Under such conditions, travel times per unit distance are entirely equal, leading to a variance of the delay travel time ratio of 0. Consequently, in the linear estimation model for variance, the constant term should be 0. In this study, the diference between TTD and TTD is 0, rendering the output of the variance estimation accurate, i.e., 0. Hence, the linear estimation model for variance does not possess a constant term.In the ideal state, the real-time estimation method for σ 2 is expressed as follows: where δ is the coefcient for variance estimation.In practical applications, it will be calibrated using sampled data from three trafc states: (i) under ideal free-fow conditions, (ii) under reliable critical conditions, and (iii) under unreliable congestion conditions, denoted as δ f , δ s , and δ j , respectively.

S-TTR Estimation Method.
Te utilization of the BPR function yields the expected value μ of the delay travel time ratio for vehicles in the road network.From the equilibrium curve of the TTD and the total number of vehicles, a linear estimation model for the variance σ 2 of the delay travel time ratio is introduced.Tis model is illustrated in equations ( 6) and ( 9).In both equations, it is necessary to determine the fow  q a (dt) on links during the interval dt, the total number of vehicles  N a (dt) traveling on each link, and the total number of vehicles  N(dt) on the road network U. Section detectors facilitate the dynamic acquisition of these pertinent parameters.Tis acquisition subsequently enables the calculation of the expected value μ and variance σ 2 for the delay travel time ratio of road network vehicles.Te workfow of the S-TTR estimation method is illustrated in Figure 5.

T-TTR Estimation Method.
In this subsection, we present a method for estimating TTR across an entire urban road network using sparse trajectory data.Urban road networks are complex, with trafc demand and supply imbalances between regions and variations in reliability at diferent times.Consequently, TTR displays both spatial and temporal unevenness.Our approach designs a third-order tensor defning the network's x-axis regions, y-axis regions, and time intervals.Leveraging temporal correlation of data, missing elements in spatial region tensors can be flled via a context-aware tensor methodology.Te method not only considers the spatial correlation of TTR among neighboring regions but also acknowledges the temporal correlation of TTR over diferent time intervals.For an Nth-order tensor x ∈ R I 1 ×I 2 ×•••×I N with missing data, the optimization objective function for tensor completion employing canonical polyadic decomposition is established as follows: min f w A (1) , A (2) ⟦A (1) , A (2) , w defned as follows: If the optimal factor matrices calculated from equation (10) are denoted as A (1) , A (2) , • • • , A (N) , the missing data of the original tensor x can be estimated using the following equation: (1 − w) * ⟦A (1) , A (2) Based on this, the complete form of the original tensor x can be computed using the following equation: (1) , A (2) where the frst part corresponds to the known data in the original tensor, while the second part represents the estimated values for the missing data in the original tensor.
To accurately estimate the TTR on a road network, missing data can be flled in using tensor completion theory.Tis involves using a specifc algorithm called high accuracy low-rank tensor completion (HaLRTC) [24] to impute the missing data.Te concept of tensors is introduced in this subsection, and a new approach to modeling TTR called the T-TTR estimation method is proposed by integrating the HaLRTC algorithm.
Given a sparse tensor X of size n 1 × n 2 × n 3 (sparse due to missing entries), with the indices corresponding to observed elements denoted as (i, j, k) ∈ Ω, a tensor S of the same size

