Effects of Spatial Interaction on Spatial Structure: a Case of Daycentre Location in Malmo

Geographical Review of Japan Vol. 66 (Ser. B), No. 2, 156-172, 1993

Effects of Spatial Interaction on Spatial Structure: A Case of Daycentre Location in Malmo

Jun YAMASHITA*

Abstract

The locational structure effects on spatial interaction in distance-decay models have been discussed since the 1970s. This discussion has led many geographers to obtaine distance-decay parameters affected not by spatial autocorrelation but by friction of distance. As BENNETTet al. (1985) stressed, however, we should recognize that spatial structure and spatial interaction are interdependently related. Thus, the present study explores spatial interaction effects on spatial structure. First, using the SIMODEL developed by Williams and FOTHERINGHAM(1984), a distance-decay parameter was estimated for intra-urban trips travelled by pensioners to daycentres in Malmo, Sweden. In addition, location patterns of those daycentres and spatial autocorrelation between them were identified by the nearest neighbour measure and Moran's coefficient, respectively. Second, through solving a location-spatial interaction model, effects of spatial interaction on spatial structure were examined in three cases of distance-decay para meter. It was proven that the three cases of distance-decay parameter caused different location patterns. Combined with previous studies addressed to the spatial structure effects on spatial interaction, the interdependency between spatial interaction and spatial structure was ex plicated.

Key Words: spatial interaction, spatial structure, spatial autocorrelation, location-spatial interac tion model, elderly care facility.

ates2. In short, the spatial structure effects or I. INTRODUCTION the map pattern effects are defined as locational structure effects of origins and destinations on Relationship between spatial interaction and parameter estimates. As OKUNO (1981, p. 166) spatial structure has provoked controversy mentioned, these effects result from spatial after the development of gravity-type spatial autocorrelation within the parameter estimates. interaction modelsl. Especially, spatial structure To obtain distance-decay parameters affected effects on the spatial interaction have been the not by the spatial autocorrelation, but solitarily main subject of discussion over the last few by friction of distance, some spatial analysts decades. In the middle of the 1970s, CURRYand have presented several solutions. CLIFF and his colleagues (CURRY, 1972; CURRYet al., 1975; ORD (1981) suggested the spatial differenc SHEPPARDet al., 1976), CLIFF and his associates ing method which can weed straightforwardly (CLIFF et al., 1974, 1975, 1976), and JOHNSTON the spatial structure effects from auto (1975) intensively discussed the spatial struc correlated interactions, and some modified stat ture effects on the distance-decay parameter istics which rectify autoregressive errors. In ad calculated through gravity-type spatial interac dition, FOTHERINGHAM(1983) modified the pro tion models. Thereafter, some researchers, in duction constrained model so as to develop the cluding SHEPPARD(1978, 1979), and GRIFFITH competing destinations model, which might and JONES (1980), continuously mentioned the remove autocorrelation effects through embedd spatial structure effects on parameter estim ing the accessibility term in the production

* Graduate student at Department of Social and Economic Grography , University of Lund, Solvegatan 13, S-223 62, Lund, Sweden Effects of Spatial Interaction on Spatial Structure 157 constrained model. within an urban area to avoid sampling error. To the contrary, little attention has been There are three large cities, namely, Stockholm, given to the spatial interaction effects on spa Gothenburg, and Malmo, in Sweden. One of tial structure. As BENNETTet al. (1985) pointed these cities, Malmo, was selected as the study out, the spatial structure not only affects spatial area. Besides, the present study intended to interaction, but also has the reaction from it. In solve location-allocation problem of facilities sum, we should recognize that the spatial inter providing services for users without exclusion action and the spatial structure are inter and rejection. This was the reason why public dependently related. Thus, the present study facilities, namely daycentres, were selected as explores the spatial interaction effects on spa the study objective. As a result, the intraurban tial structure using the location-spatial interac interaction travelled by elderly people to day tion model. centres was applied to the production con strained model. Malmo municipality has pro II. METHODS vided various social services for 340,000 cit izens, of which 48,000 were elderly people and Two steps were taken here as analytical almost all the elderly people were pensioners. A methods. First, using the production constrain variety of services has been provided to those ed model, a distance-decay parameter was cal pensioners at daycentres. For example, break ibrated amongst intra-urban trips to elderly fast, lunch and supper are served at restaurants care facilities. In addition, location pattern of within daycentres. Amusements, such as bil those facilities and the spatial autocorrelation liards, playing cards and chess, are offered amongst them were identified by the nearest there. The pensioners can also participate in neighbour measure (CLARK and EVANS, 1954) study circles, such as foreign language, textiles, and Moran's coefficient (MORAN,1950), respect and painting. As illustrated in Figure 1, Malmo ively. Second, through solving a location comprises 38 regions, in which 14 daycentres spatial interaction model based on the produc principally provide their services for pension tion constrained model, effects of spatial inter ers living within the jurisdictional boundary3. action on spatial structure were examined in In this study area, the central business district different cases of the distance-decay parameter. (CBD) is located around the old town and the 1. The Production constrained model The production constrained model developed by WILSON (1971) is formulated as follows:

Tij=AiOiWƒÁjexp(ƒÀdij) (1)

Ai=[ƒ°jWƒÁjrexp(ƒÀdij)]-1 (2)

Oi=ƒ°jTij(3)

where

Tij: the interaction between regions i and j, Ai: the balancing factor,

Oi: the total outflows at region i, Figure 1. Study area in Malmo. Wj: the attraction term, Note. 1. Fridhem 2. Ribersborg 3. Dammfri 4. Kronprinsen 5. Kronborg 6. Gamla Staden 7. dij: the distance between regions i and j, Davidshall 8. Radmansvangen 9. Mollevangen 10. ƒÀ: the distance-decay parameter, Sodervarn 11. Heleneholm 12. Sofielund 13. Augustenborg 14. Rorsjostaden 15. Varnhem 16. ƒÁ: the mass parameter. Vastra Sorgenfri 17. Ostra Sorgenfri 18. Rulltofta 19. Kirseberg 20. Segevang 21. Mellanheden 22. Limhamn Calibration of the distance-decay parameter 23. Djupadal 24. Sibbarp 25. Solbacken 26. Kroksback 27. Sodertorp 28. Kulladal 29. Lindeborg 30. Eriksfalt in this production constrained model requires a 31. Nydala 32. Gulivik 33. Hidby 34. Lundangen 35. large amount of aggregated interaction data Rosengarden 36. Hoja 37. Almgarden 38. Skravlinge. 158 J. YAMASHITA city faces the North Sea on its northern bound were provided through interviews with dir ary so that the built-up area has been expand ectors at each daycentre. Since public transporta ing southward since the Second World War. In tion dominates daily activities of pensioners accordance with the extension of the city, popu (see also, HERBERT and PEACE, 1980; SMITH, lation has been growing on the urban fringe, 1984), the separation term (dij) was determined and in turn the potential value of the elderly by time distance between centroids at each population gently declines from region No. 2 region derived from a bus time table in Malmo4. towards the urban fringe (see Figure 2). In this Here, distance within the same region was meas figure, we find four regions with high elderly ured by one-half time distance from a centroid population, namely regions No. 2, 13, 15 and 24. to the nearest one5. With the help of Multi In the production constrained model, the in dimensional scaling (MDS) method, initial lat teraction term (Tij) was measured by the tices were reassembled to an appropriate inter number of elderly visitors from their residential action space with less stress (GATRELL,1979)6. sites to the daycentres. Out of the 14 day Fortunately, a powerful and reliable calibration centres, 12 were in operation and the other two program for the various spatial interaction were under renovation in February, 1992 (see models, SIMODEL, was developed by WILLIAMS Figure 3). Thereby, the number of trips (Tij) was and FOTHERINGHAM(1984). This program en counted at the 12 daycentres through a ques abled us to offer proper parameter estimates for tionnaire survey. In geographical studies on both distance-decay and mass parameters. Fi shopping behaviour, the attraction term (Wj) nally, some measures of the goodness-of-fit was often represented by total floor area at each tested the performance of the SIMODEL. retail store. From analogy of those studies, the BAXTER(1983) argued, however, that there is no total floor area at each daycentre was used as single acceptable measure of goodness-of-fit on the index of the attraction term. The floor data the spatial interaction models and recom mended the use of several different measures for examining performance by different spatial interaction models. KNUDSENand FOTHERINGHAM (1986) supported this argument and stress ed that the coefficient of determination, which is often used as an index of goodness-of-fit of spatial interaction models, is not a suitable meas ure for these models because its response to errors diminishes with decrements in error level. This is the reason why the standard root mean square error (RMS) and the dissimilarity Figure 2. Distribution of elderly population in index (C) were chosen as measures for good Malmo (1992). ness-of-fit along with thecoefficient of determina tion (r2)7. 2. Measures of location pattern and spatial autocorrelation As an index of location pattern of daycentres, the nearest neighbour measure was applied to this study. CLARKand EVANS(1954) formulated this measure as follows:

R=[1/n•~(ƒ°irij]/(1/2D1/2) (4)

where

n: the number of points, Figure 3. Locations of daycentres in Malmo (1992). rij: the nearest neighbour distance between Effects of Spatial Interaction on Spatial Structure 159

regions i and j, the geographic distribution exhibits some pat D: density of points, namely D=n/a (here a tern. The null hypothesis on no spatial auto indicates area). correlation was examined by the t-test of z - value8. The range of this measure is between null and 2.149. If the value is naught, all points are clus 3. Location-spatial interaction model tered around a point. Conversely, distribution As mentioned in the preceding section, we of points is perfectly regular if the measure has should recognize that the spatial interaction is the maximum value. Points are distributed ran interdependently related to the spatial struc domly if the measure is equal to one. Unfortu ture. However, we would expect that the spatial nately, out of the three types, namely regular, interaction changes faster than the spatial random and clustered types, only random dis structure (BENNETT et al., 1985, pp. 632-633; tribution can be statistically tested through ex KONAGAYA,1990, p. 27). The location-allocation amining whether it is fitted to the Poisson dis spatial interaction model presented below tribution. Thus, location patterns of daycentres allows the disclosure of effects of the spatial were identified only in terms of regular, interaction on the spatial interaction in a short random or clustered. Finally, the nearest neigh time period. In the location-spatial interaction bour measure was calculated amongst day model, the spatial interaction model prob centre locations reconstructed by MDS. abilistically allocates population at each CLIFF and ORD (1981, p. 178) suggested that if region to each facility in accordance with the interval data are given, Moran coefficient per value of the exogenous distance-decay para forms better than the Geary ratio. Since the meter. This population allocation is regarded as attraction term W; refers obviously to interval spatial interaction between each region and op data, Moran's coefficient (I) was employed in timal locations. On the other hand, the location this study, and it is formulated as follows (see spatial interaction model estimates optimal MORAN, 1950, in detail): locations, which is equivalent to estimation of I =[(n/2A)•~(ƒ°iƒ°jƒÂijZiZj)]/ƒ°iZi2) (5) locational configuration of destinations, or in a whereƒÂij narrow sense, to the spatial structure. In con sequence, the location-spatial interaction model : binary variable. If region i is connected reveals the effects of the spatial interaction on with region j, then this variable has one; the spatial structure. otherwise, null, The effects of spatial interaction on spatial Zi: xi-x, structure were examined by means of the fol Zj: xj-x, lowing strategy in the present study. First, xi: value of the attraction term at region i, using three cases of the distance-decay para n: the number of regions, meter, optimal locations were explored in the A: the total number of connections, namely location-spatial interaction model. Second, the A=1/2•~ƒ°iKi, nearest neighbour measure and Moran's coeffi Kj: the number of regions faced with region cient of 14 daycentres were calculated in the i. three cases, and from comparison of the nearest As GRIFFITH and JONES (1980) mentioned, if neighbour measures and Moran's coefficients, Moran's coefficient approaches one, positive relationship between spatial interaction and spatial autocorrelation is detected, and then the spatial structure was revealed. geographic distribution is likely to show a clus In short, location-allocation problems are tered pattern. Conversely, if Moran's coeffi solved by minimizing or maximizing an object cient approaches minus one, negative spatial ive function subject to a set of constraints. autocorrelation is detected, and then the geo Heretofore, some geographers have developed graphical distribution tends not to cluster. As various probabilistic location-allocation models Moran's coefficient approaches -(n-1)-1, no incorporating spatial interaction models (see spatial autocorrelation is detected, and hence also, LEONARDI, 1981a, 1981b; HAYNES and 160 J. YAMASHITA

FOTHERINGHAM, 1984; BIRKIN and WILSON, the distance-decay parameter: -1.5, -0.15 and 1986a, 1986b; GHOSH and RUSHTON, 1987; -0 .015. As HODGSON(1978) revealed, popula FOTHERINGHAMand O'KELLY, 1989). According tion is allocated to nearest facilities in deter to BEAUMONT(1987, p. 29), an objective function ministic location-allocation models, and such of such location-spatial interaction model is for allocation is found in probabilistic location mulated as follows and it was applied to the allocation models with the distance-decay framework of the present study: parameter less than -1.5. According to FOTHERINGHAMand O'KELLY(1989, p. 151), facil Minimize ƒ°iƒ°jSijcij (6) ity locations are decentralized in the location subject to Sij=AiOiWƒÁjexp(ƒÀdij) (7) spatial interaction models with the distance decay parameter of less than -0.2, while clus Ai=[ƒ°jWƒÁjexp(ƒÀdij)]-1 (8) tered in those with a distance-decay parameter

