When to Measure Accessibility? Temporal Segmentation and Aggregation in Location-Based Public Transit Accessibility

Abstract

Accessibility analyses are important for public transit (PT) planning, as they reveal possible deficits in PT services. Since such accessibility analyses are highly time-dependent, the Modifiable Temporal Unit Problem (MTUP) should be considered. In this context, one approach is to calculate accessibility continuously for an entire day and aggregate it appropriately. However, this approach is complex and computationally intensive and is therefore rarely, if ever, applied. Instead, practitioners and researchers rely on simplified methods without considering temporal effects in detail. This paper bridges this gap by developing a simple yet representative method to account for the temporal variability of PT services. For this purpose, we calculate and compare PT accessibility for different time windows and through different aggregation methods for Germany. The results show that PT analyses between 9–11 a.m. were most representative. Alternatively, the time windows between 7–9 a.m. and between 1–3 p.m. adequately reflected accessibility. The median was suitable for aggregating individual time intervals into a representative value for the PT service throughout a day, while the maximum or mean value distorted the results. For practical planning purposes, we therefore recommend using the 9–11 a.m. time window.

1. Introduction

To ensure an efficient improvement and expansion of the public transit (PT) system, it is essential to identify appropriate measures and evaluate their impact [1]. In this context, accessibility is a valuable instrument for assessing existing and planning future PT services [2]. Accessibility as a metric allows conclusions to be drawn regarding social participation [3,4] and the attractiveness of PT services as a correlate of PT usage [5,6]. With this information, planners can improve the PT system to lay the foundation for a more just and sustainable transport system.

Accessibility describes the ease with which people can reach opportunities by PT [7]. It can be viewed from four perspectives, as defined by Geurs and van Wee [8]: location-based, infrastructure-based, utility-based, and person-based. In urban planning, location-based accessibility is usually used [9], taking into account the quantity and spatial distribution of opportunities, as well as the travel time or travel costs to reach them [8].

Such measures require information about the location and characteristics of the opportunities, as well as the associated travel times. Accordingly, it becomes evident that consideration of both the spatial and temporal dimensions of the data is essential. The former is described by the Modifiable Areal Unit Problem (MAUP), which addresses two effects of spatial association [10,11]: (1) The scale effect refers to the level of aggregation and describes how analyses are influenced by data at different spatial resolutions. (2) The zoning effect considers the zonal arrangement of spatial units and its impact on spatial analyses. A multitude of studies have examined the MAUP and found an impact of a factor of two or more on accessibility measures (see Pereira et al. [11]; Hewko et al. [12]; Stępniak and Jacobs-Crisioni [13]).

The temporal dimension of accessibility measures is considered within the Modifiable Temporal Unit Problem (MTUP) [14]. The MTUP analyzes the impact of several key factors on the outcomes of accessibility measures, including the selection of an upper limit for travel time, the options for temporal aggregation, and the choice of starting time for analysis. In this context, Cheng and Adepeju [15] observed the following three effects:

(1)

The (temporal) boundary effect refers to the duration of temporal processes. Regarding location-based accessibility measures, this effect occurs when measures depend on a defined maximum travel time [16]. For example, accessibility varies for a maximum travel time of 30 min compared to 60 min. The boundary effect is the subject of numerous studies, which have found that increasing the travel time threshold results in more than a linear increase in accessibility depending on the region and the opportunities analyzed (see Pereira [16]; El-Geneidy et al. [17]; Tomasiello et al. [18]). However, the analysis of the boundary effect is beyond the scope of this study.

(2)

The aggregation effect considers the units of time for calculation (e.g., minutes) and the aggregation level used for reporting the results (e.g., hours). For example, the trips of a timetable planned to the minute can be aggregated to the number of departures per hour. With regard to the aggregation effect, research has thus far concentrated on the resolution of the intervals into which the study period is divided. Stępniak et al. [19] compared the precision and calculation effort of resolutions between 1 min and 60 min. Results were aggregated to the 1 h level using different aggregation methods, such as the arithmetic average and a harmonic-based average, depending on the method used to calculate the location-based accessibility. However, the authors did not compare these aggregation methods as part of the aggregation effect.

(3)

The segmentation effect describes the changes in results when considering different starting times [16]. For instance, calculating PT accessibility for a 2 h interval that starts at 7 a.m. shows different results than the same interval starting at 10 a.m. This is due to the temporal variation of PT service and the composition of travel time. In addition to the actual time spent traveling by PT, the overall travel time consists of accessing the first station, waiting at this station, possible transfers, and egressing from the final station. The number of trips offered, and thus the waiting and transfer times, may vary depending on the day of the week and the time of day. Given a constant time budget, longer waiting times result in shorter in-vehicle travel times and thus fewer accessible opportunities. Thus, the timing of the analysis can influence the results, even when all other conditions remain constant. Due to the timetable, the choice of starting time and time span is of particular relevance in accessibility analysis involving PT.

The present paper contributes to the existing literature on the MTUP in location-based accessibility analyses for PT services. For this purpose, we measured location-based accessibility for all of Germany and evaluated the influence of different starting times and aggregation methods. Since we evaluated accessibility for an entire country, we can differentiate our results by city size and region type. In light of our research, we are able to give recommendations on how and when location-based accessibility should be assessed. We aim to achieve a simple approach to calculating accessibility that reflects the PT supply of a complete day and is easy to use in research and practice, while accounting for temporal segmentation and aggregation. This allows for precise accessibility analyses even when time, computational resources, and data are limited.

