Hexagon-Based Adaptive Crystal Growth Voronoi Diagrams Based on Weighted Planes for Service Area Delimitation

International Journal of Geo-Information

Article Hexagon-Based Adaptive Crystal Growth Voronoi Diagrams Based on Weighted Planes for Service Area Delimitation

Jue Wang * ID and Mei-Po Kwan ID Department of Geography and Geographic Information Science, Natural History Building, 1301 W Green Street, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA; [email protected] * Correspondence: [email protected]

Received: 22 May 2018; Accepted: 26 June 2018; Published: 29 June 2018

Abstract: Delimiting the service area of public facilities is an essential topic in spatial analysis studies. The adaptive crystal growth Voronoi diagrams based on weighted planes are one of the recently proposed methods to address service area delimitation, and consider the geographic distribution of the clients the facilities in question serve and the characteristic of the socioeconomic context, while at the same time mitigate the modifiable areal unit problem when dealing with the socioeconomic context, since the method is based on continuous weighted planes of socioeconomic characteristics rather than arbitrary areal units. However, in the method, the environmental and socioeconomic contexts are rasterized and represented by regular square grids (raster grid). Compared with a raster grid, the hexagon grid tiles the space with regularly sized hexagonal cells, which are closer in shape to circles than to rectangular cells; hexagonal cells also suffer less from orientation bias and sampling bias from edge effects since the distances to the centroids of all six neighbor cells are the same. The purpose of this study is to compare the raster grid and hexagon grid for implementing adaptive crystal growth Voronoi diagrams. With the case study of delimitating middle school service areas, the results are compared based on the raster grid and hexagon grid weighted planes. The findings indicate that the hexagon-based adaptive crystal growth Voronoi diagrams generate better delineation results compared with the raster-based method considering how commensurate the population in each service area is with the enrollment capacity of the middle school in the service area and how accessible the middle schools are within their service areas. The application of hexagon-based adaptive crystal growth Voronoi diagrams may help city managers to serve their citizens better and allocate public service resources more efficiently.

Keywords: service area delimitation; spatial optimization; adaptive crystal growth Voronoi diagrams; hexagon grid; MAUP; socioeconomic context

1. Introduction Delimiting the service areas of public facilities like schools and ﬁre stations is an essential topic in spatial analysis. Geographers and urban planners have considerable theoretical and practical interest in this topic. The methodology of service area delimitation can be applied to address a wide range of issues as evidenced by the literature on the delineation of market areas [1–6], school catchment areas [7–11], police districts [12–15], political districts [16–19], and service areas of healthcare facilities [12–15]. There are many existing methods for delimiting service areas, and they can be categorized into mixed-integer programming [6,18,20–25] and heuristic approaches (e.g., genetic algorithm [26], simulated annealing, and tabu search [17,27–30]). Among these methods, Voronoi diagrams, named after Georgy Voronoi, are a popular and promising delimiting method for partitioning space into several subregions based on the distance to a set of seed points speciﬁed beforehand [31].

ISPRS Int. J. Geo-Inf. 2018, 7, 257; doi:10.3390/ijgi7070257 www.mdpi.com/journal/ijgi ISPRS Int. J. Geo-Inf. 2018, 7, 257 2 of 20

Voronoi diagrams are widely used in the fields of geography and urban planning because of their computational efficiency and effectiveness for delineating the service area of public facilities [32–36]. Based on the ordinary Voronoi diagram, many advanced Voronoi diagrams have been developed to address the complexity of real-world geographic problems, for instance, the weighted Voronoi diagrams that consider the varying service capacities of public facilities [37–39], the city Voronoi diagrams that take into account the effect of transport networks and travel time [40], and the adaptive additively weighted Voronoi diagrams that partition a space into subregions with predefined size while considering the position and weight of seed points [38]. More recently, Wang et al. [41] proposed the adaptive crystal growth Voronoi diagrams based on weighted planes, which consider the geographic distribution of the clients the facilities in question serve and the characteristic of the socioeconomic context. The adaptive crystal growth Voronoi diagram approach has several advantages when compared with other heuristic approaches. It distributes demand over continuous space by weighted planes with fine resolution and, thus, more realistically approximates the spatial distribution of demand. It does not rely on population or demand data based on arbitrary administrative units such as census tracts and thus mitigates the modifiable areal unit problem (MAUP) to a certain extent. The approach takes into account not only the transport network but also walkable areas and natural barriers when evaluating accessibility. The method allows for real-time adaptive growth speed of each service area in order to balance the service load according to their capacity for all facilities. In spatial optimization studies, the representation issues of demand have been extensively discussed [42–46]. There are many ways to spatially represent the demand for service, which include point representation [45], regional representation [47], and area representation [48–50]. Murray et al. [46] explored the different configurations of coverage modeling and evaluated the effectiveness of various coverage modeling approaches. The representation of demand for service is one of the critical issues for spatial optimization studies and may influence the quality of the spatial delineation results [42]. For approaches based on an areal representation of demand, there are three kinds of regular polygon grids for covering the land surface without any gap or overlap: equilateral triangles, squares, and hexagons [51]. The equilateral triangular grid is not widely used because the triangles have two different orientations [52] and complex neighbor relationships. For geographic information science (GIS) modeling and spatial analysis, the square grid is widely used due to its symmetric, orthogonal coordinate system which simplifies the calculation and analysis based on the grid as well as its convenience for transformations between different spatial resolutions [52]. Moreover, the pixels of remotely sensed images are based on square grids (or raster grids), so it is more feasible to utilize square grids to integrate other spatial data with remote sensing data. The hexagon grid data structure is not widely used in GIS and spatial analysis although it has been utilized in different application fields (e.g., the spatial configuration of cell signal coverage) because of its many merits. Different from the equilateral triangle and square grid structure, the hexagon grid is the most complex regular polygon that can fill a land surface without any gap or overlap [52]. Because hexagonal cells are closer to a circle than rectangular cells in terms of their shape [53], they are the most compact regular polygons that tile the land surface, so they “quantize the plane with the smallest average error” and “provide the greatest angular resolution” [54]. Therefore, the hexagon grid can achieve higher accuracy in representing the spatial features of a land surface than can equilateral triangles and square grids from the perspective of spatial analysis [55]. In addition, from the perspective of visualization and cartography, the boundaries of hexagon cells create less distraction when distinguishing spatial patterns because human eyes have a strong response to the horizontal and vertical lines that are naturally generated by equilateral triangles and square grids [51]. Furthermore, the hexagon grid suffers less from orientation bias and sampling bias from edge effects since the distances to the centroids of all six neighboring cells are the same. Tessellated hexagon grids are widely used in ecological research, but less utilized in the spatial optimization problem for several possible reasons: (1) The geometry and spatial configuration of the hexagon gird are more complicated than those of the raster grid and, thus, they increase computational burden. (2) Compared with raster ISPRS Int. J. Geo-Inf. 2018, 7, 257 3 of 20 data, there is no mature structure for hexagon grid computation that is ready for use by researchers. (3) Remote sensing data, which is one of the most common data resources, is typically stored as pixels in the regular lattice (raster), so utilizing a hexagon grid to process these data involves the conversion from the raster to the hexagon grid data format, which may introduce errors and render data processing more difficult. (4) Processing spatial data with different resolutions is always needed in spatial analysis. Using different resolutions for a raster grid is relatively straightforward because the composition and decomposition of the raster grid maintain congruency and alignment [52], while it is more complicated for a hexagon grid. In Wang et al.’s [41] study, the demand for service and the environmental contexts are rasterized and represented by regular square grids (raster grid). Thus, the purpose of this paper is to compare the raster and hexagon grids for implementing adaptive crystal growth Voronoi diagrams. Delimitation results of middle school service areas in the study area based on the raster grid and hexagon grid weighted planes will be compared. The objective of the adaptive crystal growth Voronoi diagrams is to achieve the minimized difference of supply/demand ratio among all the facilities while considering the travel time by actual configuration of accessibility-related context features (e.g., road network, walkable area, and natural barriers). Therefore, whether the population in each service area is commensurate with the enrollment capacity of the corresponding middle school and whether the middle schools are accessible within their service areas are examined. The results indicate that the hexagon-grid-based (hereafter hexagon-based) adaptive crystal growth Voronoi diagrams (HACG) produced superior results to the raster-based one considering how commensurate the population in each service area is with the enrollment capacity of the corresponding middle school (supply/demand ratio) and how accessible each middle school is within its service area (travel time) by addressing both the socioeconomic context and accessibility-related context features when delimiting the service areas of schools. The remainder of the paper is structured as follows. Section2 introduces the new hexagon-based adaptive crystal growth Voronoi method for delimiting service area based on socioeconomic factors as well as the conventional raster-based method. Furthermore, a case study is presented in Section3 to delimit the service area of 34 middle schools in the Haizhu district in Guangzhou, China based on the accessibility-weighted and population-weighted planes. In addition, the delimitation results of the hexagon-based method are compared with those of the raster-based method to show the advantage of the hexagon-based adaptive crystal growth Voronoi diagrams. The paper follows with Section4, which includes a discussion of the advantages and limitations of the hexagon-based method. Section5 concludes the paper and also suggests some future research directions.