Oi=ƒ°ijSij (9) of more than -0.025. This is the reason why the distance-decay parameter of -0.015 was (i=1,2,•c,n.; j=1,2,•c,m.) selected in the present study. Besides, the loca where tion-spatial interaction model with a distance decay parameter of -0.15 was selected because Sij: the interaction between regions i and j, results of the application of the SIMODEL pre cij: the travel costs between regions i and j, sented in the following section exhibit the es Ai: the balancing factor, timated distance-decay parameter of approx Oi: the total population in region i, imately -0.15. Wj: the attraction term at each centre, The present study intended to obtain a set of dij: the distance between regions i and j,ƒÀ optimal locations which was purely affected by : the distance-decay parameter,ƒÁ: the spatial interaction. In other words, our at the mass parameter, tention should be directed to solitary effects of n: the number of regions, distance-decay parameters (ƒÀ) on locations of m: the number of centres, and m

L1, L2 and L3, respectively. Though TEITZ and BART algorithm was develop

(2) Evaluate Z0=ƒ°iƒ°jSijcij(i=1,2,•c,n.). In ed to offer a heuristic solution, in this paper it this example, was iterated until we obtained an exact solu tion. Z0=S11c11+S12c12+S13c13 In the location-spatial interaction model, the +S21c21+S22c22+S23c23+ specific number of daycentres was to be es +Sn1cn1+Sn2cn2+Sn3cn3. (10) timated. However, Malmo local government has no criteria for the required number of day (3) Select an alternative location (L') which does not include in the initial set. Let L' be centres. Since no normative number of day region No. 4 (Kronprinsen) in the example. centres was specified, ten to 18 daycentres were calculated on the basis of the proportion of the (4) A location in the initial set, LS, is sub stituted by the alternative location selected in expenditures for elderly care within the muni step 3, and Z'(=ƒ°iƒ°jS'ijc'ij) is calculated. Sup cipal budget in 1991. Finally, parallel with an increasing number of posing that L1 is substituted by L', Z' is evalu ated as: gerontological studies, geographers have been concerned with, and contributing to, this realm Z'=S12c12+S13c13+S14c14 of study (see also, ROWLES,1986 WARNES,1990). +S22c22+S23c23+S24c24+ As WARNES (1990) pointed out, however, most •c +Sn2cn2+Sn3cn3+Sn4cn4. (11) of those geographers have principally focused on distribution or migration of elderly people so (5) Compare Z0 with Z'. that research on accessibility to elderly care (6) If the greater improvement in the object facilities has been ignored. Thereby, the present ive function is found, the initial location select study may supplement this missing study ed in step 4, namely Ls, is replaced by L'. Thus, domain in the geography of the elderly. the new initial set comprises L' and L1, L2, •c, Lj except for Ls. In the example, the new initial set III. RESULTS of three locations includes L2 (Ribersborg), L3

(Dammfri) and L4 (Kronprinsen). If the iteration Results of the applications of the SIMODEL is not ended, return to step 3. Then, another and of the location-spatial interaction model are alternative location is selected as L', like region No. 5 (Kronborg). presented in a series of figures and tables below.

(7) If greater improvement in the objective 1. Results of the application of the SIMODEL function is not found, LS is not replaced by L'. If Although time distance was transformed by the iteration is not terminated, return to step 4 MDS, initial locations of daycentres were very and select another location in the initial set. For similar to final ones (see Figure 4). This similar example, L2 (Ribersborg) is selected and sub ity was confirmed by a low value of stress, stituted by L' (region No. 4, Kronprinsen), and which indicated that only 8.6 percent of the then Z' is written as: initial locations disagreed with the final ones. Z'=S11c11+S13c13+S14c14 The disagreement between initial and final +S21c21+S23c23+S24c24+•c locations was mainly found in the urban fringe. +Sn1cn1+Sn3cn3+Sn4cn4. (12) Especially, in the southern part of the study area, the final locations were relocated out (8) If an iteration of steps 2-7 brings no wards. To the contrary, regions in the north substitution, terminate the procedure. eastern urban fringe were replaced towards RUSHTONand KOHLER (1973) showed that this CBD. From the findings above, the final loca algorithm converges faster to an optimal solu tions did not widely differ from the initial ones. tion and is more accurate than others, such as Thus, all results of the application of the SI algorithm developed by COPPER (1963) or by MODEL and of the location-spatial interaction MARANZANA(1964). This is the reason why model are illustrated in the initial map meas TEITZ and BART algorithm was selected here. ured by physical distance, while all analyses 162 J. YAMASHITA

Figure 4. Results of Multidimensional scaling.