The paper is structured as follows: Section 2 provides a brief overview of the literature on PT accessibility analyses and the associated timeframes. Section 3 describes materials and methods. The results are then presented in Section 4. Finally, Section 5 discusses the results and offers recommendations for considering temporal aspects in PT accessibility analysis.

2. Literature Review

This paper builds on previous research examining the impact of the MTUP on PT accessibility. To gain a comprehensive understanding of these issues, we have conducted a comprehensive database search on the MTUP in the context of accessibility analyses as well as location-based PT accessibility. The latter focused on studies that provide information about the temporal frame used. Based on this, we first present a description of the location-based accessibility measures employed, followed by an analysis of studies on PT accessibility and different approaches to dealing with the MTUP.

2.1. Location-Based Accessibility

Geurs and van Wee [8] defined three approaches to assessing location-based accessibility. While measuring the travel time to the closest opportunity (proximity indicators) is the most straightforward [19], cumulative opportunities and gravity-based accessibility are used more frequently [9]. Cumulative opportunities indicate the number of opportunities accessible within a defined travel time [8]. This measure is particularly beneficial in the context of complementary opportunities of a specific type, such as specialized health care with different medical specialists [19]. As long as the travel time does not exceed a defined threshold, the exact travel time to access an opportunity is less important. Another approach is offered by gravity-based measures as a development of Hansen’s accessibility model [20]. In this case, opportunities are weighted based on travel distance, travel time, or generalized costs (see Klar et al. [1]; Lunke [21]) using different functions [8]. Compared to cumulative opportunities, gravity-based measures offer a more realistic representation of the interplay between opportunities and the transport system. However, gravity-based accessibility is more difficult to interpret and communicate [8]. Therefore, cumulative opportunities are mainly used in practice [18].

Location-based accessibility involves several interrelated components, such as land use and the transport system [8]. In the context of PT accessibility, the temporal component is of particular interest. PT travel times depend on timetables and vary between different days of the week as well as across the course of a day [22], and thus relate to the segmentation effect. For example, Liao et al. [23] found significant variations in PT travel times depending on the time of day in different cities. This directly impacts accessibility measures, as reduced PT services result in longer travel times and thus lower accessibility. When selecting both the day of the week and the time of the day, the segmentation effect of the MTUP must therefore be considered and match the study’s use case.

2.2. Approaches to Dealing with the MTUP in the Context of PT Accessibility

In the literature, there are various methods that address the MTUP. Table 1 provides an exemplary overview of location-based PT accessibility measures and their approach to accounting for temporal variation. Most studies have focused on job accessibility (see Boisjoly and El-Geneidy [24]; Lunke [21]), which is one of the most common applications of accessibility studies [25]. However, other opportunities, such as supermarkets [26] and health care [22], were also included to consider other purposes.

When accounting for temporal variation, the choice of the time window is of high importance for PT accessibility measures. Thus far, even studies with similar use cases have not used consistent approaches to select an appropriate time window. For example, job accessibility may be calculated at 6 a.m., 7 a.m., or 8 a.m. and may be based on a single departure time or a time span of 1 or 2 h (see Pereira [16]; Tomasiello et al. [18]; Lee and Miller [22]; Lunke [21]). Furthermore, some studies have not mentioned the day of the analysis (see Klar et al. [1]; Boisjoly and El-Geneidy [24]) or have not considered temporal variation at all (see Chen et al. [27]; Kinigadner et al. [28]; Viguié et al. [29]). As a consequence, the results are not only difficult to compare, but the findings of the studies themselves are also affected.

Table 1. Consideration of the MTUP in PT accessibility analyses.

Reference	Accessibility Measure	Opportunities	Day	Departure Time	Time Span	Method of Aggregation
Single departure time
El-Geneidy et al. [17]	CumOpp	Jobs	Weekday	7 a.m.		One single value
Li et al. [30]	CumOpp	Senior centers	Monday, Saturday	8:30 a.m., 2 p.m.		Multiple single values
Lee and Miller [22]	CumOpp	Jobs Health care	Thursday, Sunday	8 a.m., 1 p.m., 6 p.m., 9 p.m.		Multiple single values
Time span
Fan et al. [31]	CumOpp	Jobs	Weekday	5 a.m.	16 h	Weighted average of 1 h intervals, distinction between peak and off-peak hours
Farber et al. [26]	Travel time	Supermarkets	Monday	6 a.m.	16 h	Arithmetic average and standard deviation of 1 min intervals
Klar et al. [1]	CumOpp GravBased	Canadian Places Dataset	N/A	7 a.m.	2 h	Median of 1 min intervals
Lunke [21]	GravBased	Jobs	N/A	7:30 a.m.	1 h	Median of 1 min intervals
Owen and Levinson [32]	CumOpp	Jobs	Weekday	7 a.m.	2 h	Arithmetic average, maximum, standard deviation, variance and coefficient of variation of 1 min intervals
Pereira [16]	CumOpp	Jobs	Weekday	7 a.m.	2 h	Median of 15 min intervals
Pereira et al. [11]	CumOpp	Jobs, schools	N/A	7 a.m.	12 h	Median of 20 min intervals
Stępniak et al. [19]	CumOpp, GravBased, Travel time	City council, nurseries, theatres, specialized health care, hospitals, secondary schools, population	Tuesday	2 a.m., 7 a.m., 10 a.m., 10 p.m.	1 h	Arithmetic or weighted average of different intervals (1, 5, 15, 20, 30, 60 min)
Tomasiello et al. [18]	CumOpp	Jobs	N/A	6 a.m.	2 h	Median of 1 min intervals
Combination
Boisjoly and El-Geneidy [24]	CumOpp	Jobs	N/A	5 a.m., 6 a.m., 7 a.m., 8 a.m. 9 a.m. 12 p.m.	1 h 3 h 17 h	Multiple single values Arithmetic average Value from 12 p.m.
El-Geneidy et al. [33]	GravBased	Jobs	N/A	5 a.m., 6 a.m., 7 a.m., 8 a.m. 9 a.m. 12 p.m.	1 h 3 h 17 h	Multiple single values Arithmetic average Value from 12 p.m.

