Explorations into the relationship between spatial ...

Environment and Planning A, 1980, volume 12, pages 187-201

Explorations into the relationship between spatial structure and spatial interaction^ DA Griffith Department of Geography, State University of New York at Buffalo, Buffalo, New York 14260, USA

K G Jones Department of Applied Geography, Ryerson Polytechnical Institute, Toronto, Ontario M5B 1E8, Canada Received 25 June 1979

Abstract. This paper explores the relationship between spatial structure and spatial interaction at the intraurban level. To examine this relationship an experimental framework is designed based on the application of a doubly constrained entropy-type gravity model to journey-to-work data for twentyfour Canadian urban areas. The study demonstrates that distance-decay exponents are strongly influenced by geographic structure and the geometry of origins and destinations. As such, both the influence of map pattern and the friction of distance should be explicitly incorporated into spatial interaction models. The paper also explores the impact of city size and the nature of the economic base of the urban area upon distance-decay exponents. Introduction

The attenuating effect of distance has long been recognized as an important factor of spatial interaction. Finite geographic landscapes impose two restrictive constraints on this distance-decay factor though. On the one hand, interactees are not completely free to move amongst areal units, but rather their movements are channelled by the underlying spatial configuration of these units. On the other hand, the number of interactees supplied by origins is enhanced or diminished in accordance with attributes displayed by neighboring origin locations. Similarly the number of interactees received by destinations is enhanced or diminished in accordance with attributes displayed by neighboring destination locations. The former phenomenon may be labelled a geographic infrastructure effect, whereas the latter phenomenon can be called a spatial interdependence or autocorrelation effect. Hence spatial interaction is channelled within a landscape through the latent spatial structure, or the way in which geographic infrastructure governs the pattern of spatial interdependencies. Relationships between the convolution of these two constraints and both the rate of distance decay and the estimation of a distance-decay parameter have been the topic of a recent controversy (Johnston, 1973; 1975; 1976; Curry, 1966; 1972; Cliff et al, 1974; 1975; 1976; Curry et al, 1975; Sheppard et al, 1976; Sayer, 1977). The purpose of this paper is to help resolve this controversy. In other words, the basic research problem being addressed here may be stated as follows: is the rate of distance decay in spatial interaction models independent of the spatial structure associated with corresponding origins and destinations? The nature of this problem may be exemplified by an experiment in which the relationship between distance decay and spatial structure is to be articulated. First consider the classical gravity model of spatial interaction, which may be written as Fn = KP?Pfd? ,

(1)

f This paper was presented at the annual meetings of the Association of American Geographers held in New Orleans during April 1978.

188

D A Griffith, K G Jones

where Fjj Pu Pj

is the flow from origin areal unit / to destination areal unit /; are the respective 'populations' of origin areal unit / and destination areal unit /; dfj is the 'distance' separating areal units / and /; and K, a, j3, 7 are parameters. Next place all flows Fq (i = 1, 2,..., n; j = 1, 2,..., n) into an origin-destination matrix F, and denote the observed 'population' vector for the origins by PQ, the observed 'population' vector for the destinations by Pd, and the interareal unit distance matrix by D. Spatial autocorrelation parameters pQ and pA can then be attached to their respective vectors P0 and P d . Now spatial-structure effects can be removed from these two vectors, resulting in the corresponding vectors P 0 and P a , by conducting the following mathematical manipulations: PI = ( I " PoW)P0

and

P*d = (I -

PdW)Pd

,

(2)

where matrix W is the stochastic version of a connectivity matrix C depicting the spatial arrangement of origins/destinations. In equations (2) the expressions I - p 0 W and I —pdW are linear spatial operators associated with spatial autocorrelation. Moreover their inverses are mechanisms which induce geographic distributions to show map patterns. Similarly spatial-structure effects can be removed from the flows matrix, F, by operating both on its rows and its columns. Destination effects can be extracted by the following mathematical transformation, where T denotes matrix transpose: (I-pdW)FT. Then, to account for the origin effects, d-PoW)[(I-p d W)F T ] T = ( I - p 0 W ) F ( I - p d W ) T = F* .

(3)

Consequently a synthetic gravity model from which the 'pure' friction-of-distance parameter, c, could be calculated may be written as F*

=K(P*)a(P*)bdf.

(4)

Given equations (2), (3), and (4), the experiment concludes by first calibrating equations (1) and (2)-(4) and then testing the following null hypothesis: H0: c - 7 = 0 . Failure to reject this hypothesis implies that no relationship exists between the rate of distance decay and the prevailing spatial structure. Rejection of this hypothesis permits one to draw the inference that a relationship does exist. An entropy model of journeys to work The general form of t h e gravity model utilized in this study may be written as T9 = AiOtBjDj

exp(-7ty),

where Tjj is a measure of t h e interaction between origin i and destination /"; A h Bj are balancing or normalizing factors for the origin i and t h e destination / respectively; Ot is the total number of trips generated from origin i; Dj is t h e total number of trips terminating in destination /; ctj is a measure of t h e distance, or the cost of travel, between origin i and destination/; and, 7 is the distance-decay parameter (Wilson, 1970).