2. Adaptive Crystal Growth Voronoi Diagrams This section introduces the conventional adaptive crystal growth Voronoi diagrams based on raster weighted planes as well as the new method based on hexagonal weighted planes. In addition, examples of the raster-based and hexagon-based methods for service area delineation are presented to compare these methods.

2.1. Raster-Based Adaptive Crystal Growth Voronoi Diagrams The raster-based adaptive crystal growth Voronoi diagrams (RACG), first proposed by Wang et al. (2014), are an innovative method for delimiting the service areas of public facilities that takes into account the socioeconomic context while mitigating the MAUP. With the middle school service area delineation problem as the case study, the study area was tiled into rectangular cells (raster grid) and two weighted planes (accessibility weighted plane and population distribution weighted plane) were generated based on the raster grid. As illustrated in Figure1a, on the weighted planes, middle schools were represented as seed points (the cells that intersect with the location of middle schools) with their service capacity as the attribute. The accessibility of the research area (e.g., road network, walkable area, and natural barriers) was simulated as different weight values on the accessibility weighted ISPRS Int. J. Geo-Inf. 2018, 7, 257 4 of 20 plane, while the distribution of the population was represented as cells with weights on the population distribution weighted plane. In the method, the Voronoi diagrams are grown simultaneously from all seed point cells (e.g., the location of middle schools), and the service area of each middle school will growISPRS to Int. its J. Geo-Inf. neighbor 2018, 7 cells, x FORcycle PEER REVIEW by cycle. On the one hand, the growth speed at each 4 of 20 location (cell) isarea, adjusted and natural based barriers) on the attributewas simulated of that as different cell. For weight instance, values roads on the speed accessibility up the weighted growth while rivers orplane, lakes while block the the distribution growth. Onof the the population other hand, was the represented further growthas cells ofwith any weights grown on area the can be either encouragedpopulation distribution or constrained weighted in real plane. time accordingIn the method, to the the service Voronoi capacity diagrams of middle are grown schools and the populationsimultaneously coverage from of all their seed grownpoint cells areas (e.g., (supply/demand the location of middle ratio). schools) For, and the the purpose service of area illustration, of the distributioneach middle of theschool population will grow to is its simplified neighbor cells and cycle represented by cycle. On as cellthe one values hand, in the Figure growth2 with speed a range at each location (cell) is adjusted based on the attribute of that cell. For instance, roads speed up the of 1 to 3. As shown in the figure, crystal growth starts from the seed point cells to their neighbor cells, growth while rivers or lakes block the growth. On the other hand, the further growth of any grown and thearea growth can be continues either encouraged cycle by or cycle constrained (Figure in2 re(a1)al time illustrates according one to the crystal service growth capacity cycle). of middle Different from otherschools cells, and the the growth population speed coverage of transport of their grown network areas cells(supply/de is fastermand due ratio). to theFor the higher purpose accessibility of they enableillustration, (Figure the2(a1,a2)), distribution while of the growth population is constrained is simplified by and natural represented barrier as cells cell values because in Figure it is normally 2 difficultwith to passa range natural of 1 to 3. barriers As shown such in the as figure, rivers crys andtal growth lakes (Figurestarts from2(a2,a3)). the seed point As the cells crystal to theirgrows, the populationneighbor coveragecells, and the of growth each grown continues area cycle is calculatedby cycle (Figure in real 2(a1) time illustrates at each one crystal crystal growth cycle. cycle). Different from other cells, the growth speed of transport network cells is faster due to the Accordinghigher to theaccessibility population they enable coverage (Figure of their2(a1,a2)), grown while areas, growth the is constrained growth speed by natural is adaptively barrier cells adjusted based onbecause specific it is criteria. normally For difficult example, to pass if thenatural population barriers such size as of rivers any grownand lakes area (Figure is 10% 2(a2,a3)). larger As than any other grownthe crystal area, grows, the further the population growth coverage of this grown of each areagrown will area be is temporallycalculated in stoppedreal time at until each its crystal population size is nogrowth more cycle. than 10%According larger to than the population any other covera grownge area of their in order grown to areas, balance the thegrowth population speed is among all grownadaptively areas. The adjusted crystal based growth on specific concludes criteria. when For example, all the cellsif the arepopulation grown size and of assigned any grown to area the grown is 10% larger than any other grown area, the further growth of this grown area will be temporally area of one seed point (Figure2(a4)). The delineation results would be the service area of each seed stopped until its population size is no more than 10% larger than any other grown area in order to point (e.g.,balance middle the population school) considering among all grown both areas. the distribution The crystal growth of population concludes and when the allaccessibility the cells are of the service region.grown and assigned to the grown area of one seed point (Figure 2(a4)). The delineation results would Notebe thatthe service there arearea two of each kinds seed of neighborpoint (e.g., cells middle on theschool) raster-based considering weighted both the planes:distribution 4 orthogonal of neighborpopulation cells sharing and the an accessibility edge (or of “von the service Neumann region. neighborhood”) and 4 diagonal neighbor cells sharing a cornerNote (orthat “Moorethere are neighborhood”).two kinds of neighbor Because cells onthe the distanceraster-based to weighted the centroid planes: of 4 diagonal orthogonal neighbor neighbor cells sharing an edge (or “von Neumann neighborhood”) and 4 diagonal neighbor cells cells is largersharing than a corner that (or to orthogonal“Moore neighborhood”). neighbor cells, Because errors the willdistance be introducedto the centroid if the of accessibilitydiagonal to these twoneighbor types ofcells neighbor is larger cells than are that not to treatedorthogonal differently. neighbor Therefore,cells, errors two will region-growing be introduced if algorithmsthe are introducedaccessibility for to the these crystal two types growth of neighbor Voronoi cells diagram: are not treated a 4-domain differently. region-growing Therefore, two algorithm region- that only considersgrowing thealgorithms 4 orthogonal are introduced neighbor for cells the crystal and an growth 8-domain Voronoi region-growing diagram: a 4-domain algorithm region- that takes into accountgrowing both algorithm kinds of that neighbor only considers cells. Figure the 4 3orthogonala,b illustrate neighbor these cells two and raster-based an 8-domain region-growing region- growing algorithm that takes into account both kinds of neighbor cells. Figure 3a,b illustrate these algorithms. In Wang et al.’s (2014) method, the 8-domain region-growing algorithm was used for the two raster-based region-growing algorithms. In Wang et al.’s (2014) method, the 8-domain region- crystal growthgrowing of algorithm road network was used cells for due the to crystal their high grow accessibility,th of road network while thecells 4-domain due to their region-growing high algorithmaccessibility, was utilized while for the the 4-domain crystal region-growing growth of other algorithm cells. was utilized for the crystal growth of other cells.

