Performance of Accessibility Measures in Europe

Performance of Accessibility Measures in Europe

Siamak Baradaran*
Farideh Ramjerdi
Royal Institute of Technology

Abstract

Although there is no universally acknowledged definition of accessibility, various indicators with different theoretical backgrounds and complexities have been proposed and implemented in empirical investigations. Consequently, results from these models are widespread and reflect more or less the modeler's aim and point of view. Given the importance of accessibility measures as tools in planning, the aim of this paper is to elicit an understanding of the mechanism behind their diversity. In this paper, accessibility measures are classified according to their underpinning theories, complexity in constructions, and demand on data. The classifications comprise travel-cost, gravity, constraints-based, utility-based, and composite approaches. While simpler models are less demanding on data, they fail to address the subject in a theoretically rigorous manner. The paper also summarizes issues that are important in modeling accessibility. We compare the performance of some conferred accessibility measures in a European context and examine the effects of functional forms of the deterrence variable and agglomeration effect.

Introduction

Trade and flows of commodities and information are recognized as important factors behind economic growth and increased welfare. It is in this context that various researchers have related accessibility between supply and demand of goods and services to economic growth (see Lundqvist 1978; Bruinsma and Rietveld 1998). As a result, accessibility indices are among the most prevailing measures used by planners and politicians to bolster their everyday propositions. Attempts to foster accessibility from national governments, policymakers, and planners have mostly been limited to local or nationwide improvement of the transportation infrastructure. Less attention and resources have been offered to border regions and international accessibility because of geographical and political borders between countries.

After introduction of the European Economic Community (EEC) in the 1960s, more and more countries entered the common market. Furthermore, the Maastricht Treaty of 19911 intensified economic activities between member states and transformed Europe into a huge market. Inspired by the principles of equity and efficiency, which require that all member countries benefit from the new common market, incentives to improve the European transportation infrastructure and accessibility have grown (Vickerman 1995). Clear evidence of this is development of the Trans-European Network (TEN) projects. It is hoped that construction of new highways and high-speed railroads will overcome disparities between the EEC member states, but an evaluation of the present level of accessibility indicators in Europe is needed to gauge the impact of these measures.

Gould (1969, 64) states "accessibility . . . is a slippery notion . . . one of those common terms which everyone uses until faced with the problem of defining and measuring it." Although there is no universally acknowledged definition of accessibility, various indicators with different theoretical backgrounds and complexity have been proposed and implemented in empirical investigations (see, e.g., Ingram 1971; Morris et al. 1978; Handy and Niemeier 1997). Recognizing the value of accessibility measures as planning tools, it is important to understand the mechanism behind their diversity. This paper first presents a summary of different accessibility indicators and clarifies their underpinning theories and corresponding properties. It then addresses issues important in measuring accessibility. The following section discusses some conferred measures applied to major European cities. Similarities and differences between these measures are then evaluated in the Analysis of Results section. Finally, some conclusions are presented.

A Review of Accessibility Indicators

The two most fundamental questions concerning accessibility measures are for whom and for what, and the most straightforward description of accessibility is the state of connectivity. A location is assumed to be accessible if it is connected to other locations via a link to a road or railroad network (see, e.g., Bruinsma and Rietveld 1998) or to an airport or harbor. Accessibility described as connectivity does not need to have a binary form (that the location is connected or not). The extent of accessibility can also be calculated as the number of different links and modes to which the specific location has access. Despite the simplicity of the outline of such indicators, the obscurity of accessibility as a measure of connectivity is apparent.

Different accessibility indicators can be employed to describe and summarize characteristics of the physical infrastructure (e.g., accessibility to certain links, the network, or specific mode or modes). These conventional indicators, often referred to as objective or process indicators, reveal the level of service of the infrastructure network from the suppliers' perspective, regardless of their utilization. On the other hand, the importance of recognizing perceived accessibility by individuals as the real determinant of behavior is emphasized by many researchers, and it is argued that proof of access lies in the use of services. The inherent conflict between the choice of process indicators (objective indicators) and outcome indicators (perceived measures that reflect behavior) gives rise to a great range of indicators with different degrees of behavioral components.

Comprehension of differences between accessibility indicators necessitates classification. The criteria adopted for such classification is based on the discussion above, starting with the group of measures that address the supply side. The other groups of measures are perceived measures that represent the behavioral component. This approach to the classification of accessibility measures has been used by many researchers (see, e.g., Koenig 1977; Morris et al. 1978). Five major theoretical approaches for measurement of accessibility indicators can be found in the literature:

  1. travel-cost approach,
  2. gravity or opportunities approach,
  3. constraints-based approach,
  4. utility-based surplus approach, and
  5. composite approach.

Approaches 1 to 3 have been acknowledged by Arentze et al. (1994) and others, while Miller (1998; 1999) and Miller and Wu (1999) categorize approaches 3 and 4 and derive a new composite indicator (5).

Travel-Cost Approach

The first class of accessibility indicators embodies those measuring the ease with which any land-use activity can be reached from a location using a particular transportation system (Burns and Golob 1976). These indicators have been utilized to indicate performance of the transportation infrastructure (Guy 1977; Breheney 1978). The common aspect for this class of accessibility indicators is determined by their configuration, where the indicator is simply some proxy of transport cost (network or Euclidean distance, travel time, or travel cost). A simple functional form for this class of measures is presented by equation 1.

uppercase a subscript {lowercase i} equals summation lowercase j is an element of uppercase l (1 divided by the function (lowercase c subscript {lowercase i j}))

where

Ai is the measure of accessibility at location i,

L is the set of all locations, and

f(cij) is the deterrence function and cij is a variable that represents travel cost between nodes i and j.