(5)

The relationship between spatial structure and spatial interaction

189

The Furness method was employed to calibrate equation (5) for journeys to work in twenty-four selected Canadian urban centers (1) (Hall, 1976; Ewing, 1974; 1978; Evans, 1970; Kadas and Klafszky, 1976). These urban areas are either Census Metropolitan Areas or Census Areas. The accompanying data sets were compiled on the basis of those census tracts falling inside the boundaries of these urban places, essentially ignoring those outlying census tracts that might be in the labor-force catchment areas of these places. In most cases the data values for these latter tracts were zero. Thus each data set included an origin-destination flows matrix and two column vectors. The first vector was for row totals, and indicated the number of workers residing in census tract i (i = 1, 2, ..., n). The second vector was for column totals, and indicated the number of workers employed in census tract/ (/ = 1, 2, ...., n). Meanwhile simple straight-line measures between census-tract centroids were used for inter-census-tract distances. The results of calibrating equation (5.) with these data sets appear in table 1. Clearly with an average percentage of variance accounted for of 83-7, these calibrations furnished an acceptable set of models. As an aside, these aforementioned data sets were supplemented with connectivity matrices that represented geographic configurations of census tracts. A similar study has been released by Hutchinson and Smith (1977), but, although their findings generally were consistent with those trends found in table 1, the approach they used differed. Hutchinson and Smith used an attraction-constrained entropy model, included all census tracts lying in the labor catchment areas of urban Table 1. Parameters of the journey-to-work models for selected Canadian cities. City

Brantford Calgary Edmonton Halifax Hamilton Kingston Kitchener-Waterloo London Oshawa Ottawa-Hull Peterborough Regina St Catharines Saint John St John's Sarnia Saskatoon Sault Ste Marie Sherbrooke Thunder Bay Trois Rivieres Victoria Windsor Winnipeg Average (

Distance-decay exponent -0-074 -0-225 -0-284 -0-419 -0-234 -0-354 -0-253 -0-249 -0-228 -0-192 -0-227 -0-257 -0-241 -0-182 -0-262 -0-193 -0-297 -0-188 -0-351 -0-196 -0-382 -0-132 -0-180 -0-218 -0-242

Mean trip length (km)

Average origin/destination weight

r2

2-7 5-0 4-3 3-1 5-1 1-9 3-4 3-7 3-6 5-7 2-4 3-0 4-5 3-8 2-2 3-4 3-1 3-7 2-1 3-9 2-4 4-9 4-3 5-1

1-110 2-140 2-396 2-557 2-435 1-502 2-326 1-811 1-752 2-692 1-455 1-605 2-806 1-571 1-388 1-311 1-779 1-562 1-607 1-680 2-089 1-511 1-623 2-193

0-839 0-848 0-780 0-837 0-799 0-860 0-888 0-862 0-870 0-796 0-938 0-900 0-929 0-771 0-842 0-922 0-903 0-935 0-783 0-853 0-895 0-585 0-719 0-731

3-6

1-871

0-837

' The data set excluded seven urban centers. Toronto, Montreal, and Vancouver were excluded for reasons of their size and the associated costs of computer time. Guelph, Chicoutimi-Jonquiere, and Sudbury were excluded because of incomplete data files. Quebec City was omitted on the basis of the lack of spatial continuity of census tracts.

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places, used minimum-path arterial road distances, and obtained parameter estimates by the golden-section search technique (Baxter and Williams, 1975). Estimates of the balancing factors At and Bj associated with equation (5) are of special interest here. The interpretation of these factors has been addressed by a number of authors. Wilson (1970) suggests that balancing factors are related to accessibility. Kirby (1970) maintains that these normalizing factors can be viewed as mean values of the inverse of the deterrence function. Ferragu and Sakarovitch (1970) have suggested that the At and Bj values relate to the residual effects once an overall 'cost of movement' factor has been taken into account. More recently Cesario (1977) has viewed normalizing factors as propensity measures both for emissiveness of origins and attractiveness of destinations, and holds that ultimately these factors can be related to socioeconomic correlates of behavior patterns. In keeping with this last interpretation, these values take on a special significance in this study because by definition they depend upon the rate of distance decay that is present. Moreover the question to be addressed may be phrased as follows: do the attraction and emission weights display a geographic pattern? In turn equation (5) focuses attention on the notion that the number of interactees supplied by an origin is enhanced or diminished in accordance with the propensity of emissiveness of its neighboring origin locations, and on the idea that the number of interactees received by a destination is enhanced or diminished in accordance with the propensity of attractiveness of its neighboring destination locations. Spatial structure of journey-to-work components Spatial structure has been loosely defined in the opening section of this paper as the convolution of geographic interdependence and spatial configuration. In this section, measures of geographic interdependence exhibited by origin and destination attributes will be calculated for the selected Canadian urban centers. Also a quantitative description of spatial configuration will be developed and surveyed for geometric patterns. The purpose of deriving these sundry numerical measures is to be able to relate them to corresponding distance-decay exponents (see table 1). In turn the evaluation of these relationships permits a conclusion to be drawn regarding whether or not the rate of distance decay in spatial interaction models is independent of spatial-structure components. These numerical measures also shed light upon whether or not attraction weights At and emission weights Bj consistently display geometric patterns over urban areas. Spatial structure may be defined more specifically as the geometric linkages between areal units over which forces pulsate that bring about spatial interrelationships. Various methods can be employed to measure features of geographic landscapes generated by geometric linkages. One prominent feature is physical in nature and is concerned with the arrangement of areal units. Two questions can be posed here. What is the overall level of physical structure? How does a given areal unit's position relate to the positions of other areal units? Answers to these two questions can be obtained from an analysis of a connectivity matrix depicting areal-unit juxtapositions —this sort of analysis is explained in detail in Tinkler (1972; 1974; 1976) and Taaffe and Gauthier (1973). Such a matrix was alluded to in the introduction of this paper. The principal eigenvalue, Xp, of this matrix provides an index of physical structure. Matrix theory indicates that for a given network of areal units the maximum value for \ p would be n - 1, where n denotes the number of areal units, and the minimum value would be 1. Thus an index summarizing physical structure can be constructed by dividing Ap by n - 1. The purpose of this division is to remove a size bias present in the magnitude of the principal eigenvalue. The maximum value Ap = n - 1 would be achieved when every unit is in juxtaposition


