Data-driven exploration of traffic speed patterns to identify potential road links for variable speed limit sign implementation

ABSTRACT

The focus of this study is to identify potential road links suitable for implementing a variable speed limit (VSL) sign by analyzing real-world traffic speed data collected over one year in Charlotte, North Carolina, United States. Exploratory and bivariate analyses were conducted to examine variations in traffic speed patterns during weekdays and weekends across eight specific timespans. The results revealed that road links with lower posted speed limits consistently experienced traffic speeds exceeding the posted speed limits. The mean traffic speeds are generally close to the posted speed limits for road links with higher posted speed limits while the 85^th percentile speeds exceeded the posted speed limits, indicating a potential need for VSL sign implementation. The road links with 40 mph or 50 mph posted speed limits have a unique pattern compared to road links of other posted speed limit clusters. The mean traffic speed on these road links decreased as the standard deviation increased. The findings contribute to an improved understanding of traffic speed patterns and provide valuable insights for implementing a VSL sign.

KEYWORDS:

• Variable speed limit
• traffic speed
• speed limit

Introduction

Traffic engineers use the 85^th percentile of free-flow speed (FFS) distribution as a standard to set the speed limit at a safe speed for promoting uniform traffic flow and minimizing crashes on roads (Elvik, 2010; FHWA, 2020). Speed limits in the United States have been centralized at the state level since 1995 (Albalate & Bel, 2012). The change followed the controversial debates that led to the abolition of the national speed limit of 55 mph, typically provided on freeways, in 1994 (Speed Limits by State, 2022). With advancements in vehicle safety features and infrastructure, states’ have gradually raised their maximum speed limits on freeways. These limits are determined by general statutes as well as local and state ordinances and are enforceable even if the speed limit is not posted.

Minimizing the speed differential or speed variation below a certain level based on road functional class, time of day, and day of the week is crucial for smooth traffic flow (Tagar & Pulugurtha, 2021a). Furthermore, reducing the speed differential and harmonizing vehicular speeds can improve mobility and operational performance (Abdel-Aty & Yu, 2013; Kalambay & Pulugurtha, 2023). Intelligent transportation systems-based solutions such as variable speed limit (VSL) signs make roads more efficient from a traffic operation perspective.

A VSL sign can maintain speed variations below thresholds by dynamically adjusting speed limits in response to changing traffic conditions, thereby preventing excessive speed differences between vehicles. Additionally, the VSL sign can harmonize vehicular speeds by encouraging all drivers to adhere to synchronized speed limits, reducing congestion and enhancing overall traffic flow (Abdel-Aty & Yu, 2013).

Although traffic simulation-based analysis is a valuable tool for studying traffic speed variation and assessing the effectiveness of VSL sign implementation, it has inherent limitations in accounting for real-world factors. A data-driven approach using real-world data might be more appropriate, especially when examining traffic patterns from a macroscopic perspective. By relying on real-world data, a more accurate and comprehensive understanding of traffic speed patterns can be achieved, allowing for more effective planning and decision-making at the city level.

Not all road links consistently experience high-speed variations. The authorities can focus on improving the infrastructure, implementing traffic management measures, and designing more efficient transportation systems on road links with high-speed variations. This can lead to reduced congestion and travel times, enhanced road safety, and improved traffic flow as well as the efficient allocation of resources.

The focus of this study is to identify road links with abnormal (high) speed variation and indicate the likely need for a VSL sign to guarantee smooth traffic. The objectives of this study are: (1) to compute and evaluate traffic speed measures by day of the week and time of the day, (2) to explore their association with the posted speed limit or reference speed by the road functional class, and, (3) to provide guidance on identifying road links susceptible to variations in traffic speeds for implementing VSL signs. The city of Charlotte, North Carolina, United States is considered as the study area. Like many urban areas, the city has grown substantially posing unique challenges associated with the efficient and safe movement of people and goods. Therefore, identifying the road links with consistent high-speed variations in cities like Charlotte, North Carolina, U.S.A. can help address the mobility and speeding-related issues that did not exist before.

The findings from this study serve as valuable insights and guide speed management interventions on roads by the posted speed limit. They assist practitioners with prioritizing road links for implementing VSL signs and efficiently allocating resources. Practitioners can also apply the methodology, examine the correlations between the standard deviation (SD) of speeds and 15^th percentile speed (15%ile), and 85^th percentile speed (85%ile), and reference speed, and make informed decisions pertaining to the implementation of VSL signs.

Literature review

In the United States, most research on the impact of speed limit changes on freeways has been conducted at the federal level. In 1974, the maximum speed limit on roads was reduced to 55 mph (Albalate & Bel, 2012). This legislation was prompted by the oil crisis affecting the overall economy in the ‘70s (Srinivasan et al., 2006). Although some research found that speed limit reductions sometimes have an insignificant effect on crash frequencies (Chen et al., 2013), many others indicate that the raised speed limits contribute to an increase in total number of crashes (Alhomaidat et al., 2020; Kwayu et al., 2020). Contrarily, Tagar and Pulugurtha (2021a) observed a decrease in the frequency of total, non-incapacitating and possible injury, and property damage-only crash frequencies at entry ramp speed-change lanes after the increase in the freeway-posted speed limit. The predictor variables influencing the merging speed change lane crash severity risk are different for cloverleaf, diamond, and other interchange types (Tagar & Pulugurtha, 2021b).

Fixed speed limits can unnecessarily decrease operating speeds, even when not needed (for example, during off-peak hours), resulting in lower traffic operational performance. Therefore, VSL may help regulate traffic based on real-time traffic conditions or pre-determined speed control algorithms on different road links. A VSL is typically designed to coordinate with ramp metering, hard shoulder running, and queue warning systems and is increasingly being explored as part of active traffic management (ATM) systems (Abdel-Aty & Yu, 2013). Besides the operational benefits, past research shows that VSL signs reduce overall speeds and, therefore, crash frequency (De Pauw et al., 2018). Levin et al. (2019) found that a VSL sign is an effective countermeasure for preventing speed-related crashes and controlling congestion, especially in work zones.

