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Path-Based Accessibility

Path-Based Accessibility

Svante Berglund*
Royal Institute of Technology


This paper explores the development of an accessibility measure based on daily travel patterns. In contrast to traditional zone-based measures, distance is calculated using a predefined travel matrix. The travel pattern for each zone is used as a weight in the accessibility measure. This path-based accessibility measure is implemented in a computer program that is closely coupled to a transport-oriented geographic information system. The measure is demonstrated in an application for two Swedish counties. The properties of the measure are evaluated and compared with standard accessibility measures used in the planning process. This paper shows that there are differences between traditional measures and the suggested path-based measure and differences in accessibility between socioeconomic groups with different travel patterns. It is concluded that path-based accessibility measures could be very useful to analyze accessibility for high-mobility groups.


Accessibility implies the ability to physically travel to a resource at a fixed location. The introduction of new technologies, such as electronic commerce, has complicated the definition of presence, but in this article we are concerned with physical presence as a result of travel to a supply source. Because accessibility is a crucial positive outcome of the transportation system,1 how it is measured is important.

Accessibility measures (AMs) can be categorized in many different ways, but in the recent literature there is a tendency to discriminate between zone-based and individual AMs (see, e.g., Hanson 1995, Kwan 1998, and Miller 1999). As the labels indicate, zone measures try to capture the overall accessibility for a zone, while individual measures try to capture the accessibility of individuals based on detailed characteristics of space, available time, and means to overcome space. One of the main advantages of individual measures is that they can take into account the fact that most individuals face a mandatory daily travel pattern, such as to and from work. Zone-based measures neglect the importance of mandatory travel patterns on accessibility.

In the simplest form, zone-based measures result in one figure of accessibility for each zone, which may become a target of criticism. In practice, however, different accessibility scores are calculated based on gender, socioeconomic status, etc., but these scores are still averages across a number of individuals. Disaggregating population data is one way of obtaining more realistic accessibility figures using zone-based measures.

Individual measures, on the other hand, may lead to as many values of accessibility as there are individuals in the study area. Individual measures are conceptually attractive, but face difficulties from an operational standpoint (Hanson 1995). One of the most notable difficulties with obtaining individual measures is collecting data because revealed preference data cannot be used. Information on time constraints and mandatory activities cannot be obtained from a single travel survey question, but result from a series of questions. Although conceptually the two measures are very different, their mathematical formulation can be identical (Hanson 1995). The conflict is between conceptual elegance and implementation. One way of increasing the realism of aggregated zonal measures is to use detailed population data. Another option is to add mandatory travel pattern information on a zonal level, thus maintaining the operational advantages of zonal measures while bringing in components from individual AMs. This latter approach will be developed in subsequent sections.

The paper is organized as follows. In the next section we take a look at different approaches to measuring accessibility. In the third section, an alternative AM is defined where a mandatory travel pattern is taken into account. In the forth section, data for an empirical example are presented and implementation of the AM in a GIS software is described. Then an analysis of the properties of the suggested AM and comparisons with more established AMs are presented. Finally, the last section provides concluding remarks.

Accessibility Measures

Regardless of the type of AM, two components are always present-representation of travel cost (in a wide sense) and representation of opportunities at the destination. Travel cost could be represented as a simple 0/1 variable or defined in detail using a parameterized function. Similarly, description of the opportunities can range from a simple description of the resource location to detailed address-coded registers of a multitude of opportunities. Population or number of work places are frequently used as measures of opportunitites.

Individual space-time accessibility measures (STAMs) (Miller and Wu 2000) have gained increasing popularity recently (see, e.g., Kwan 1998 and Miller 1999). This is partly due to GIS developments that include programming facilities and techniques for visualizing individual behavior. Examples of implementation of individual AMs in GIS can be found in Miller (1999), Miller and Wu (2000), and Kwan (1998). Despite the fact that most implementation of individual AMs are recent, the theories behind those AMs are mature and originate from Hägerstrand's space-time framework2 (Hägerstrand 1970; see also Lenntorp 1976). In the space-time framework, the mobility of the individual is constrained by transportation resources available, which affect access to opportunities and encourage combining activities with other people.

