University of Pennsylvania
The character of accessibility as measuring the .situation of a location in a region rather than its intrinsic qualities is emphasized throughout this paper. A brief characterization lays the basis for a sketch of data requirements, a specification of operational definitions, and a review of earlier findings. The idea of accessibility under competition is developed with several formulations, which are then compared through a synthetic example. Concluding comments suggest some guidelines and future directions.
This paper will not attempt to serve as a general review of the literature on accessibility or of general practice in measuring and using it. Rather, it is an attempt to crystallize my own experience and thinking on the subject and to present a somewhat normative view of how the term accessibility should be defined and used. The ideas presented here are an extension of my much earlier "Notes on Accessibility" (Harris 1966). This note enjoyed limited circulation, but was never published in a journal. Here I also present a view of spatial competition approached through accessibility measures.
The following first three sections of the paper discuss the general nature of accessibility, the data requirements for its calculation, and possible exact definitions. Conclusions from earlier work are given. Then these definitions are expanded to deal with spatial competition, and a synthetic example is presented with some procedural suggestions. A concluding section discusses applications of these measures in a more general context.
The Oxford English Dictionary defines access, a noun, as amongst other things: "the habit or power of getting near or into contact with. . . ." Clearly there is a mechanism governing ease of access. In dealing with locational matters, I focus on the influence of separation or distance in reducing access, which is thus universally applicable and of graded difficulty. Other impediments to access might require additional treatment.
Access is between entities, and most usually between actors, but we may conveniently in many cases replace entities with locations, usually assuming that these contain aggregates of entities and actors. The appropriateness of this aggregation must be constantly reviewed.
Accessibility is a measure of ease of access, which must be further defined. Generally, access is symmetrical: if A has access to B, then B has access to A; however, its measurement may be asymmetrical. Most common measures, scoring separation in space, define inaccessibility, or the opposite of ease of access. For the common-sense definition of accessibility I will focus on declining functions of separation and discuss them more fully in the second following section.
Access is not in general one-sided; we should not say that a given community has "good access" without specifying "access to what." For a pair of entities or locations we may define a measure of access, depending on their separation, but such a single quantity does not have much analytic power. A set of single measurements with one end fixed, such as the distance from the central business district (CBD), permits us to compare localities with respect to their centrality or their removal from some single center of interest. If we convert distance to some actual costs of access, we get a measure that may vary over time and may then provide a changed ranking of localities by centrality. If we aggregate the access measures to the CBD over the region, we can compare its centrality with that of other single facilities such as an airport, sports stadium, parks, or outlying recreational areas for which we might make similar aggregations.
These approaches help us understand the nature of accessibility, but they do not capture its essence. Most metropolitan locational decisions consider the variation across localities not only of immediate local conditions, or of the accessibility to single facilities, but also of situational variables related to the entire region. Thus there are many suburban communities with virtually identical local conditions, but with differing proximity to employment opportunities of different types and to other significant facilities. Useful and meaningful accessibility measurements provide a way to secure a synoptic view of locational qualities that result from nonlocal influences.
This view depends on three factors that our calculations will have to bring together. We imagine a beholder taking a view of the region from one location after another. First, we select a target being viewed as it is distributed over all locations in the region. Second, we identify those variations in cost of access between the viewing point and other locations that will influence choices. And third, we decide how a view will evaluate these costs as diminishing the importance of less accessible targets. I will propose that accessibility be measured, zone-by-zone, by a weighted average of access from each zone in the region to some target of opportunity in all other zones.
There are thus three essential elements needed to implement this conception: a distribution of one or more targets, a measurement of separation between zones, and a definition of the functional form of this weighted moving average that can reflect variations in attitudes toward interaction. In the next two sections I discuss the data required to support these ideas and the formal statement of a functional relationship.
As to data, measuring and computing accessibilities requires: first, a system of subareas that subdivide a larger defined region (preferably exhaustively); second, one or more sets of measurements of the pairwise separation of the subareas; and third, areally distributed data sets of people, activities, and entities of interest. I will limit the possible choices in this discussion, but many changes and extensions are possible.