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Journal of Advanced Transportation is defned as a binary tensor consisting of elements 0 and 1. Specifcally, Te objective function of tensor completion problem can be formulated as follows: where x.Te specifcs are as follows: In the objective function, the parameters α 1 , α 2 , and α 3 need to satisfy the condition: Te aforementioned model yields the reliability estimate  X i for region i.Due to the nonuniform distribution of data in road network regions, some regions have a small amount of data or even lack data.To address this, a variable a is introduced, which is related to penetration rate, time window length, and trafc fow.When the data quantity θ i in road network region i is less than or equal to a, the data for that region is treated as missing (assumed to be 0), as outlined: Due to the temporal correlation present in the TTR data of road network, an exponential smoothing method is introduced to update the aforementioned model.Tis further enhances the infuence of known data within the time window on estimated data.Te fundamental formula for exponential smoothing is as follows: where S t represents the smoothed value at time t, y t represents the actual value at time t, and a is the smoothing constant, ranging between 0 and 1. Te model update formula is as follows: where c represents the number of road network regions with the least amount of data, and S  X i denotes the estimate obtained through the updated exponential smoothing method.
Te alternating direction method of multipliers (ADMM) is suitable for solving distributed convex optimization problems, known for its fast processing speed and good convergence properties.By leveraging the ADMM framework, we can derive iterative update formulas for tensors B 1 , B 2 , and B 3 , and the estimated tensor x, thus obtaining the HaLRTC estimation method.Building upon the HaLRTC estimation method and combining it with the • Te trafc volume x a for each link.
• Te delay d a (x a ) for each link.
• Te travel time t a (x a ) for each link.
Step 3: estimating σ 2 • Te actual value of the total distance traveled by the vehicle for the corresponding travel time TTD (TTS).
• Desired value of the total distance traveled by the vehicle for the corresponding travel time TTD (TTS).
Step 4: dynamic road network travel time reliability estimates Step 1: sectional detector estimation • Flow q a (dt) in various links of the road network.
• Total number of vehicles N a (dt) traveling in each link.
• Total number of vehicles N (dt) traveling on road network U.
In the algorithm, the operator D a q /ρ (•) is given a specifc defnition.Taking the example of a matrix X with dimensions m × n mentioned above, we have D a q /ρ (X) � U a q /ρ V T , where  a q /ρ � diag(max (σ i − a q /ρ, 0)); the symbol "fold q (•)" denotes the operation of reshaping a matrix back into a tensor, which is the reverse of unfolding.

Analysis of the Efectiveness of the S-TTR Method.
Tis subsection utilized the VISSIM to test the efectiveness of the S-TTR estimation method.A case study was conducted on a 3 × 3 standard square microsimulation road network, as shown in Figure 6.Te trafc composition, vehicle speed distribution, and carfollowing model parameters were calibrated using data from the Huangpu district of Shanghai, which has similar scale and road network conditions.Te simulation road network had an intersection spacing of 500 meters, lane width of 3.5 meters, greenbelt width of 4 meters, and zebra crossing width of 8 meters.Each entrance lane consisted of two lanes, one for straight and left-turning vehicles and the other for straight and right-turning vehicles.Te vehicle type used for the simulation was the "small car," each with an expected speed of 40 kilometers per hour, and their speeds follow a normal distribution in VISSIM.Te intersections were two-phase signalized intersections with a signal cycle of 60 seconds, and the green light interval was set to 5 seconds.Trafc fow is evenly distributed throughout the road network by assigning a fxed fow rate to each entrance lane and using a trafc assignment model.Tere are 25 input paths at the edge of the network and 21 in the middle, assuming that vehicles do not take detours.To achieve balance distribution, the road network is adjusted by removing four-turns and U-turn routes and adjusting trafc fows based on the number of turns.Straight movements are assigned a weight of 1, one-turn movements 0.5, two-turn movements 0.25, and three-turn movements 0.125.
We analyzed the impact of signal cycles on trafc network efciency and reliability.Tree scenarios were considered, with signal cycles of 60 s, 90 s, and 120 s at each intersection.A comparative analysis was conducted to assess the efectiveness of the S-TTR estimation method.Te accuracy of the S-TTR method was evaluated by comparing the actual ground truth of network reliability (measured as the delay travel time ratio of each vehicle) using complete vehicle data to the estimated values obtained using the S-TTR method.Tis study analyzed simulation data to establish a threshold for delay travel time ratios within the network.Table 3 shows the percentage of the probability distribution for vehicle delay travel time ratios.
Te graph in Figure 7 shows network reliability based on diferent delay travel time ratio thresholds.Te pattern remains consistent for all thresholds, with a rapid drop in reliability shown by the solid line.Te 75th percentile threshold, with a delay travel time ratio pc 0 of 0.6899, effectively represents the overall trend of reliability variation.Tis threshold was selected for the simulation.
We simulated various vehicle inputs and collected data for reliability estimation experiment calibration.Results are presented in Table 4. Te relationship between reliability true values and estimated values is compared in Figure 8.
According to Figure 8(a), when the TTS initially increases, the trafc fow in the road network is smooth with no congestion.Both estimated and true reliability values are consistent and equal to 1.However, when the TTS exceeds a certain threshold, congestion starts to occur, and the trafc fow in the network approaches the critical fow state.In turn, both estimated and true reliability values start to decrease steadily.Nevertheless, the estimated values are higher than the true values, and the larger error region is in zone A. As the TTS continues to increase, trafc fow in the network becomes congested, causing signifcant delays for vehicles.Tis leads to a rapid decline in network reliability and unreliable travel times for vehicles, with delay ratios exceeding the threshold.
In Figure 8(b), there is a trend in the change in reliability values that is similar to Figure 8(c).Te estimated values are slightly smaller than the true values, and there is a larger error region that is concentrated in zone B. Additionally, in Figure 8(c), the phenomenon of underestimation is even more evident, with a larger error region concentrated in zone C.