Note: CumOpp = cumulative opportunities; GravBased = gravity-based; Travel time = travel time to the closest facility; N/A = information not available.

Table 1. Consideration of the MTUP in PT accessibility analyses.

Reference	Accessibility Measure	Opportunities	Day	Departure Time	Time Span	Method of Aggregation
Single departure time
El-Geneidy et al. [17]	CumOpp	Jobs	Weekday	7 a.m.		One single value
Li et al. [30]	CumOpp	Senior centers	Monday, Saturday	8:30 a.m., 2 p.m.		Multiple single values
Lee and Miller [22]	CumOpp	Jobs Health care	Thursday, Sunday	8 a.m., 1 p.m., 6 p.m., 9 p.m.		Multiple single values
Time span
Fan et al. [31]	CumOpp	Jobs	Weekday	5 a.m.	16 h	Weighted average of 1 h intervals, distinction between peak and off-peak hours
Farber et al. [26]	Travel time	Supermarkets	Monday	6 a.m.	16 h	Arithmetic average and standard deviation of 1 min intervals
Klar et al. [1]	CumOpp GravBased	Canadian Places Dataset	N/A	7 a.m.	2 h	Median of 1 min intervals
Lunke [21]	GravBased	Jobs	N/A	7:30 a.m.	1 h	Median of 1 min intervals
Owen and Levinson [32]	CumOpp	Jobs	Weekday	7 a.m.	2 h	Arithmetic average, maximum, standard deviation, variance and coefficient of variation of 1 min intervals
Pereira [16]	CumOpp	Jobs	Weekday	7 a.m.	2 h	Median of 15 min intervals
Pereira et al. [11]	CumOpp	Jobs, schools	N/A	7 a.m.	12 h	Median of 20 min intervals
Stępniak et al. [19]	CumOpp, GravBased, Travel time	City council, nurseries, theatres, specialized health care, hospitals, secondary schools, population	Tuesday	2 a.m., 7 a.m., 10 a.m., 10 p.m.	1 h	Arithmetic or weighted average of different intervals (1, 5, 15, 20, 30, 60 min)
Tomasiello et al. [18]	CumOpp	Jobs	N/A	6 a.m.	2 h	Median of 1 min intervals
Combination
Boisjoly and El-Geneidy [24]	CumOpp	Jobs	N/A	5 a.m., 6 a.m., 7 a.m., 8 a.m. 9 a.m. 12 p.m.	1 h 3 h 17 h	Multiple single values Arithmetic average Value from 12 p.m.
El-Geneidy et al. [33]	GravBased	Jobs	N/A	5 a.m., 6 a.m., 7 a.m., 8 a.m. 9 a.m. 12 p.m.	1 h 3 h 17 h	Multiple single values Arithmetic average Value from 12 p.m.

Note: CumOpp = cumulative opportunities; GravBased = gravity-based; Travel time = travel time to the closest facility; N/A = information not available.

Based on the studies reviewed, there are two approaches to calculating PT accessibility taking into account temporal constraints: (1) using one or multiple single departure times, or (2) a time span.

2.2.1. One or Multiple Single Departure Times

The most straightforward way to calculate location-based accessibility by PT is to use a single departure time. This approach is particularly useful when examining accessibility by trip purposes, which allows conclusions to be drawn about the most relevant travel times. Trips to work, for example, are usually undertaken at a similar point in time within the morning peak hour on a weekday (see El-Geneidy et al. [17]; Conway et al. [34]). However, measures using a single departure time neglect variations in accessibility over time.

A further development and first attempt to capture the effect of segmentation is the comparison of multiple single departure times. Lee and Miller [22] analyzed the impact of new PT services on job and healthcare accessibility in Columbus, Ohio, for four different departure times throughout the day. Depending on the type of opportunity and the maximum travel time, accessibility was found to be relatively stable across these departure times. The accessibility of senior centers in Philadelphia, Pennsylvania, differed between the morning peak hour at 8:30 a.m. and 2 p.m. due to temporal and spatial differences in service frequency and congestion [30]. Stępniak et al. [19] compared the accessibility of different opportunities in Szczecin, Poland, for four different 1 h time spans with different PT frequencies and find significant variations in the results.

By comparing multiple departure times, the previous studies have provided an idea of the temporal variation of accessibility at different times of the day. Nevertheless, they are limited to the selected time frames. The mentioned studies only compared single results and did not aim to make a statement about overall accessibility for the whole day.

This approach of using (multiple) single departure times is relatively easy to apply. It provides a rough overview of the accessibility of a location. In addition, the comparison of individual departure times allows for a first assessment of the variability of PT services. Due to the small number of times examined, temporal aggregation is not advisable for reasons of distortion. Thus, the approach can only be used representatively for very specific questions if the trip purpose can be linked to a few particular points in time. In these cases, the departure time can be based on the trip purpose and the opening times of the destinations. Another limitation is that the start time of the analysis is independent of the actual departure time at the station. Since PT travel times depend on fixed timetables, this can lead to unrealistic waiting times and, thus, inaccurate analysis results.