(a) (b)

Figure 1. Illustration of raster-based (a) and hexagon-based (b) weighted planes. The numbers in the Figure 1.gridIllustration cells are the of weights raster-based representing (a) andthe hexagon-basedspatial distribution ( bof) weighteda socioeconomic planes. factor The (e.g., numbers in the gridpopulation). cells are The theblue weights cells represent representing elements of the the spatialroad network; distribution the black ofcells a represent socioeconomic natural factor (e.g., population).barriers; the red, The green, blue and cells yellow represent cells are elements three seed ofcells the representing road network; public service the black facilities cells (e.g., represent natural barriers;middle schools). the red, green, and yellow cells are three seed cells representing public service facilities (e.g., middle schools). ISPRS Int. J. Geo-Inf. 2018, 7, 257 5 of 20 ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 5 of 20

(a1) (b1)

(a2) (b2)

(a3) (b3)

(a4) (b4)

Figure 2. Illustration of the adaptive crystal growth Voronoi diagrams with raster-based (a1–a4) and Figure 2. Illustration of the adaptive crystal growth Voronoi diagrams with raster-based (a1–a4) and hexagon-based ( b1–b4)) weighted planes. planes. The The weights weights for for all all grown areas (served population) of the finalﬁnal delineationdelineation resultsresults are are concluded concluded as as follows: follows: Red: Red: 93, 93, Green: Green: 88, and88, and Yellow: Yellow: 82 for 82 the for raster-based the raster- basedmethod; method; Red: 87, Red: Green: 87, Green: 88, and 88, Yellow: and Ye 88llow: for 88 the for hexagon-based the hexagon-based method. method.

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2.2.ISPRS Hexagon-Based Int. J. Geo-Inf. 2018 Adaptive, 7, x FOR Crystal PEER REVIEW Growth Voronoi Diagrams 6 of 20

2.2.Different Hexagon-Based from Adaptive the raster Crystal grid, Growth which coversVoronoi theDiagrams land surface with continuous rectangular cells arranged in columns and rows, the hexagon grid tiles the space with regularly sized hexagonal cells, whichDifferent are from the mostthe raster complex grid, regularwhich covers polygons the land that surface can fill with the landcontinuous surface rectangular without any cells gap arranged in columns and rows, the hexagon grid tiles the space with regularly sized hexagonal cells, or overlap [52]. Because hexagonal cells are closer in shape to circles than to rectangular cells, they which are the most complex regular polygons that can fill the land surface without any gap or overlap are more compact in space than rectangular cells [53,55]. Furthermore, the hexagon grid suffers less [52]. Because hexagonal cells are closer in shape to circles than to rectangular cells, they are more from orientation bias and sampling bias from edge effects since the distances to the centroids of compact in space than rectangular cells [53,55]. Furthermore, the hexagon grid suffers less from all six neighbor cells are the same [55]. To the contrary, in a rectangular grid, the distances to the orientation bias and sampling bias from edge effects since the distances to the centroids of all six centroidsneighbor of cells the are 4 neighbor the same cells[55]. thatTo the share contrary, a corner in a arerectangular longer than grid, the the distancesdistances to the other centroids neighbor cells.of the Since 4 neighbor hexagon cells cells that have share only a corner one kind are oflonger neighbor than the cells distances that share to other the same neighbor edge cells. and haveSince the samehexagon distance cells tohave their only centroids, one kind only of neighbor one region-growing cells that share algorithm the same is needededge and for have a hexagon the same grid: a 6-domaindistance to region-growingtheir centroids, only algorithm, one region-growing which reduces algorithm the complication is needed infor defininga hexagon neighbor grid: a 6- cells (illustrateddomain region-growing in Figure3c). algorithm, which reduces the complication in defining neighbor cells (illustratedSimilar in to Figure raster-based 3c). adaptive crystal growth Voronoi diagrams, the distribution of the socioeconomicSimilar to variable raster-based is represented adaptive crystal as cell growth weights Voronoi in the diagrams, hexagon the grid. distribution For illustration, of the thesocioeconomic socioeconomic variable context is (e.g.,represented population as cell distribution) weights in is the simplified hexagon and grid. represented For illustration, in Figure the 1b withsocioeconomic a value range context of 1 (e.g., to 3. population Figure2(b1–b4) distribution) present is simplified an example and of represen the hexagon-basedted in Figure 1b adaptive with crystala value growth range Voronoiof 1 to 3. diagrams Figure 2(b1–b4) based on present weighted an example planes. Asof the illustrated hexagon-based in Figure adaptive2(b1), the crystal crystal growthgrowth starts Voronoi from diagrams the seed based point on cells weighted based on planes. a 6-domain As illustrated region-growing in Figure 2(b1), algorithm, the crystal and the growth growth continuesstarts from cycle the by seed cycle. point Similar cells tobased the raster-basedon a 6-domain method, region-growing the growth algorithm, speed of and transport the growth network cellscontinues is faster cycle due by to cycle. its higher Similar accessibility to the raster-bas and growthed method, is constrained the growth speed by natural of transport barrier network cells such ascells rivers is faster and lakes. due to The its higher crystal accessibility growth speed and isgr adaptivelyowth is constrained adjusted by according natural barrier to the cells real-time such as total weightrivers ofand each lakes. grown The area crystal based growth on the speed same iscriteria adaptively as for adjusted the raster-based according adaptiveto the real-time crystal growthtotal weight of each grown area based on the same criteria as for the raster-based adaptive crystal growth Voronoi diagrams illustrated above. As shown in Figure2, the total weight of the grown area in the Voronoi diagrams illustrated above. As shown in Figure 2, the total weight of the grown area in the final result of the hexagon-based method (Red: 87; Green: 88; Yellow: 88) is more balanced than the final result of the hexagon-based method (Red: 87; Green: 88; Yellow: 88) is more balanced than the total weight of the grown area in the raster-based method (Red: 93; Green: 88; Yellow: 82), which total weight of the grown area in the raster-based method (Red: 93; Green: 88; Yellow: 82), which indicates that the hexagon-based adaptive crystal growth Voronoi diagrams may work better in service indicates that the hexagon-based adaptive crystal growth Voronoi diagrams may work better in areaservice delimitation area delimitation problems. problems.