191

with all other units. The minimum value Xp = 1 would be achieved when each areal unit is juxtaposed with only one other unit. By assuming a minimally integrated network of areal units (that is, a linear arrangement of units), the real minimum value of the principal eigenvalue would be approximately 2. Consequently a physicalstructure index close to one signifies a compact arrangement having maximum connectivity, whereas an index value close to zero signifies a dispersed arrangement having minimum connectivity. The principal eigenvector of the connectivity matrix provides a measure of relative positioning for each areal unit. Matrix theory guarantees that all the elements of this vector will be nonnegative real numbers. Standardizing these individual numbers by the sum of their values produces an index whose range lies between zero and unity. As the index value approaches one, the corresponding areal unit is more centrally located within the arrangement of units. As this value approaches zero, the corresponding unit is more peripherally located within the arrangement. Therefore the principal eigenvector serves as a configuration index. Connectivity matrices were constructed for the selected Canadian cities, and principal eigenvalues and eigenvectors were extracted from each of these matrices. Results for this part of the analysis are presented in table 2. Here the physicalstructure index has been multiplied by 100 so that its range lies between 0 and 100. Clearly smaller cities, such as Peterborough and Brantford, display a more fully articulated arrangement than do larger cities, such as Hamilton and Winnipeg. This finding is not surprising since, as the number of areal units increases, the probability of each unit being connected to all other units will tend to decrease. Consequently the physical-structure index reflects the degree and manner of partitioning an urban area has undergone. Next the geographic distributions of the principal eigenvector values were surveyed for geometric patterns. This type of probing can be accomplished through the use of two additional indices (thoroughly discussed in Moran, 1948; Geary, 1954; Cliff and Ord, 1973). The Moran coefficient, M, is the ratio of covariation between juxtaposed values to the variation amongst values for a given phenomenon. Similarly the Geary ratio, G, is the ratio of the sum of squared deviations between juxtaposed values to the variation amongst values for a given phenomenon. These two indices measure the nature and degree of spatial autocorrelation displayed by a geographical distribution. As M approaches 1 and G approaches 0, positive spatial autocorrelation is detected, and thus the geographic distribution under study exhibits a geometric pattern in which similar values tend to cluster. As M approaches-1 and G approaches 2, negative spatial autocorrelation is detected, and hence the geographic distribution under study exhibits a geometric pattern in which dissimilar values tend to cluster. As M approaches —{n - 1 ) _1 and G approaches 1, zero spatial autocorrelation is detected, and hence the geographic distribution under study fails to exhibit a geometric pattern of any type. Because these two indices rely upon the connectivity matrix C, with entries c^-, identifying properties of this matrix becomes an increasingly useful task. Returning to the principal eigenvectors, on the average the selected Canadian cities showed a moderate to strong tendency for similar configuration-index values to cluster (see table 2). This finding is not surprising since one would expect, given the geometry of most urban areas, that the most highly connected census tract(s) would be surrounded by the next most highly connected tracts. In turn these tracts would be located near ones displaying a slightly smaller degree of connectivity. This ring pattern would continue until the periphery of an urban area is reached. The boundary effect along a periphery would almost guarantee a reduction in connectivity. For each of the twenty-four Canadian cities, evidence exists which indicates that