2.2.2. Time Spans

Another approach to dealing with the MTUP is the use of time spans, which aims to reduce the aforementioned impact of imprecise waiting times. Therefore, the study period is divided into shorter, equally long time intervals, ranging from 1 min (see Klar et al. [1]) to 60 min (see Fan et al. [31]). Stępniak et al. [19] recommended 15 min intervals to achieve relatively precise results while keeping computational time acceptable. Accessibility is then calculated for each interval and aggregated for the whole study period. For example, Pereira [16] calculated job accessibility for 15 min intervals between 7 a.m. and 9 a.m. and used the median value of these time intervals to express the overall accessibility. Owen and Levinson [32] used different aggregation methods, such as arithmetic average and maximum, to aggregate 1 min intervals to the 2 h level. As described by the aggregation effect of the MTUP, different ways of temporal aggregation can lead to different analysis results. Therefore, the aggregation method as well as the duration of time spans should be selected according to the study’s use case.

The literature presented in Table 1 provides multiple examples of how time spans can be used to deal with the temporal variation of accessibility over the course of a day. One option is the analysis of longer time spans to consider differences in supply. Hence, Pereira et al. [11] calculated job and school accessibility between 7 a.m. and 7 p.m. for Rio de Janeiro, Brazil, based on the median of 36 intervals of 20 min. However, there was no further explanation or investigation of that specific choice of time frame and its impact on the results. This is especially relevant because the time span under consideration exceeded the morning peak hour, which is most commonly used when it comes to job accessibility. Travel to school is also usually concentrated on a few times of the day. However, this procedure includes several time frames with little significance for the use case, which may distort the result. Due to the aggregation method, the impact of different departure times throughout the day is not recognizable.

Using a different approach, Farber et al. [26] analyzed the average and standard deviation of travel times to the closest supermarket in Cincinnati, Ohio, across a 16 h time span. In contrast to Pereira et al. [11], in the case of supermarket accessibility, it seems more appropriate to consider a longer time period as it corresponds to the opportunities’ opening hours. Ideal accessibility is indicated by a low average and a low standard deviation, referring to short travel times with little variability. This method considers the segmentation and aggregation effect of the MTUP but is relatively complicated as it requires the calculation and interpretation of two combined values.

Fan et al. [31] calculated the demand-weighted average accessibility of jobs in Minneapolis and Saint Paul, Minnesota, for a 16 h time span. They determined accessibility for all peak hours and non-peak hours and then multiplied it by the share of trips made within the corresponding time intervals. This approach resulted in a single value representing accessibility over the course of a day while accounting for the MTUP. But in this study, the weighting was based on single departure times for every hour. This coarse resolution may have led to a lack of precision. However, in combination with a finer resolution, the demand-weighted average can be considered a good approximation of accessibility throughout the day.

There are also combined approaches that use both single departure times and time spans. El-Geneidy et al. [33] calculated relative and competitive accessibility to different job categories across various time spans throughout a day for the Greater Toronto and Hamilton Area, Canada. While these results cannot directly be transferred to general location-based measures, the underlying travel times for different times of the day showed significant differences, indicating variations in accessibility. The study by Boisjoly and El-Geneidy [24] used the same spatial and temporal scale. Even though it was not possible to fully capture the fluctuations throughout the day due to different resolutions of the single time spans, the analysis considered variations between peak and off-peak hours as well as during the morning hours. A high correlation between the accessibility of different time spans was found in this study.

2.3. Problem Statement and Scope of the Paper

In summary, the following findings emerge from the literature reviewed: (1) The studies examined have considered the effect of segmentation only to the extent that accessibility is calculated for different departure times throughout the day. However, the single accessibility values were not combined and did not allow conclusions to be drawn about a representative value for the whole day. (2) The effect of aggregating several times to a single value was mainly determined by the median, mean, maximum, or standard deviation in the literature. In particular, the demand-weighted mean seems to be a viable option to illustrate accessibility by means of a representative value. However, this method is computationally intensive and time-consuming. Considering both effects requires numerous accessibility calculations and subsequent comparisons. Due to this additional effort, the MTUP is often not sufficiently addressed. To our knowledge, analyses that are easy to carry out without neglecting temporal segmentation and aggregation have not been investigated yet.

Against this background, the aim of this paper is to develop an easy-to-use but representative measure for the calculation of PT accessibility. We descriptively evaluate the use of different time spans and aggregation methods, considering the differences in PT services between urban and rural areas. Our analysis allows us to draw profound conclusions on how best to simplify the temporal variance of accessibility in order to obtain an accessibility measure that represents a full day of PT accessibility. Thus, we answer the question of when to measure accessibility.

3. Materials and Methods

To examine the effects of segmentation and aggregation of the MTUP in the context of PT accessibility, the following steps were applied (see Figure 1). The first step was to prepare the data, which included calculating PT accessibility. Then, we used descriptive analysis to evaluate the demand for motorized trips and PT accessibility. Finally, we combined these two steps to analyze the effects of the MTUP.

This chapter includes a thorough description of the data and its preparation. First, the calculation of PT accessibility using the cumulative opportunities approach is discussed. Then, the different spatial types are presented in order to carry out regionally differentiated evaluations. Finally, the methodology is described in detail.

3.1. Calculation of Location-Based Accessibility with PT

To evaluate the impact of temporal differences in PT service on the results of location-based accessibility measures, we used the approach of cumulative opportunity accessibility. Initially, we outlined the required input data, its preparation, and the assumptions made. Next, we described the specific calculation of PT accessibility.