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Journal of Advanced Transportation Errors were found in the estimation method in zones A, B, and C when the road network was close to saturation.A comparison of true and estimated reliability values was conducted through 180-minute simulations with a 60second statistical time window.Table 5 shows the results of the error analysis that used fve diferent random seeds and averaged the fndings.
Based on the above analysis results, it is clear that the S-TTR estimation method proposed in this study has an average absolute error of 0.0568, 0.0617, and 0.0759 for road network signal periods of the 60 s, 90 s, and 120 s respectively.Tese errors are all below 10%, indicating that the S-TTR estimation method can accurately estimate the reliability of the road network using data collected from section detectors.

Analysis of the Applicability of the S-TTR Method.
Tis subsection aims to examine how diferent detector deployment strategies can afect the accuracy of road network TTR.Terefore, a regular 4 × 4 grid network is used as an example.Each intersection entrance has four lanes while the exit roads have three.Each lane measures 3.5 meters in length, and the total distance between any two intersections is 500 meters.Te road network layout is based on these specifcations.Te network uses fxed-time signal control with four signal phases, each with its own set of signal lights.To ensure unbiased experimental results and account for variations in vehicle inputs on diferent roads, each road segment has a consistent trafc volume of 2000 passenger car units per hour (pcu/h).Trafc is made up of 91% passenger cars, 3% large trucks, and 6% buses.To maintain a safe and efcient fow of trafc, expected speeds are set at 50 km/h for passenger cars, 40 km/h for large trucks, and 30 km/h for buses.
In order to study the infuence of diferent detector deployment schemes on the accuracy of road network TTR, fve detector distribution schemes have been set up.Tese schemes consider the comprehensiveness and density of detector placements in the road network to assess TTR.Specifc detector deployment schemes are as follows: Five diferent detector deployment schemes are presented in Figure 9. Te frst scheme places a detector on each link in the east-west direction, but not on any north-south links.Tese detectors monitor vehicle data within 500 meters of deployed links.Te second scheme also places a detector on each link, but removes the four detectors on the lower right square of the network.Te third scheme places a detector on each link throughout the network, monitoring vehicle data within 500 meters.Te fourth scheme places two detectors on each link throughout the network, monitoring vehicle data 250 meters.Te ffth and fnal scheme places three detectors on each link throughout the network, monitoring vehicle data within 167 meters.Trough simulation experiments, Table 6 shows a comparison between estimated TTR values and ground truth values for diferent schemes, as well as the impact of detector deployment on estimation accuracy.
To refect the sparsity of the data, the original data were randomly screened with penetration rates of 5%, 10%, and 20%, respectively.After applying the same data processing methods as the original data, the estimated TTR values and actual values for the fve detector deployment schemes were obtained.Te obtained results of the TTR estimation and actual values are presented in Table 7.
To visually depict disparities between original TTR and post-random screening reliability (at penetration rates of 5%, 10%, and 20%), we assessed the efects of detector deployment on accuracy and reliability fuctuations.Te fndings, regarding accuracy impact and reliability shifts following penetration screening, are detailed in Tables 8 and 9.
Table 8 reveals that post random screening at penetration rates of 5%, 10%, and 20%, TTR data trends are generally aligned with the original data.Notably, under Scheme 5, the estimated reliability values are closest to the actual values, indicating minimal impact on the estimation accuracy of this detector scheme.
Table 9 illustrates that, notwithstanding the variations between actual and estimated reliability values, the estimations exhibit consistency subsequent to penetration screening under diverse schemes.Te simulated network data reasonably refect the accuracy of the simulation results.
Conclusively, the data analysis and scheme evaluation underscore that detector deployment uniformity, completeness, and density infuence reliability estimation accuracy.Enhanced uniform, complete, and dense deployment leads to more accurate TTR estimation.