Table 2. Spatial-structure characteristics for selected Canadian cities. City

Physicalstructure index

Spatial-structure indices configuration index M

G

distribution of workers

distribution of jobs

M

M

G

G

entropy-model origin weights

entropy-model destination weights

M

M

G

1-2144 ^~0"800 T0429 0-2732** "0-6503** "(K2822 0-3970*** 34-2838 0-2423'* "¥6394* Brantford 0-0404 1-0881 0-1810*** 0-6733*** 0-6054*** 0-3632*** 0-7578*** 7 0676 0-7570*** 0-3837*** Calgary 0-3595*** 0-8310** 0-2357*** 0-9865 0-5174*** 0-3346*** 0-0590 6 5421 0-8434*** 0-3027*** Edmonton 0-8805 0-4925*** 0-7260** 0-3814*** 0-2297*** 0-4740*** 0-0939 11 6850 0-6969*** 0-2962*** Halifax 0-3320*** 0-6691*** 0-7468*** 5 4764 1-1517*** 0-4916*** 0-2286*** 0-6673*** 0-2186*** 0-7570*** Hamilton 0-8425 0-2424** 0-7437 0-5244*** 0-4042*** 0-0889 0-2604** 26 6656 0-3651*** 0-6245** Kingston 0-9175 0-4781*** 0-3991*** 0-0117 0-1479* 0-9616 0-2819*** Kitchener -Waterloo 12 6090 0-8784*** 0-2112*** 1-0749 0-0172 1-0761 0-5452*** 0-2068*** 0-4852*** 10 5733 0-6427*** 0-3331*** -0-0796 London 0-1295 0-8189 0-0631 0-8690 0-1426 1-0332 0-6639*** Oshawa 23 7257 0-4505*** 0-4827*** 0-8958 1-5845*** 0-4757*** 0-5375*** 5 6585 0-6999*** 0-4073*** 0-0726 0-3892*** 0-0619 Ottawa-Hull 35 0110 0-3229*** 0-6081** -0-1645 0-0150 1-1320 0-2415** 0-3619*** 0-9031 0-1147 Peterborough 0-2852*** 0-8687 0-5618*** 0-3667*** 0-3315*** 0-1004 0-8540 Regina 16 5761 0-3054*** 0-6012*** 1-0362 0-8732 0-3304*** 0-7606** St Catharines 10 3179 1-0260*** 0-1535*** -0-0147 0-0099 0-4014*** 1-1050 18 2068 0-3631*** 0-7359* 1-1187 0-0194 0-3857*** Saint John 0-0226 0-2161** 0-3734*** 1-1348 0-7954 -0-1581 0-2080** 0-3765*** St John's 0-1172 0-3200*** 29 6270 0-3896*** 0-4659*** 0-8708 -0-0054 0-5989** 0-4399*** 0-3751*** -0-0958 Sarnia 0-0454 31 1734 0-3089** 0-5141*** 22 7533 0-4999*** 0-3989*** 0-1181 0-9183 0-1570* Saskatoon 0-1598* 0-7844 0-3298*** 0-4940*** 1-0938 0-0088 0-2975*** 0-6302** 28 1682 0-3611*** 0-8118 1-0409 0-2295** -0-1450 Sault Ste Marie 0-7413 -0-0818 0-9649 0-2099** 0-7054* Sherbrooke 24 4361 0-4866*** 0-5133*** 0-0494 0-3032*** 0-9072 0-0705 0-8317 24 0599 0-6453*** 0-5860*** -0-0546 0-2579*** 0-9043 0-1699* Thunder Bay 0-0651 25 5905 0-3396*** 0-8191 0-1834* 0-6956* 0-4347*** Trois Rivieres 0-2491** 0-9617 0-2993*** 0-3055*** 0-1328*** 0-2651*** 0-4271*** 0-1106 1-4996*** Victoria 0-0130 13 5380 0-5466*** 0-3877*** 1-0580 0-9430 0-0647 0-5197*** 0-1493*** Windsor 12 •1828 0-6778*** 0-3914*** 0-0350 0-4659*** 0-3684*** 0-8384** 0-2187*** 0-7077*** 5 •6685 0-9748*** 0-2311*** 0-1159** 0-8666* 0-3197*** Winnipeg 0-1158 0-9717 0-3535 18 3999 0-5823 0-4746 0-0336 0-8831 0-3489 0-4833 Average * Significantly different from the value indicating no autocorrelation [that is, Af = - ( « - l)" 1 ; G = 1], given a two-tailed setup, at the 10% level ** at the 5% level; *** at the 1% level. Note: the standard errors of M and G have been calculated under the assumption that each of the 120 variables in question is normally distributed, A cursory examination of these distributions has suggested that approximately 70% of them do not deviate significantly from a normal curve.

G 0-4721*** 0-3153*** 0-4070*** 0-3372*** 0-1568*** 0-7272 0-4808*** 0-3836*** 0-4327*** 0-5435*** 0-4708*** 0-5551*** 0-7206*** 0-3236*** 0-5187*** 1-0160 0-4914*** 0-6667** 0-5900*** 1-0063 0-6178** 0-1862*** 0-1926*** 0-5907*** 0-5084


193

positive spatial autocorrelation is contained in the map pattern of a configuration index. Because M is very close to 1, and G is very close to 0, St Catharines, Winnipeg, and Kitchener exhibit the most striking geometric patterns. In contrast, since the G values for Saulte Ste Marie and Trois Rivieres are not significantly different from 1, and since certain index values for Brantford, Peterborough, and Saint John are significant at a level greater than 1%, these cities exhibit the most obscure geometric patterns. Nevertheless, all twenty-four patterns are quite pronounced. Other prominent features of the journey-to-work phenomenon include the geographic distributions of workers, of jobs, and of entropy-model origin and destination weights. M and G also have been used to summarize those geometric patterns displayed by these four distributions. Once again results for this part of the analysis are presented in table 2. On the average the selected Canadian cities showed a very weak tendency for similar numbers of workers or jobs to cluster. In fact very few index values furnish evidence for the presence of any type of nonzero spatial autocorrelation. What tendency exists seems to be more prominent in the distribution of jobs than in the distribution of workers. Although Brantford, London, and Sault Ste Marie each yield index values for the distribution of workers in which a slight tendency has been observed for dissimilar numbers of workers to cluster, none of these three sets of values are significant. Hamilton exhibited a somewhat marked pattern, with both of its index values being significant at the 1% level, in which similar numbers of workers tended to cluster. Meanwhile, although Brantford and St John's each yielded index values for the distribution of jobs in which a tendency has been detected for dissimilar numbers of jobs to cluster, neither of these two sets of values is significant. Contradictory evidence has been obtained for Ottawa-Hull and Victoria. Both of these cities yielded nonsignificant M values in which a slight tendency was observed for similar numbers of jobs to cluster, as well as G values that are significant at the 1% level and provide evidence for the clustering of dissimilar numbers of jobs. Presumably these two cities have the most complex geometric patterns of job distributions. The most striking geometric patterns are exhibited by Calgary, Halifax, Hamilton, and Winnipeg. However, more than half of the sets of index values are not significant, meaning that most geometric patterns displayed by the spatial distribution of numbers of jobs are obscure. The final two salient components of journeys to work are the trip-making propensity of origin i per unit of destination accessibility weights, and the tripdrawing propensity of destination / per unit of origin accessibility weights associated with a doubly constrained entropy-type gravity model. Values for these weights have been calculated in accordance with the model discussed in the preceding section and are summarized in table 1. Except for the Oshawa origin weights and the Sarnia destination weights, the index values tend to be significant, primarily at the 1% level, furnishing evidence that on the average the selected Canadian cities show a moderate to strong tendency for similar weights to cluster. Although Sarnia yielded index values for the distribution of destination weights in which a slight tendency has been observed for dissimilar weights to cluster, these index values are not significant. Osiiawa origin weights and Thunder Bay destination weights yielded inconsistent index values, but the G values suggesting the presence of negative autocorrelation are not significant. Presumably these two cities have the most complex geometric patterns for the spatial distribution of origin and destination weights respectively. Brantford, Sherbrooke, Thunder Bay, and Trois Rivieres exhibited the most subtle geometric patterns of origin weights, and Kingston, Peterborough, Sault Ste Marie, and Thunder Bay exhibited the most subtle geometric patterns of destination weights, since each of these cities has either an insignificant index value or no index value that is significant at the 1% level. Meanwhile Calgary, London, and Windsor displayed the most striking

