The use of spatial separation in the measurement of transportation accessibility

Transpn. Res.-A. Vol. 29A, No. 6, pp. 421427, 1995 Copyright 0 1995Elsevier Science Ltd Printed in Great Britain. All

Pergamoo

THE USE OF SPATIAL SEPARATION IN THE MEASUREMENT OF TRANSPORTATION ACCESSIBILITY * JAMES A. POOLER Department of Geography, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5A5 (Received I April 1994; in revisedform 31 January 1995)

Abstract-In a recent article, Alien er al. (1993) [ Transpn Res. 27B, 439-4491 introduce what they call a new transportation accessibility measure. The present paper raises some questions about the history of the measure and its novelty. It is argued that accessibility indices have a longer history than the authors suggest and that the proposed index of accessibility, being the average distance among a set of locations, is neither new, nor worthy of the claims made for it. In addition, some questions are raised about the associated empirical results.

1. INTRODUCTION

Accessibility indices are well known for their important role in the fields of trip distribution and retail shopping modelling, location analysis, industrial siting, urban and regional planning, regional economics and so on. The common thread in the use of accessibility indices is in the concern with the location of sites and/or facilities relative to one another, or relative to exogenous features. Discussions of accessibility indices are in, among others, Stewart and Warn& (1958), Hansen (1959), Isard (1960), Neft (1966) Ingram (1971), McCalden (1972, 1975), Vickerman (1974), Dalvi and Martin (1976), Johnston (1979), Morris, Dumble and Wigan (1979), Rich (1980), Weibull (1980) Kellerman (1981), Keeble, Owens and Thompson (1982), Batty (1983), Chisholm (1985), Pooler (1987), Wrigley (1988), Brocker (1989), and Martin and Williams (1992). Typically, the indices discussed in the literature cited above, weight the locations according to their size, as in the common Stewart-War& (1958) measure Ai =

2

SjSib

i= 1,2,...,n

,=I j#l

(1)

or the Hansen (1959) measure Ai = f: S’eXp(-hij) ,=I ifi

i=1,2 Y-*.1 n

(2)

where the accessibility A at location i varies directly with the sixes S of the other locationsj, and inversely with the spatial separation s between i andj. Size is measured with respect to quantities such as retail floor area, population, retail sales, etc., while spatial separation is measured with respect to distance, travel cost, travel time and other similar, spatial impedance variables. Allen et al. (1993) take a different tack from that outlined above in their approach to accessibility. They develop their measure around a less common but well known approach which omits the size variable and focuses instead on the separation variable. Following *The author acknowledges gratefully the comments of the referee. 421

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Ingram (1971) they take, as a point of departure, a measure where accessibility at a given location i is defined as the sum of the spatial separations from all other locationsj to that location Ai = 2

Sij

i= 1,2 )...) n.

(3)

j=l l#i

All other things equal, in a uniform, bounded spatial distribution of points, a centralized location will tend to have a lower Ai value than a peripheral one, with eqn (3). The present note argues that the index which Allen et al. (1993) develop from eqn (3) reproduces the mean separation among the set of points or locations under consideration. The argument does not need to be demonstrated or proven, in-as-much as the authors themselves describe their index as an average travel time. However, in their paper it is portrayed as a new index of accessibility. In the first instance, this paper presents a brief consideration of accessibility indices and a review of some other common uses of the mean separation or distance. In the second instance, the proposed index is examined and, in addition, some questions are raised about the findings of the regression analysis in which the index is employed. 2. A BRIEF REVIEW OF ACCESSIBILITY

INDICES

Allen et al. state (1993, p. 439) that access indices have a history of some 30 yr. The path of investigation actually reaches back at least 50 or 60 yr. With respect to the type of accessibility measure given in eqns (1) and (2) above, the concept extends back to the 1930s. Reilly’s (1931) ‘law’ of retail gravitation is based on the idea that the proportion of trade ‘won’ by competing stores or towns is in direct proportion to their size and in inverse proportion to the square of the distance between them. The law may be written