Analysis of the Efectiveness of the T-TTR Method.
In this subsection, simulation experiments were conducted using a minimal road network unit consisting of a 3 × 3 grid.Specifc data for simulated road networks remain consistent with Section 4.1.Within VISSIM, nine nodes were chosen to defne the road network region, and region numbers were assigned sequentially from top to bottom and left to right, ranging from 1 to 9. Te selected road network nodes and their corresponding numbers are depicted in Figure 10.
Each node is designated to represent a small road network.Te travel time of each vehicle within the road network is represented by the diference between its end time and start time.Time windows are established based on the start time of the vehicle, with a total of 10,800 seconds divided into 12 windows of 900 seconds each.
Assuming uniform signal timings for the nine intersections, the vehicle composition is exclusively passenger cars, and the trafc fow input adheres to the daily trafc volume variation pattern of the network.Sudden incidents are disregarded, and the delay of each vehicle within a node is considered the delay within the small road network.By calculating the ratio of delay to travel time for each vehicle, the reliability of individual vehicles is determined.Te count of vehicles with reliability values equal to or below the threshold indicates the number of vehicles with reliable travel within the road network.Dividing the count of reliable vehicles by the total number of vehicles within each node yields the TTR of that node (small road network).Te acquired TTRs for diverse time windows across various road network regions are presented in Table 10.
Te 80% rule advises that substances with a nonmissing portion constituting less than 80% of the total sample should be excluded.Adhering to this principle, the experiment in this subsection employs a 20% penetration rate, assuming that the count of road network regions with missing data is no more than 2. Employing the proposed T-TTR estimation method on sparse data yields TTRs for distinct road network regions, as summarized in Table 11.From the 6th to the 12th time window, comprehensive data imputation results in an average error of 0.0059, clearly demonstrating the efectiveness of the T-TTR method in accurately estimating road network TTR.

Analysis of the Applicability of the T-TTR Method.
Tis subsection conducts a case study to analyze the estimation results of the T-TTR method at various data penetration rates and discusses the impact of data sparsity on the estimation technique.We employs simulated sparse trajectory data from VISSIM trafc data.Realistically, vehicle data collection across diferent road network regions is often uneven.Four comparison groups based on data uniformity, are proposed as follows: Control Group: Uniform data removal from VISSIM data.A certain percentage of VISSIM data is uniformly removed, resulting in a specifed penetration rate (e.g., 80% data removal for a 20% penetration rate).
Experimental Group 1: Random data removal from VISSIM data.Non-uniform data removal is employed to simulate random data gaps.
Experimental Group 2: Road network reliability estimated using the HaLRTC method.Tensor completion is applied to randomly missing data using the HaLRTC estimation method.
Experimental Group 3: Building upon Experimental Group 2 data, data from a specifc road network region meeting defned criteria are treated as missing data with a value of 0. Tensor completion is then performed using the T-TTR estimation method.Considering the limited availability of real-world trajectory data, typically exhibiting low penetration rates, representations are made for 5%, 10%, and 20% data penetration rates.Table 12 presents the accuracy of road network reliability obtained by the three methods across varying penetration rates.
Table 12 presents fndings from a uniform data analysis, indicating that diferent penetration rates yield varying results among Experimental Groups 1, 2, and 3. Notably, Experimental Group 3 and 2 outperformed Group 1, while the Control Group exhibits the lowest values.Remarkably, Experimental Group 3, applying the T-TTR estimation method, displays heightened precision as the penetration rate rises.At 5% penetration rate, Experimental Groups 3 and 2 exhibit comparable accuracy.Tis underscores that a more uniform data distribution within the road network enhances the fdelity of TTR to actual conditions.Te proposed T-TTR     Journal of Advanced Transportation method excels particularly when penetration rates surpass 5%.
For assessing the infuence of data penetration on the T-TTR method, Figure 11 illustrates the ground truth and chromaticity maps of road network TTR at 5%, 10%, and 20% penetration rates.Red hues indicate heightened congestion, with reliability close to 0 denoting severe congestion.Te fgure demonstrates escalating congestion as time windows extend.Te value trend of the T-TTR method aligns with the ground truth, confrming its efectiveness.Analyzing the infuence of diferent penetration rates on the T-TTR method, the errors for penetration rates of 5%, 10%, and 20% are calculated and summarized in Table 13 for error analysis.13 demonstrates a clear relationship: higher penetration rates result in more accurate road network TTR values from the T-TTR estimation method.Figure 12 presents a comprehensive comparison of Experimental Groups 1, 2, and 3 with penetration rates of 5%, 10%, and 20%.Te graph confrms that, at the same penetration rate, the T-TTR method consistently outperforms Experimental Groups 1 and 2, showcasing superior accuracy.Moreover, higher penetration rates within each group yield enhanced accuracy.Tis underscores that in practical scenarios, higher GPS coverage and richer vehicle trajectory data yield more precise estimations of road network TTR.Incorporating enhancements of the T-TTR method over HaLRTC yields smaller errors, afrming the utility of spatiotemporal correlations.Consequently, the T-TTR method excels in estimating road network TTR compared to the HaLRTC method.