Research on congestion reduction is extensive in the literature. Pulugurtha et al. (2015) found evidence of the influence of travel time and traffic speeds on upstream and downstream road links. Pulugurtha and Pasupuleti (2010) integrated travel time (traffic speed) data from the regional travel demand model and the crash data to assess link reliability as a function of congestion components. However, only a few recent studies have utilized a data-driven approach to identify potential road links suitable for VSL sign implementation. Some examples include the studies conducted by He (2016), Silvano et al. (2020), Yasanthi and Mehran (2020), and Duvvuri et al (2020, 2021). which are summarized in Table 1, highlighting their specific areas of focus, the data and methodology employed, and their key conclusions.

Table 1. Recent research on speed variation using a data-driven approach.

Authors	Focus	Data and methodology	Key conclusions
He (2016)	To investigate if higher FFS will lead to a less congested freeway; estimate the impact of varying FFS on the performance of a freeway system	Data: Numerical examples based on observation of the reality Method: Mathematical modeling/approach based on a numerical approximation model called Cell Transmission Model	With the increasing FFS, the average delay of vehicles decreases at the beginning and then increases; the actual bottleneck capacity increases at the beginning and then decreases with the continuous increase of FFS
Silvano et al. (2020)	To estimate the FFS and the impact of speed limit changes and road characteristics on FFS distribution	Data: Dataset of speed observations from urban roads with different characteristics Method: Parametric probabilistic latent approach	Land use, on-street parking, and the presence of sidewalks influence FFS; changes in posted speed limit impact the distribution of the free-flow vehicles and the speed distribution of the constrained vehicles
Yasanthi and Mehran (2020)	To investigate driver behavior in terms of variations in the desired driving speeds under adverse road-weather conditions	Data: Road-weather data and traffic data including traffic speeds from weigh-in-motion traffic counter Method: Linear and non-linear regression models	Slight, moderate, and heavy snow will respectively reduce the FFS of light vehicles traveling in the shoulder lane by 0.2%, 3.4%, and 0.8%, and by 0.3%, 0.4%, and 1.5% when traveling in the median lane
Duvvuri et al. (2020)	To examine the influence of historical speed patterns on the degree of speed variations during peak hours	Data: Traffic speeds using probe data and road characteristics Method: K-means clustering and forest-based classification	Traffic volumes and average speeds of upstream and downstream road links influence the average speed of the corresponding road link

Main guiding principles of VSL sign implementation

Traditional traffic control strategies and management (especially of freeways) include VSL signs, whereby changes in the speed limit are conveyed to drivers by displaying the current speed limit on overhead variable message signs (Arora & Kattan, 2022). VSL systems have three different applications, namely, congestion, weather, and work zones. A congestion-related VSL system, which is the focus of this research, relies on changeable speed limit signs, speed sensors, and communications equipment to transmit data.

Different VSL strategies are used to control congestion, decrease travel times, and improve the operating speeds. Khondaker and Kattan (2015) provided an overview of VSL control strategies. They divided these strategies into reactive rule-based and proactive approaches. The reactive rule-based control strategies were mainly formulated as simple reactive rule-based logic to harmonize speed variations and stabilize traffic flow. These strategies adjust speed limits based on pre-selected thresholds of traffic flow, occupancy, or mean speed. VSL signs implemented using these strategies were found effective in reducing speed variations and improving traffic safety.

Proactive VSL control strategies are more advanced since they can predict future traffic, anticipating traffic breakdowns before they occur (Hegyi, A., De Schutter, B., & Hellendoorn, J. (2005). Khondaker & Kattan, 2015). In addition, the closed-loop feedback control used in proactive VSL strategies better controls the discrepancy between demand prediction and real demand. Hegyi, A., De Schutter, B., & Hellendoorn, J. (2005). developed a VSL control strategy, called Specialist, consisting of shock wave detection, control scheme generation, resolvability assessment, and control scheme application. Lu et al. (2010) proposed a VSL control strategy based on a speed limit tracking model to consider the transient effects of VSL. They filled in the gaps of previous research that merely considered VSL effects on steady-state speed values. Recently, Vrbanić et al. (2021) conducted a comprehensive survey of VSL and ramp metering control algorithms, including the most recent reinforcement learning-based approaches, and recommended establishing a multi-agent VSL and merging system based on agent cooperation to maximize the system performance.

Limitations of past research

Past studies commonly used traffic simulation-based analysis of VSL signs and speed variations. However, this approach has limitations in capturing and incorporating real-world factors. While simulations offer valuable insights into traffic speed variation, their ability to fully account for real-world complexities like behavioral and spatial effects is inherently limited. This study uses real-world traffic speed data, enabling and accounting for a more accurate and comprehensive understanding of traffic speed patterns in the VSL implementation decision-making. In this study, speed variations, computed as the difference between the average vehicle speed and the posted speed limit are used to identify road links with consistent high-speed variations. Consequently, these identified road links become potential locations for a VSL strategy, aiming to dynamically adjust posted speed limits in response to real-time traffic conditions.

Study area and data

Selected road links (Figure 1) in the city of Charlotte, North Carolina, United States were considered for this study.

Figure 1. Study area.

The link-level traffic speed data was obtained from the National Performance Management Research Data Set (NPMRDS) website. The data collection period is from July 2021 to June 2022. The raw traffic speed data collected for each link includes a date-time stamp, recorded speeds at 5-minute intervals, and reference speed. The reference speed is an indicator of the FFS of the corresponding link.