This class of measures has a number of advantages. They are

  • easy to understand because of the simplicity of model construction,
  • quite easy to calculate, and
  • less demanding on data than other indicators.

The following are the most critical disadvantages of indicators within this class:

  • they neglect variations in the quality of locations,
  • they neglect variations in the value of time among travelers,
  • they are highly sensitive to the choice of demarcation area (see, e.g., Bruinsma and Rietveld 1999), and
  • they do not consider the behavioral aspects of travelers (see Hensher and Stopher 1978).

Gravity or Opportunities Approach

Indicators based on spatial opportunities available to travelers are among the first attempts to address the behavioral aspects of travel. A great number of accessibility indicators are in this class. The potential to opportunities or the gravity approach is undoubtedly the most utilized technique among accessibility indicators (see, e.g., Dalvi and Martin 1976; Linneker and Spence 1991; Geertman and Ritsema Van Eck 1995; Bruinsma and Rietveld 1998; Brunton and Richardson 1998; Kwan 1998; and Levinson 1998). An early attempt was made by Hansen (1959), who claimed that accessibility is the "potential of opportunities for interaction" or literally "a generalization of population-over-distance relationship" (p. 73). The concept of potential to opportunities is closely associated with the gravity models based on the interaction of masses and has been extensively discussed by Rich (1978). Equation 2 shows a simple form for this class of accessibility indicators.

uppercase a subscript {lowercase i} equals summation lowercase j is an element of uppercase l (uppercase w subscript {lowercase j}) divided by the function (lowercase c subscript {lowercase i j}, lowercase beta)

where

Wj represents the mass of opportunities available to consumers, regardless of if they are chosen or not,

function (lowercase c subscript {lowercase i j}, lowercase beta) is the deterrence function,

cij is a variable that represents travel cost between nodes i and j, and

lowercase beta is the travel-cost coefficient usually estimated from a destination choice model.

Advantages of this class of accessibility measures are

  • ease of comprehension,
  • ease of calculations,
  • they are less demanding on input data than other indicators that reflect behavioral aspects, and
  • the ability differentiate between locations.

Some disadvantages of this class of indicators are their

  • sensitivity to the choice of demarcation area,
  • deficiency in treatment of travelers with dispersed preferences, and
  • ambiguity in what the magnitude of indicators express (dimension problem).

Constraints-Based Approach

Despite the popularity of potential accessibility indicators, they have some weak points. One weak point with gravity models is that they do not address time constraints facing individuals. The constraint-oriented approach was developed by Hagerstrand (1970) within the space-time framework and is based on the fact that individual accessibility has both spatial and temporal dimensions. Opportunities or potential to opportunities for an individual are not only constrained by the distance between them, but also by the time constraints of the individual.

Miller (1999, 2) defines Potential Path Space (PPS) by stating that: "The space-time prism delimits all locations in space-time that can be reached by an individual based on the locations and duration of mandatory activities (e.g., home, work) and the travel velocities allowed by the transportation system." Assume an individual located at time t1 in node (X0, Y0). Again assume that at time t2 the individual has to be back at the same node. Then the available time for all activities is given by t = t2 - t1. Figure 1 shows the contained volume by two cones that represents the space-time prism or PPS.

The projection of PPS on the two-dimensional XY-space represents the potential path area (PPA) that corresponds to the potential area that an individual can move within, given the time budget.

Lenntorp (1976; 1978) developed a so-called program evaluating the set of alternative sample paths (PESASP) to calculate the number of feasible paths between nodes, given the activity schedules and space-time constraints. The number of feasible activity schedules simulated by the program represents a measure of accessibility. In other studies, modified space-time prisms have been employed to indicate the individual accessibility based on various travel speeds, multistop trip chaining, and changes in activity schedules (see Hall 1983 and Arentze et al. 1994).

A frequently adopted indicator within this class is the cumulative opportunity measure or the so- called isochronic indicators that estimate accessibility in terms of opportunities available within predefined limits of travel cost, C (Dunphy 1973; Sherman et al. 1974; Breheny 1978; Hanson and Schwab 1987).

This class of indicators addresses some of the limitations of the earlier models by:

  • consideration of the temporal dimension of human activities, which leads to indicators that account for the individuals time constraints, and
  • the recognition of multipurpose activity behavior by a space-time prism.

Wang (1996) points out four weak points with this approach:

  • assuming a constant speed in all directions is not realistic and variable speed makes the model exceedingly burdensome to handle;
  • the planar space defined as PPA is too abstract-a large PPA is not necessarily better than a small one, if the smaller PPA contains more potential locations;
  • the activity schedules are usually incomplete and do not cover the whole spectrum of activities; and
  • even though a time budget is introduced, the individual's travel behavior is not fully addressed in this class of measures.