Conclusion
Tis study defnes road network TTR as the likelihood that the ratio of delay to travel time for all vehicles in the network is below a specifed threshold.Utilizing road network and vehicle data from Huangpu District of Shanghai over four days, the study confrms that the delay-to-travel time ratio follows a normal distribution.Building on this, the study presents a calculation method for road network TTR and outlines parameter setting strategies.Given the normal distribution of delay-to-travel time ratios, study introduces the S-TTR and T-TTR estimation methods.Simulation experiments yield the following insights: (i) Te S-TTR estimation method accurately gauges road network TTR using only partial trafc data.It captures reliability fuctuations efectively, particularly when network trafc is not saturated.(ii) Te nonuniform and incomplete deployment of detectors leads to a reduction in the precision of the S-TTR method, while the uniform and comprehensive deployment of detectors contributes to an enhancement in estimation accuracy.(iii) Te T-TTR method, rooted in tensor completion theory, is contrasted with the spatially and temporally optimized HaLRTC algorithm.T-TTR effectively flls gaps in data with high precision.
Data uniformity enhances road network TTR accuracy.Te T-TTR method is notably more accurate when penetration rates surpass 5%, yielding more precise outcomes.
Tis study does not account for adaptive signal control or trafc guidance efects on road network TTR.Future research aims to address these limitations by integrating  crowd-sourced vehicle trajectory data to enhance the research method, achieving more precise road network TTR estimation through data fusion from a wider range of sources.
(ii) Two methods for estimating road network TTR have been introduced, using two diferent data sources.Tese methods are called the S-TTR (Travel Time Reliability of Network Based on Trafc Section Data) and T-TTR (Travel Time Reliability of Network Based on Sparse Trajectory Data) estimation methods.(iii) Simulation experiments have validated the efectiveness of both S-TTR and T-TTR estimation methods.

Figure 1 :Figure 2 :
Figure 1: Schematic diagram of trafc network and trajectory data collected in Huangpu, Shanghai, China, from May 25 to May 31, 2020.

the tensors B 1 ,
B 2 , and B 3 are all of size n 1 × n 2 × n 3 .Te matrix B 1(1) of size n 1 × (n 2 n 3 ) represents the unfolding of tensor B 1 in mode 1.Similarly, the matrix B 2(2) represents the unfolding of B 2 in mode 2, and the matrix B 3(3) represents the unfolding of B 3 in mode 3.Te optimization model has two constraints: the frst one ensures that the estimated tensor x and the original tensor x have equal elements over the set Ω; the second one sets the intermediate variables B 1 , B 2 and B 3 equal to the estimated tensor

vehicle delay and travel time ratio threshold pc 0 ••
Te parameters α and β of the BPR function for each link.

Figure 5 :
Figure 5: Te production of road network TTR estimated by section detection data.
Phase 1 allows for an East-West straight and right turn for 26 seconds, Phase 2 for an East-West left turn for 26 seconds, Phase 3 a North-South straight and right turn for 25 seconds, and Phase 4 for a North-South left turn for Trafc Volumn (pcu) Duration of Simulation (s) Variations in Trafc Volume at the Entrance Ramp

Figure 8 :
Figure 8: Comparison chart of TTS and reliability true values and estimated values.(a) Depicts a signal cycle length of 60 seconds; (b) depicts a signal cycle length of 90 seconds; (c) depicts a signal cycle length of 120 seconds.

Figure 11 :
Figure 11: Road network TTR at diferent penetration rates using the T-TTR estimation method.

Table 2 :
Distribution goodness of ft of delay and travel time ratio.

Table 3 :
Percentile of delayed travel time ratio probability distribution.

Table 4 :
Values of parameters for reliability estimation model.

Table 5 :
Values of parameters for reliability estimation model.

Table 6 :
Comparison of estimated and true road network TTR for diferent schemes.

Table 7 :
Estimations of TTR for various schemes under penetration rate screening.
y x Figure 10: Diagram of road network nodes and network IDs.

Table 10 :
TTR in road network regions under complete data.

Table 11 :
TTR in road network regions with 20% penetration rate sparse data.

Table 12 :
Analysis of diferent experimental groups with diferent penetration rates.

Table 13 :
Errors of T-TTR estimation method at diferent penetration rates.