Conventional analysis focuses principally on metropolitan regions, divided into traffic analysis zones, census tracts, or aggregations of these. Valid measurements of separation include airline distance, route distances, travel time, cost, lack of safety or convenience, amenity, and weighted combinations of these. These measurements may vary by mode and time of day, and according to personal choice procedures for routes. (Measuring these quantities between the centroids of subareas introduces virtually unavoidable error. A special and important case is within-area travel; its nominal zero cost is often replaced by an estimated average.)
Data needed by subarea may include at least one of the following: jobs, establishments, workers at their residences, households, dwellings, vacant land, or facilities serving shopping or recreation, as well as those serving public health and safety. These categories can, and often should, be subdivided into more narrowly defined strata, including those defined by race, income, gender, family size, and the like.
Much of the foregoing information is readily available in transportation studies, but in both these and land-use studies very few items of data are deployed in any significant detail, even when most or all of it is stored in a geographic information system (GIS). Most land-use studies make limited use of large matrices of zone-to-zone time and cost. Transportation analysis pays little attention to details of housing types and probably too little attention to detailed aspects of ridership. Yet with increasing frequency these two types of studies are becoming more interdependent, and demands of equity are side-by-side with those of pollution control in calling for more detailed analyses.
Given the very large computational load in both transportation analysis and accessibility computations, it is desirable to focus on relatively few variables for these particular activities. More work will be needed to determine what accessibility computations capture all the variables that differentially affect locational choices. Analysis of those choices, in turn, may influence the way in which transportation demand analysis interprets travel behavior. After a reasonable period of further study, the scope and detail of accessibility calculations may possibly be reduced without impairing its potential power.
The idea of accessibility as a weighted moving average of access to targets or "opportunities" may be illustrated in a very simple way, which incidentally defines a technique that can easily be adapted to the use of GIS.
Suppose that we are talking about the accessibility of various locations to retail trade customers. Imagine that we have a circular disc with a radius of one mile. We place the center of that disc on the map centroid of a zone of concern. We tally up all the customers in locations on the map within the circle. This tally represents the total accessibility to customers of that location. If we divide by the total of all customers in a relevant region, we have an average accessibility which is defined by the proportion of all customers who are within one mile of the center under study. If we were to use a larger circle, we would have a different average, the first perhaps applicable to food shopping and the second to apparel. (The graphic illustration of a circle can have as a radius only a map distance, but calculations could be based on actual time or cost.)
What we have described is a simple weighted average: every customer, in or out of the circle, is a weight; the accessibilities of customers within the circle are all 1 and those outside are all 0. The weighted sum is simply the count of those customers within the circle, and the sum of the weights is all customers. This generalizes to a series of concentric rings, to which the center has declining accessibilities, measured by their inverse radii and weighted by their populations. It can also be extended to deal with, say, purchasing power instead of customers. In each case, the result, when the weighted sum has been divided by the total target, which is the sum of the weights, including these with 0 access, is a kind of proportionate accessibility to the total "market."
We can now move the disc in any direction, centering it on another zone centroid, and we get a new average. The reader may object, and rightly, that there is a likelihood of error when we deal with areas and their centroids, rather than with the "precise" location of individuals or houses. With millions of houses in large cities, the aggregation by area is a practical necessity that may be mitigated but not eliminated.
I now seek a more flexible measure of access between pairs of locations, which as I have suggested ought to be a declining function of their separation. Unlike the GIS approach just discussed, this function should be continuous. Accessibility has a close connection with the earlier gravity models, which with their implicit connection to Newtonian gravitation used first an inverse square of distance and later a general negative power of distance. Distance itself was generalized to a composite cost, which might become route distance, time, or monetary cost, or some combination of such variables as impede access. This definition of access has been modified in later practice to a negative exponential function, which I will use. The two definitions are equivalent, because if we use the logarithm of composite cost in the exponential function, it reduces to the negative power function. (The negative power function has a singularity when the cost equals 0, while the negative exponential varies from 0 to 1.) More complicated functions may be employed.