Utility-Based Surplus Approach

This class of accessibility indicators is another attempt to include individual behavior characteristics in accessibility models. Utility-based indicators have their roots in travel demand modeling. Ben-Akiva and Lerman (1979, 654) states: "accessibility logically depends on the group of alternatives being evaluated and the individual traveler for whom accessibility is being measured." In that sense, the shortcoming of gravity-based indicators becomes obvious, as all individuals within the same zone will experience the same amount of accessibility, regardless of the differences between their perceived utility of alternatives. Ben-Akiva and Lerman (1979, 656) continue: for any single decision, the individual will select the alternative which maximizes his/her utility," uppercase u superscript {lowercase n} subscript {lowercase j given lowercase i} Thus a simple definition of accessibility uppercase a superscript {lowercase n} subscript {lowercase i} is:

uppercase a superscript {lowercase n} subscript {lowercase i} equals maximize over lowercase i, lowercase j is an element of uppercase l (uppercase u superscript {lowercase n} subscript {lowercase j given lowercase i})

where

n is a mutually exclusive and collectively exhaustive individual member of I,

j is the destination {lowercase j equals (one, two, ..., lowercase j, ..., lowercase l); for all of the lowercase j not equal to lowercase i} and

i is the node for which the accessibility is calculated;

and

uppercase u superscript {lowercase n} subscript {lowercase j given lowercase i} equals (lowercase v superscript {lowercase n} subscript {lowercase j}) minus (lowercase c superscript {lowercase n} subscript {lowercase i j}) plus (lowercase epsilon subscript {lowercase i j})

where

vjis some measure reflecting the attraction of the alternative j, observable to the modeler,

cij is the cost of travel between i and j, and

lowercase epsilon subscript {lowercase i j} is the stochastic, random, and unobservable part of the utility lowercase epsilon = 0 for the individual but unknown for the modeler).

By assuming that the random variables are independent and identically distributed according to the extreme value distribution, the accessibility of location i for individual n is:

uppercase a superscript {lowercase n} subscript {lowercase i} equals maximize over lowercase i, lowercase j is an element of uppercase l (uppercase u superscript {lowercase n} subscript {lowercase j given lowercase i}) equals (1 divided by lowercase mu) times natural logarithm [summation over lowercase j is an element of uppercase l (lowercase e superscript {lowercase mu times ((lowercase v superscript {lowercase n} subscript {lowercase j}) minus (lowercase c superscript {lowercase n} subscript {lowercase i j}))]

where lowercase mu is a positive scale parameter.

The measure of accessibility defined in this way is in monetary units, which enables the comparison of different scenarios. Williams (1977) noted that utility-based accessibility is linked to consumer welfare. McFadden (1975) and Small and Rosen (1981) showed how this measure can be derived in the discrete choice situation for the multi-nomial logit (MNL) model when income effect is not present. For examples of investigations on utility-based accessibility measures see papers by Niemeier (1997) and Handy and Niemeier (1997).

The advantage of this class of indicators is that they are supported by relevant travel behavior theories. Some disadvantages of this class of indicators are:

  • modeling of utility-based accessibility indicators demands extensive data on locations and individuals' travel behavior and their choice sets, and
  • the assumption of nonpresence of an income effect is restrictive.

Composite Approach

Representation of the multiple-purpose property of trips is lacking in the utility-based measures. These drawbacks have been discussed by some researchers. Among them Miller (1998; 1999) summarizes the disadvantages of these measures and derives new measures by combining the space-time and the utility-based models into a composite model. Miller's work has Weibull's (1976) axiomatic approach as its starting point. Miller calls these models space-time accessibility measures (STAMs), which are based on the assumption of uniform travel speed.

STAMs are based on the utility of performing a series of discretionary activities (e.g., shopping, visiting), given the mandatory activities (e.g., work). The following utility function, u(.), defined by Burns (1979) and Hsu and Hsieh (1997), is employed as the base:

lowercase u subscript {lowercase i j} times (lowercase a subscript {lowercase k}, uppercase t subscript {lowercase k}, lowercase t subscript {lowercase k}) equals (lowercase a superscript {lowercase alpha} subscript {lowercase k}) times (uppercase t superscript {lowercase beta} subscript {lowercase k}) times (lowercase e superscript {negative lowercase lambda times (lowercase t subscript {lowercase k})})

where

ak = attractiveness of discretionary activity location k,

lowercase alpha is the parameter for the attraction mass,

uppercase t subscript {lowercase k} equals the inclusion of ((lowercase t subscript {lowercase j}) minus (lowercase t subscript {lowercase i}) minus (lowercase t subscript {lowercase k}); if greater than zero); and (zero else) is the available time for participation in activities [T = f(t)],

ti,tj = stop times for mandatory activity i and start time for mandatory activity j,

tk = [d (xi, xk) + d (xk, xj)] /s is the required travel time from/to the mandatory activities,

xi = location vector of mandatory activity i,

d(xi,xk) = distance from activity location i to activity location k,

s = constant velocity of travel,

lowercase beta is the coefficient for available time, and

lowercase lambda is the travel time coefficient.

Based on these formulations Miller (1999) defines three different STAMs as:

uppercase a m subscript {1} equals (1 divided by lowercase lambda) times natural logarithm [summation over lowercase k is an element of uppercase l times exponential ((lowercase a superscript {lowercase alpha} subscript {lowercase k}) times (uppercase t superscript {lowercase beta} subscript {lowercase k}) times (lowercase e superscript {negative lowercase lambda times (lowercase t subscript {lowercase k})})]

uppercase a m subscript {2} equals summation over lowercase k is an element of uppercase l (lowercase b subscript {lowercase k})

uppercase a m subscript {3} equals maximum over lowercase k [lowercase b subscript {lowercase k}]

where

lowercase b subscript {lowercase k} equals 0 if lowercase a subscript {lowercase k} or uppercase t subscript {lowercase k} equal 0; else exponential [lowercase lambda times ((lowercase alpha divided by lowercase lambda) times natural logarithm (lowercase a subscript {lowercase k})) plus ((lowercase beta divided by lowercase lambda) times natural logarithm (uppercase t subscript {lowercase k}) minus (lowercase t subscript {lowercase k})]

AM1 corresponds to the user-benefit approach while AM2 and AM3 correspond to the locational benefits approach. AM2 considers the whole choice set while AM3 assumes that an individual only considers the choice that maximizes her utility. Miller and Wu (1999) develop this approach further to incorporate a departure-based, discrete time network flow model. While this approach aims at avoiding the problems of the other accessibility measures, its main disadvantage is related to the vast data requirement.