(a) (b) (c)

Figure 3. The 4-domain (a) and 8-domain (b) region-growing algorithms for the raster grid; and the Figure 3. The 4-domain (a) and 8-domain (b) region-growing algorithms for the raster grid; and the 6-domain (c) region-growing algorithm for the hexagon grid. 6-domain (c) region-growing algorithm for the hexagon grid. 3. Middle School Service Area Delimitation 3. Middle School Service Area Delimitation In this application, adaptive crystal growth Voronoi diagrams were generated using both raster- basedIn thisand application,hexagon-based adaptive weighted crystal planes growth of the same Voronoi study diagrams area with were the same generated socioeconomic using both raster-basedcontext. The and results hexagon-based of the HACG weighted will be planescompared of the with same those study of the area RACG with thein delimiting same socioeconomic service context.areas of The middle results schools of the HACGusing adaptive will be compared crystal growth with thoseVoronoi of the diagrams RACG inbased delimiting on accessibility- service areas ofweighted middle schools and population-weighted using adaptive crystal planes. growth The Voronoi delineation diagrams results based of HACG on accessibility-weighted and RACG are andcompared population-weighted in four different planes. spatialThe resolutions delineation of the results weighted of HACG plane. and RACG are compared in four different spatial resolutions of the weighted plane.

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3.1. Study Area Because of the shortage of educational resources, parents in developing countries face key issues for their children’s education. education. Under Under these these circumstan circumstances,ces, delimitating delimitating the the service service areas areas of ofschools schools is iscrucial crucial for for ensuring ensuring an anequal equal and and fair fair distribution distribution of educational of educational resources resources among among the the population. population. In Inthis this study, study, the the Haizhu Haizhu District District in in Guangzhou, Guangzhou, China China was was selected selected as as the the study study area becausebecause Guangzhou is one of the largest cities in China thatthat faces the problems of bothboth rapidrapid populationpopulation growth and shortage in educational resources. The Haizhu District in Guangzhou is an island district 2 (Figure4 4).). ItIt hashas anan areaarea of of 90 90 km km2 and a population of about 1.56 million in 2010 based on the sixth nationwide populationpopulation census census of of China. China. There There are are 34 middle34 middle schools schools and 18and census 18 census tracts intracts the district.in the Asdistrict. an island, As an the island, Haizhu the DistrictHaizhu is District relatively is relatively independent independent of other partsof other of Guangzhou; parts of Guangzhou; therefore, choosingtherefore, itchoosing as the study it as the area study would area help would to reduce help to boundary reduce boundary effects and effects edge and effects edge in effects the analysis. in the Further,analysis. its Further, large population its large population and the considerable and the considerable number ofnumber middle of schools middle make schools it an make ideal it locationan ideal forlocation experimenting for experimenting with middle with school middle service school area service delineation. area delineation. The delineation of the service areas of middle schoolsschools is an allocation problem that spatially allocates demand to service facilities. This allocation problem is suitable to be solved by the adaptive crystal growth Voronoi diagramdiagram approachapproach becausebecause ofof thethe followingfollowing meritsmerits ofof thethe method:method: (1) The adaptive crystalcrystal growth growth Voronoi Voronoi diagrams diagrams distribute di thestribute population the population and environmental and environmental characteristics overcharacteristics a continuous over surfacea continuous (covered surface with (covered rectangular with or rectangular hexagon grids). or hexagon They thus grids). eliminate They thus the inﬂuenceeliminate the of the influence boundaries of the of boundaries administrative of administrative areas (e.g., census areas (e.g., tracts) census on delineation tracts) on resultsdelineation and mitigateresults and the mitigate MAUP to the a certainMAUP extent to a certain (although extent the (a scalelthough of the the grid scale structure of the grid (different structure grid (different size) may affectgrid size) the results).may affect (2) Thethe results). approach (2) takes The intoapproach account takes not onlyinto account the transport not only network the transport but also walkablenetwork areasbut also and walkable natural barriers areas and when natural evaluating barriers accessibility. when evaluating (3) The approachaccessibility. allows (3) forThe real-time approach adaptive allows growthfor real-time speed adaptive of eachservice growth area speed in orderof each to service balance area the in service order load to balance according the service to their load capacity according for all facilities.to their capacity This is for not all possible facilities. using This other is not methods. possible using (4) Wang other et methods. al.’s [41] study(4) Wang has et justiﬁed al.’s [41] that study the adaptivehas justified crystal that growth the adaptive Voronoi diagramcrystal growth approach Voronoi performs diagram better whenapproach compared performs with otherbetter relatedwhen methodscompared for with school other service related area methods delineation. for school That serv studyice usesarea delineation. rectangular grids,That study and the uses current rectangular study extendsgrids, and the the method current by study using extends a hexagon-grid-based the method by frameworkusing a hexagon-grid-based and may obtain furtherframework improvement and may inobtain the results.further improvement in the results.

(a)

Figure 4. Cont.

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(b)

Figure 4. The location of the study area—Haizhu, Guangzhou (the background map was abstracted Figure 4. The location of the study area—Haizhu, Guangzhou (the background map was from Google Maps)—and the illustration of the population-weighted plane (a) and accessibility- abstracted from Google Maps)—and the illustration of the population-weighted plane (a) and weighted plane (b). accessibility-weighted plane (b).