Once we have chosen a measure of access between pairs of points, it remains to define the measure of accessibility as a weighted sum or average of these measures. For any future behavioral analysis that we may attempt, the appropriate weights would be the targets of behavioral interest-such as jobs or shops for resident workers or shoppers, or workers or customers for business establishments. Behavioral considerations such as willingness to travel or completeness of information influence the choice of parameters for any declining function of distance, but the analysis of behavior itself goes beyond the measurement of accessibility.
I will now examine in some more detail the relation of accessibility to some other behavioral concepts used in land-use and transport analysis, at the same time providing a more precise definition of accessibility itself.
Let's first set out a useful example of a definition of accessibility, designated by Wilson as "Hansen Accessibility," from Hansen's seminal paper on "How Accessibility Shapes Land Use" (1959). Hansen accessibility for a given subarea i, to all other subareas j, each containing a sub-population Wkj of some total population of opportunities Wk, is given by:
Where the impedance function f is now usually specified, with C a generalized cost and b a non-negative parameter, by:
fij = exp(-bCij) (2)
When b is small there is little impediment to access, so that accessibility is high, and vice versa.
The foregoing may be modified in a simple way that facilitates both computations and interpretation. We define the population or target of interest in each subarea as a proportion of the total population. Hence using lower case variables:
wkj = Wkj/Wk (3)
The a's now represent an average accessibility and correspond inversely to a kind of average cost, which can be readily calculated as an average cost incurred in accessing activity k from location i under the current value of b:
cik = -ln(aik)/b (6)
This average approaches 0 as b becomes very large, and becomes larger as b approaches 0. This result corresponds with the fact that b represents a measure of unwillingness to travel.
It should be clear that using a normalized population involves only a change of scale in the accessibilities and does not, in a behavioral analysis, affect most comparisons between locations of the results from computing the values of a with the same b, k, and set of costs.
It is also important to note the relation between Hansen accessibility as in either definition above, and the logit or multinomial logit model of discrete choice theory. Discrete choice is based on a concept of utility, either as a weighted sum of the logarithm of variables contributing to utility or as the product of the exponentiated variables. These are equivalent as measurements of utility, but the exponential form leads to more appropriate definitions of the probability of choice. (This form has some similarity with the Cobb-Douglas production function.) Since C is a measure of disutility, it carries a negative coefficient (-b) that would be reweighted if a specific accessibility variable were used in a behavioral analysis.
With a view to further exploration, this definition of accessibility can be related to the singly constrained gravity model, as discussed by Wilson (2000) in a limited context. This model derives the number of trips T between subareas i and j by allocating the number of originating trips at i, Oi, in proportion to the number of destinations at j, Dj, and in inverse proportion to the impedance or cost separating the two zones. In order to ensure that the correct number of trips is distributed from each origin, a proportionality factor (call it G) is introduced:
Tij = GiOiDjfij (7)
Pj.| i = Djfij/Ai (9)
Thus the proportion of trips leaving i for j is exactly j's share of the total Hansen accessibility of i.
Up to this point I have summarized, with some elaboration, the basic ideas of my earlier Note. There are two empirical findings of which we may also take account. First, I found that for fixed k and C, the results of measuring Hansen accessibilities over different b-values were closely related. Accessibilities calculated with intermediate values could be expressed with great accuracy as linear combinations of more extreme values. This finding merits further theoretical and empirical investigation.
The second finding was based on a brief exploration of accessibility in Hartford, Connecticut, at the census tract level, using five classes of employment, two modes of travel, and two values of b, corresponding to short and long trips. These 20 measures of accessibility for over 100 tracts were subject to principal component analysis. The first component defined a general accessibility that accounted for over 80 percent of the variance. Three other much smaller components accounted for nearly all the rest of the variance. They measured the difference between auto and transit accessibility, accessibility for short and long trips, and accessibility to manufacturing employment verses all other employment. This analysis is carried out for populations in an urban area and is shown in tables 1 and 2.