Mandatory travel patterns, such as going to and from work and picking up children, play an important role in space-time theory. The implications for accessibility of mandatory travel patterns are twofold. On the one hand, a mandatory travel pattern restricts mobility and prevents the individual from reaching certain opportunities, on the other hand, a mandatory travel pattern brings the individual to places that may provide opportunities and reduce the need for special purpose trips.

Possibilities for overcoming distance and other obstacles to mobility differ among individuals depending on where they live and work as well as on their mobility resources. All these restrictions define an area, called the potential path area (PPA), that a specific individual can cover given the set of constraints. Despite its conceptual simplicity, the functional form of the travel impedance for individual space-time measures may be a complex sequence of conditions, depending on how many restrictions in space and time are taken into account. The PPA simply defines a subset of the total study area that should be taken into account when measuring accessibility for an individual. This is in contrast to standard measures where even distant opportunities can contribute to accessibility, although to a limited extent.

A next step is to determine the utility of opportunities that can be reached. Here, a weighting scheme is necessary. A similar accounting of distance to opportunities can be applied in both individual and aggregate zonal AMs. The simplest alternative is to put equal weight on all opportunities within a cutoff value of distance in aggregate AMs and let the PPA define the cutoff value for individual measures of distance (cumulative AMs). Another alternative is to use a gravity-based weight function. Accessibility measures based on gravity principles adopt a weighting scheme according to some aggregate travel behavior. Formally, gravity-based measures can be written as follows:

lowercase a subscript {lowercase i} equals summation over lowercase j (lowercase x subscript {lowercase j} function (lowercase t subscript {lowercase i j})

where ai is the accessibility of zone i with regard to the supply of x across all zones j, and tij is the distance or some other measure of the travel impedance between i and j. The shorter the distance the better. Common alternatives for f(tij) is the exponential function and the power function. Cumulative opportunity measures can be written in the same form as gravity measures by using

function (lowercase t subscript {lowercase i j}) equals 1 for lowercase t subscript {lowercase i j}) is less than uppercase t; ) 0 otherwise

where T is the cutoff value. Cumulative opportunity measures are simpler to use compared with gravity measures, because they do not require estimation of parameters.

A third alternative is to use an AM based on random utility theory. The most widely used model of this type is the logit model from which the logsum is derived:

lowercase a subscript {lowercase i} equals logarithm [summation over lowercase j ((lowercase x subscript {lowercase j}) times exponential (negative lowercase beta times lowercase t subscript {lowercase i j}))

In equation (3), the utility is simply a function of distance as in previous measures and of the opportunities of zone j. Logit models can handle time constraints in the choice set and constrained models have been successfully used by Thill and Horowitz (1997). An application to accessibility where the logsum is used in a time-space framework can be found in Miller (1999). In an article by Richardson and Young (1982), the properties of the logsum as an accessibility measure are explored for linked trips.

The formulation of the functional form of the distance function has no doubt attracted the most interest in the literature. In some respects, perceptions of opportunities at the destinations are critical. At one extreme you may find opportunities characterized by "the more the better" and at the other extreme "one is enough." In the first alternative an additive indicator is appropriate, and in the second case a maxitive indicator is required (see Weibull (1980) for a discussion on additive and maxitive indicators).

There are several problems with zone-based AMs. We must remember that accessibility analysis does not differ from any other zone-based analysis of spatial data. The resulting accessibility will depend on how and to what scale we have aggregated our data and zones (i.e., the modifiable areal unit problem). By using zones we cannot explicitly take individual time constraints into account. Zone-based measures also fail to analyze interactions between individuals, which is one of the strong arguments for individual measures.

Aggregate Path-Based Accessibility

In order to take advantage of the information present in a predefined travel (to work) matrix, an AM will be developed wherein the accessibility of each zone is weighted by a travel matrix. This is illustrated in figure 1 (left) where the housing area is denoted h, alternative destinations (e.g., for shopping trips) are denoted s, and the travel distance is equal across all alternatives. The AM used in association with figure 1 (left) will be a standard aggregate AM as in equation (1)