As NPMRDS does not provide posted speed limit data, they were deduced from the road characteristics data gathered from the city of Charlotte Open Data Portal (City of Charlotte, 2022). This dataset contained the posted speed limits for all selected links in the study area.

The road characteristics at the link level were obtained from shapefiles available on the NPMRDS website. These data include functional class, link length, and the number of through lanes.

NPMRDS provides the annual average daily traffic (AADT) as the solely available traffic characteristic. This AADT data is obtained in a geospatial format (shapefile) and covers road links belonging to interstates, US and NC routes, secondary roads, ramps, and most non-state roads maintained in North Carolina.

The data used for the exploratory data analysis is summarized in Table 2. Values in bold in Table 2 indicate the highest data share within sets of variable attributes. Regarding the categories of road links, 50.37% belong to the ‘Other principal arterials’ road functional class. In terms of link length, 48.89% of the data consists of road links that are at most 0.5 miles long. About 52.83% of road links have two through lanes. The AADT of 40.89% of the road links is less than or equal to 17,500 vehicles per day. Moreover, 45.81% of the road links have posted speed limits of 35 or 40 mph. The reference speed for 26.23% of the road links is between 40 mph and 50 mph.

Table 2. Data summary.

Variable	Attribute	#	%
Road category
Functional class	1: Interstate and freeway/expressway	335	41.26
	2: Other principal arterial	409	50.37
	3: Minor arterial, collector, and local	68	8.37
Road geometry
Length	1: ≤ 0.5 mile	397	48.89
	2: > 0.5 mile & ≤ 1 mile	229	28.20
	3: > 1 mile & ≤ 1.5 miles	112	13.79
	4: > 1.5 miles	84	9.12
# of through lanes	1	47	5.79
	2	429	52.83
	3	195	24.01
	>4	141	17.37
Traffic-related
AADT	1: ≤ 17500	332	40.89
	2: > 17500 & ≤ 35000	187	23.03
	3: > 35000 & ≤ 52500	121	14.90
	4: > 52500 & ≤ 70000	101	12.44
	5: > 70000	71	8.74
Speed-related
Speed limit	1: 25 mph or 30 mph	78	9.61
	2: 35 mph or 40 mph	372	45.81
	3: 45 mph or 50 mph	154	18.97
	4: 55 mph or 60 mph	112	13.79
	5: 65 mph or 70 mph	96	11.82
Reference speed limit	1: ≤ 40 mph	29	3.95
	2: >40 mph & ≤50 mph	213	26.23
	3: >50 mph & ≤60 mph	194	23.89
	4: >60 mph & ≤70 mph	204	25.12
	5: >70 mph	169	20.81

Methodology

This section outlines the methodology employed for exploring traffic speed patterns to identify links suitable for VSL signs. This methodology includes:

1. Traffic speed data merging and clustering

2. Exploratory data analysis

3. Identification and mapping of road links susceptible to high-speed variations

The third step in this methodology is complemented by the choice of the most appropriate model for the speed distribution of traffic speeds on road links with high-speed variations. For illustration purposes, five candidate distributions were fitted to the traffic speeds of a road link exhibiting high-speed variations. Through the analysis of different distribution models, valuable insights were obtained, aiding in selecting the distribution that most effectively captures the observed traffic speed patterns. The identification of the most appropriate model for traffic speed distribution enables improved management of traffic flow on similar road links, thereby facilitating enhanced traffic control and optimization strategies.

The analysis is divided into two phases. In the first phase, the analysis was conducted without clustering road links per posted speed limits. This initial phase provided a general overview of the data and allowed for a broad understanding of the speed patterns. In the second phase, the analysis was conducted considering clusters of posted speed limits. This phase enabled a more detailed examination of traffic speed patterns within those clusters.

The analysis was performed across the days of the week by distinguishing between weekdays and weekends. Additionally, the analysis was conducted considering eight specific timespans (12 AM to 3 AM, 3 AM to 6 AM, 6 AM to 9 AM, 9 AM to 12 PM, 12 PM to 3 PM, 3 PM to 6 PM, 6 PM to 9 PM, 9 PM to 12 AM), allowing for a sufficient detailed analysis of traffic patterns or speed variations at the city level. By incorporating both temporal dimensions, the potential variations in speed behavior throughout the day and the week were captured. The timespans were identified based on local traffic patterns (minimal variation within a timespan) and may vary for other cities/towns.

Traffic speed data merging and clustering

The data merging process involved combining the traffic speed data at the link level with all the relevant road and traffic characteristics. This merging was accomplished using the spatial join feature in ArcGIS Pro.

After the merging process, the resulting dataset was divided into different clusters based on posted speed limits. Five clusters were considered, categorized as follows: 25 mph or 30 mph, 35 mph or 40 mph, 45 mph or 50 mph, 55 mph or 60 mph, and 65 mph or 70 mph. This clustering facilitated a more focused analysis by grouping road links with similar posted speed limit ranges.

Subsequent analysis was conducted on both the entire dataset and the individual cluster datasets. While a comprehensive understanding of the overall trends and patterns was obtained by examining the entire dataset, analyzing the clustered datasets provided insights into the specific characteristics and behaviors within each posted speed limit category.

Exploratory data analysis

Exploratory data analysis was conducted to gain insights into the main characteristics of the traffic speed data, such as central tendency, dispersion, and distribution. The analysis focused on the following summary statistics: mean, SD, minimum (Min), maximum (Max), 15%ile, and 85%ile.

A bivariate analysis was conducted to assess the relationship between pairs drawn from the summary statistics. Bivariate analysis can reveal how variations in mean speed, SD and percentile speeds relate to each other. This information is essential for analyzing traffic flow dynamics and providing insights into potential congestion or safety issues. For example, a road link with a high mean speed and a high SD indicates that there is substantial variability in individual vehicle speeds around the average. However, a high mean speed coupled with a low SD indicates that most vehicles are traveling at similar speeds, contributing to a more uniform and stable traffic flow.