These two findings suggest that, although in principle scores or hundreds of measures of Hansen accessibility can be defined, the intrinsic structure of urban activity distributions and their transportation connections limits the dimensionality of its significant variation, perhaps to as few as 5 or 10 composite measures. This possibility could be explored not only in its own right in connection with locational modeling, but as a powerful means of defining and comparing different urban structures.
Anticipating the behavioral applications of these measures, I now discuss a subtle but crucial modification. In many instances, accessibility is not measured correctly if we fail to take into account the competition from other subareas for access to the target population. For instance, when considering locating a new shopping center, a developer will measure the accessibility to customers-yet if a location has high accessibility to customers, but is well served by other nearby centers, it will not be attractive. In some sense, the most attractive locations will have the greatest difference between accessibility to customers and accessibility to other shopping locations. The reverse case in this instance is not so clear. A residential area accessible to shops will not be so adversely affected by the closeness of other residences unless this leads to egregious overcrowding in the shops. Other cases are more symmetrical. The value of accessibility to jobs from home is diminished by the accessibility of the same jobs to other residents. Conversely the value to an employer of accessibility to workers is diminished insofar as the nearest workers have access to many other jobs.
In the event that a market and source of supply are in perfect spatial balance, the accessibilities to each should be similar in every location, and no site would offer opportunities for greater competitive advantage than other sites to either suppliers or demanders. (It is not clear that this concept of balance would apply under all definitions of impedance or cost, or to all levels of unwillingness to travel, as indicated by the level of the parameter b.) I distinguish three basic approaches for operationalizing this concept, all giving somewhat similar results.
First, we may directly compare the accessibilities, forming either their difference or their ratio. A particular new location is more advantageous to the supplier or the market, depending on which has the lower accessibility from this location. The behavior of locators following this rule would modify the relative accessibility in this location so as move the two sides of the market toward spatial balance. Considering only the accessibilities applying to these two activities, an area favorable for the location of one is unfavorable for the other. We may thus define two new accessibility variables. (From this point, we will usually assume that all accessibilities and target populations are normalized, without using the lower case representation.) The first of these new variables is the accessibility to population 1, discounted by the proximity of population 2, while the second is the inverse of this:
Ai3 = Ai1/Ai2 (10a)
Ai4 = Ai2/Ai1 (10b)
The second approach is one developed by Shen (1998). He calculates the accessibility of each of two activities, which we again designate as 1 and 2, from every subarea. He then recalculates the accessibility on the basis of one of the two new variables defined by
W3j = W1j/Aj2 (11a)
W4j = W2j/Aj1 (11b)
Call activity 1 employment and activity 2 workers at home. Then activity 3 will be employment discounted for access to workers at home, with activity 4 being workers at home, discounted by their proximity to employment. If we are to treat the two possible new accessibilities as a weighted average access, then the new activity variables must be normalized to sum to 1, but it is perhaps preferable to use an unnormalized variable in this case. The result would be a new measure that would vary around unity as does the first approach. This general approach may be extended to other pairs of variables, so long as the universes' activity totals are equal, which is true if both are normalized.
As a third approach, we can use the two balancing factors of a doubly constrained gravity model, as defined by Wilson (1970). In this model, trips between (say) home locations and work locations are to be distributed in proportion to the number of workers at each type of location, and in inverse proportion to the impedance between locations. However, ensuring that the totals at each location are exactly satisfied by the sums of trips requires two sets of balancing factors. We define these using a modification of the standard notation with H and B replacing A and B, and with trips, origins, destinations, and impedance factors as above:
Tij = HiOiBjDjfij (12a)
The balancing factors H and B are vectors unique to a multiplicative factor and are not readily comparable in raw form; I adjust them so that their geometric means are equal. The reciprocals of the balancing factors are modified accessibilities of the types discussed in the two previous possible procedures, in which two distributions interact. Indeed, as pointed out by a referee, the previous method as proposed by Qing is equivalent to the first iteration of one way of determining H and B. In practice, such modified accessibilities fall on both sides of unity, and their interpretation as average costs requires a special approach. In every case, they may be taken to be costs, either positive or negative, that modify the measured average cost of separation. This economic interpretation is clarified below, and may be extended by analogy to the second procedure above.