The calculation of the cumulative opportunity accessibility was based on a maximum travel time of 30 min. We did not compare different thresholds and therefore did not evaluate the boundary problem, as there is already extensive literature on that topic. All populated 100 × 100 m² grid cells [35,36] serve as both starting points for the inhabitants and destination points for the opportunities. In order to reduce aggregation error, we used the centroid according to residential and settlement area [37] instead of the geometric centroid [13]. Since the quality of open-source data on points of interest across Germany is insufficient [43,44] and nationwide workplace data are not available at a detailed spatial level, we used population data from the 2011 census to describe existing opportunities.

The nationwide timetable data stem from DELFI [38], which compiles and publishes timetable data of German PT operators, including all available PT modes (mainly bus, rail, tram, and metro). Our analysis includes approximately 238,000 stops and 836,000 trips over the entire day. Since the data come directly from the operators, a high level of quality and accuracy can be expected. Unfortunately, not every operator has provided timetable data for the selected date. Hence, to reduce bias, we performed cross-validation by comparing the covered PT stops with a nationwide dataset of PT stops [38,39] and stops from OpenStreetMap [40]. Since none of the previous datasets are complete, we only evaluated regions where there was a high degree of correspondence between the datasets. Finally, we were able to calculate the cumulative opportunity accessibility for 367 out of 400 counties (75.7 out of 80.3 million inhabitants in census data [35]) with sufficient timetable data.

The overall PT travel time consisted of accessing the first station, waiting for the service, traveling by PT, transferring between services (if necessary), and leaving the last station. We considered walking to the first station, from the last station, and between stations when transferring at a speed of 70 m/min based on German guidelines [45]. Since 99% of PT trips in Germany under 100 km consist of a maximum of two transfers [41], we set the allowed number of transfers to two. Furthermore, we considered a maximum walking time to and from the station of 19 min and a minimum of 3 min for transfers between services. Thereby, transfers were possible between all trips and modes operating within 7 min of walking time. We supposed that PT users had extended knowledge of the scheduled departures and added a waiting time at the first station of 1 min to the calculated travel time. We performed the calculation for all populated 100 × 100 m² grid cells in Germany [35,36] for each minute throughout Thursday, 22 September 2022, and selected the maximum accessibility of each 1 h interval. As for the technical software, we used the R-package tidytransit (version 1.4) [46] to compute the optimal travel times between stations and the routing plus API [47] to compute walking distances.

3.2. Distinction of Different Regions

Since cumulative opportunity accessibility does not only depend on the time span analyzed but also on the settlement structure and the means of PT available, we differentiated region types in our analysis. To minimize the interference of the settlement structure and the means of transit available, we formed four homogeneous region types based on the Germany-specific spatial typology ‘RegioStaR’ [42].

The first group consists of all German metropolises (RegioStaR-category 71). These are the 16 biggest cities in Germany, each with more than 500,000 inhabitants. They are usually characterized by a dense street and PT network, while there are multiple means of PT (regional train, metro, tram, bus) available. These metropolises show a high PT mode share [41]. Overall, about 18% of the German population resides in these cities [35].

All other large cities in Germany with over or around 100,000 inhabitants make up the second category (RegioStaR-category 72). While these cities often feature less diverse facilities than metropolises, they are still a fixture for regional transportation. They are either part of a larger metropolitan area (e.g., in the Ruhr area) or the center of a region. While there is typically less PT infrastructure available, there are still regional trains, trams, and buses in operation. In Germany, about 14% of the population resides in these 66 cities [35].

Medium-sized cities with a population of more than approximately 25,000 make up the third category (RegioStaR categories 73 and 75). These cities are either in the vicinity of large cities or metropolises or resemble the central city of a rural area. Most of these cities are connected to their surroundings by regional trains, but otherwise, they depend on bus services. All in all, 31% of the German population lives in one of the approx. 1500 medium-sized cities [35].

The last group contains small towns with less than 25,000 inhabitants (RegioStaR categories 74, 76, and 77). These small towns can either be in urban or rural areas and predominantly rely on regional bus services to connect to trains or bigger cities. Typically, these bus services run infrequently, and the mode share of PT is low [41]. This group of 9400 towns accounts for 36% of the German population [35].

3.3. Methodology

To evaluate the impact of segmentation and aggregation effects on cumulative opportunity accessibility, we first assessed the temporal variation of demand to set the basis for the supply evaluation. Therefore, we analyzed the temporal distribution of departures in motorized traffic in the German national household travel survey (MiD 2017, [41]). With the aggregation of car and PT travel, we could estimate the desired departure times for motorized trips. We assumed that when there was no PT connection to the desired destination, people used their car or travelled by taxi. Hence, in an optimal case, PT accessibility should be high when demand is high.

To analyze this relationship, we then compared the temporal variation of PT accessibility with the demand for motorized travel. This comparison not only showed the level of fit between PT service and demand, but it also allowed us to evaluate the differences in PT accessibility across different time spans. With that, we were able to obtain a first insight into the influence of temporal segmentation on the results. To further enhance the understanding of the differences, we mapped PT accessibility for exemplary regions.

Next, the paper assesses different time spans and aggregation methods. As a reference for the evaluation, we used a weighted average of the PT accessibility throughout the day. We weighted the PT accessibility of each hour with the demand for motorized trips calculated from the household travel survey. With that, we hypothesized that PT accessibility is more valuable when demand for motorized trips is higher and less valuable when demand is lower. This approach corresponds to the methodology used by Fan et al. [31].