Correlation coefficients from the bivariate analysis were computed to quantify the strength and direction of the relationship between two variables. They range from −1 to 1, with values closer to −1 or 1 indicating a stronger correlation. Coefficients close to 0 indicate a weak correlation. For the purposes of this study, strong correlations are indicated by coefficient values greater than 0.5 or lower than 0.5. For example, a positive correlation coefficient greater than 0.5 suggests a strong positive relationship, implying that as one summary statistic increases, the other also tends to increase.

Identifying road links suitable for the implementation of VSL signs

The primary objective of this task was to identify road links that consistently exhibit high or very high variations in traffic speed throughout the study period. To assess the level of speed variation, the difference between the average speed (referred to as the mean) and the posted speed limit was computed. Average speeds considerably exceeding the posted speed limit can cause traffic congestion and safety issues. Therefore, adjusting the posted speed limit in real-time based on traffic conditions can help optimize the traffic flow and improve traffic safety.

In this study, the levels of speed variation were categorized into four distinct groups. They are:

• Low-speed variation. This category includes road links with a speed variation between 0 and 5 mph. Road links in this range exhibit relatively small deviations from the posted speed limit.

• Moderate-speed variation. Road links with speed variations ranging from > 5 mph to ≤ 10 mph fall into this category. These links experience moderate fluctuations in speed compared to the posted speed limit.

• High-speed variation. Road links with speed variations > 10 and ≤ 15 mph are classified as having high-speed variation. These links demonstrate significant deviations from the posted speed limit.

• Very high-speed variation. This category encompasses road links with speed variations exceeding 15 mph. These links exhibit substantial variations from the posted speed limit, indicating a considerable level of speed fluctuation.

To visualize and analyze the speed variation patterns, maps were generated for the five clusters of road links. These maps provide an overview of speed variations during weekdays and weekends, covering the eight specific timespans. By examining the maps for each cluster at different timespans, insights can be gained regarding the consistency and magnitude of speed variations across different days and times.

Results

In this section, results are presented according to the steps mentioned in the methodology. The examination of traffic speed variation across months and times of the day, along with exploratory and bivariate analyses, identification of the road links with high-speed variations, and fitting candidate distributions to traffic speeds are presented and discussed.

Variation of traffic speeds across months and times of the day

Figure 2 shows the variation of the mean speed interval for all road links across months and times of the day for both weekdays and weekends. Overall, for corresponding times of the day, the intervals of mean speeds are consistently higher by one increment (5 mph) on weekends compared to weekdays. Similar trends were observed in variations of SDs during weekdays and weekends.

Figure 2. Variation of the mean speed interval across months and times of the day.

Exploratory data analysis of traffic speeds

Table 3 summarizes statistical measures of traffic speeds for both weekdays and weekends across different the timespans. The lowest mean traffic speeds were recorded from 3 PM to 6 PM and 12 PM to 3 PM for weekdays and weekends, respectively, measuring 39.6 mph and 47.3 mph. These speeds are associated with the highest SDs. Additionally, the lowest speeds (Min) were found to be for the same timespans. The 85%ile speeds across the different timespans are quite uniform, particularly during weekends, ranging from 71 mph to 74 mph. The maximum speed of 99 mph was recorded from 3 PM to 6 PM and 12 AM to 3 AM for weekdays and weekends, respectively.

Table 3. Summary statistics of traffic speeds (mph).

Timespan	Mean	SD	Min	Max	15%ile	85%ile
Weekday
12 AM − 3 AM	63.2	8.6	30	90	55	71
3 AM − 6 AM	63.6	9.6	31	88	53	73
6 AM − 9 AM	48.0	17.9	13	93	26	69
9 AM − 12 PM	48.1	18.0	12	85	26	69
12 PM − 3 PM	44.7	19.1	9	90	22	68
3 PM − 6 PM	39.6	19.4	3	99	18	65
6 PM − 9 PM	48.0	18.2	12	89	26	69
9 PM − 12 AM	57.8	14.2	19	85	40	71
Weekend
12 AM − 3 AM	61.4	11.8	24	99	48	72
3 AM − 6 AM	64.4	9.1	33	88	55	73
6 AM − 9 AM	60.9	14.1	23	84	43	74
9 AM − 12 PM	52.3	18.4	14	87	29	72
12 PM − 3 PM	47.3	19.7	9	93	24	71
3 PM − 6 PM	48.1	19.4	10	93	25	71
6 PM − 9 PM	52.0	17.9	14	86	30	71
9 PM − 12 AM	58.2	14.7	17	86	39	72

Exploratory data analysis of traffic speeds by posted speed limit clusters

Tables 4– 8 summarize statistics of traffic speeds by posted speed limit clusters.

Table 4. Summary statistics of traffic speeds (mph) for 25 mph or 30 mph posted speed limit road links.

Speed limit cluster	Timespan	Mean	SD	Min	Max	15%ile	85%ile
25/30 mph	Weekday
	12 AM − 3 AM	51.7	9.6	30	66	40	62
	3 AM − 6 AM	50.8	9.3	31	64	39	61
	6 AM − 9 AM	36.0	11.9	14	57	23	51
	9 AM − 12 PM	36.4	13.0	13	59	22	53
	12 PM − 3 PM	32.8	13.0	10	57	19	50
	3 PM − 6 PM	29.6	11.7	9	53	17	44
	6 PM − 9 PM	35.8	12.1	14	58	22	51
	9 PM − 12 AM	42.8	12.2	19	62	28	57
	Weekend
	12 AM − 3 AM	47.1	11.5	24	64.5	33	60
	3 AM − 6 AM	52.2	8.6	33	65	42	62
	6 AM − 9 AM	46.6	12.3	23	66	32	61
	9 AM − 12 PM	39.2	13.3	15	62	24	57
	12 PM − 3 PM	35.1	13.0	12	59	21	52
	3 PM − 6 PM	35.3	12.9	12	59	21	52
	6 PM − 9 PM	38.0	12.8	14	60	24	54
	9 PM − 12 AM	42.7	13.0	17	63	27	58

Table 5. Summary statistics of traffic speeds (mph) for 35 mph or 40 mph posted speed limit road links.