We may define two new variables, U and V, as follows:
Ui = ln(Hi)/b (13a)
Vj = ln(Bj)/b (13b)
These variables, when used in the calculation of T, show how U and V modify the costs, C, and illustrate the relation between the doubly constrained gravity model and the transportation problem of linear programming, or the Hitchcock Problem:
Tij = OiDjexp[-b(Cij-Ui-Vj)] (14)
We may interpret U and V as offsets to interaction costs, in the metric of C; these are analogous to the dual variables required to clear the market under the behavioral assumptions of this model. Trips from one origin are distributed over many destinations, unlike the case in linear programming, where the number of different active origin-destination pairs is strictly limited. If U or V is negative this indicates a locational disadvantage and if positive an advantage. With some stretch of the imagination, we may regard the H's and B's as inverse Hansen accessibilities, so that, for example, a low balancing factor corresponds to high competitive accessibility, which leads to a high positive offset.
In computing the doubly constrained gravity model, I find it useful to normalize both O and D, each to sum to unity. (An adjustment akin to normalization is necessary whenever the two populations are originally unequal in size.) Then as a result T, which does not enter directly into their definition, would in fact be normalized so that its double summation over i and j is also unity. The computation of the doubly constrained model is degenerate if any of the O's and D's are nonpositive.
The previous formulations of accessibility and the effects of competition were examined in a series of computations based on a simple hypothetical metropolitan area. I assumed an array of 35 square zones, 5 rows by 7 columns, with the central business district in the center of the lowest row of zones. Most data reported below are presented as if mapped in this array. Costs or impedances were computed as the Euclidean distances between zone centroids; no effects of congestion or mode choice were examined. The unit of distance or impedance in the computations is the separation of two adjacent zones. This seems to correspond with an actual distance of about three miles. I arbitrarily assigned three classes of workers-400,000 low income, 400,000 middle income, and 200,000 high income-to places of employment and residence, according to a pattern that was intended to be somewhat realistic. Calculations were all done with normalized employment, so that accessibility measures correspond directly with average impedances or costs. Values of b in the 0.25 to 3.0 range were employed, and results for selected values are reported in detail.
The following was the general scheme of the accessibility calculations. There are eight populations located in the model metropolis: home and workplace for each of three classes and for their totals. These populations were examined in pairs for each given b-value; there are 28 pairs, a few are of more substantive interest than the rest, but most showed similar behavior. For each pair of populations eight measures were calculated: simple accessibility and each of the three competitive measures-all of these four with respect to each member of the pair, three of them in competition that was felt through the other member. The corresponding average impedances were calculated for each accessibility measure. These calculations were the basis for a simple statistical analysis. The total output of these computations involved 5 b-values, 28 pairs of populations, and 8 types of accessibility in 2 forms, always for 35 zones: or a total of 78,400 "observations" or numbers. There was limited redundancy but a great deal of collinearity.
From the design of this experiment, it is not possible to examine the relationship of measures across modes of travel or types of impedance measures. I will ignore the relationships of accessibilities to a given population under different b-values, which tend to be linearly dependent. Similarly, I do not examine the relationships between accessibilities to different populations under various b-values, where a principal component analysis would show a somewhat less striking collinearity, but a strong dominant component with a variety of modifying factors based on different locational patterns (see Harris 1966). My principal focus is on the relationships among the three measures of accessibility under spatial competition and the stability or instability of these relations across pairs of populations. The results of this investigation lead to tentative recommendations as to the practical treatment of spatial competition in the broader context of a more extensive spatial analysis.
The process of analysis and the results are illustrated in the following tables:
The basic analysis is supported principally by data in table 4, but the features of the analysis will be outlined by considering all the tables consecutively.
Thus the most interesting finding to emerge here is the fact that the first measure of competitive accessibility, despite its lack of attention to explicit structure, may be adequate in many analyses. This would prove to be a significant advantage, because it makes it possible to bypass the very large number of pairs of populations whose competitive interaction might be considered important in location. Using either the Shen method or the doubly constrained gravity model requires calculating a new set of measures for relevant pairs of activities, and in the second of these cases, many iterations may be required. Identifying the most important pairs of locators, computing numerous competitive accessibilities, and using them in a large-scale analysis present formidable difficulties.