(4) where bi, and b, represent the amount of trade drawn to towns x and y from location i; S, and S,, are the populations of the two towns; and dix and diy are their respective distances to i (Taylor, 1977). Later, the astrophysicist Stewart (1941, 1942), published similar ideas on the “influence of population at a distance”, leading ultimately to the development of the concept of the population potential at a point, given in eqn (1) above (reviewed in e.g. Rich 1980; Pooler 1987). A still earlier thread of investigation is to be found in Ravenstein’s (1885) ‘laws’ of population migration. Ravenstein’s work does not relate directly to accessibility but does contain very similar elements of thought. The second law notes, in part (p. 199), that: “The inhabitants of the country immediately surrounding a town of rapid growth, flock into it; the gaps thus left in the rural population are filled up by migrants from more remote districts, until the attractive force of one of our rapidly growing cities makes its influence felt, step by step, to the most remote corner of the kingdom. Migrants enumerated in a certain centre of absorption will consequently grow less with the distance proportionately to the native population which furnishes them.” Early reviews of the concepts of accessibility and potential are in Carrothers (1956), Haggett (1965) and Olsson (1965). Ravenstein is discussed in Grigg (1977). Closely related early work is also to be found in Von Thunen’s 1826 work (in Hall, 1966). The use of accessibility measures of the type given in eqn (3) can be traced back into the 1950s. Shimble (1953), within the context of graph theory, defines the accessibility of a vertex in a graph with respect to the sum of the distances at that vertex, as

The measurement of transportation accessibility

Ai = f:

dii

i= 1,2,...,n

423

(5)

j=l j#i

where du is the shortest path from vertex i to vertex j. In order to measure the overall graph, Shimble defines another elementary measure Ai = 2 2 dij i=l j=l I#<

i= 1,2,. ..,nj=1,2

, . . . , n.

(6)

Equation (S), representing the simple sum of the distances, is described usually as a measure of the compactness of a graph. Shimble’s (1953) work in graph theory was applied later by geographers to the study of transportation networks, and is well known (e.g. Haggett and Chorley, 1969). The work cited by Allen et al. (1993) with respect to indices of the form of eqns (3) or (5) is Ingram (1971). In citing this reference the authors apparently did not trace the source back to its original roots. Prior to stating equations, Ingram (1971, p. 101) notes that “ ...the following definitions are implicit in the analysis of connectivity matrices in applications of graph theory to the study of networks”. The source then cited in Ingram (1971), with respect to graph theory, is Haggett and Chorley (1969). In turn, the relevant source cited in Haggett and Chorley is Shimble (1953). It appears that the work of Allen et al. (1993) has its original roots in Shimble. 3. SOME ALTERNATIVE

USES OF MEAN DISTANCES IN SPATIAL ANALYSIS

Mean distances are employed commonly in quantitative spatial analysis. For example, in central place theory, geographers have long been concerned with the mean distances among communities at different levels in the hierarchy of settlements (e.g. King, 1984). The original work in this area, translated into English in 1967 (Baskin, 1966), was published in 1933 (Christaller, 1933). This theory, and the concern with mean distances in it, is very well known in geography, regional science, planning, and civil engineering. Similarly, another well known instance of the use of a spatial average is found in the work of Janelle (1969) who investigates the historically declining travel times among urban centers. The study quantifies the idea of “time-space convergence”, that is, that the average travel time among sets of urban communities is decreasing as transportation technology improves. Another related field of investigation concerns itself with the distributions of distances among geographical point patterns in bounded spaces. These are termed finite distance distributions and, as de Smith (1977, p. 332) notes “The set of distances defined by the spacing of randomly selected pairs of points in a bounded region represents a (statistical) distribution whose frequency varies with line length”. Geometric solutions to questions of mean trip length are provided in this approach. de Smith, for example, notes that Smeed (1971) determines, for trips from the city centre in a circular town, that “...the average length of trips is given by 0.725a, where a is the radius of the city”. A common perspective in spatial analysis is to take the point of view that unweighted mean distances are of less interest than those which take into consideration the distribution of supply and demand variables. In trip distribution modelling for example, the observed mean travel distance or travel cost is used almost universally in calibration (Pooler, 1994a). In elementary trip models the mean separation is employed as a calibration target such that the model mean distance is constrained to match the observed mean distance in the observed trip pattern. Different spatial systems exhibit different, observed mean travel distances, but this is not normally of much interest. An exception to the rule is in the case where the value of the average distance constraint in a trip model is manipulated deliberately, in order to examine alternative properties of the trip distribution being investigated. Such models fall into the class of “relaxed trip models” (Pooler,