Speed limit cluster	Timespan	Mean	SD	Min	Max	15%ile	85%ile
35/40 mph	Weekday
	12 AM − 3 AM	60.4	7.4	43	71	51	68
	3 AM − 6 AM	59.9	9.3	39	73	48	70
	6 AM − 9 AM	39.4	14.2	14	64	23	57
	9 AM − 12 PM	39.0	14.9	12	65	22	58
	12 PM − 3 PM	35.2	14.8	9	63	19	55
	3 PM − 6 PM	30.2	13.1	7	57	16	46
	6 PM − 9 PM	39.3	15.2	12	66	22	59
	9 PM − 12 AM	51.5	13.8	24	71	34	67
	Weekend
	12 AM − 3 AM	57.3	11.5	32	73	42	70
	3 AM − 6 AM	61.2	8.5	42	73	51	70
	6 AM − 9 AM	54.7	13.9	27	74	37	70
	9 AM − 12 PM	43.1	16.1	14	70	25	63
	12 PM − 3 PM	37.1	15.7	9	66	20	58
	3 PM − 6 PM	38.3	15.7	10	67	21	58
	6 PM − 9 PM	43.1	15.5	15	69	25	62
	9 PM − 12 AM	51.9	14.4	23	72	34	68

Table 6. Summary statistics of traffic speeds (mph) for 45 mph or 50 mph posted speed limit road links.

Speed limit cluster	Timespan	Mean	SD	Min	Max	15%ile	85%ile
45/50 mph	Weekday
	12 AM − 3 AM	61.2	9.6	39	75	48	71
	3 AM − 6 AM	61.0	10.3	39	75	47	72
	6 AM − 9 AM	44.8	18.0	13	74	25	69
	9 AM − 12 PM	45.3	17.9	13	74	26	69
	12 PM − 3 PM	42.0	18.9	9	74	22	68
	3 PM − 6 PM	35.7	17.7	5	70	18	61
	6 PM − 9 PM	42.8	16.6	14	71	25	65
	9 PM − 12 AM	55.9	15.0	26	77	37	72
	Weekend
	12 AM − 3 AM	59.4	12.6	33	77	43	72
	3 AM − 6 AM	62.2	10.0	40	76	49	73
	6 AM − 9 AM	58.8	15.6	27	80	39	75
	9 AM − 12 PM	50.8	19.2	15	80	29	74
	12 PM − 3 PM	46.0	20.2	10	78	24	73
	3 PM − 6 PM	46.8	19.8	11	78	25	73
	6 PM − 9 PM	50.3	18.8	15	79	29	73
	9 PM − 12 AM	56.4	15.7	25	78	37	73

Table 7. Summary statistics of traffic speeds (mph) for 55 mph or 60 mph posted speed limit road links.

Speed limit cluster	Timespan	Mean	SD	Min	Max	15%ile	85%ile
55/60 mph	Weekday
	12 AM − 3 AM	63.9	6.4	46	81	58	70
	3 AM − 6 AM	64.4	5.9	49	79	58	71
	6 AM − 9 AM	55.1	13.5	18	93	39	67
	9 AM − 12 PM	58.8	9.4	31	85	49	67
	12 PM − 3 PM	55.3	12.8	20	90	40	67
	3 PM − 6 PM	47.8	17.6	3	99	26	65
	6 PM − 9 PM	58.5	11.1	27	89	46	68
	9 PM − 12 AM	63.1	7.0	43	81	56	70
	Weekend
	12 AM − 3 AM	64.3	7.0	45	83	58	71
	3 AM − 6 AM	65.2	7.0	46	84	58	72
	6 AM − 9 AM	65.8	7.0	46	84	59	73
	9 AM − 12 PM	62.7	8.8	36	87	54	71
	12 PM − 3 PM	58.1	12.4	23	93	44	69
	3 PM − 6 PM	58.1	12.4	23	93	44	69
	6 PM − 9 PM	61.5	9.1	35	86	52	70
	9 PM − 12 AM	63.6	6.9	44	82	57	70

Table 8. Summary statistics of traffic speeds (mph) for 65 mph or 70 mph posted speed limit road links.

Speed limit cluster	Timespan	Mean	SD	Min	Max	15%ile	85%ile
65/70 mph	Weekday
	12 AM − 3 AM	69.2	7.0	47	90	63	76
	3 AM − 6 AM	71.2	6.2	53	88	64	77
	6 AM − 9 AM	70.2	5.5	52	87	65	75
	9 AM − 12 PM	70.1	4.9	54	84	65	75
	12 PM − 3 PM	70.0	4.9	54	84	65	75
	3 PM − 6 PM	68.0	7.0	43	89	62	74
	6 PM − 9 PM	70.6	5.7	52	87	65	76
	9 PM − 12 AM	70.8	5.6	55	85	65	76
	Weekend
	12 AM − 3 AM	69.8	8.6	31	99	63	77
	3 AM − 6 AM	71.1	6.3	53	88	64	77
	6 AM − 9 AM	73.5	4.5	62	84	69	78
	9 AM − 12 PM	72.7	4.6	58	85	68	77
	12 PM − 3 PM	72.0	4.7	56	86	67	76
	3 PM − 6 PM	72.3	4.6	58	85	68	77
	6 PM − 9 PM	72.0	5.1	56	86	67	77
	9 PM − 12 AM	71.6	5.3	56	86	66	77

From Table 4, the mean traffic speeds consistently exceed the 25 mph or 30 mph posted speed limit throughout the day. Furthermore, the 85%ile speeds are much higher, indicating that drivers on 25 mph or 30 mph posted speed limit roads tend to drive at speeds twice as high as the posted speed limit. Safety measures like enforcement, rather than traffic operational measures, may be more beneficial on these road links. Moreover, no statistical differences were observed in these traffic speed measures between weekdays and weekends.