Based on the demand-weighted accessibility, we examined the segmentation effect for different time intervals. Although we have calculated and analyzed accessibility for each hour, in this paper, we focused on the four intervals of 7–9 a.m. (“morning peak”), 9–11 a.m. (“morning off-peak”), 1–3 p.m. (“end of school”), and 7–9 p.m. (“evening”). We believe that these four intervals are diverse enough to reflect the differences in supply and demand without adding complexity. With regard to the aggregation effect, we compared the demand-weighted accessibility with the average, median, and maximum accessibility throughout the day, as well as with the accessibility of a single time span.

For the evaluation of the different time spans and aggregation methods, we applied an indicator proposed by Friedrich et al. [48] to evaluate the quality of transport demand models. They proposed the scalable quality value (SQV) to analyze the quality of a transport demand model by estimating the difference between modeled and measured values based on the following formula:

$$SQV={\displaystyle \frac{1}{1+\sqrt{\frac{{\left(m-c\right)}^2}{f·c}}}}$$

(1)

with

• c observed/measured value
• m modeled value
• f scaling factor

The SQV ranges from 0 (poor fit) to 1 (good fit). However, it does not show whether the model values overestimate or underestimate the measured values. In our paper, the demand-weighted average represents the observed/measured values c as the best measure of accessibility. The aggregation by average, median, or maximum of the day, and the value of a single time span yielded the modeled values m. The scaling factor f took a value of 10⁴ due to the magnitude of the PT accessibility. This allowed us to examine how well the different aggregations represent demand-weighted accessibility.

4. Results

The results section describes the MTUP effects of segmentation and aggregation on PT accessibility. The structure follows the methodology outlined in the previous section (see also Figure 1): First, we measured the distribution of motorized demand and then analyze the accessibility for different time intervals and regions. Based on this, we assessed the influences of segmentation and aggregation via the SQV.

4.1. Demand for Motorized Trips

PT is most effective when supply meets demand. This means that demand is a key factor in evaluating the accessibility of PT services. Assessing when people use motorized transport therefore also makes it possible to assess when high PT accessibility is needed. Figure 2 shows the motorized trips made between Monday and Friday by hour and by region type. Overall, there were only minor differences between the region types. As a result, we only discuss the general pattern of the graph.

At night and in the early morning hours, there was little demand. Starting at around 4 a.m., demand rose to the morning peak at around 7 a.m. This consisted mainly of journeys to school and work and accounts for around 9% of all journeys. In the morning between 8 a.m. and midday, demand flattened off to around 6% of the daily motorized trips each hour. This was followed by the afternoon peak at 4 p.m., which was similar in size to the first peak of the day. These were mainly trips home, for leisure, or shopping. Finally, demand decreased towards the night.

4.2. PT Accessibility

The analysis of demand raises the question of whether the supply of PT matches the demand. To answer this question, we examined the differences in accessibility for different time periods in the respective region types. We calculated the 25%, 50%, and 75% quantiles of all cells in each region type and indexed them to their median accessibility between 7 a.m. and 8 a.m.

Figure 3 shows the resulting index graph of PT accessibility by region type. In general, the 25% and 75% quantiles followed the same trend as the median. Accordingly, the temporal deviation of cells with high and low accessibility was similar.

In all region types, there was a morning peak at around 7 a.m. and a subsequent decline towards 11 a.m. During this period, PT accessibility followed the distribution of the demand for motorized trips. In contrast, the afternoon hours showed two peaks of PT accessibility: The first one was at 1 p.m., when school usually ends in Germany, which was not apparent in the demand distribution. The other peak corresponded to the afternoon peak of the demand.

In addition to the general trends in all regions, each individual region type had specific characteristics. Metropolitan areas, for example, showed a wide interquartile distance, indicating that accessibility by PT varied greatly between the cells. In addition, the index fluctuated only slightly over the course of the day, which suggests constant PT service, even in times of lower demand for motorized trips. In contrast to the other region types, the morning peak was earlier, and the afternoon peak was as late as 8 p.m., e.g., there was good PT service even in off-peak hours.

The PT index for large cities was similar to that of the metropolises, except that the bandwidth between the quantiles was smaller. This indicates greater similarity between the cells of large cities. In addition, there was no evening peak. Instead, the service remained constant throughout the afternoon and decreased from 7 p.m.

In contrast to the two aforementioned region types, medium cities showed more pronounced peaks, especially in school transport at 1 p.m. This indicates that the service targets individual user groups, such as pupils. PT accessibility declined earlier in the evening, so that, for example, around one-quarter of the cells had almost no PT service from 8 p.m. onward.

Similar to metropolises, small towns exhibited a high variance in PT accessibility. The size of a city therefore does not appear to be related to the variability of PT services. As the very pronounced peaks in the morning and at midday show, there was an even stronger orientation towards individual user groups. Between these peak times, around 25% of cells had no PT service at all. In addition, the service decreased rapidly after the afternoon peak at 4 p.m.

In addition to the general distribution of accessibility over the course of the day, we analyzed local variations. To this end, we exemplarily examined one area for each region type (see maps in Figure 3): Munich (metropolis); the Ruhr area, including Bochum, Gelsenkirchen, and Herne (large cities); Esslingen in the Stuttgart region (medium cities); and Rothenburg in Franconia (small towns).