3.2. Weighted Planes 3.2. Weighted Planes Two types of weighted planes, namely an accessibility-based plane and a population-based Two types of weighted planes, namely an accessibility-based plane and a population-based plane, plane, were generated for each cell structure (raster grid and hexagon grid). In order to explore the were generated for each cell structure (raster grid and hexagon grid). In order to explore the scale scale effects (how the weighted planes in different scales will affect the delineation results), four effects (how the weighted planes in different scales will affect the delineation results), four different different spatial resolutions were used for both raster-based and hexagon-based methods; there are spatial resolutions were used for both raster-based and hexagon-based methods; there are thus one thus one accessibility-based plane and one population plane for each cell structure in each spatial accessibility-based plane and one population plane for each cell structure in each spatial resolution. resolution. The accessibility-weighted planes take into account the effect of the transport network The accessibility-weighted planes take into account the effect of the transport network and water bodies and water bodies (as natural barriers) on the accessibility of the schools in the study area. The (as natural barriers) on the accessibility of the schools in the study area. The population-weighted population-weighted planes represent the distribution of population based on distinct types of planes represent the distribution of population based on distinct types of residential buildings in the residential buildings in the study area. The raster and hexagon cells in these weighted planes have study area. The raster and hexagon cells in these weighted planes have the same size in one spatial the same size in one spatial resolution to ensure consistency and comparability of the results. Four resolution to ensure consistency and comparability of the results. Four different spatial resolutions different spatial resolutions of the raster-based and hexagon-based weighted planes are generated to of the raster-based and hexagon-based weighted planes are generated to compare the delineating compare the delineating performance. Specifically, these four spatial resolutions include cell sizes of performance. Speciﬁcally, these four spatial resolutions include cell sizes of 50, 100, 150, and 200 m2. 50, 100, 150, and 200 m2. For the accessibility-weighted planes, illustrated in Figure4a, seed point cells are assigned a cell For the accessibility-weighted planes, illustrated in Figure 4a, seed point cells are assigned a cell value of 10 with a growth speed of 1 cell/cycle. The natural barrier cells (e.g., rivers and lakes) are value of 10 with a growth speed of 1 cell/cycle. The natural barrier cells (e.g., rivers and lakes) are assigned a value of 0 with a growth speed of 0 since they are difﬁcult to pass without a road or bridge. assigned a value of 0 with a growth speed of 0 since they are difficult to pass without a road or bridge. The transport cells are assigned a value of 1 to 2, and the growth speed varies with their average travel The transport cells are assigned a value of 1 to 2, and the growth speed varies with their average speed. The growth speed of the primary transport network (e.g., highways, freeways, and primary travel speed. The growth speed of the primary transport network (e.g., highways, freeways, and trunk roads) cells is 6 pixels/cycle for an average driving speed of about 30 km/h, while the growth primary trunk roads) cells is 6 pixels/cycle for an average driving speed of about 30 km/h, while the speed of the secondary transport network (e.g., secondary trunk roads, collector roads, and local roads) growth speed of the secondary transport network (e.g., secondary trunk roads, collector roads, and cells is 4 pixels/cycle for an average driving speed of about 20 km/h. For all other cells, including local roads) cells is 4 pixels/cycle for an average driving speed of about 20 km/h. For all other cells, walkable area cells, the growth speed is 1 pixel/cycle for an average walking speed of about 5 km/h. including walkable area cells, the growth speed is 1 pixel/cycle for an average walking speed of about The cell attributes of the accessibility-weighted plane are listed in Table1. 5 km/h. The cell attributes of the accessibility-weighted plane are listed in Table 1.

Table 1. Cell values and weights of the accessibility-weighted and population-weighted planes.

Cell Type Cell Value Growth Speed Natural Barrier Cells 0 0 Primary Transport Network Cells 1 6 pixels/cycle Accessibility-weighted plane Secondary Transport Network Cells 2 4 pixels/cycle Walkable Area Cells 3 1 pixel/cycle Low-Rise Residential Cells 4 1 pixel/cycle Medium-Rise Residential Cells 5 1 pixel/cycle Population-weighted plane High-Rise Residential Cells 6 1 pixel/cycle Seed Point Cells (Middle Schools) 10 1 pixel/cycle

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Table 1. Cell values and weights of the accessibility-weighted and population-weighted planes.

The geographic distribution of the population is one of the major considerations for school service area delimitation in addition to accessibility. Thus, another type of weighted plane for the adaptive crystal growth Voronoi diagrams in this study is based on population distribution. As discussed in Wang et al. (2014), due to the modiﬁable areal unit problem when using census tracts as the zonal scheme, creating better representations of the continuous distribution of the population is crucial for better results in service area delineations. Therefore, the population-weighted planes in this study were created based on census and land use data, which were used to estimate the population distribution of the research area using the same method in Wang et al. (2014). In the method, the continuous population distribution was simulated by distributing the population from census tracts to each cell in the weighted plane based on the distribution and attributes of residential areas. Residential areas are classiﬁed according to an urban land use map and urban planning data into three categories: low-rise residential areas, medium-rise residential areas, and high-rise residential areas. With maximum likelihood estimation, population density is 0.0680 persons/m2 for low-rise residential areas, 0.1135 persons/m2 for medium-rise residential areas, and 0.0937 persons/m2 for high-rise residential areas. Because four different spatial resolutions are used in this study, the cell weights of the population were calculated separately by multiplying the population density with the unit cell size. For instance, with the resolution of 100 m2/cell, the weights of the cells for the three types of residential areas are 6.8 persons/cell (low-rise residential areas), 11.35 persons/cell (medium-rise residential areas), and 9.37 persons/cell (high-rise residential areas) on the weighted plane. Residential cells are assigned a value of 4 (low-rise residence cells), 5 (medium-rise residence cells), or 6 (high-rise residence cells) for the three types of residential areas with a growth speed of 1 cell/cycle. Table1 lists all the possible cell values and growth speeds on the weighted planes, and Figure4b illustrates the population-weighted plane.

3.3. Crystal Growth Conﬁguration In order to make sure that the results of the raster-based and hexagon-based adaptive crystal growth Voronoi diagrams are comparable, the growth rules for both methods are uniﬁed as follows:

(1) The seed cells for the raster-based method are the raster cells that intersect with the locations of the 34 middle schools, while the seed point cells for the hexagon-based method are the hexagon cells that intersect with the locations of the 34 middle schools. In the crystal growth process, the service area of each facility grows from the seed point simultaneously. (2) In each crystal growth cycle of the raster-based method, the 8-domain region-growing algorithm was used for the crystal growth of road network cells due to their high accessibility, while the 4-domain region-growing algorithm was utilized for the crystal growth of other cells. Regarding the hexagon-based method, the 6-domain region-growing algorithm was used. (3) If a neighbor cell has already been labeled as a grown area or it is a barrier cell, it will be skipped. Otherwise, the neighbor cell will be labeled as a grown area of the speciﬁc middle school. Whenever a particular cell is incorporated into the grown service area of a particular seed, it becomes part of that service area. In other words, the calculation is based on the rules of “ﬁrst ISPRS Int. J. Geo-Inf. 2018, 7, 257 10 of 20

come, first served.” If the growth patterns of two service areas come into contact with each other at one place, the growth at that border location will stop for both service areas because the cells beyond that point are already incorporated into either of the service areas. If a cell is claimed by two or more different service areas at the same cycle during the crystal growth process, this cell will be randomly assigned to one of the service areas. (4) The speed of crystal growth for each cell is determined by the cell value and its growth speed, which was defined on the accessibility-weighted plane. Therefore, the speed is 1 cell per crystal growth cycle for seed point cells, walkable area cells, and residential area cells, while the speed is 4 cells per cycle for secondary transport network cells (the average travel speed on secondary transport network is about 4 times the speed of walking) and 6 cells per cycle for primary transport network cells (the average travel speed on primary transport network is about 6 times the speed of walking). (5) The growth speed of each grown area is adaptively adjusted in real time to optimize the population in each service area because the population in each service area should be commensurate with the enrollment capacity of each middle school. Therefore, the enrollment capacity of middle schools and the corresponding proportions of enrollment capacity out of the total enrollment capacity in the study area are considered as benchmarks for adjusting the crystal growth speed of each service area. The population of each grown area is calculated based on the population-weighted plane in real time. If the proportion of the population of a grown area (Pi) is larger than the proportion of its enrollment capacity (PECi) by a defined value ω (e.g., ω = 10%), the growth of the corresponding middle school will be restricted:

EC = i ∗ ( ≤ ≤ ) PECi N 100% 1 i N (1) ∑k=1 SCk

N AW = ∑ k=1Wk (2) W P = i ∗ 100% (3) i AW

where N is the number of middle schools, ECi is the enrollment capacity of middle schools, PECi is the proportion of enrollment capacity of each middle school out of the total enrollment capacity in the study area, Wk is the population in each grown area, and Pi is the proportion of the population of each grown area out of the total population in all the grown areas. If Pi > PECi ∗ (1 + ω)(1 ≤ i ≤ N), then the growth of middle school i will be restricted until Pi ≤ PECi ∗ (1 + ω)(1 ≤ i ≤ N). ω is the parameter that defines the strictness of the crystal growth constraint, and it is calibrated to achieve the best result. Generally speaking, the lower the ω value, the stricter the growth control; the higher the ω value, the less restrictive the crystal growth. For the specific case study of middle school service areas, the capacity of the service area of each middle school is controlled in the following manner. Equations (1)–(3) are used to calculate the proportion of the population of a grown area (Pi) and the proportion of its enrollment capacity (PECi) among all middle schools in the study area. Both values are calculated for each grown area after each growth cycle. For any grown service area, if the Pi is larger the PECi by a defined value ω (e.g., ω = 10%), the growth of the corresponding middle school will be restricted. In the calculation, the ω value will iterate from 0.001 to 0.1 in steps of 0.005 to find the best setting for the delineation considering the supply/demand ratio of the results. (6) The crystal growth will finish if all the accessible cells are grown and assigned to the grown area of one seed point (middle school). ArcMap was used for the data preprocessing and the hexagon grid generation. The simulation of the service area delineation was implemented with an algorithm we developed based on Python 2.7 and the open source package of Geospatial Data Abstraction Library [56]. The calculation was performed on a Windows 10 desktop computer (CPU: Intel Core i7-6700K 4 GHz; RAM: 16 GB). ISPRS Int. J. Geo-Inf. 2018, 7, 257 11 of 20

3.4. Hexagon Data Structure The raster-based weighted planes for RACG are generated based on the widely utilized raster data structure. Nevertheless, there is no mature data structure for hexagon grid computation to the best of our knowledge. The hexagon tessellation implemented by ArcGIS is in the vector format (collection of polygon features), which leads to very low computational efficiency. To address the limitation, we designed a hexagon data structure (we named it “Hexter”) in a similar format to a raster grid but specifically for the crystal growth algorithm to increase the computational efficiency in this study. As illustrated in Figure5a, the original hexagon grid is indexed as “ColID-RowID”, in which “ColID” represents the index of the column (ranging from “A” to “ZZZ”) in the grid, while “RowID” indicates the index of the row (ranging from “1” to “N”) in the grid. With the indexing, the hexagon ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 11 of 20 grid in the vector format is converted to a raster-like data structure (see the Hexter table in Figure5b) withrelationship their “ColID” used in as the the crystal columns growth and “RowID” algorithm as is the the rows neighbor in the cells, Hexter we table. defined With the the neighboring change of therelationship data format, of the the Hexter spatial based relationship on the inoriginal the table relationship is also changed. of the hexagon Since the grid. only Illustrated spatial relationship in Figure used5b, for in the the cells crystal in the growth odd columns algorithm (e.g., is the C, E, neighbor and G), cells,the neighbors we defined are the the neighboringcells in their relationshiptop, bottom, ofleft, the right, Hexter bottom-left, based on and the originalbottom-right relationship (e.g., cells of the with hexagon indices grid. “C-2” Illustrated and “C-7” in in Figure Figure5b, 5), for while the cellsfor the in thecells odd in the columns even columns (e.g., C, E, (e.g., and B, G), D, the and neighbors F), the neighbors are the cells are in the their cells top, in bottom,their top, left, bottom, right, bottom-left,left, right, top-left, and bottom-right and top-right (e.g., (e.g., cells cells with with indices indices “C-2” “H-2” and “C-7” and “H-7” in Figure in 5Figure), while 5). for The the rows cells indo thenot eveninfluence columns the neighboring (e.g., B, D, and relationship F), the neighbors of cells. are Note the that cells the in theirHexter top, table bottom, does left, not right,represent top-left, the andspatial top-right relationship (e.g., cellsof hexagon with indices cells. It “H-2” is only and used “H-7” to read in Figure the vector-formatted5). The rows do nothexagon influence grid theinto neighboringthe memory relationshipin the format of of cells. a table Note for thatcrystal the grow Hexterth computation. table does not After represent the computation, the spatial relationship the crystal ofgrowth hexagon results cells. (cell It isvalues) only used will tobe readprojected the vector-formatted to the original hexagon hexagon grid grid by into matching the memory the “ColID- in the formatRowID”. of a table for crystal growth computation. After the computation, the crystal growth results (cell values) will be projected to the original hexagon grid by matching the “ColID-RowID”.

(a) (b)

Figure 5. The original hexagon grid in the vector format (a) and transformed hexagon grid in the Figure 5. The original hexagon grid in the vector format (a) and transformed hexagon grid in the “Hexter” data structure (b) with an illustration of the neighbor cells definition. The illustrated center “Hexter” data structure (b) with an illustration of the neighbor cells deﬁnition. The illustrated center cells are colored in blue while their immediate neighbor cells are colored in yellow. cells are colored in blue while their immediate neighbor cells are colored in yellow.