If the alternatives (which we assume) are equal, all alternatives can be chosen with equal probability. If we add information about a mandatory trip (e.g., a trip to and from work), we have a new activity pattern to consider, figure 1 (right). In this setting the available shops will not be indifferent to the traveler in the example. With path-based measures, it is possible to calculate the extra travel time the activity requires given the two initial activities at i and j. This is an important aspect not taken into account by other types of accessibility measures. The extra travel time caused by going to k (s3) is given as

lowercase t subscript {lowercase k given lowercase i j} equals ((lowercase t subscript {lowercase i k}) plus (lowercase t subscript {lowercase k j}) minus (lowercase t subscript {lowercase i j})

where i is allowed to be equal to j, which means a trip was not made or that job and home are in the same zone. In this case there will be no difference between a traditional zone-based AM and a (non-) path-based AM. A modified distance measure like this can be found in Richardson and Young (1982). If ij and k is along the road from i to j, the extra time equals the time consumed by activity l, and will be denoted by tl. The total extra time consumed by activity l at an arbitrary k will equal (lowercase t subscript {lowercase k given lowercase i j}) plus (lowercase t superscript {lowercase l} subscript {lowercase k}). If it is impossible (or difficult) to obtain some reasonable estimate of lowercase t superscript {lowercase l} subscript {lowercase k}, we could use some other stop penalty. The probability of making an additional trip or stop is not modeled in the application below. Changes in that respect, however, will not alter the fundamental properties of the path-based AM. Just changing the distance measure by taking one possible trip pattern into account will not add much realism to our AM. To gain something more, we must weight the AM by incorporating information on probabilities for mandatory destinations using a trip pattern that is not evenly spread across all destinations. The next component is, thus, a travel matrix Fij, where i represents the residential zone and j represents the work zone. This matrix could be observed from a travel survey or estimated by some model.

lowercase omega subscript {lowercase i j} equals ((uppercase f subscript {lowercase i j}) divided by (uppercase f subscript {lowercase i}))

where uppercase f subscript {lowercase i} equals summation over lowercase j (uppercase f subscript {lowercase i j}), (summation over lowercase j (lowercase omega subscript {lowercase i j}) equals 1), we can then write a path-based AM weighted by the trip pattern lowercase omega subscript {lowercase i j} :

lowercase a subscript {lowercase i} equals [summation over lowercase j (lowercase omega subscript {lowercase i j})] times [summation over lowercase k (lowercase x subscript {lowercase k}) times function (lowercase t subscript {lowercase k given lowercase i j})]

We noted above that lowercase omega subscript {lowercase i j} is a predefined travel matrix that could be obtained from a survey (as available in Sweden) or be the results of an earlier estimation. But, if the matrix is estimated, it may originate from a process like Fij = Fi × Pj|i. If we substitute the right side of (5) into (6) and use the assumed model for Fij we will obtain:

lowercase a subscript {lowercase i} equals [summation over lowercase j [((uppercase f subscript {lowercase i}) times (uppercase p subscript {lowercase j given lowercase i})) divided by (uppercase f subscript {lowercase i})]] times [summation over lowercase k ((lowercase x subscript {lowercase k}) times function (lowercase t subscript {lowercase k given lowercase i j}))]

which will simplify to

lowercase a subscript {lowercase i} equals [summation over lowercase j (uppercase p subscript {lowercase j given lowercase i})] times [summation over lowercase k ((lowercase x subscript {lowercase k}) times function (lowercase t subscript {lowercase k given lowercase i j}))]

One important determinant of our AM will be the number of trips outside the residential zone. If the travel pattern only consists of within-zone trips, lowercase omega subscript {lowercase i j} will be zero except for the diagonal. Then our AM will equal traditional zone-based AMs. If the travel pattern consists of trips between any pair of zones, the path-based accessibility score will be equal to or higher than scores of traditional AMs. The usefulness of the suggested AM will, thus, depend on zone size because the share of within-zone trips can be expected to be proportional to the zone size. If we disaggregate lowercase omega subscript {lowercase i j} into groups that can be expected to have different mobility characteristics, the analytical power will increase. Segmentation can be made with regard to socioeconomic status or education. Yet another alternative is to transpose the weight matrix and obtain an accessibility score for the work zones.3

Our suggested AM is still a zone-based measure and suffers from the same problems as other aggregate AMs (mentioned in the previous section). For example, using a path-based measure of this type will not capture interactions between individuals. What could be done is to impose a complex weighting scheme and argue that the realism of our AM has increased. This, however, would not alter the fundamental properties of zone-based AMs (e.g., we still do not capture interactions between individuals). Instead, the argument for our measure is that we maintain the operational properties of aggregate AMs while adding information on one important daily activity-trips to work.