The mean traffic speeds for road links with posted speed limits of 35 mph or 40 mph, 45 mph or 50 mph, 55 mph or 60 mph, and 65 mph or 70 mph in Tables 5–8 are relatively close to the posted speed limits. However, the 85%ile speeds are higher than the posted speed limits on these roads. This indicates that many drivers exceed the posted speed limits, suggesting a need for safety measures like enforcement or traffic operational measures like speed management strategies to align these drivers with the intended posted speed limits. Also, this observation suggests that posted speed limits should be increased for certain road links to better accommodate drivers, albeit on the condition that their safety is maintained.

While the mean traffic speeds generally align closely with the posted speed limits, certain road links display significant deviations in their mean traffic speeds. These deviations, whether exceptionally low or exceptionally high, can go unnoticed when examining the aggregated mean traffic speeds presented in Tables 5–8.

The variations in mean traffic speeds on specific road links can provide valuable insights into the effectiveness of current posted speed limits. If certain road links consistently exhibit significantly lower mean traffic speeds, it could indicate potential issues such as congestion, road conditions, or other factors impeding the traffic flow. Conversely, road links with consistently higher mean traffic speeds might indicate a need for stricter enforcement or speed management measures to ensure the safety of road users.

It is crucial to consider and analyze individual road links separately to understand their speed patterns and identify any areas where the existing posted speed limits might require adjustments. By examining the data at an increasingly granular level, transportation authorities can make better-informed decisions regarding posted speed limit modifications or targeted interventions to address specific road links that deviate significantly from the aggregated mean traffic speeds.

Bivariate analysis of traffic speed measures

The bivariate analysis consisted of pairwise correlations of traffic speed measures, including Mean, SD, 15%ile, 85%ile, reference speed (RSP in Figure 3), and a posted speed limit (SPL in Figure 3). The bivariate analysis was conducted with and without considering clusters of road links based on their posted speed limits. Figure 3 displays the correlation coefficients of the six traffic speed measures for weekdays and different timespans.

Figure 3. Correlations of traffic speed measures for weekdays.

Note: *, **, and *** indicate that the corresponding coefficients are significant at the 95%, 99%, and 99.9% confidence levels, respectively.

The bivariate analysis for weekends exhibited similarities to weekdays in terms of correlation coefficients and statistical significance levels. The analysis revealed significant correlations between the mean traffic speeds and several other factors. A strong correlation existed between the mean traffic speed and the 15%ile speed, indicating a relationship between the overall mean traffic speed and the lower end of the speed distribution (say, FFS). Similarly, a strong correlation existed between the mean traffic speed and the 85%ile speed, indicating a relationship between the mean traffic speed and the upper end of the speed distribution. Additionally, a strong correlation was observed between the mean traffic speed and the posted speed limit, suggesting that the imposed speed restrictions influence the mean traffic speed. These findings highlight the importance of considering these factors when assessing and analyzing traffic patterns and speed-related dynamics.

A weak correlation was observed between the mean traffic speed and the SD of traffic speeds. This suggests that the mean traffic speed has a limited influence on the variation or dispersion of speeds within the traffic flow. Furthermore, a weak correlation was observed between the mean traffic speed and reference speed. This implies that the mean traffic speed does not strongly dictate the speed at which traffic flows under uncongested conditions. In fact, a weak correlation was observed between the SD of traffic speeds and any other traffic speed measure. This suggests that the variation in speeds within the traffic flow does not strongly correspond to other measured traffic speed parameters.

It is important to note that this first bivariate analysis was from a heterogeneous traffic speed dataset that does not differentiate road links based on their characteristics. This lack of distinction introduces potential limitations in interpreting the correlation coefficients and their corresponding statistical significance levels. For instance, the analysis combines data from road links with varying posted speed limits, ranging from very low speed to very high speed. As a result, the correlations observed may be influenced by the inherent characteristics of the road links rather than solely reflecting the relationships between the traffic speed measures.

Table 9 presents the correlation coefficients between traffic speed measures for weekdays from 12 PM to 3 PM. These correlation coefficients provide insights into the relationships between different speed measures and the posted speed limits assigned to the clusters.

Table 9. Correlations between traffic speed measures for weekdays (12 PM to 3 PM).

Measure 1	Measure 2	Speed limit cluster (mph)
		25/30	35/40	45/50	55/60	65/70
Mean	SD	−0.24*	−0.11*	−0.64***	−0.40***	−0.13
Mean	15%ile	0.99***	0.98***	0.99***	0.97***	0.95***
Mean	85%ile	0.98***	0.98***	0.99***	0.98***	0.96***
Mean	RSP	0.69***	0.87***	0.83***	0.82***	0.78***
Mean	SPL	−0.06	−0.08	0.17*	0.21*	0.17
SD	15%ile	−0.37***	−0.27	−0.73***	−0.58***	−0.41***
SD	85%ile	−0.08	0.06	−0.54***	−0.20*	0.15
SD	RSP	−0.10	0.02	−0.50***	−0.07	−0.17
SD	SPL	0.16	−0.05	−0.11	−0.28***	0.05
15%ile	85%ile	0.95***	0.93***	0.97***	0.91***	0.83***
15%ile	RSP	0.68***	0.83***	0.82***	0.74***	0.75***
15%ile	SPL	−0.07	−0.07	0.16*	0.25**	0.14
85%ile	RSP	0.68***	0.86***	0.83***	0.86***	0.75***
85%ile	SPL	−0.07	−0.09	0.16*	0.17	0.19
RSP	SPL	−0.07	−0.03	0.17*	−0.03	0.28***
			≥0.5		>0 & <0.5
			≤ − 0.5		> −0.5 & <0

Note: *, **, and *** indicate that the corresponding coefficients are significant at the 95%, 99%, and 99.9% confidence levels, respectively. RSP is reference speed and SPL is posted speed limit.