were submitted by the time distance trans tected but weak. Conversely, the mass para formed by MDS. meter has plus sign and is higher than the Through the questionnaire survey, 1,806 distance-decay parameter. In the middle row of trips to the 12 daycentres were observed and Table 1, three measures of goodness-of-fit are this figure was equal to 3.8 percent of the total presented. The coefficient of determination (r2) elderly population. Those person trips were ap is high enough to satisfy the t-test at 0.01 level plied to the SIMODEL and its results are sum of confidence. This indicates that the SIMODEL marized in Table 1. In this table, the distance well explains the actual interactions. This high decay parameter has a minus sign but moderate explanation in the model was confirmed by low value so that the distance-decay effect was de values of both dissimilarity index (G) and the standard root mean square error (RMS). In the bottom row of Table 1, the nearest neighbour Table 1. Statistics on trips to daycentres in Malmo. measure shows approximately 1.5, which indic ates that daycentres were regularly dis tributed in the study area. This regular distri bution was supported by Moran's coefficient. Moran's coefficient has a minus sign and is low enough to reject the significant test at 0.01 level, which indicates that there was no spatial autocorrelation among the 12 daycentres and

*coefficient is significant at 0 .01 level of confidence. that they were not clustered. From the observa Effects of Spatial Interaction on Spatial Structure 163 tions above, it became clear that if destinations 2. Results of the location-spatial interaction are regularly distributed and are not spatially model autocorrelated to each other, the distance-decay parameter has moderate value. In other words, Results of the location-spatial interaction it might be inferred that the spatial structure model are presented in the following two sub seldom influences spatial interaction if destina sections. First, under the different number of tions are not clustered. daycentres, some statistics of distance and popu lation, and optimal locations are compared amongst the three cases of the distance-decay parameter with -1.5, -0.15, and -0.015 (here

Table 2. Measures of distance and population under the different number of daycentres in Malmo.

a) ƒÀ=-1.5

b) ƒÀ=-0.15

c) ƒÀ=-0.015

Note: SD indicates standard deviation, and CV indicates coefficient of variation. The total population is 47,654. 164 J. YAMASHITA

Figure 5. Optimal locations and population allocation of daycentres (ƒÀ=-1.5).

Figure 6. Optimal locations and population allocation of daycentres (ƒÀ=-0.15). Effects of Spatial Interaction on Spatial Structure 165

Figure 7. Optimal locations and population allocation of daycentres (ƒÀ=-0.015).

after, these parameters are abbreviated as ƒÀ1, ƒÀ2, pensioners travelled farther than other two and ƒÀ3, respectively). Second, location pattern cases. To the contrary, in the case of ƒÀ3, facilities of 14 daycentres and spatial autocorrelation might be clustered so that pensioners did not amongst them are examined in each case. Be need to travel farther than the other two cases. sides, allocations of population are compared As well as the total distance, mean distances of amongst the three cases. ƒÀ1 are longer than the other two. The mean distances are around 14.0 minutes, while being 1) Overviews around 13.50 in the case of ƒÀ2, and around 11.0

Statistics of distance and population are sum in the case of ƒÀ3. Coefficients of variation in marized in Table 2. The total travel costs in the distance have the largest value in the case of ƒÀ2 case of ƒÀ1 are the lowest of the three. In this case than the other cases. The coefficients of varia the costs decline with increments in daycentres, tion are around 0.58, whereas around 0.53 in while increasing in the case of ƒÀ3. As a result, both cases of ƒÀ1 and ƒÀ3. Besides, fluctuations in the costs of ƒÀ2 are around 360,000. This vari coefficients of variation are small in each case ation in fluctuation of the total travel costs from the case of ten daycentres to the 18. The amongst the three cases also results from differ stabilities in coefficients of variation suggest ence in increasing rate of travelled distance that decline in mean distance coincided with because the population allocated to daycentres decrease in the standard deviation of distance. declined at the same rate. Since the total population was fixed at 47,654 In the three cases, the total distance gradu and the number of daycentres is the same in the ally increases with increments in daycentres. three cases, description on mean population Comparing the three cases, we find that the case was omitted11. In the three cases, the standard of ƒÀ1 has the highest values. In this case, the deviations of population decline with incre distribution of daycentres might be scattered. ment in facilities. Compared with the three In other words, distances between daycentres cases, the case of ƒÀ3 is the lowest of the three, might be long. This is also the reason why which indicates that population at each region 166 J. YAMASHITA