Further Issues in Accessibility Models

The following discussion summarizes a chapter in Bruinsma and Rietveld (1999), in addition to some further issues.

Measurement of Spatial Separation

The degree of spatial separation between locations can be measured several ways. Common proxies are travel distance, travel time, and generalized travel cost. Travel distance and travel time are usually easy and straightforward to calculate, while operation with generalized travel cost is more cumbersome. In the case of generalized travel cost, other than the calculation of distance-dependent costs, information associated with costs of vehicle use, fares, taxes, and so forth, are needed. Since such data is not readily available at the disaggregate level, mean values must be used, which implies further assumptions.

The calculation of travel time is usually based on a shortest path algorithm. A more precise method is use of a route choice simulation procedure, which is especially necessary for congested networks. However, the procedure is data demanding and requires trip-matrices as well as volume-delay functions. In the case of public transport, waiting, transfer, and auxiliary times are also relevant in addition to in-vehicle time and fares.

The functional form of the deterrence variable is also important. For instance, we know that the perception of utility (disutility) derived from waiting time is not equal to the in-vehicle time.

Furthermore, the deterrence variable does not necessarily have to be linear in construction.

Measurement of Attraction Masses

Earlier in this paper, two important questions were raised, accessibility for whom and to what. While the first question is answered by the choice of the model (e.g., individual or aggregate), the choice of attraction mass responds to the second question. The mass of attraction in accessibility models represents the potential utility for opportunities at a destination,2 or in other words, the utility an individual can derive by visiting a specific location or a set of locations. The choice of appropriate interaction mass is crucial for the determination of accessibility. In large-scale accessibility studies and in the absence of other attributes, population is often used as the interaction mass variable. Other possible proxies are percentage of gross domestic product, number of employees, volume of sales, etc.

Choice of Demarcation Area

Arbia (1989) divides the problems related to choice of demarcation area into two subproblems. The first is related to the effects of scale while the second corresponds to zoning problems.3 A third problem arises as a consequence of the choice of total study area.

The scale problem is related to the number of units represented in the study area. Inclusion or exclusion of units will affect the results of the accessibility model. The zoning problems relate to the way locations are presented. Expressing locations as nodes that correspond to urban centers will cause aggregation problems, that is, all individuals in the same zone will have the same level of accessibility (Ben-Akiva and Lerman 1979). Furthermore, the underlying assumption is that all locations presented by that node have similar accessibility measures (Bruinsma and Rietveld 1998). That also complicates the calculation of internal accessibility measures. However, the use of geographic information system (GIS) and disaggregated census data can reduce these difficulties. In this case other problems might arise, like definition problems concerning the grid resolution and issues related to the modifiable areal unit problem.

The choice of total study area is also an important problem that needs attention. With the determination of the study area, one will consequently decide which areas should be excluded. The choice of a closed study area will ignore the effects from outside, which in many cases can be questionable (Bruinsma and Rietveld 1998).

Unimodality versus Multimodality

Uni- versus multimodality is also a relevant consideration in modeling accessibility. For instance, for a work trip, a range of travel modes can be appropriate. In case of trips by air, we can easily imagine that the traveler actually faces two additional mode choices. One has to determine travel modes to the airport of departure and from the airport of disembarkation. Multimodality can partially be handled in accessibility models. In a travel-cost approach or gravity approach, multimodality can be embedded in the calculation of travel time or cost for all modes. These can be presented separately or by the assumption that the traveler might choose the fastest or the least expensive among alternative modes. In the case of utility-based and composite accessibility models, multimodality can be brought to the model by the construction of a nested destination/mode choice model.

Time of Day

Differentiation between accessibility measures at different times of day is necessary when the level of service varies during the day or when traffic congestion is a factor. The variation in accessibility for different times of day can be reproduced by the construction of separate accessibility models for different time periods. However, in many cases, especially in the case of long-distance trips, these variations could be small and may have only a minor impact on accessibility measures.

Agglomeration Effects

The magnitude of opportunities offered at a location also encompasses opportunities available in surrounding locations within the individuals' travel constraints. Inclusion of agglomeration effects is a complicated task. However, since agglomeration effects have a direct impact on the utility derived from the opportunities, the easiest way of approximating these effects is through transformation of the attraction mass variable.

A pre-set degree of spatial dependence can be embedded in a variable by means of spatial transformation. Different techniques can be used to realize these transformations, which can simply be called spatial averaging (see Anselin 1992). One transformation technique is termed the spatial window average.

uppercase w superscript {asterisk} subscript {lowercase i} equals ((uppercase w subscript {lowercase i}) plus (summation over lowercase j is an element of uppercase l (uppercase psi subscript {lowercase i j times (lowercase d)}) times (uppercase w subscript {lowercase j}))) divided by (1 plus (summation over lowercase j is an element of uppercase l (uppercase psi subscript {lowercase i j times (lowercase d)})))

where uppercase w superscript {asterisk} subscript {lowercase i} is the transformed mass variable representing the attraction mass of node i (agglomeration effects included) compared with Wi the mass variable at node i and uppercase psi subscript {lowercase i j times (lowercase d)} is a spatial weight from a contiguity matrix4 up to distance d. This formulation is not suitable when the mass is in monetary units.