3.5. Results 3.5. Results In the study, middle school service areas were delimited by RACG and HACG in four different In the study, middle school service areas were delimited by RACG and HACG in four different spatial resolutions (R1: 50 m2; R2: 100 m2; R3: 150 m2; R4: 200 m2) with different settings ranging spatial resolutions (R1: 50 m2; R2: 100 m2; R3: 150 m2; R4: 200 m2) with different ω settings ranging from 0.001 to 0.1. In order to compare the results, two types of measurements are utilized: (1) how from 0.001 to 0.1. In order to compare the results, two types of measurements are utilized: (1) how commensurate the population in each service area is with the enrollment capacity of the corresponding middle school; and (2) how accessible each middle school is within its service area. The root-mean-square error (RMSE) and maximum error (Max. Error) are employed to compare the commensurateness between the enrollment capacity of a school and the population of its service area. Figure 6 shows the RMSE of the differences between the proportion of enrollment capacity and the proportion of the population in the service areas delimited by RACG and HACG. It can be seen from the figure that, with different spatial resolutions, the HACG (solid lines in the figure) generally has smaller RMSE compared with the RACG (dashed lines in the figure). In addition, the finer the spatial resolution, the better the results considering the RMSE. In the figure, all lines present a U shape. The best results for different methods and different resolutions are generally achieved when has a value between 0.015 and 0.065. If is larger or smaller than these values, the RMSE increases. For the HACG, the best observed result is achieved when the spatial resolution is 50 m2 and is 0.055 or ranging from 0.015 to 0.035. Similarly, the best observed result is achieved when the spatial resolution is 50 m2 and is between 0.015 and 0.035 for the RACG. However, comparing the two

ISPRS Int. J. Geo-Inf. 2018, 7, 257 12 of 20 commensurate the population in each service area is with the enrollment capacity of the corresponding middle school; and (2) how accessible each middle school is within its service area. The root-mean-square error (RMSE) and maximum error (Max. Error) are employed to compare the commensurateness between the enrollment capacity of a school and the population of its service area. Figure6 shows the RMSE of the differences between the proportion of enrollment capacity and the proportion of the population in the service areas delimited by RACG and HACG. It can be seen from the figure that, with different spatial resolutions, the HACG (solid lines in the figure) generally has smaller RMSE compared with the RACG (dashed lines in the figure). In addition, the finer the spatial resolution, the better the results considering the RMSE. In the figure, all lines present a U shape. The best results for different methods and different resolutions are generally achieved when ω has a value between 0.015 and 0.065. If ω is larger or smaller than these values, the RMSE increases. For the HACG, the best observed result is achieved when the spatial resolution is 50 m2 and ω is 0.055 or ranging from 0.015 to 0.035. Similarly, the best observed result is achieved when the spatial resolution ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 12 of 20 is 50 m2 and ω is between 0.015 and 0.035 for the RACG. However, comparing the two curves of 2 HACGcurves andof HACG RACG and with RACG the spatial with resolution the spatial of resolution 50 m , the delimitationof 50 m2, the results delimitation of the hexagon-based results of the methodhexagon-based generally method perform generally better with perform lower better values with of RMSE. lower Figurevalues7 of illustrates RMSE. Figure the Max. 7 illustrates Error of thethe differencesMax. Error betweenof the differences the proportion between of enrollment the proportion capacity of enrollment and the proportion capacity of and the the population proportion in the of servicethe population areas delimited in the service by the areas RACG delimited and HACG by withthe RACG different and resolution HACG with and differentω settings. resolution Similar withand the settings. results of Similar RMSE, with the HACG the results (solid of lines RMSE, in the the figure) HACG generally (solid lines has smaller in the Max.figure) Error generally compared has withsmaller the Max. RACG Error (dashed compared lines inwith the figure)the RACG with (dash differented lines spatial in the resolutions. figure) with For bothdifferent RACG spatial and HACG,resolutions. the finerFor both the spatialRACG resolution,and HACG, the the smaller finer the the spatial Max. Error.resolution, According the smaller to the the figure, Max. the Error. best observedAccording results to the for figure, RACG the are best achieved observed when resultsω is ranging for RACG from are 0.001 achieved to 0.015, when while the is bestranging observed from results0.001 to for 0.015, HACG while are the achieved best observed when ω resultsis ranging for HACG from 0.02 are to achieved 0.08. Considering when is both ranging the RMSE from 0.02 and Max.to 0.08. Error, Considering the best observed both the resultRMSE is and obtained Max. Erro whenr, ωtheis best ranging observed from 0.02result to is 0.035 obtained for HACG, when while is itranging is achieved from when0.02 toω 0.035is 0.015 for forHACG, RACG. while With it the is achieved best observed whenω values, is 0.015 the for HACG RACG. has With much the lower best RMSEobserved and Max. values, Errors the comparedHACG has with much the lower RACG, RMSE as shown and Max. in the Errors figure. compared According with to the RACG, results, bothas shown the Max. in the Errors figure. and According RMSE of theto the differences results, both between the theMax. proportion Errors and of RMSE enrollment of the capacity differences and thebetween proportion the proportion of population of enrollment in the service capacity areas and are the lowest proportion when usingof population the hexagon-based in the service method areas afterare lowest calibration when ofusing the parameterthe hexagon-basedω. Therefore, method the resultsafter calibration indicate that of the the parameter hexagon-based . Therefore, method performsthe results better indicate considering that the thehexagon-based enrollment capacity method of performs middle schoolsbetter considering and the population the enrollment in their servicecapacity areas. of middle schools and the population in their service areas.

Figure 6. The root-mean-square error (RMSE) of the differences between the proportion of enrollment Figure 6. The root-mean-square error (RMSE) of the differences between the proportion of enrollment capacity and the population proportion of service areas with different ω settings. RACG: raster-based capacity and the population proportion of service areas with different ω settings. RACG: raster-based adaptive crystal growth method. HACG: hexagon-based adaptive crystal growth method. adaptive crystal growth method. HACG: hexagon-based adaptive crystal growth method.

Figure 7. The maximum error of the differences between the proportion of enrollment capacity and the population proportion of service areas with different ω settings.

ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 12 of 20

curves of HACG and RACG with the spatial resolution of 50 m2, the delimitation results of the hexagon-based method generally perform better with lower values of RMSE. Figure 7 illustrates the Max. Error of the differences between the proportion of enrollment capacity and the proportion of the population in the service areas delimited by the RACG and HACG with different resolution and settings. Similar with the results of RMSE, the HACG (solid lines in the figure) generally has smaller Max. Error compared with the RACG (dashed lines in the figure) with different spatial resolutions. For both RACG and HACG, the finer the spatial resolution, the smaller the Max. Error. According to the figure, the best observed results for RACG are achieved when is ranging from 0.001 to 0.015, while the best observed results for HACG are achieved when is ranging from 0.02 to 0.08. Considering both the RMSE and Max. Error, the best observed result is obtained when is ranging from 0.02 to 0.035 for HACG, while it is achieved when is 0.015 for RACG. With the best observed values, the HACG has much lower RMSE and Max. Errors compared with the RACG, as shown in the figure. According to the results, both the Max. Errors and RMSE of the differences between the proportion of enrollment capacity and the proportion of population in the service areas are lowest when using the hexagon-based method after calibration of the parameter . Therefore, the results indicate that the hexagon-based method performs better considering the enrollment capacity of middle schools and the population in their service areas.

Figure 6. The root-mean-square error (RMSE) of the differences between the proportion of enrollment ISPRScapacity Int. J. Geo-Inf. and 2018the population, 7, 257 proportion of service areas with different ω settings. RACG: raster-based13 of 20 adaptive crystal growth method. HACG: hexagon-based adaptive crystal growth method.