is the most evenly allocated to each daycentre. that Figure 6 showed a multiple nuclei struc In the case of ƒÀ1, inversely, population is also ture with the four circles around regions No. 3, assigned to a specific facility. Because standard 13, 15, and 2412. This may indicate that the deviations of ƒÀ1 are the highest and the mean medium value of the distance-decay parameter distances decline at same rate amongst the strongly affected trips by pensioners within the three cases, coefficients of variation have the four circles, while functioning weakly amongst highest value in this case. Focusing on fluctua the four circles. tions in the coefficients of variation in each From the observations above, the distance case, we find that the statistics of ƒÀ1 increase decay parameters evidently had a large in with increments in daycentres, while fluctua fluence on location pattern of facilities: high tions in the other two cases are mostly small. parameter permitted the regular distribution of This indicates that the increments of facilities destinations, the medium produced the random do not contribute to diminution of variance in and the low caused the clustered. In other population in the case of ƒÀ1. Conversely, decline words, the spatial interaction affected the spa in mean population coincides with reduction in tial structure. Those relationships are sharply variance in population in the other two cases. confirmed through the examination of the near From the interpretation of the statistics est neighbour measure and Moran's coefficient above, it was inferred that daycentres may be in the following subsection. regularly distributed in the case of ƒÀ1, while 2) The case of 14 daycentres clustered in the case of ƒÀ3. In consequence, the case of ƒÀ2 may exhibit a transitional pattern In Table 3, the nearest neighbour measure between the other two cases. and Moran's coefficient are presented in the Optimal locations of daycentres in the three case of 14 daycentres. The nearest neighbour cases are presented in Figures 5 to 7. These measure is approximately 1.5 in the case of ƒÀ1, figures support the inference mentioned above. which indicated that the 14 daycentres were Apparently daycentres are regularly dis regularly located. In the case of ƒÀ2, daycentres tributed in the case of ƒÀ1 (see Figure 5). With are randomly distributed because the measure increments in facilities, new daycentres are prin has almost one. Conversely, the measurement is cipally added in the intermediate area where greater than null and less than one in the case few facilities are allocated in the case of ten of ƒÀ3. This suggests that daycentres are clus daycentres. The case of ƒÀ2 is illustrated in tered around a core region. Figure 6. In this case, daycentres are randomly The clustered distribution in the case of ƒÀ3 distributed and make four circles around re was also confirmed by Moran's coefficient. This gions No. 3, 13, 15, and 24. New daycentres are statistic is high enough to satisfy the t-test at chiefly allocated to a region adjoining to one of 0.01 level so that daycentres were spatially these circles from the case of ten daycentres to autocorrelated to each other. To the contrary, the 18. In the case of ƒÀ3 daycentres are clustered Moran's coefficients are low enough to reject around region No. 8 (see Figure 7). With incre the significance test in both cases of ƒÀ1 and ƒÀ2 ments in daycentres, new facilities are allocated so that there was no spatial autocorrelation to adjoining regions around the cluster. As a amongst the optimal locations. result, Figures 5 and 7 confirmed the findings Compared with the actual interaction (see by FOTHERINGHAM and O'KELLY (1989). As men also Table 1), we could predict that the existing tioned in the preceding section, they showed 12 daycentres would be distributed randomly that facilities are clustered in the centre of because the distance-decay parameter was study, when the distance-decay parameter is -0 .15723. However, the location pattern of the small. On the other hand, facilities are decen 12 daycentres was identified as regular distri tralized in the location-spatial interaction bution by the nearest neighbour measure. This model with large distance-decay parameter as contradiction between the prediction and the well as the deterministic location-allocation observation also results from the effect of mass models (p. 317). Moreover, it is worth noting parameter, which was assumed as one in the Effects of Spatial Interaction on Spatial Structure 167

Table 3. Nearest neighbour measure (R) and Moran's coefficient (1) on the different distance-decay parameters (is) in the case of 14 daycentres.

*coefficient is significant at 0.01 level of confidence. location-spatial interaction model. From the findings above, the relationship be tween distance-decay parameters and location patterns of destinations was reconfirmed: a high parameter leads to regular distribution of destinations, medium to random, and low to clustered. Besides, we showed the relation be tween location pattern and spatial autocorrela tion which had been found by GRIFFITH and

JONES (1980): if being clustered, destinations are spatially autocorrelated to each other; other wise, there is no spatial autocorrelation amongst them. Consequently, we disclosed the trinity amongst the spatial interaction, the spa tial structure and the spatial autocorrelation. In the preceding subsection, we inferred that population might be allocated to nearest day centres in the case of ƒÀ1 , and that it might be evenly assigned to all daycentres in the case of