Empirical Example, Data, and Program

In order to illustrate our measure we provide one application with an observed travel pattern and one application with an estimated travel pattern. For the empirical example we used two sets of data-one from the Stockholm region and one from the county of Jämtland about 600 kilometers (km) northwest of Stockholm (see maps in figures 2, 3, 4). For characteristics of the two regions, see table 1. The regional division is based on small area marketing statistics zones of varying size. In the city centers, the zones consist of just a few blocks, while in the periphery the largest zones are over 100 km2. The two application areas are different in two important aspects: 1) for the Stockholm region we used an estimated matrix as the travel pattern weight (lowercase omega subscript {lowercase i j}), while we used a matrix from a total survey for Jämtland; and 2) Stockholm is an urban region with more than 1.7 million inhabitants with a dense population, while Jämtland is rural and sparsely populated.

One of the contributions of the AM put forward in this article is the weighting of the travel paths. As shown in the section on aggregate path-based accessibility, this can be done using observed or estimated travel flows of a compulsory trip pattern. In the application for Jämtland, we used a matrix obtained from a total survey, the 1990 census-the last year in which data with mode choice are available on a geographically detailed level. Our data set contains variables for gender, education, and mode. To restrict the empirical example, only the car mode was considered. For the Stockholm region, we used estimated matrices for men and women as weights for trips by car.

Two different types of opportunities were selected: one where "more is better" (additive) and another where "one is enough" (maxitive). For the additive opportunity, the number of jobs in retail trade was used. As the maxitive opportunity, pharmacies were used. Pharmacies were selected because this type of opportunity is independent of the size or number of opportunities.4 In this study, alternative ways of distributing prescription drugs were not taken into account.5 Access to retail trade and access to pharmacies were measured to the centroid of the STAMs that contains the relevant opportunity.

Network data were obtained from the Swedish road administration and the Swedish Institute for Transport and Communications Analysis (SIKA). In the sparsely populated region we used free flow travel times, while in the Stockholm region we used travel times from the afternoon peak hour. In our example, we have used a precalculated travel time matrix. Another alternative is to include the shortest path algorithm in the calculation of the AM and avoid storage of the travel time matrix. This might be an alternative for GISs that cannot handle matrices, but is not a restriction in our case.

We defined the general form of our AM in terms of one opportunity (xj) and one impedance function f(tk|ij) or f(tij). In the applications presented below, we used the simple formulation from equation 2 (cumulative opportunity) and the logsum from equation 3. For the cumulative opportunity measure, we used a cutoff time of 25 minutes. The reason behind choosing cumulative opportunity is that, despite its shortcomings, this is a frequently used measure in applied work. An alternative measure is the logsum, which is a natural alternative in association with transport models. The logsum is a parameterized AM and needs estimation of the parameters of a logit model. This model was estimated using data from the national travel survey6 (RVU 94) where information on secondary trips was available. In order to concentrate on the AM, a simple model with travel time as impedance and number of workplaces (wk) in retail trade at k as attraction was estimated, uk|ij = 0.3603log wk -0.2265tk|ij, where uk|ij is the utility of going to k conditioned on a trip from i to j. Most secondary trips are short, and the destination is either close to home or close to work, consequently our parameter is rather high (-0.2265). We used these estimates for both applications.

In a nested logit model, the secondary trips will most likely be in a nest below the destination choice. In such a model structure, different levels should be coupled by the inclusion of a logsum term. It is, however, not likely that someone would choose their place of work with regard to the service supply along the road between home and work. We have, thus, not included any logsum term from the secondary trips into the utility function of the destination choice model.