After computing the correlations for different days of the week and timespans, it was observed that the levels or magnitudes of the correlations and their significance are generally consistent across weekdays and weekends, regardless of the time of day. This suggests that the relationship between traffic speed measures and posted speed limit clusters remains relatively stable throughout the different days of the week, including weekends.

Significant correlations were observed between each of the mean, 15%ile and 85%ile speeds, and the reference speed, regardless of the posted speed limit cluster. These correlations indicate a robust relationship among these traffic speed measures and are statistically significant, confirming their reliability. When the mean traffic speed is high, both the 15%ile and the 85%ile speeds are likely to be high as well. This indicates a general trend for speeds to align and vary together within a given posted speed limit cluster.

Road links subject to a 45 mph or 50 mph posted speed limit have a distinct traffic pattern when compared to the road links that fall within the other posted speed limit clusters. Specifically, these road links demonstrate a notable finding: the SDs of traffic speeds exhibit strong statistical correlations with the mean traffic speed, the 15%ile speed, the 85%ile speed, and the reference speed. However, what sets these correlations apart is the fact that they are negative, rather than positive.

Strong negative correlations between the SDs of traffic speeds and the mean traffic speed, 15%ile speed, 85%ile speed, and reference speed indicate an inverse relationship between these variables on road links with a 45 mph or 50 mph posted speed limit. This means that as the SD of traffic speeds increases, the mean traffic speed, speeds at the lower 15%ile and the higher 85%ile points, and the reference speed tend to decrease. Conversely, these speed measures tend to increase when the SD of traffic speeds decreases.

Understanding the implications of these negative correlations can offer valuable insights for traffic management and road safety. The strong negative correlation between the SD of traffic speeds and the mean traffic speed suggests that road links with higher speed variations tend to have lower mean traffic speeds. This indicates a potential discrepancy in driving behaviors and a higher likelihood of speed fluctuations within this posted speed limit cluster.

Additionally, the negative correlations between the SD of traffic speeds and the 15%ile and 85%ile speeds and the reference speed can provide useful information about the speed distribution within the 45 mph or 50 mph posted speed limit cluster. These correlations suggest that as the variation in traffic speeds increases, the lower and higher percentile traffic speeds and the reference speed tend to decrease. This insight can help identify areas where speed management interventions may be necessary to address the speed distribution and potential risks associated with this posted speed limit cluster.

Identifying road links with high-speed variations

Figure 4 presents maps of road links exhibiting various speed variations during weekdays from 9 AM to 12 PM, divided according to posted speed limit clusters. A negative speed variation indicates that the mean traffic speed falls below the specified posted speed limit.

Figure 4. Speed variations for weekdays between 9 AM and 12 PM.

The variations in traffic speed, specifically those categorized as high- and very high-speed variations, were examined. As stated previously, high-speed variations refer to speed fluctuations exceeding 10 mph and up to 15 mph, while very high-speed variations encompass fluctuations exceeding 15 mph.

In this study, an important assumption asserts that road links, particularly freeway sections, exhibiting consistently high or very high-speed variations are prime candidates for VSL implementation. Table 10 showcases road links within the 65 mph or 70 mph posted speed limit cluster. These road links are of particular interest as they represent crucial sections of freeways that require careful analysis and potential consideration for VSL sign implementation.

Table 10. 65/70 mph posted speed limit road links of high-speed variations.

Time of the day	TMC code
Weekday
12 AM − 3 AM	N/A
3 AM − 6 AM	125N10200
6 AM − 9 AM	125N16944
9 AM − 12 PM	N/A
12 PM − 3 PM	N/A
3 PM − 6 PM	N/A
6 PM − 9 PM	N/A
9 PM − 12 AM	N/A
Weekend
12 AM − 3 AM	N/A
3 AM − 6 AM	N/A
6 AM − 9 AM	125N10200, 125N16944
9 AM − 12 PM	125P10200, 125N16944
12 PM − 3 PM	125N10200, 125P10200, 125N16944
3 PM − 6 PM	125N10200, 125P10200, 125N16944
6 PM − 9 PM	125N10200, 125N16944
9 PM − 12 AM	N/A

Note: N/A indicates the absence of any road link exhibiting high-speed variation.

The analysis of road links with high-speed variations is intended to identify sections of the road network where drivers experience notable changes in traffic speed. These variations can be attributed to many factors, such as traffic congestion, road design, intersections, or other localized conditions. By pinpointing these areas, transportation authorities and urban planners can gain insights into the factors contributing to traffic speed fluctuations and devise strategies to improve traffic flow, enhance road safety, and optimize overall transportation efficiency.

Moreover, the examination extends to road links with very high-speed variations, which exhibit even more pronounced changes in speed. Such variations often indicate the presence of critical bottlenecks, hazardous conditions, or other factors that significantly impact traffic. Identifying road links with very high-speed variations is crucial for targeted interventions and engineering solutions to mitigate potential risks and enhance the overall quality of road infrastructure.