194


geometric patterns of origin weights, and Calgary, Hamilton, and Oshawa displayed the most striking geometric patterns of destination weights. Each of these last six spatial distributions yielded a relatively high M value accompanied by a relatively low G value, with all of these index values being significant at the 1% level. This last speculation may be attributed to the scale-of-analysis problem. In other words, the size of areal units in smaller urban centers may mask geometric patterns that are comparable with those uncovered in larger urban centers. The next question to be addressed in this study revolves around the ideas of whether or not the selected Canadian cities, when discriminated amongst on the basis of work characteristics, demonstrate geometric patterns having latent urban-system dimensions. This question complements those asked in this section. City classification: the Canadian urban system Evidence presently exists to support the notion that the Canadian urban system can be classified into distinct groups based upon economic characteristics (Maxwell, 1965; King, 1966; Britton, 1973; Marshall, 1975). In the present study twelve employment categories, seven centrality indices, four manufacturing variables, and two physical attributes were used to classify the thirty-one major urban centers in Canada. Measures for these variates were obtained for the year 1971. The method used to classify these cities by use of this data set was a principal-components analysis. The results of this analysis appear in table 3. Simple structure was attained by use of the varimax criterion. Three general city categories were identified by this procedure. Together they accounted for 99% of the variation exhibited by the thirty-one cities. Membership in these groupings was determined by the component-loadings pattern. First, since a correlation coefficient is considered to be significantly different from zero at the 1% level, given a sample of twenty-five variables (N = 25) and a two-tail setup, if it exceeds 10-505 |, only those component loadings greater than 0-505 in absolute value have been considered meaningful. This particular information is used to label the three urban classes. Second, because city membership in a single group is desirable for analytical purposes, the cutoff point was increased to 10-7001 in order to eliminate multiple memberships. The loadings greater than 10-7001 are in italics in table 3. This particular information is used to study the relationship between city class and distance decay. Group 1 comprises major 'heartland' metropolises and major 'hinterland' centers. Group 2 consists of intermediate manufacturing centers. Group 3 appears to contain smaller local service centers that also exhibit a pronounced manufacturing sector. In order to examine the relationships among these three urban classes, the distancedecay exponent appearing in table 1, and the structural indices appearing in table 2, numerical indices denoting membership in each group need to be constructed. Three dummy variables, Xl3, Xl4, and Xl5, have been created to designate group 1, group 2 and group 3 centers respectively. The question of whether or not city type induces differential effects upon distance decay can be answered through the application of the analysis-of-variance technique. Moreover, is the total variation displayed by the distance-decay parameter mainly due to variation occurring between the three urban groupings? Or is this total variation primarily due to variation occurring within these three urban groups? Analysis-ofvariance results appear in table 4. Clearly, since the F ratio is not significant (significant values at the 5% and 1% level a r e ^ 2 , 2 1 , 0 0 5 ~ 3-47 a n d F 2 , 2 , 0.01 = 5-78),

the data set in question furnishes no evidence to suggest that the average distancedecay exponent value is different for these three urban groups. In other words, the basic source of variation comes from within the urban groups not from between them. Thus city type does not produce a differential effect upon the deterrence of distance.

195


Table 3. Principal-components loadings for city classification. City

Unrotated pattern C\

Varimax rotated pattern

Ci

C3

905 -0-084 414 Brantford 0-071 883 459 Calgary -0-010 622 783 ChicoutimiJonquiere 451 -0-022 891 Edmonton 583 -0-021 810 Guelph 278 0-005 959 Halifax 386 0-004 915 Hamilton 508 -0-143 845 Kingston 017 -0-001 980 KitchenerWaterloo -0-052 942 327 London 0-025 831 555 Montreal 052 -0-009 988 Oshawa 375 -0-006 924 Ottawa-Hull 593 -0-077 Peterborough 797 448 0-043 Quebec 891 257 Regina 0-292 917 374 St Catharines -0-116 888 162 Saint John -0-188 962 296 -0-019 St John's 951 559 -0-219 795 Sarnia 735 0-538 411 Saskatoon 381 -0-135 910 Sault Ste Marie 476 0-228 849 Sherbrooke 460 -0-234 Sudbury 846 404 -0-154 Thunder Bay 898 562 0-021 825 Toronto 380 0-621 Trois Rivieres 685 521 0-015 853 Vancouver 219 0-034 972 Victoria 197 -0-037 Windsor 962 497 866 0-043 Winnipeg 1-056 23-760 5-891 Eigenvalue Cumulative percentage of variance accounted for 76-6 95-6 99-0

C\

Ci

400

901

968

223

174

948

964

260

220

942

902 940

419 315

290

945

723

644

920 990 774 939

390 135 603 331

C3 161 092 263 006 248 085 051 118 155

201

957

966 870

244 326

412 604

870 788

907

408

216 152 424 332 318 398

972

016 017 132 046 195 070 358 178 008 057 051

632

758

895 820 939 909

105 446 020 090 010

990 984 874 849 979

126 534 176 462 491 193

019 129 065 054

15-901

12-916

1-890

51-3

93-0

292

793

99-0

Variables used in the analysis (N = 25) Others: Disposable income Employment: Management Processing Total population Teaching Chemicals Checks cashed Clerical Food and beverages Building permits Sales Textiles Shipments Printing Machinery Value added Services Fabrication Newspaper circulation Note: Q denotes component /.