In addition to the spatial differentiation, we selected four exemplary time windows based on the distribution of demand over the course of the day, as described in Section 4.1. This allowed us to consider and compare time windows with different conditions, e.g., PT supply and trip purposes. Moreover, 2 h intervals were used to compensate for small deviations in PT service:

• The morning peak time (7–9 a.m.) is often used in the literature to measure the accessibility of workplaces. It is characterized by a high number of trips to school and work.
• The morning off-peak time (9–11 a.m.) has a considerably reduced PT service and correspondingly lower accessibility than the peak hour.
• The end of school/afternoon peak time (1–3 p.m.) has a similarly high level of PT service as the morning peak time and serves, among other purposes, the trips from school to home.
• In the evening (7–9 p.m.), PT services decrease significantly, especially in small towns.

Each map shows the average accessibility of the respective 2 h interval for the populated 100 × 100 m² grid cells compared to the reference accessibility (median accessibility of the grid cells for each region type between 7 a.m. and 8 a.m.). Green areas have higher than median accessibility, gray areas match the median, and red areas are below. In general, accessibility was high in city centers and along major PT axes, while suburbs and remote settlement areas showed lower accessibility compared to the reference. Over the course of the day, metropolises and large cities showed little variation in accessibility, indicating a continuous PT service. In medium cities, the city center also had high accessibility, but in the peripheral areas (e.g., northeast of Esslingen city center), there was a change from high accessibility in the morning to lower accessibility in the remaining time periods. This effect was more pronounced in small towns such as Rothenburg, where accessibility in the center was high, mainly during the morning and midday school hours. Outside these times, and especially in peripheral locations, PT accessibility, e.g., often even the availability of a PT service, seemed to be a game of chance.

4.3. Effects of Segmentation and Aggregation

Based on the calculation of motorized demand and PT accessibility in the previous sections, we can now study the influence of segmentation and aggregation on the results. Therefore, we use the demand-weighted accessibility, calculated as the average of hourly PT accessibility weighted by motorized demand. Matching PT service and demand should be a good representation of PT quality. However, this approach has two disadvantages: First, accessibility has to be calculated for every hour, which is very time- and resource-consuming. Second, information on demand is necessary to weight different time windows. As a result, the question arises whether indicators that are easier to measure can serve as a proxy for the demand-weighted average PT accessibility.

Based on this question, we investigated the following two topics: (1) Which time period is particularly representative in the course of the day (influence of segmentation), and (2) How do the results vary for different aggregation methods if hourly accessibility is available, but demand data are missing (influence of aggregation)? For our analyses, we evaluated the similarity to the demand-weighted average accessibility using the SQV as described in Section 3.3.

4.3.1. Segmentation

In terms of segmentation, we identified which of the previously described time spans is most representative of the demand-weighted PT accessibility in Germany. Figure 4 shows the cumulative distribution of the SQV for the four previously selected time spans, differentiated by region type. The SQV is plotted on the x-axis, with lower values indicating low resemblance and values close to 1 indicating high resemblance. The cumulative proportion of inhabited 100 × 100 m² grid cells is plotted on the y-axis. A later rise in the graph indicates that the selected time window better matches the demand-weighted average PT accessibility. In that case, many cells have a high SQV. Conversely, an early rise occurs when a larger proportion of cells have low SQV values, reflecting a lower match to the reference.

In terms of segmentation, there was only a slight difference between the region types, as was the case when examining demand. The different time periods were also relatively similar. The morning peak interval (7–9 a.m.) reflects the situation best. The periods from 9–11 a.m. and from 1–3 p.m. were almost identical and mostly close to the morning peak. Only the evening period from 7–9 p.m. is an exception and not suitable for evaluating the daily PT accessibility.

4.3.2. Aggregation

The following section illustrates the ability of different aggregation methods to represent the demand-weighted average of PT accessibility. Similar to the different time spans in the segmentation section, the region-specific SQV was calculated for four different methods: accessibility of the single time interval from 9–11 a.m. (from the segmentation section), as well as median, average, and maximum accessibility over the course of the day. As shown in Table 1, these are commonly used in the literature and do not require any data apart from the previously calculated accessibility for each hour.

Figure 5 shows the comparison of the different aggregation methods, which, as can be seen, have a large influence on the result. Overall, the maximum (best accessibility during the day) and the average (average accessibility during the day) do not reflect the demand-weighted PT accessibility well. In metropolises, large cities, and medium cities, the cumulated SQV graphs of these aggregation methods rose early, indicating a low resemblance to the demand-weighted average. The maximum performed similarly poorly in small towns, while the average was in the range of the other two aggregation methods.

For the most part, the single time interval from 9–11 a.m. and the median (“middle” accessibility during the day) were interchangeable in their ability to resemble the demand-weighted average. With a high number of cells having high SQV values (late rise of the graph), these aggregation methods are well suited to simplify the accessibility analysis. Especially in metropolises, most cells showed SQV values over 0.8, with the median performing slightly better. While the difference between the single time span and the median was hard to identify in large and medium cities, the single time span performed better in small towns. However, for small towns, the approximation was somewhat worse, with about 70% of the grid cells having an SQV above 0.8. This was partly due to the large variance in PT services (see Figure 3).

5. Discussion and Conclusions

Location-based accessibility provides information about social participation [3,4] and the attractiveness of PT services [5,6]. Thus, it helps to evaluate the quality of PT and improve its services efficiently. However, it is important to consider the dependency of PT on timetables to assess the impacts of these measures correctly. Studies have used different resolutions and aggregation methods to account for the temporal variability of PT services and the associated MTUP. They often select a single point in time (see El-Geneidy et al. [17]) or a short time span of 1 or 2 h (see Lunke [21]). In the latter case, aggregation methods such as the median are used to obtain the accessibility for the whole study period (see Tomasiello et al. [18]). Occasionally, time spans are longer (see Pereira et al. [11]), or the studies include multiple periods distributed throughout the day (see Stępniak et al. [19]). However, studies rarely address the timing of their analyses in detail (see Chen et al. [27]), which raises the question of whether the chosen time window is representative of PT services and to what extent the MTUP affects the results. Currently, the literature lacks a detailed discussion of the impact of the chosen time period (segmentation) and aggregation method (aggregation) on the representativeness of the PT accessibility analysis.