ƒÀ3 (see also Table 2). This inference is confirmed Figure 8. Population allocation to 14 daycentres below. Figure 8 illustrates population alloca in the three cases of distance-decay tion (Sij) in the case of the 14 daycentres (see parameters. also Figures 5 to 7 in terms of locations of Note. Top: population at each daycentre location is allocated to its own facility. Bottom: population at daycentres). Apparently, population is allocated each region is allocated to all the 14 daycentres, to nearest daycentres in the case of ƒÀ1. In this though some allocations are omitted in this figure because they do not exceed 0.15 percent of the total case, the allocation pattern is much similar to population. that in deterministic location-allocation models. This suggests that the deterministic location allocation problems are regarded as one of vari model generates interaction (Sij), and simultan ations in the location-spatial interaction prob eously determines locations of facilities (j). lems. In the case of ƒÀ2, the nearest centre rule is While the interaction defines the population alleviated so that population in each region is allocation at each facility, the locations of facil allocated to some daycentres. In the case of ƒÀ3, ities specify location pattern of them. In con population in each region is assigned to all sequence, the location pattern dominates spa daycentres because of no distance-decay effects tial autocorrelation amongst the facilities. If the there. distance-decay parameter is high, population at All findings derived from the applications of each region is allocated to nearest facilities, the location-spatial interaction model are sum which are regularly distributed and are not marized in Table 4. An exogenous distance autocorrelated to each other. If the distance parameter (ƒÀ) in the location-spatial interaction decay parameter is medium, population at each 168 J. YAMASHITA

Table 4. Relationship between spatial interaction and spatial structure.

region is assigned to some facilities. In this case, lar distribution of destinations, which are not the facilities are randomly distributed and are autocorrelated. not autocorrelated to each other. If the distance (2) Medium degree of effect causes random decay parameter is low, population at each distribution, where spatial autocorrelation is region is allocated to all facilities, and then the not detected. facilities are clustered around a centre of the (3) Low effect results in clustered locations, study area. Besides, the spatial autocorrelation where autocorrelation is detected. is detected amongst the facilities. Combined Compared with the actual trips, we found a with previous studies addressed to the spatial disagreement with this trinity: while the dis structure effects on the spatial interaction, there tance-decay parameter had a medium value, the fore, the present study explicated the inter location pattern of the 12 daycentres was identi dependency between the spatial interaction and fied as regular type by the nearest neighbour the spatial structure. measure. However, this contradictory between the prediction and the observation also resulted IV. DISCUSSION AND CONCLUDING from the effect of mass parameter which was REMARKS assumed as one in the location-spatial interac tion model. In this study, the influence of spatial interac Since the map pattern effects on parameter tion on the spatial structure was explored in the estimates were already proven in many previ location-allocation model incorporating the ous studies, the interdependency between the production constrained model. Before the loca spatial interaction and the spatial structure was tion-spatial interaction model was solved explicated in this study. However, from the through TEITZ and BART algorithm in the three comparison of the results of ISHIKAWA(1981) cases of the distance-decay parameter, the with those of the present study, we found a actual trips travelled by pensioner to day difference in the relation between the distance centres in Malmo were analyzed by the SI decay parameter and the locational config MODEL. As a result, the distance-decay para uration. In the present study, the nearest neigh meter had a moderate value. By the nearest bour measure declined with decrements in the neighbour measure and Moran's coefficient, it distance-decay parameter, while ISHIKAWA was shown that the 12 daycentres in operation (1981) did not show such coherence between were regularly distributed and were not spati the locational configuration and the spatial ally autocorrelated to each other. Thus, it was interaction13. In his article the production term shown that if daycentres are regularly dis Oi was not assumed that it was normally dis tributed and are not spatially autocorrelated, tributed in the study area, which might be pos the distance-decay parameter has medium sibility leading to the incoherence of the rela value. tion between the distance-decay parameter and Through solving the location-spatial interac the locational pattern. However, it is difficult to tion problem, we found that the spatial interac assume the normal distribution of the produc tion evidently influenced the spatial structure tion term of interurban interaction. Thereby, and that there was the trinity amongst the the difference in the relation between the dis spatial interaction, spatial structure and spatial tance-decay parameter and the nearest neigh autocorrelation: bour measure which might be affected by dif (1) A high distance-decay effect leads to regu ferent interaction data is also debatable further. Effects of Spatial Interaction on Spatial Structure 169

Acknowledgements n: the number of destinations (namely, centres). The range of this measure is between null and The author would like to thank Professor Olof one. If it takes null, spatial interaction model WARNERYD at University of Lund and Professor explains perfectly actual interactions. The dis Takashi OKUNOat the University of Tsukuba for their similarity index is formulated as follows: valuable comments. He also extends his gratitude to Mrs. Yasuko NAKAMURAfor her computer carto G=50•~ƒ°iƒ°j[(Tij/ƒ°iƒ°jTij)-(Tij/ƒ°iƒ°jTij)] (15) graphic assistance and to an anonymous referee for where helpful comments and suggestions on earlier drafts of Tij: the observed flow from region to centre j, this paper. Tij: the estimated flow from region region i to (Received Oct. 9, 1993) center j, (Accepted Dec. 18, 1993) (i=1,2,•c,n.; j=1,•c,m.), m: the number of origins, Notes n: the number of destinations (namely, centres).