Retail sales Plants Employees Retail stores Land area Population density

Table 4. Analysis of variance of distance-decay parameters for city classifications. Source of variation Between classes Within classes Total

Sum of squares 1

0-223 x 10" 1-177 xlO" 1 l-400xl0_1

Degrees of freedom 2 21 23

Mean squares 1

0-112x 10" 0-056 x 10"1

F ratio 1-991

196


This basic implication of no difference between average city-class distance decay, coming from table 4, appears to be a robust one. The distribution of members for each of these three groups appears to conform to a normal distribution. This contention has been tested with the Shapiro-Wilk statistic (Shapiro and Wilk, 1965), whose respective value for each of the three urban classes is 0 • 8764, 0 • 9033, and 1 • 0000. None of these three values is significantly different from 1 -0000 at the 5% level, resulting in a failure to reject the null hypothesis that each group comes from a normal distribution. Further, no evidence exists to doubt that the homogeneity-of-variance assumption is satisfied. In this instance an F ratio value of 1 -835 was calculated, with the critical value beingi7107j2,o-35 ~ 3-08. Finally three empirical tendencies in urban-system characteristics are noteworthy. Returning to tables 1 and 2, as city size increases, the distance-decay exponent tends to fluctuate at random. However, as mean trip length increases, the distance-decay exponent tends to become larger in absolute value. Likewise, as city size increases, the average origin/destination weight tends to increase. More than likely this third urban-system characteristic may be attributed to the increase in geometric complexity that accompanies an increase in size. Relationships between spatial structure and distance decay Thus far three categories of spatial structure have been explored. The first type involved a distance-decay mechanism and dealt with the manner in which geographic flows are structured. The second type had to do with geometrical patterns resulting from the way in which areal units are structured. The third type referred to structurings arising from variable covariations among urban centers. The major objective of this section is to investigate prominent patterns of relationships between the first category and the last two categories. Because this portion of the analysis is primarily exploratory, those variables compiled earlier have been subjected to a battery of multivariate techniques in an attempt to massage these data and illuminate the relationships in question. First a principal-components analysis was carried out. The input data included the distancedecay parameters for a doubly constrained gravity model, presented in table 1, the eleven spatial-structure indices, presented in table 2, and a set of dummy variables denoting the city classification groups, presented in table 3. Since principal-components analysis is a linear transformation permitting the extraction of linear combinations of the original variables, these variables needed to be modified so that they were approximately pairwise linearly related. In this case the base variable used in all cases to implement this modification was the distance-decay exponent. Consequently the input data consisted of the distance-decay parameter, the fourth power of the physical-structure index, the inverse of M for the physical-configuration index and the origin and destination weights, the square of G for the distribution of workers and jobs and the origin weights, the logarithm of G for the physical-configuration index, M for the distribution of workers, the square of M for the distribution of jobs, the inverse of the square of G for destination weights, and dummy variables for city classification groups 1 and 2 of table 3. A dummy variable for city classification group 3 was not included because the absence of its members in groups 1 and 2 already denoted their presence in this third group. Obviously these transformed variables are not amenable to simple substantive interpretations. The principal-components analysis of these fourteen variables will be used to evaluate the following three hypotheses: (1) distance decay is independent of the spatial structure of journey-to-work elements; (2) there exists a fundamental geometric dimension relating to the geographic distribution of workers/jobs; and

197


(3) intraurban spatial structure is independent of the interurban system. Each of these hypotheses is expressed in null form. Results of the principal-components analysis are presented in table 5. Only important components are included in this table; the criterion being used to identify them was whether or not their accompanying eigenvalues exceeded unity. Based on this criterion five components were extracted that together accounted for 77% of the variance displayed by the fourteen original variables. Then a varimax rotation was utilized in order to obtain a simple structure solution which simultaneously maximized the covariation accounted for by each of these five components. Consequently the distribution of variance accounted for was shifted from 32%, 16%, 11%, 10%, and 8%, to 25%, 17%, 14%, 12%, and 9%. Clearly the only major allocation change brought about by this rotation was experienced by the first component. Finally, component loadings greater than or equal to 10-4001 (indicated by italic numbers in table 5) were used to label these five components. This critical value was selected because it roughly corresponds to the critical value of a correlation coefficient, with twenty-two degrees of freedom, for a two-tail test having a 5% level of significance. The following labels have been attached to these five components in accordance with the generic names of those variables having a high loading on each component: Component 1 (Cx): interface between intraurban and interurban systems—physicalstructure index (X2), city classification (Xl3, X^), and spatial structure of the physical-configuration index (X3, X4) and origin weights (X-9)\ Component 2 (C2): origin geometry—city classification (X14), and spatial structure of the distribution of workers (X5, X6) and origin weights (X9); Table 5. Principal-components loadings for spatial structure and distance decay (IV = 24). Variable