If an analysis is made using methods based on the theory of discrete choice in a multinomial logit model, the variable influencing utility might be the ratio of two other variables. In the actual fitting, a log-linear model is used. Thus the ratio of competitive accessibilities does not appear, and the influence of the difference of the logarithms of simple accessibilities is merged across pairs. Ten different accessibilities generate 45 different pairs, but all 55 variables can be represented by the logarithms of the 10 original accessibilities.
Stated differently, variables that might not be expected to influence some particular behavior will in fact influence it because of indirect effects. If it is desired to separate direct and indirect effects, at least in part, then a more explicit form of spatial competition must be introduced. This is only the beginning of a far more intricate process, owing to the collinearity of many important influential variables in spatial analysis.
The analysis of location involves far more than the examination of sites and their immediate vicinities-contrary to the suggestion of much planning practice and of the customary applications of GIS. The specification of location within an urban region can be accomplished with the designation of rings and sectors. However, this is vacuous to anyone (like a computer) who cannot immediately associate these designations with the contents of these segments, and with their connections with the rest of the region, and is consequently invariant over time and circumstance. The character of these subregions may be specified by variables like density and population composition, but these are again local and are in fact the result of the connections within the region interacting with local conditions.
Accessibility is a set of measures of varied form and content that makes it possible to overcome local myopia. For this, it must be defined clearly and used carefully. Accessibility is a quality of places that varies from place to place independent of any local conditions except connections with the rest of the region. It is not an intrinsic attribute or property of actors or classes of people and activities. For example, the accessibility of an area to jobs does not depend on the fact that some or most of its residents are discriminated against in employment. This dependency is defined by the class of jobs being examined. Thus accessibility's fundamental source is the distribution of properly specified activities over the region, but it also depends on the costs of the means of interaction between places, on the assumed willingness or actual capacity to employ those means, and on the separation from the place of measurement from the target activity to be accessed.
Important issues of equity and discrimination can be addressed purely through considerations of accessibility. For example, we might want to study the ability of low-income families to access low- and middle-income employment. Every zone has a measurable accessibility to these targets. We could form an average accessibility, weighted by the low-income population of each zone. Then what? The same measurement for high-income families' access to high-income jobs might show a lower average accessibility, because members of these families travel further to their jobs. A more sophisticated analysis is needed, showing the relative importance of accessibility in residential choice and the role of discrimination or the lack of transport alternatives (following Shen) in making these choices.
There is a danger in confounding the effects of accessibility and related variables. For example, density is closely correlated with accessibility, yet often one cannot be used as a proxy for the other. When accessibility runs ahead of this expected relationship, growth may be anticipated, and vice versa. Thus in a more complex model, with many locational decisions, these two variables may play different roles, and these roles may seem to shift over time as other variables change. This is only one example of the complexities of collinearity in urban analysis.
Special attention must be paid to the relationship between accessibility and actual place of work in residential location choice. Some working-class neighborhoods are concentrated like company towns around employment opportunities, and generalized accessibility plays little part in the locational choices of its residents. Conversely, many upper-income residential areas are far from employment in the CBD, with low accessibility. There is, however, a large population that seems to make location choices on the basis not only of housing prices and neighborhood variables, but on a mixture of accessibility and closeness to an actual job. Aggregated and cross-sectional studies are not adequate to sort out these decision processes, and suitable detailed longitudinal studies are required, with analyses that include accessibility.
All of these examples suggest the importance of a new and more flexible and imaginative use of accessibility measures, to which this paper has attempted to make one of many possible contributions.
Part of this research was supported through a contract from the U.S. Department of Housing and Urban Development (HUD) with the University of Pennsylvania. The author appreciates the help of HUD and wishes to thank the referees for many helpful suggestions.
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Britton Harris, 114 West Rittenhouse Street, Philadelphia, PA 19144-2714. Email: firstname.lastname@example.org.