The application platform is a transport-oriented GIS, TransCAD7 (TC). Beside the standard GIS tool box, TC contains routines for transportation analysis, such as different modeling tools. TC also provides an internal matrix database format (lacking in most GISs), which simplifies our application. The program8 that computes the accessibility is written in TC's internal programming language (Caliper script) and integrated in a "tool box" where different AMs are available (see Berglund 1999). The usage of a native GIS programming language makes it possible for us to offer a close integration between GIS and the computational routines. The program can only run within TC.9

Comparative Analysis of AMS

Using aggregate path-based accessibility measures, accessibility with regard to spatial location (which is traditional) and impacts of socioeconomic status (education) and mobility pattern (based on groups) will be analyzed.

In order to explore some of the properties of the path-based AM, it is compared with existing and well known AMs. Such AMs are the nonpath-based equivalent of the AMs selected for this study. In previous studies, comparisons between different AMs were made using correlation coefficients (see, for example, Kwan 1998). The fact that two AMs are correlated does not indicate quality but may provide an intuitive sense of their properties. Remember that the case with no compulsory trips will yield the same value of path-based accessibility as the corresponding traditional AMs. Thus, low mobility groups are expected to have a path-based accessibility similar to standard zone-based accessibility.

In standard AMs, the only factor that determines accessibility is the location of the zone in relation to the opportunities. This might imply a continuous pattern of accessibility. Given equal access to mobility resources, the differences between socioeconomic groups will be negligible. For path-based measures, the resulting accessibility will also depend on the travel pattern associated with the population in each zone and its socioeconomic composition. It is well known that different socioeconomic groups have different mobility patterns and that different travel time sensitivities are obtained when estimating models.

When we weight the AM with the travel pattern, we expect to discover inequalities in accessibility that are difficult to uncover using other types of AMs. This will also result in less continuous patterns of accessibility, and adjacent zones will show different accessibility depending on socioeconomic composition. We can check this by using a test for the degree of similarity between adjacent zones (spatial autocorrelation). The most widely used test for global spatial autocorrelation is Moran's I (Moran 1948; Cliff and Ord 1972). The value of Moran's I will be in the range +1 to -1. Moran's I will be positive when neighboring areas have similar attributes and negative when the attributes are dissimilar. The hypothesis is that the path-based measures score lower than the conventional AMs.


Let us first look at the correlation coefficients in tables 2 through 5 (table 2, table 3, table 4, table 5). The first two letters of the code in the "variable" column of tables 2 through 5 refer to the type of AM, where CU = cumulative opportunity and LS = logsum. Letters 3 and 4 refer to the opportunity: RT = retail trade and PH = pharmacy. Letter 5 refers to gender: M = men, W = women. In tables 2 and 3, the last letter in the code indicates educational level: L = low, I = intermediate, and H = high. Finally, AA is the traditional zonal measure that is unweighted. Three questions are now considered.

Is there a difference between the weighted measures and the traditional ones, i.e., to what extent are the traditional AMs (in bold face in tables 2 and 3) correlated with the weighted AMs?

The coefficients with regard to retail trade range from 0.788 to 0.922 (cumulative opportunity) and 0.511 to 0.833 (logsum). The differences are more obvious for accessibility with regard to pharmacies, with overall lower coefficients indicating less similarity between the path-based measures and the zone-based measures. The same pattern holds for the AMs weighted by estimated matrices. The maps in figures 5 and 6 illustrate the difference between traditional AMs and weighted AMs. For the weighted AM, a larger area in the central region obtains high accessibility scores while the scores for the unweighted AM declines toward the periphery. A notable difference between the two types of measures can be found in the northeastern part of the region where the weighted AM scores high while the unweighted is quite low. This pattern can be attributed to the fact that the most important commuting flows (or commuting probabilities as estimated by the model) move toward areas where pharmacies can be reached.

Does accessibility differ between groups depending on travel pattern, i.e., to what extent is the path-based AM for different groups correlated?

Looking at the correlation between cumulative opportunity measures of retail trade (table 2, upper left) with different weights, the answer would probably be, "they are not very different." Taking weights 1 to 6 into account, the coefficients range from 0.866 to 0.976. Looking at the parameterized measure (the logsum, table 2, lower right), the answer is different. The same 6 groups (8 to 13) yield correlation coefficients ranging from 0.371 to 0.864. Turning to the example with accessibility to pharmacies, the differences are more pronounced for the cumulative opportunity measures and less obvious for the logsum. In the example with estimated matrices as weights (tables 4 and 5), we find that the differences between men and women are very small, and it appears that our model that generated the weight matrix has not been able to capture differences between genders. One reason is that our AM does not take mode choice into account, which would seriously affect the accessibility for women.