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1994b). In these types of models one can identify, among other things, the minima and maxima of the constraint on the mean travel distance. Pooler (1993) defines the trip distribution associated with the minimum value of the mean distance (which just allows the constraints on the origins and destinations to be satisfied) as the “structural spatial interaction”. Such an approach to modelling manipulates the mean distance in order to investigate properties of the trip distribution. In empirical analyses where access to facilities is a concern, the norm is to employ measures of accessibility in which the sizes of origins or destinations, as well as the distances, are taken into account. Reviews of the indices (e.g. Rich, 1980; Pooler, 1987; Martin & Williams, 1992) discuss several such measures of accessibility. Indices of access at a point are considered ultimately to be measures of relative location and it is often difficult to quantify this concept in a meaningful manner without reference not only to distances, but also to the relative locations and sizes of populations, facilities, and so on. With respect to indices intended to describe accessibility over an area, an alternative to the approach proposed by Allen et al. (1993) is to construct a multivariate index, including variables such as traffic, network, and population densities, in addition to travel time. This approach creates a truly multivariate “index” in the traditional sense. Considerable additional work has been published concerning the quantitative analysis of distances and spatial patterns. Some relevant sources with respect to point patterns include Rogers (1974), Bartlett (1975), Getis and Boots (1978), Diggle (1983) and Upton and Fingleton (1985). Related work on the use of distances to measure the properties of human spatial distributions and patterns has been carried out also by Bachi (e.g. 1957, 1962, 1976) and Neft (1966). Researchers have concerned themselves also with conceptual and perceptual spaces (e.g. Golledge & Stimson, 1987), with alternative distance metrics in different forms of space (e.g. de Smith, 1977), and with the role of distance in network analysis and transportation geography (Berry & Marble, 1968; Haggett & Chorley, 1969; Eliot-Hurst, 1974). There exists also a body of work on centrographic measures (e.g. Kellerman, 1981) and another on spatial statistics (e.g. Cliff & Ord, 1981; Ripley, 1981; Barber, 1988; Griffith, 1988). The conclusion to be drawn is that in the wealth of material on quantitative analysis in the human spatial sciences, the common thread of the spatial separation variable-and in particular the mean distance-appears quite often and with little fanfare. 4. THE ACCESSIBILITY

INDEX

Having presented the accessibility given by eqn (3), Ingram (1971), under the heading “Accessibility Measures Based On Average Distance,” goes on to suggest a measure of the average distance at a point of the form ,$ dii Ai = jti

i= 1,2 ,...,

n

n.

(7)

He (1971, p. 102) notes that “The use of average distances, rather than sums of distances..., facilitates comparison with the literature on area1 moments” (see Neft, 1966). Ingram (1971) does not define n explicitly but clearly intends it to refer to the n-l distances to each single j, rather than the n points. In developing their accessibility measure, Allen et al. (1993) first propose the normalization of Ai in eqn (5) above. This produces the mean spatial separation, as follows

A;

z-21 n-

sij

1 j=, j#i

i= 1,2 ,...,

N.

In eqns (7) and (8), At and Ai represent the observed mean spatial separation between any given i and the js. Allen et al. (1993) call the average spatial separation in eqn (8) a normalized integral access measure.