The above formulation is highly sensitive to the definition of contiguity. As an example, if we define contiguity by masses within a distance d from a location, then the above formulation will underestimate a large agglomeration with many surrounding settlements compared with another with few surrounding settlements. An approach to correct for this problem is to average the mass of agglomeration (nominator) by a fixed number, K, for all locations i. This implies all nodes have the same degree of neighborhood (K-1).

uppercase w superscript {asterisk} subscript {lowercase i} equals ((uppercase w subscript {lowercase i}) plus (summation over lowercase j is an element of uppercase l (uppercase psi subscript {lowercase i j times (lowercase d)}) times (uppercase w subscript {lowercase j}))) divided by uppercase k

Dimension Problem

The dimension problem arises because almost all accessibility indicators (except utility-based and composite measures) present the accessibility of locations as nondimensional values that are not comparable with each other. These nonmonetary values complicate the evaluation of infrastructure improvements. A method that can be used for comparison of different accessibility measures is ranking. By dividing each accessibility measure by the highest accessibility measure, indicators will become normalized in a way that makes them suitable for comparison.

A Study of Accessibility Measures of European Cities

The aim of our study is to understand the built-in mechanism of some of the accessibility models discussed earlier, while looking at accessibility measures of European cities with road infrastructure. Even though the discussed accessibility models are operational, not many of them have been applied in large-scale studies. In large-scale accessibility studies, the unavailability of illustrative and homogeneous data is always a limiting factor. Consequently, one's choice is limited to more simple and straightforward models. For this reason, the empirical study presented here is based on the first and the second class of the models (travel-cost and gravity type), with consideration of the agglomeration effect. Furthermore, variations in accessibility caused by different assumptions about the deterrence variable will be examined.

Data

To decrease the problems associated with the choice of the demarcation area, all of Europe was chosen as the study area (except for Turkey due to the absence of appropriate data). Accessibility indicators are calculated for more than 4,500 cities with a population greater than 10,000, located in 44 European countries connected to each other by the road infrastructure. The data source is a modified GIS data layer containing urban centers in Europe5 that includes population data. Travel distance and travel time variables are used as proxies to the spatial separation variable. These are calculated using a digitized road network from three different sources implemented in a GIS-database. The sources for the road network data are:

  1. the IRPUD road network,6
  2. the digitized road network for Sweden,7 and
  3. the digitized road network for Finland.8

Travel distance and travel time are calculated using the shortest path algorithm in TransCAD.9 The calculation of travel distance is based on the length attributes of the links, while travel time is based on different link speeds, commonly assumed for different link categories. Hence, the effect of congestion is not taken into account in this study. Car ferry links are penalized by an additional travel time of 45 minutes.

The calculation of internal accessibility measures is necessary. In the absence of appropriate data, the internal travel distances and travel times are calculated with the assumption that cities are circular,10 based on the following equations:

lowercase t subscript {lowercase i i} equals ((lowercase d divided by 4) divided by 40)

where

lowercase d equals ((square root of uppercase o divided by lowercase pi) divided by 2)

and

uppercase o equals (population divided by density)

where d is the diameter of the city. An average travel speed of 40 kilometers per hour has been assumed for all internal trips.

Selected Accessibility Models

One group of accessibility models based on the travel-cost approach and two groups of gravity-based models will be examined in this work. In all model groups, an internal accessibility measure is included. For each model group, three deterrence functions will be examined:

  1. linear in travel time (t),
  2. exponential in travel time, and
  3. Box-Cox transformed travel time.11

The first group of measures is based on the travel-cost approach where the measure of accessibility can be interpreted as the level of connectivity of the nodes as:

lowercase a subscript {1} equals (1 divided by lowercase t subscript {lowercase i i}) plus (summation over lowercase j is an element of uppercase l (1 divided by lowercase t subscript {lowercase i j})), lowercase i is not equal to lowercase j

lowercase a subscript {2} equals (1 divided by lowercase e superscript {lowercase beta times (lowercase t subscript {lowercase i i})}) plus (summation over lowercase j is an element of uppercase l (1 divided by lowercase e superscript {lowercase beta times (lowercase t subscript {lowercase i j})})), lowercase i is not equal to lowercase j

lowercase a subscript {3} equals (1 divided by (lowercase e superscript {lowercase delta times ((lowercase t superscript {lowercase theta} subscript {lowercase i i} minus 1) divided by lowercase theta)})) plus summation over lowercase j is an element of uppercase l times (1 divided by lowercase e superscript {lowercase delta times ((lowercase t superscript {lowercase theta} subscript {lowercase i j} minus 1) divided by lowercase theta)}), lowercase i is not equal to lowercase j

where

tii is the internal travel time at i, and

tij is the travel time between locations.