Will the map of accessibility be more heterogeneous with path-based AMs?

In table 6, Moran's I for the AMs are presented. Table 6 shows a mixed pattern. For accessibility to retail trade, except for women in the Jämtland application, the traditional AMs are spatially more homogeneous and show a more continuous accessibility surface. Access to pharmacies shows the opposite pattern. This is not surprising since location of pharmacies is a 0/1 variable (a zone either has one or not and no zone has more than one) for the cumulative opportunity measure. Hence, there will be zones with an accessibility score of 1 or of 0 (remember, pharmacies are assigned a maxitive AM). The very low value for Moran's I is not a surprise in this case. Taking the opportunities along a path into account will even out the accessibility between the zones. Again we can see a similar pattern between the two applications.

Low and High Accessibility Mobility Patterns

Using path-based AMs, it is possible to detect differences in accessibility related to differences in mobility patterns. From an initial calculation, one zone was selected (see figures 7 and 8) with quite different accessibility for two groups-men with low and high education.10 The zone under consideration has no pharmacies, is quite distant from everything else (this is a sparsely populated area), and is separated from the regional center by a lake. To reach more qualified service from this zone, a trip is necessary.

The two groups under consideration have quite different mobility patterns. The most significant difference is that for the group with high education the rate of commuting out of the residential zone is 85 percent, while it is 56 percent for the group with low education. The commuting patterns (see figures 7 and 8) indicate a stronger concentration of the commuting flows to the service centers for the highly educated group than for the less educated group. In this case, with a low local level of service, commuting to service centers will yield high accessibility.


In this paper an accessibility measure has been presented where accessibility is calculated with regard to a mandatory travel pattern for each zone. It is shown that there are quite large differences in accessibility between groups with different travel patterns if an observed matrix is used as a weight. In our example with estimated matrices the differences between groups were negligible. It is of course difficult to capture details in travel patterns by a model. The differences between traditional AMs and the path-based AMs are not as evident for the cumulative opportunity measure as for the logsum. The same pattern holds for the AMs weighted by estimated matrices.

The pattern of similarities between adjacent zones shows a mixed result. For access to retail trade, neighboring zones can have very different accessibility scores depending on the mandatory travel pattern. For the case of pharmacies (using a maxitive AM), the path-based AMs show a more smooth pattern.

When could a path-based AM be useful?

For low-mobility groups who work close to home, the path-based component will not change the accessibility score much and will not be very useful (but not less useful; see the aggregate path-based accessibility discussion). For high-mobility groups, a path-based AM can capture accessibility obtained along the daily travel path and, thus, is useful. A situation where a path-based AM could be useful is in transition regions outside urban areas where part of the population is active in sectors where jobs are found locally (mainly traditional sectors of the labor market) and others find their employment within sectors located in the urban center. If an estimated matrix is used as a weight, the model must be able to capture differences depending on socioeconomic status.


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Address for Correspondence and End Notes

Svante Berglund, Department of Infrastructure and Planning, Royal Institute of Technology, S-100 44 Stockholm, Sweden. Email: svante@infra.kth.se.

1 Many other effects, such as pollution, accidents, and consumption of land, are negative.
2 This is frequently illustrated using the space-time prism (see, e.g., Lenntorp 1976).
3 This option is available in the software developed for this paper but is not used in the application below.
4 It is not reasonable to regard a destination with two pharmacies as twice as good as a destination with one pharmacy. For retail trade in general, it could be a reasonable assumption that a large destination (e.g., a shopping mall) constitutes a more attractive alternative than a small one (e.g., a single store).
5 In some sparsely populated areas, drugs are distributed by a local shop or post office the day after an order has been placed.
6 This travel survey is sample based.
7 Available at http://www.caliper.com.
8 The program is available from the author on request.
9 Since this AM goes over a loop that is n × n × n (see equation 6), where n is the number of zones and GIS programming languages are not very computationally efficient, we also wrote an alternative program in FORTRAN.
10 For this zone, the accessibility score was about twice as high for the highly educated group compared with the group with low education.

Updated: Saturday, May 20, 2017