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Allen et al. (1993) then go on to propose their index. They suggest that a new accessibility index is definable by finding the average of the values of the A : terms. In their words (p. 441) one can “... develop a new accessibility index for a given area by integrating the integral accessibility index over the points within the area”. This is given as

E=;$

A;.

I=1

(9)

1 nn =p cc n(n - 1) i=, j=, sij j#l

and is described as the new transportation accessibility measure. The new measure is the average spatial separation among the unweighted set of locations under consideration. As such, it is a well known property of a point pattern. In spite of this, Allen et al. (1993, p. 441) describe E as “... a normalized integration of A; (the normalized integral access measure) over location i, or a normalized double integration of the relative accessibility (the spatial separation)...“. This is an obscure manner of describing the mean distance among a set of points. With respect to the measures of access described in the second section above, the mean separation E is obtainable by dividing the Shimble measure (eqn 6) by the n(n - 1) distances under consideration. 5. INTERPRETATIONS

OF REGRESSION

RESULTS

A secondary purpose of the Allen et al. (1993) paper is to investigate the relationship between accessibility and employment growth rates for 60 United States Metropolitan Statistical Areas (MSAs). The authors calculate E in eqn (9) for each of the cities and then employ E as one of a series of independent variables in three multiple regressions, with employment growth as the dependent variable in each case. They state (p. 446) in regard to the regression results that “... accessibility has very significant impacts on employment growth rates...“. Intuitively, the relationship between these two variables does not seem obvious. One wonders about the extent to which gross, average travel times within a city have a meaningful impact on the growth or decline of employment in that city. This raises the question of the extent to which the reported significant relationship is confounded with other variables. Although the authors report that their analysis shows that multicollinearity is not a problem, the Pearson product-moment correlation between their reported E, and the log of the 1980 city populations (World Atlas, 1981) is positive and significant at the 0.05 level (two-tailed, r = 0.26). This is not high enough to present a serious multicollinearity problem but it is a significant relationship. Further explorations of additional data suggest other interesting relationships. It is well known, for example, that jobs and people are migrating to the sun belts. The Pearson correlation between the calculated mean travel time E and the observed mean annual temperature in 37 of the 60 metropolitan areas (those 37 for which data were available readily-U.S. Fact Book, 1976) is significant at the 0.02 level (two-tailed, I = 0.41).* This significant correlation exists between two variables where one would not normally expect to see such a relationship. Average urban travel time (or distance) is certainly not expected *The subset of 37 cities includes: Akron, OH; Albany, NY; Allentown, PA; Anaheim, CA; Atlanta, GA; Baltimore, MD; Birmingham, AL; Boston, MA; Buffalo, NY; Charlotte, NC; Chicago, IL; Cincinnati, OH; Cleveland, OH; Columbus, OH; Dallas, TX; Dayton, OH; Denver, CO; Detroit, MI; Fort Lauderdale, FL; Gary, IN; Grand Rapids, MI; Greensboro, NC; Hartford, CT; Houston, TX; Indianapolis, IN; Kansas City, MO, Los Angeles, CA; Louisville, KY; Memphis, TN, Milwaukee, WI; Minneapolis, MN; Nashville, TN; Newark, NJ; New Orleans, LA; New York City, NY; Oklahoma City, OK; Orlando, FL; Philadelphia, PA; Pittsburgh, PA; Portland, OR; Providence, Rl; Richmond, VA; Rochester, NY; Sacramento, CA; St. Louis, MO; Salt Lake City, UT; San Antonio, TX; San Diego, CA; San Francisco, CA; San Jose, CA; Seattle, WA; Syracuse, NY; Tampa, FL; Toledo, OH; Tulsa, OK; Washington, DC; and Youngstown, OH. Three outliers, Honolulu, Miami and Phoenix were omitted from the analysis. These cities have very warm annual average temperatures in comparison to their average travel times.

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