The second group of measures is based on the gravity approach models (Hansen type) and are:

lowercase b subscript {1} equals (lowercase p subscript {i} divided by lowercase t subscript {lowercase i i}) plus (summation over lowercase j is an element of uppercase l (lowercase p subscript {j} divided by lowercase t subscript {lowercase i j})), lowercase i is not equal to lowercase j

lowercase b subscript {2} equals (lowercase p subscript {i} divided by lowercase e superscript {lowercase beta times (lowercase t subscript {lowercase i i})}) plus (summation over lowercase j is an element of uppercase l (lowercase p subscript {j} divided by lowercase e superscript {lowercase beta times (lowercase t subscript {lowercase i j})})), lowercase i is not equal to lowercase j

lowercase b subscript {3} equals ((lowercase p time lowercase i) divided by (lowercase e superscript {lowercase delta times ((lowercase t superscript {lowercase theta} subscript {lowercase i i} minus 1) divided by lowercase theta)})) plus summation over lowercase j is an element of uppercase l times (lowercase p subscript {j} divided by lowercase e superscript {lowercase delta times ((lowercase t superscript {lowercase theta} subscript {lowercase i j} minus 1) divided by lowercase theta)}), lowercase i is not equal to lowercase j

where p is population.

The last group of measures also belongs to the gravity type with the agglomeration effect included as:

lowercase c subscript {1} equals (lowercase p superscript {*} subscript {i} divided by lowercase t subscript {lowercase i i}) plus (summation over lowercase j is an element of uppercase l (lowercase p superscript {*} subscript {j} divided by lowercase t subscript {lowercase i j})), lowercase i is not equal to lowercase j

lowercase c subscript {2} equals (lowercase p superscript {*} subscript {i} divided by lowercase e superscript {lowercase beta times (lowercase t subscript {lowercase i i})}) plus (summation over lowercase j is an element of uppercase l (lowercase p superscript {*} subscript {j} divided by lowercase e superscript {lowercase beta times (lowercase t subscript {lowercase i j})})), lowercase i is not equal to lowercase j

lowercase c subscript {3} equals ((lowercase p superscript {*} subscript {i}) divided by (lowercase e superscript {lowercase delta times ((lowercase t superscript {lowercase theta} subscript {lowercase i i} minus 1) divided by lowercase theta)})) plus summation over lowercase j is an element of uppercase l times (lowercase p superscript {*} subscript {j} divided by lowercase e superscript {lowercase delta times ((lowercase t superscript {lowercase theta} subscript {lowercase i j} minus 1) divided by lowercase theta)}), lowercase i is not equal to lowercase j

where lowercase p superscript {asterisk} subscript {lowercase i} is the transformed population of location calculated as:

[lowercase p superscript {asterisk} subscript {lowercase i} given (lowercase t subscript {lowercase i j} less than or equal to 1 hour)] equals (lowercase p subscript {lowercase i} plus (summation over lowercase j not equal to lowercase i [(uppercase psi subscript {lowercase i j times (lowercase d)}) times (lowercase p subscript {lowercase j}))] divided by (uppercase k)

A location j is assumed to be a neighbor of location i if tij is less than or equal to one hour. The choice of one hour as the threshold is related to the time constraint a traveler faces making a roundtrip during a working day.12

Conventionally, parameters in the models of accessibility should be estimated, but due to the absence of appropriate data for the whole study area, parameters from a Swedish study are used13 (Baradaran 2001). These are:

lowercase beta equals 0.00329, lowercase delta equals 0.07014, and lowercase theta equals 0.545.

Analysis of Results

Relationships between different aspects of the selected measures are analyzed by examination of correlations and other deviation measures and by comparisons of accessibility maps.

Examination of Correlations and Other Deviation Measures

The similarities and differences between models are investigated by construction of the correlation14 table (see table 1).

Examination of the correlation table shows that measures in the third group of models (group c, which includes the agglomeration effect) are quite different from the first two groups (group a and b). Within the first two groups, measures based on linear construction of the deterrence variable (a1 and b1) are highly correlated with each other, while having lower correlation with other measures based on nonlinear construction of the deterrence variable. Similarly, measures based on nonlinear construction of the deterrence variable are highly correlated with each other, while they have lower correlation with measures based on linear construction of the deterrence variable. Group c measures that includes agglomeration effects have higher correlation with the linear measures (a1 and b1).

Similarities and differences between the models have also been analyzed using dispersion and skewness statistics shown in table 2. The second column in table 2 represents a dispersion measure, lowercase phi, which is constructed as follows:

lowercase phi equals (standard deviation divided by mean accessibility)

This measure describes the degree of dispersion of the calculated accessibility measures. This measure is of course dependent on the area of the study. Hence, it is not the magnitude of this measure that is crucial, but the degrees of similarity or dissimilarity among these measures that provides the necessary information. Table 2 shows that group c measures that include agglomeration have a much higher lowercase phi -value than other measures. This suggests that measures that include agglomeration are different from the rest. Among other measures, the nonlinear measures (a2, a3, b2, and b3) have the lowest lowercase phi -values, suggesting that a nonlinear transformation of the deterrence variable has a kind of smoothing effect on the accessibility measures.

The last column in table 2 represents the skewness15 of measures estimated from different models. Skewness helps identify the degree of asymmetry of a distribution around its mean. Positive skewness indicates that the asymmetrical tail is protracted toward more positive values while negative skewness indicates the opposite. Again we can see that the skewness of the linear measures (a1 and b1) and group c measures represent cumulative processes (because they are positive) while the nonlinear measures (a2, a3, b2, and b3) show declining processes (because they are negative).