Unrotated pattern C\

C2

C3

Varimax rotated pattern C4

Cs

Distance-decay parameter X,(y) 0-391 -0-735 0-166 0-044 0-419 Physical-structure index X2(k*) 0-818 0-157 0-073 0-195 -0-052 Physical-configuration index X3(M~l) 0-780 0-263 0-159 0-424-0-041 X*(lnG) 0-601 0-364 0-366 0-357 0-108 Distribution of workers X5(M) -0-725 0-519 0-105 0-070 0-231 X6(G2) 0 - 5 0 8 - 0 - 3 9 8 - 0 - 4 4 8 0-110-0-113 Distribution of jobs Xn{M2) -0-568 0-383 -0-294 0-234 -0-241 X8(G2) 0-034 -0-512 0-699 -0-110 -0-201 Entropy-model origin weights l X9(M~ ) 0-090 0-683 0-470-0-112 0-174 Xl0(G2) 0-165 0 - 1 8 3 - 0 - 0 8 1 - 0 - 7 0 8 0-526 Entropy-model destination weights Xn (M_1) 0-035 -0-004 0-421 -0-477 -0-595 Xn (G~2)-0-448 -0-283 0-333 0-319 0-347 City classification X13 -0-775 -0-362 0-060 0-200 0-028 Xl4 0-818 0-061 -0-248 -0-183 0-042 Eigenvalue 4-407 2-318 1-559 1-347 1-140 Cumulative percentage of variance accounted for 31-5 48-0 59-2 68-8 76-9

Ci

C2

C3

C4

c5

0-051 -0-339

0-865

0-055-0-173

0-810 -0-260

0-087

0-084

0-924-0-126 0-844 0-188

0-052 -0-096 -0-061 0-120 -0-050 -0-054

0-029

-0-323 0-756 -0-361 -0-067 -0-232 0-177-0-764 0-096 0-012-0-130 -0-280 -0-054

0-209 -0-656 -0-306 -0-151 0-075 0-643 -0-255 0-562

0-427 -0-068

0-679-0-120 0-139 0-112

-0-023 -0-300

0-047 -0-022 0-040 0-869 0-374 0-421 -0-368 -0-269

-0-693 0-206 0-560 -0-485 3-512 25-1

2-339 41:8

0-254 0-130 0-897-0-053

0-117 -0-471 -0-134 0-030 0-469 0-004 1-975 55-9

1-633 67-6

1-312 76-9

198


Component 3 (C3): journey to work--distance-decay parameter (X^, and spatial structure of the distribution of jobs (Xl9X^) and destination weights (X12); Components (C4): interface between interurban system and origin geometry—city classification (X 13 , X 14 ), and spatial structure of the origin weights (Xl0); Component 5 (Cs): destination geometry—spatial structure of the distribution of jobs (Xs) and destination weights (Xn). From a conceptual perspective these five components do not appear to be obfuscations, but rather are meaningful dimensions of intraurban spatial structure. The first dimension plugs cities into a national urban system. The second and fourth dimensions define geometry as an important feature of geographic structure. And the third dimension suggests that spatial interaction and geometry are inseparable phenomena. These components also supply a vehicle for testing the aforementioned hypotheses. According to component 3, the first hypothesis is rejected. Therefore distance decay is not independent of the spatial structure of journey-to-work elements. According to components 2 and 4 there is no evidence for rejecting the second hypothesis. According to component 1, the third hypothesis is rejected. Therefore intraurban spatial structure is not independent of the interurban system within which it is located. Returning to component 3 and recalling that the main objective of this section is to disclose characteristics of the relationship between distance decay and spatial structure, those variables loading highly onto this component were then used in a stepwise multiple regression analysis. The results of this analysis are presented in table 6. In this instance the dependent variable was the distance-decay exponent (Xi), and the independent variables were the spatial structure of the distribution of jobs (Xy, X8) and destination weights (Xn). Together these latter three variables accounted for roughly 41% of the variation between urban centers exhibited by the distance-decay parameter. Further, the relationship between distance decay and the geometric pattern of numbers of jobs was found to be highly significant. This finding is interesting in that on average the geographic distribution of jobs showed a somewhat weak tendency for similar numbers to cluster, with several cities even displaying a tendency for dissimilar numbers to cluster, whereas on average the geographic distribution of entropy-model destination weights showed a moderate tendency for similar values to cluster, with no evidence that a tendency was present in any of the cities for dissimilar values to cluster. The regression coefficients reveal that an indirect relationship exists between distance decay and geometric pattern, with this relationship being reasonably sensitive to changes in the geometry of destinations. Moreover, smaller distance-decay exponents are affiliated with geometric patterns in which similar numbers of jobs tend to cluster, whereas larger distanceTable 6. Summary results for the stepwise regression model. Step Variable

1 2

Xn Xn

3

Xn Xs Xn

Multiple; correlation

Intercept

R

R2

increase in/? 2

0-538 0-629

0-289 0-395

0-289 0-106

-2-156 xlO"1*** -2-331 xlO" 1 ***

0-642

0-412

0-017

-2-565 xlO" 1 ***

X\2

Coefficient estimates

-7-277 -7-991 2-578 -7-191 2-207 2-379

*Sig nificantly different from zero at the 10% level; *** at the 1% level.

x KT1*** x lO-1*** x 10"3* x 10-1*** x 10"2* x 10"3

Standard error of estimate 2-431 x 10"1 2-326 x 10"1 l-347x 10"3 2-577 x 10"1 2-519 x 10"2 l-386x 10"3