Fitting candidate distributions to traffic speeds

Five candidate distributions were fitted to the traffic speeds of a road link with high-speed variations. By analyzing different distribution models, valuable insights are gained on selecting the most appropriate distribution that accurately represents the observed traffic speed patterns, facilitating better understanding and management of traffic flow on such road links.

Figure 5 exhibits the skewness-kurtosis graph, generated using the ‘fitdistrplus’ package (Delignette-Muller & Dutang, 2015) in the R programming language for the traffic speeds observed on road link 125N10200 (I-485 S/I-85 Exit 10) during the weekend interval of 3 PM to 6 PM. A nonparametric bootstrap procedure was implemented to address the uncertainty surrounding the estimated values of kurtosis and skewness derived from the computed traffic speeds. The procedure involved computing skewness and kurtosis on 1,000 bootstrap samples, with the resulting values presented on the skewness-kurtosis plot as depicted in Figure 5.

Figure 5. Skewness-kurtosis plot for traffic speeds on the road link “125N10200” during the weekend time interval of 3 PM to 6 PM.

The related goodness-of-fit plots, i.e. the density plot and the cumulative distribution function (CDF), are presented in Figures 6(a,b) respectively.

Figure 6. Goodness-of-fit plots of the candidate distributions fitted to the weekend time interval (3 PM to 6 PM) traffic speeds on the road link “125N10200”.

Table 11 presents the goodness-of-fit results of the five candidate distributions. The KolmogorovSmirnov and Cramer-von Mises goodness-of-fit statistics, Akaike’s Information Criterion, and the Bayesian Information Criterion were used in this study. The goodness-of-fit statistics measure the distance between the fitted parametric and the empirical distributions. Therefore, the lower the parameter listed in Table 11, the better the distribution’s fit. Taking the outputs of the goodness-of-fit statistics into account, the Burr distribution is the best-fitted distribution of the case study.

Table 11. Comparison of goodness-of-fit results for the weekend time interval (3 PM to 6 PM) traffic speeds on the road link ‘125N10200’.

Goodness-of-fit statistic	Gamma	Weibull	Log-normal	Log-logistic	Burr
Kolmogorov-Smirnov statistic	0.20	0.14	0.22	0.09	0.09
Cramer-von Mises statistic	40.86	17.09	55.17	5.81	5.20
Good-of-fit criteria (AIC)	22756.19	20477.35	23731.25	20176.99	19938.96
Good-of-fit criteria (BIC)	22768.62	20489.78	23743.68	20189.42	19957.61

Conclusion

The traffic speed patterns for roads with posted speed limits of 25 mph or 30 mph consistently exceed the posted limits, with the 85^th percentile speeds being twice as high. There were no differences in speed measures between weekdays and weekends. For roads with posted speed limits of 35 mph or 40 mph and higher, the mean traffic speeds are close to the limits, but the 85th percentile speeds exceed them.

The mean, 15th percentile traffic speed, 85th percentile traffic speed, and reference speed are significantly correlated regardless of the posted speed limit cluster. Higher mean traffic speeds are associated with higher 15th and 85th-percentile traffic speeds, suggesting a general trend of traffic speeds aligning and varying within a given posted speed limit cluster.

Road links with a 45 mph or 50 mph posted speed limit exhibited a distinct pattern compared to the road links that belong to the other posted speed limit clusters. The SD of traffic speeds in these road links show strong negative correlations with the mean traffic speed, 15th percentile traffic speed, 85th percentile traffic speed, and reference speed. This negative correlation implies that as the SD of traffic speeds increases, the mean traffic speed, lower percentile traffic speed, the higher percentile traffic speed, and reference speed tend to decrease. Conversely, a decrease in the SD of traffic speeds is associated with an increase in these traffic speed measures.

The negative correlations between the SD of traffic speeds and the mean traffic speed, the 15^th percentile traffic speed, the 85^th percentile traffic speed, and reference speed provide valuable insights for traffic management and road safety. Higher variations in traffic speeds in road links with a 45 mph or 50 mph posted speed limit indicate lower mean traffic speeds and a higher likelihood of speed fluctuations within this posted speed limit cluster. Furthermore, the negative correlations suggest that increasing traffic speed variations are associated with lower traffic speeds at the lower and higher percentiles and the reference speed. This information can help identify links that require VSL interventions to address the traffic speed distribution and potential risks associated with this specific posted speed limit cluster.

In conclusion, this study examined road links and their traffic speed variations. Additionally, an illustrative example was presented, highlighting the significance of selecting the appropriate traffic speed distribution for a road link with consistent speed variation. This example demonstrated the practical implications of considering speed patterns and implementing VSL signs. The study’s findings contribute to understanding of traffic speed patterns and provide valuable insights for implementing VSL signs. The road links with consistent high-speed variations can be considered as potential candidates for the implementation of VSL signs, pending additional microscopic investigation.

Disclaimer

This paper is disseminated in the interest of information exchange. The views, opinions, findings, and conclusions reflected in this paper are the responsibility of the authors only and do not represent the official policy or position of the University of North Carolina at Charlotte or other entities. The authors are responsible for the facts and the accuracy of the data presented herein. This paper does not constitute a standard, specification, or regulation.

Acknowledgments

This paper is prepared based on information collected for a research project funded by the United States Department of Transportation - Office of the Assistant Secretary for Research and Technology (USDOT/OST-R) University Transportation Centers Program (Grant # 69A3551747127). The authors sincerely thank the staff of the North Carolina Department of Transportation and the Regional Integrated Transportation Information System for their help with the traffic speed data used in this research. The National Performance Management Research Data Set used in this research is based upon work supported by the Federal Highway Administration under contract number DTFH61-17-C-00003. The authors also thank Sarvani Duvvuri for her efforts during the initial stages of the project.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the U.S. Department of Transportation.