Differences among accessibility models can also be investigated by using a numerical taxonomy. Sneath and Sokal (1973, 116) state that ". . . a coefficient of similarity is a qualification of the resemblance between the elements in two columns of the data matrix representing the character state of two operational taxonomic units in question." Two different dissimilarity coefficients are calculated. These are

  • mean absolute difference (MAD), which is a variant of Minkowski metrics16 adjusted for number of vector elements and specified as
    uppercase m a d equals (one divided by uppercase l) times [summation over lowercase i is an element of uppercase l (absolute value (uppercase a hat subscript {lowercase i}) minus (uppercase a subscript {lowercase i}))]
    where uppercase a hat subscript {lowercase i} is the accessibility measure for location i and L is the set of all locations.
  • dissimilarity index (DSI), also known as Leontief index (after multiplication by 100), specified as uppercase d s i equals (1 divided by uppercase l) times [summation over lowercase i is an element of uppercase l [(absolute value (uppercase a hat subscript {lowercase i}) minus (uppercase a subscript {lowercase i})) divided by ((uppercase a hat subscript {lowercase i}) plus (uppercase a subscript {lowercase i}))]], uppercase a hat subscript {lowercase i} is not equal to zero or uppercase a subscript {lowercase i} is not equal to zero

The results are presented in the appendix (table 3, table 4, table 5). However, due to differences in their ranges, these metrics are not directly comparable. For comparison they are normalized in the following way:

uppercase m bar equals (uppercase m minus minimum (uppercase m)) divided by [maximum (uppercase m) minus minimum (uppercase m)]

where M is the metric and uppercase m bar is its transformed form. The result of this transformation are metrics that vary from 0 to 1. To avoid zeros in the case of DSI-metric, zeros are replaced with 0.000001.

Figure 2 shows the differences between accessibility measures with respect to measure a1, using MAD and DSI metrics. The examination of different metrics points to 3 clusters for the 9 accessibility measures. One cluster is a linear deterrence variable (a1 and b1). The second cluster is a nonlinear deterrence variable (a2, a3, b2, and b3). The third cluster includes an agglomeration effect (group c measure).

The differences between the examined accessibility measures can be caused by either some key assumptions made in the calculation, such as parameters and internal travel time, or by the functional characteristics of the models.17 The examination of relationships between the selected measures by use of correlation coefficients, measures of skewness and dispersion, and other metrics (MAD and DSI) support each other. The following are some general conclusions that can be drawn from the examination of different deviation measures.

  • Differences in accessibility measures are better explained by the choice of functional form for the deterrence variable than by the choice of model approach.
  • Various methods used to evaluate the differences between measures suggest that models based on linear functional forms of the deterrence variable are not the same as measures based on nonlinear designed models.
  • Nonlinear specification of the deterrence variable decreases the level of dispersion among the measures.
  • Corrections for the agglomeration effect produce results that are significantly different from the other examined approaches.

Comparisons of Accessibility Maps

Finally, different accessibility maps are constructed using a GIS-platform by construction of TIN-models.18 Isochor polygons are the result of the TIN-model, where the magnitude of accessibility in each polygon will demonstrate its level comparable to the other polygons in its surrounding neighborhood. Each isochor surface is classified by its rank, where rank 0 corresponds to locations with the least accessibility and 100 corresponds to locations with maximum accessibility. The accessibility rank19 of each city is used as the Z-value,20 which differentiates the isochors. The dark colors represent highly ranked areas, while the bright areas are ranked lower for accessibility. The continuous range of accessibility ranks is divided into 10 equal segments. This, however, makes a visual examination of small changes on the accessibility maps difficult. For the comparison of minor differences of two accessibility maps, one can zoom in areas of interest and use finer segments.

Figure 3 shows an accessibility map of Europe based on model a1 (travel-cost approach and linear deterrence variable), while figures 4 and 5 show corresponding maps based on model a2 (travel-cost approach and nonlinear deterrence variable) and model b2 (gravity approach and non-linear deterrence variable). A comparison of these figures suggests that the accessibility maps of Europe are more sensitive to the linearity of the deterrence variable than the approaches for the calculation of the accessibility measure (travel-cost or gravity approach).

Figure 6 shows the accessibility map of Europe based on model c1 (gravity approach corrected for the agglomeration effect and the linear deterrence variable). Comparison of this figure with previous maps suggests that the correction for the agglomeration effect has changed the relative rankings of accessibility values in Europe significantly. With correction for the agglomeration effect, large agglomerations such as London, Paris, or Moscow get very high rankings compared with the rest of Europe. In fact Moscow has a significant place on this map compared with maps presented in figures 3, 4, and 5, where the agglomeration effect is not accounted for. These maps show that the most accessible part of Europe is Central Europe (around Germany) and accessibility decreases as one moves away from this area. Note that with a different scale, the relative rankings of accessibility values will change; however, the large agglomerations in Europe will have the highest accessibility values. In general, visual examination of accessibility maps confirm the results from the statistical tests.

Finally, examination of the accessibility maps of Europe suggest important issues with policy implications for the European Economic Community (EEC). One interesting observation is that accessibility measures in border regions of all the European countries seem to be much lower than internal accessibility measures. The lower level of accessibility measures in the border regions can be explained by two factors:

  • the density of cities in border regions is usually lower than for the interior of a country, and
  • accessibility in border regions is lower due to lower density of transport infrastructure in these locations.

Spiekermann and Wegener (1996) have reported similar observations in an accessibility study. One can expect that by taking congestion into account in calculating travel time, these border problems with respect to accessibility measures should become less severe, but they would not disappear. Indeed, the accessibility at border regions has emerged as an important policy issue for the EEC.

Another important observation is low accessibility in the peripheries of Europe, especially in the regions in the east and southeast. The choices of the demarcation area can at least partly explain this observation.