199

decay exponents are affiliated with geometric patterns in which dissimilar numbers of jobs tend to cluster. A parallel, but considerably less sensitive, relationship prevails between distance decay and origin geometry. In summary, conclusions reached in this section supply answers to the original class of questions posed in this paper. Origin and destination characteristics of spatial interaction do indeed contain a latent geometric dimension. Spatial structure and levels of distance decay do indeed covary. In other words, one should expect the rate of distance decay in spatial interaction models and the spatial structure associated with origins and destinations to be interdependent. These conclusions imply that Qiff et al's (1974; 1975; 1976) skepticism is unfounded and that Curry's (1966; 1972) and Johnston's (1973; 1975; 1976) conjecture, stating that distance-decay exponents for spatial interaction models measure both the influence of map pattern and the true friction of distance effects, applies both to interurban and intraurban situations. This conjecture certainly is a prime candidate to be cast in the form of a theorem and its proof. Finally, the stepwise regression results suggest that interaction modelling approaches must be revised so that spatial-structure effects are made explicit expressions in these models. Concluding comments To summarize, one controversial issue of spatial interaction modelling is whether or not the rate of distance decay is independent of the geographic structure associated with origins and destinations. In other words, do the propensity of origins to emit interactees and the propensity of destinations to attract interactees vary as the geometry of origins and destinations changes? Furthermore, do these propensities change with variations in the nature and degree of spatial autocorrelation latent in the geographic distributions of origin and destination totals? Geometry has been represented in this study by connectivity matrices depicting the configurations of origins and destinations. Spatial autocorrelation has been indexed in this study with the Moran coefficient and the Geary ratio. Spatial interaction has been modelled with a doubly constrained gravity-type entropy model. The parameters of this model have been estimated with a Furness procedure. Finally, the data base employed here consisted of twenty-four cities from the Canadian urban system. A number of interesting inferences can be drawn from this study. First, evidence is supplied which positively supports the conjecture that distance-decay exponents for spatial interaction models measure both the influence of map pattern and the actual friction-of-distance effect. Clearly this modelling problem can be circumvented by properly specifying interaction models. In other words, terms must be introduced into these models which result in an explicit incorporation of geometric patterns. Therefore specification work is required. Further, a theoretical foundation for identifying the relationship between distance decay and spatial structure is needed to complement this analysis. Curry (1972) has furnished a starting point for the intraurban case. He also has provided a beginning for the interurban case (Curry, 1977). Second, the traditional distance-decay exponent for spatial interaction is particularly sensitive to the geometry of destinations, and somewhat sensitive to the geometry of origins. One question that arises here refers to the fact that only plane-geometry notions have been dealt with. A voluminous literature exists in which more general topological relationships emerge which are attributed to geographic organization induced by a central-place superstructure. Surely this set of geometric connections needs to be explored too. Meanwhile, whereas the distance-decay exponent seems sensitive to origin/destination geometry, the doubly constrained entropy-model calibration procedure appears to do an excellent job of capturing this geometry.

200

DA^Griffith, K G Jones

Third, spatial structure generates a significant effect at the intraurban and the interurban levels. In addition the national urban system apparently affects intraurban spatial structure. One inference along this line that can be drawn from table 2 refers to urban size and spatial structure. In small urban settings, distance-decay and spatial-structure effects blend together. This blending is caused by two factors. On the one hand, the scale of analysis will often employ areal units that because of their sizes mask most of the spatial-structure effects. On the other hand, in small urban areas spatial propinquity will cause movers to be unable to discriminate clearly between the two effects. In an intermediate-sized city these two effects should be reasonably distinguishable. And the emergence of urban subcenters in large cities adds a dimension of complexity to the situation that once again results in a blending of these two effects. Another; inference regarding the relationship between a national urban system and intraurban spatial structure can be drawn from tables 3 and 4. Comparing those results presented in table 3 with ones found in the literature suggests that generic groups of cities indeed do exist and that these groupings remain reasonably stable with alterations in the underlying data base. Meanwhile, table 4 implies that the deterrence of distance is not related to the type of city in question. More specifically, the operation of distance-decay mechanisms appears to be insensitive to an urban area's economic base. One final implication concerns the experimental framework outlined in the first section. The synthetic spatial interaction model represented by equation (4) is quite difficult to calibrate. The potential presence of nonpositive values in F*, P*, andP^ means that, for example, a logarithmic regression technique cannot be employed. Similarly, because the first partial derivatives with respect to the exponents involve logarithms, a nonlinear regression technique cannot be utilized. Furthermore, the potential presence of negative values in these matrices will introduce complications into standard procedures for estimating constrained versions of the model. In conclusion, this study has helped to illuminate a resolution to the controversy concerning the relationship between distance decay and spatial structure. It also perrnits the following three questions to serve as guides for subsequent research. 1. Are the findings of this study particular to journey-to-work interactions or the Canadian urban system? 2. Is destination geometry more important than origin geometry? 3. How does the national urban system affect intraurban geometry? Another important theme is stressed by Sayer (1977). Griffith (1978) formally demonstrates that a pronounced link exists between time paths and the geometric pattern that is latent in a geographic distribution. Obviously this type of trend must be incorporated. However, the recent debate does not signify atrophy in urban and regional modelling. At least a minimal understanding of the relationship between distance decay and spatial structure must be forthcoming before the development of geometric patterns can be successfully handled. Accordingly this debate constitutes an evolutionary stage rather than a halting of progress in urban and regional modelling. Acknowledgements. We would like to express our appreciation to D Dowhal of the Ryerson Computing Center and D Lynch of the Geography Department, State University of New York at Buffalo, for the help they provided in utilizing the numerous data tapes required for this paper. We would also like to thank Statistics Canada for making available the journey-to-work data tapes. References Baxter R, Williams I, 1975 "An automatically calibrated urban model" Environment and Planning A { 7 3-20 Britton J, 1973 "The classification of cities: evaluation of Q-mode factor analysis" Regional and Urban Economics 2 333-356 '


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