Calculating PT accessibility across Germany for each inhabited 100 × 100 m² grid cell, differentiated by region type, allowed us to investigate the impact of segmentation as well as aggregation on accessibility analyses. On this basis, we are able to make recommendations for the consideration of temporal variation. First, we have analyzed the temporal distribution of supply and demand. Second, we have compared different segmentation and aggregation methods with the demand-weighted average throughout the day, which is considered to be the most detailed and representative approach.

Our comparison of demand and supply shows that demand is similar across region types, while supply is not. In metropolises, PT accessibility is fairly constant during daytime, but the smaller the city is, the more accessibility fluctuates. Particularly in small towns, there are pronounced peaks in school transport services in the morning (7–8 a.m.) and at midday (1–2 p.m.), while services are substantially lower during the rest of the day.

Using the demand data and the accessibility analysis for the entire day to calculate a demand-weighted average would be the most representative way to measure PT accessibility. However, the required demand data are often unavailable, and the approach is computationally extensive since it requires the calculation of accessibility throughout the whole day. Our analyses show that carefully selected simplifications of the calculations can provide sufficient results.

We have examined segmentation effects in accessibility analyses by comparing the results of different time windows with the demand-weighted average. Since PT services in bigger cities are more constant throughout the day, these region types show a higher resemblance to the average for the chosen time spans than smaller cities. The 7–9 a.m., 9–11 a.m., and 1–3 p.m. time intervals provide sufficient results, with 9–11 a.m. tending to perform slightly better. However, studies during peak hours (e.g., 7–9 a.m.) are not representative because supply decreases more than demand. Considering that the PT service in the morning off-peak (9–11 a.m.) is free of short supply increases that only serve selected user groups (e.g., school buses), this time interval seems to be particularly suitable for analyses of general PT accessibility.

Finally, we have compared the single time interval from 9–11 a.m. with several simplified aggregation methods. Using the overall maximum or unweighted average over the day showed little resemblance to the weighted average. Therefore, they are not suitable for PT accessibility analyses, as large deviations may occur. In contrast, the median and the single time interval performed similarly well. The median was closer to the demand-weighted average in metropolises, while the single time interval showed higher resemblance in small towns. Regarding the computational effort, the single time interval outperformed the median, since the former did not require the calculation of accessibility throughout the entire day.

Our study covers the whole country of Germany over one entire day and therefore considers the different spatial and temporal structures of the PT service. Given this broad scope, we expect our analysis to be applicable to different spatial and demographic contexts, and to allow the transfer of our results for temporal segmentation and aggregation to different PT networks. However, there are a few aspects to consider: First, due to the lack of sufficient points-of-interest or workplace data, we measured accessibility as the number of accessible residents. As a result, we overestimated the accessibility of high-density residential areas and underestimated areas that are primarily used for business or leisure. Second, the SQV metric is insensitive to differences in over- or undercounting. A difference of +10 and −10 leads to the same SQV. This may have influenced the interpretability or transferability of our analysis, but it is irrelevant to the task at hand. Third, the usage of the demand-weighted average as a reference deserves further attention. Demand affects supply, and supply affects demand. A change in supply may therefore not only influence the timetable-based accessibility but also the weighting factor. This can lead to unforeseen interdependencies in accessibility analysis. Nevertheless, the demand-weighted average seems to be the most representative measure of accessibility. Fourth, to illustrate our analysis, we have limited our selection to four time spans. These covered different characteristics of motorized demand (see Figure 2) and PT supply (see Figure 3). In our opinion, this provides a good compromise between complexity and comprehensibility. However, a different choice of time window could influence the results. Nevertheless, the 9–11 a.m. time window is already a good approximation of demand-weighted accessibility. Fifth, this paper sheds light on the impacts of temporal segmentation and aggregation in evaluating daily PT accessibility. Our aim was to analyze the measurement of general PT accessibility and make appropriate recommendations for its implementation. As such, our results are not transferable to studies considering individual trip purposes or user groups (e.g., accessibility of schools), as these applications have a fundamentally different demand structure.

Further research could address the combination of temporal segmentation and aggregation with the boundary effect, evaluate other location-based methods, or compare the impact of the MTUP on different modes of transport. Car traffic, for example, shows anti-cyclical tendencies in the temporal distribution of travel times. At night, when there is almost no PT service and streets are empty, car travel times are lowest, and therefore car accessibility is high. In the morning rush hour, when demand and PT accessibility are highest, car travel times increase due to congestion, resulting in reduced car accessibility. This cycle continues throughout the day, raising the question of whether the segmentation and aggregation effects on car accessibility are similar to those of PT accessibility. By combining car and PT accessibility, further research could also analyze potential distributional effects of enhanced PT services.

In conclusion, while most studies do not assess their choice of time frame or aggregation method, this study shows that an inappropriate temporal framework can bias the results of PT accessibility analysis. While the complex calculation of a demand-weighted average may be the most representative of general PT accessibility in theory, the calculation of PT accessibility for the time period between 9–11 a.m. should be sufficient in practice. This finding allows researchers and transport planners to conduct representative studies with simple means.