Conclusions

In this paper, five approaches for measuring accessibility were classified based on a literature review: travel-cost approach, gravity approach, constraints-based approach, utility-based approach, and composite approach. Certain properties of each class of accessibility models have been discussed as have their pros and cons. Basically, accessibility measures in these classes differ in three respects: theoretical foundation, complexity of construction, and demand on data. In general, the simpler measures are less data dependent, but they fail to adequately address the subject in a theoretically sound manner. Availability of data is usually an important factor in the choice of the appropriate measure in an accessibility study. The purpose of a study is another factor that should influence the choice of the measure. In the empirical part of this study, even with the limited number of measures, we have illustrated that the choice of the measure has an important affect on the accessibility map and hence, the focus on a particular issue.

Furthermore, some important issues relevant in modeling accessibility are summarized:

  • measurement of spatial separation,
  • measurement of attraction masses,
  • choice of demarcation area,
  • unimodality versus multimodality,
  • agglomeration effects,
  • the dimension problem, and
  • time of day.

In the empirical part of the study, accessibility measures for more than 4,500 major European cities were constructed based on the travel-cost approach and gravity approach with and without correction for the agglomeration effect. Three different functional forms of the deterrence variable were examined in each approach, one linear and two nonlinear in construction. Differences between the calculated measures were studied using statistical and visual techniques. Correlation coefficients, measures of skewness and dispersion, and different metrics, mean absolute difference and dissimilarity index, were used. Finally, accessibility maps of Europe were produced for all approaches. We can draw some conclusions by examining different deviation measures:

  • the choice of functional form for the deterrence variable explains the differences in accessibility measures more than the model approach,
  • a measure with a linear functional form of the deterrence variable is different from measures based on nonlinear functional form,
  • a nonlinear specification of the deterrence variable decreases the level of dispersion among the measures, and
  • corrections for the agglomeration effect produce significantly different results.

This study is subject to many qualifications. An important qualification relates to the availability of necessary data for the comparison and evaluation of accessibility measures by all identified approaches. The results of this study, however, illustrate the importance of understanding the performance of these measures.

Finally, examinations of the accessibility maps of Europe suggest that the choice of approach influences the relative accessibility of locations, hence, highlighting the importance of issues differently. It is therefore important to use an approach relevant to the problem. Some important issues with policy implications for the EEC can be observed from these accessibility maps. One important observation is the low accessibility measures in border regions of all the European countries compared with internal accessibility values. This can be explained by low density of settlements and transport infrastructure in border regions. Another important observation is low accessibility in the peripheries of the Europe, especially in the regions in the east and southeast.

Acknowledgment

The Swedish National Road Administration supported this research. The authors wish to thank two anonymous referees for their helpful comments and suggestions.

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Address for Correspondence and End Notes

Siamak Baradaran, Royal Institute of Technology, Department of Infrastructure and Planning, Division of Transport and Location Analysis, Fiskartorpsvagen 15-A, SE-100 44 Stockholm, Sweden. Email: sia@infra.kth.se.

1 For more information, go to http://www.facts.com/cd/v00087.htm.
2 In the case of potential, time-space, utility-based, and composite accessibility indicators.
3 The spatial arrangement of units or the modifiable areal unit problem.
4 A contiguity matrix represents the degree of neighborhood of a location with its surrounding locations.
5 The source for urban centers data is CEC-Eurostat/GISCO.
6 This digitized road network is developed by the Institute of Spatial Planing in Dortmund, Germany.
7 The source of this network is the Swedish National Road Administration (Vagverket).
8 The source of this network is the Finnish National Road Administration (VTT).
9 TransCAD is a transportation-GIS software from Caliper Corp. (www.caliper.com).
10 This formulation of internal distance has been discussed by Rich (1980) and also by Bruinsma and Rietveld (1998).
11 Box-Cox transformation implies: Box-Cox transformation implies: lowercase y equals (lowercase x superscript {lowercase theta} minus 1) divided by lowercase theta..
12 One should indeed conduct a sensitivity test to evaluate the importance of the threshold.
13 These parameters are estimated by using a multinomial-logit model with disaggregate data for long-distance trips in Sweden.
14 Correlation: corr [lowercase x, lowercase y] equals covariance [lowercase x, lowercase y] divided square root [variance (lowercase x) times variance (lowercase y)] is an element of [negative 1, 1]
15 A skewness coefficient is a measure of asymmetry of a distribution. Skew = the summation over lowercase i times [(lowercase x subscript {lowercase i} minus lowercase mu) superscript {3}] divided by lowercase sigma superscript {3}; where lowercase mu is the population mean and lowercase sigma is the standard deviation. where lowercase mu is the population mean and lowercase sigma is the standard deviation.
16 The Minkowski metric corresponds to the Minkowski inequality, specified as [summation from lowercase i equals 1 to lowercase l [absolute value (uppercase a hat subscript {lowercase i} minus uppercase a subscript {lowercase i})] superscript {lowercase r}] superscript {1 divided by lowercase r} less than or equal to [summation from lowercase i equals 1 to lowercase l [absolute value (uppercase a hat subscript {lowercase i})] superscript {lowercase r}] superscript {1 divided by lowercase r} plus [summation from lowercase i equals 1 to lowercase l [absolute value (uppercase a subscript {lowercase i})] superscript {lowercase r}] superscript {1 divided by lowercase r}
17 The use of simulated data can make the distinctions between the causes more clear.
18 A TIN (triangular irregular network) is made by constructing a network using municipality centers as nodes with links connecting them to neighboring locations.
19 The locations are ranked according to their measure of accessibility. The least accessible area is ranked to 0 while the highest ranked location has the value of 100.
20 Here Z-value is the height of each polygon perpendicular to the XY-plane.