1) Although the term, spatial structure, comprises This measure indicates the ratio of trip assigned to wrong destination in the range of unll to 100: various aspects of space, such as elements, or if the spatial interaction model completely ex ganization, order, and time scale, the present study follows the definition by FOTHERINGHAM plains the actual flows, this index points unll; to the contrary, if the model thoroughly misfits to (1983). He defined the spatial structure as loca the observed interaction, the measure labels one tional configuration of origins and destinations hundred. within a spatial system (p. 15). 8) The z-value is presented as follows: 2) SUGIURA(1986) and ISHIKAWA(1988) summarized the controversy between the three groups, and z=[I-E(I)]/[var(I)]1/2 (16) the following discussion. where 3) Two areas-Bunkeflo and Oxie- were not taken into account in this study because the built-up E(I)=-1/(n-1), area was not continuous between the urban var(1)=E(I2)-[E(I)]2, fringe in Malmo and the two areas, and because almost no trips were observed through the ques E(I2)=[4An2-8(A+D)n+12A2]/[4A2(n2-1)], tionnaire survey. D=1/2[ƒ°iLi(Li-1)]. 4) The author utilized the bus time table published by Malmo Traffic Co. Ltd. (Malmo Trafik AB) in 9) WILLIAMS et al. (1990), and WILLIAMS and KIM October, 1991. (1990a, 1990b) tackled location-spatial interac 5) Because there is no acceptable measure of the tion models with both mass and distance-decay distance within the same region, the one-half parameters. distance was employed here. 10) Malmo Statistic Yearbook 1991 (Malmo Statist 6) The stress is formulated as follows (see YOUNG, isk ARSBOK 1991) and its raw materials were 1972, pp. 80-84, in detail): employed here. 11) Here, mean population is calculated in the fol S=[ƒ°iƒ°j(dij-dij)/ƒ°iƒ°jdj2]1/2 (13) lowing formation:

where Mean population=(ƒ°iƒ°jSij)/(n•~m) (17) dij: the actual distance between regions i and j, where dij: restructured distance between regions i and j through a MDS procedure. Sij: population assigned from region i to day centre j, Apparently, the stress has a range between null n: the number of regions, and one. If this value has null, final locations are m: the number of daycentres. just the same as initial ones. 7) The formulations of the standard root mean 12) See also, TAGUCHIand NARITA(1986) in terms of square error and the dissimilarity index are pre the multiple nuclei structure in a metropolitan sented below. area. 13) In his article, the nearest neighbour measure RMS=[(Tij-Tij)2/m•~n]1/2 (14) was 1.83 and the distance-decay parameter was -0 where .103 in the regular distribution. In the random distribution, the measure was 1.09 and Tij: the observed flow from region i to centre j, the parameter was -0.061. In the clustered dis Tij: the estimated flow from region i to centre j, tribution, the measure was 0.53 and the para m: the number of origins, meter was -0.122. 170 J. YAMASHITA

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空間構造に対する空間的相互作用の影響:マルメ市におけるデイセンターの立地を事例として

山下潤*

1970年以降,空間的相互作用に対する空間構造の影ラム, SIMODELによって距離減衰パラメータを推計し響が議論されてきた。このような議論は,地理学者の関た。さらに,最近隣平均距離とモラーン係数を用いて, 心を空間的自己相関の影響を受けない距離減衰パラメーデイセンターの立地パタンーと空間的自己相関を示しタの抽出へと向けさせた。しかしながら, Bennettらた。ついで,発生制約モデルを内挿した立地配分モデル (1985)が主張するように,空間構造と空間的相互作用とを用いて,異なる3つの距離減衰パラメータごとにディは相互依存関係にあると認識すべきである。本研究の目センターの最適立地が求められ,空間的相互作用の空間的は,空間構造に対する空間的相互作用の影響を明らか構造に対する影響を吟味した。結果として, 3つの距離にすることである。まず,スウェーデンのマルメ市内で減衰パラメータが異なる立地パターンをもたらすことを高齢者によってなされたデイセンターまでのパーソント証明した。従って,空間構造による空間的相互作用へのリップのデータを用いて, Williams Fotheringham 影響を明らかにした従来の研究と合せ,空間的相互作用 (1984)によって開発されたキャリブレーションプログと空間構造は相互依存関係にあるといえる。

* 院・ルンド大学社会経済地理学部Solvegatan 13, S-223 62, Lund, Sweden