Figure 7. The maximum error of the differences between the proportion of enrollment capacity and Figure 7. The maximum error of the differences between the proportion of enrollment capacity and the the population proportion of service areas with different ω settings. population proportion of service areas with different ω settings. ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 13 of 20

The longest travel time (LTT) to a middle school from any location in its service area is used as The longest travel time () to a middle school from any location in its service area is used as a measure of the accessibility to the middle school within its service area in this study. The maximum a measure of the accessibility to the middle school within its service area in this study. The maximum LTT indicates the lowest accessibility of the middle school in its service area. As discussed above, indicates the lowest accessibility of the middle school in its service area. As discussed above, the the best observed results are achieved for both RACG and HACG when the spatial resolution is best observed results are achieved for both RACG and HACG when the spatial resolution is 50 m2 2 considering50 m considering the enrollment the enrollment capacity capacity of middle of middle scho schoolsols and and the the population population in intheir their service service areas; areas; therefore,therefore, thethe accessibilityaccessibility toto the the middle middle school school within within its its service service area area of theof the delineation delineation results results is only is compared using a spatial resolution of 50 m2. Figure8 shows the maximum LTT of the service areas only compared using a spatial resolution of 50 m2. Figure 8 shows the maximum of the service ω areasof all of middle all middle schools schools in the in studythe study area area based based on theon the delimitation delimitation results results obtained obtained with with different different LTT ωsettings. settings. For For the the HACG, HACG, the the maximum maximum varies varies with with different differentω settings: settings: itstays it stays at at high high values values of ≤ ≤ ofaround around 24 min24 min when when0.035 0.035ω 0.05 0.05, while , while it is as it low is asas 21low min as with21 min other withω values. other Regarding values. LTT Regardingthe RACG, the the RACG, lower values the lower of maximum values of maximum(around 25 min) (around are 25 achieved min) are when achievedω is smaller when than is smaller0.02 and than larger 0.02 than and 0.045. larger Comparing than 0.045. the Comparing two curve linesthe two of RACG curve andlines HACG, of RACG it is and clear HACG, that HACG it is LTT clearalways that performs HACG always better than performs RACG better with than smaller RACG maximum with smaller. maximum .

Figure 8. Maximum longest travel time ( of service areas based on the delimiting results with Figure 8. Maximum longest travel time (LTT) of service areas based on the delimiting results with different ω settings. different ω settings. The mean is the average value of the longest travel time to the middle schools within their service areas, which is used as a general indicator of the accessibility of all middle schools in their service areas based on the delimitation results. Figure 9 illustrates the mean of the service areas of all middle schools based on the delimitation results obtained with different ω settings. The figure shows that the mean for the HACG stays around 8.4 min while it is around 9.4 min for the RACG, but the mean slightly increased with the increase of the value of ω for both methods. Like the maximum , the mean obtained with the RACG is always higher than that obtained with the HACG.

Figure 9. Mean of service areas based on the delimiting results with different ω settings.

ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 13 of 20

The longest travel time () to a middle school from any location in its service area is used as a measure of the accessibility to the middle school within its service area in this study. The maximum indicates the lowest accessibility of the middle school in its service area. As discussed above, the best observed results are achieved for both RACG and HACG when the spatial resolution is 50 m2 considering the enrollment capacity of middle schools and the population in their service areas; therefore, the accessibility to the middle school within its service area of the delineation results is only compared using a spatial resolution of 50 m2. Figure 8 shows the maximum of the service areas of all middle schools in the study area based on the delimitation results obtained with different ω settings. For the HACG, the maximum varies with different settings: it stays at high values of around 24 min when 0.035 0.05 , while it is as low as 21 min with other values. Regarding the RACG, the lower values of maximum (around 25 min) are achieved when is smaller than 0.02 and larger than 0.045. Comparing the two curve lines of RACG and HACG, it is clear that HACG always performs better than RACG with smaller maximum .

ISPRS Int. J. Geo-Inf. 2018, 7, 257 14 of 20 Figure 8. Maximum longest travel time ( of service areas based on the delimiting results with different ω settings. The mean LTT is the average value of the longest travel time to the middle schools within their service areas,The mean which is used is the average as a general value indicatorof the longest of thetravel accessibility time to the middle of allmiddle schools within schools their in their serviceservice areas areas, based which on the is used delimitation as a general results. indicator Figure of the9 illustrates accessibility the of mean all middleLTT schoolsof the service in their areas service areas based on the delimitation results. Figure 9 illustrates the mean of the service areas of all middle schools based on the delimitation results obtained with different ω settings. The ﬁgure of all middle schools based on the delimitation results obtained with different ω settings. The figure shows that the mean LTT for the HACG stays around 8.4 min while it is around 9.4 min for the RACG, shows that the mean for the HACG stays around 8.4 min while it is around 9.4 min for the but theRACG, mean butLTT the slightlymean increased slightly increased with the with increase the increase of the valueof the ofvalueω for of bothω for methods.both methods. Like the maximumLike theLTT maximum, the mean , LTTthe meanobtained obtained with the with RACG the RACG is always is always higher higher than than that that obtained obtained with the HACG.with the HACG.

Figure 9. Mean of service areas based on the delimiting results with different ω settings. Figure 9. Mean LTT of service areas based on the delimiting results with different ω settings.

ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 14 of 20 The standard deviation of LTT measures the amount of variation of the longest travel time to the middle schoolsThe standard within deviation their delineated of measures service the areas. amount Figure of variation 10 illustrates of the longest the standard travel time deviation to the of the LTTmiddlefor allschools middle within schools their baseddelineated on the service delineated areas. Figure service 10 areas illustrates with the different standardω settings.deviation Similarof to thethe maximum for allLTT middle, for schools the HACG, based theon the standard delineated deviation service areas of LTT withstays different at high ω settings. values Similar of around to the maximum , for the HACG, the standard deviation of stays at high values of around 4.9 4.9 min when 0.035 ≤ ω ≤ 0.05. For all the other ω settings, it stays at low values of around 4.2 min. min when 0.0350.05. For all the other ω settings, it stays at low values of around 4.2 min. On LTT On thethe contrary, contrary, the the standard standard deviation of of the the obtainedobtained with with the the RACG RACG is always is always larger larger than than5 min. 5 min. Again,Again, the standard the standard deviation deviation of ofLTT obtained obtained usingusing the RACG RACG is is al alwaysways higher higher than than that that obtained obtained with thewith HACG. the HACG.

Figure 10. The standard deviation of of service areas for all middle schools based on the Figuredelimiting 10. The standardresults with deviation different of ω LTTsettings.of service areas for all middle schools based on the delimiting results with different ω settings. According to the calibration results of the ω settings, the best delimitation results were selected Accordingbased on how to thecommensurate calibration the results population of the inω eachsettings, service the area best is delimitationwith the enrollment results capacity were selected of basedthe on middle how commensurate school in the service the population area and how in each access serviceible the area middle is with schools the are enrollment within their capacity service of the middleareas. school Thus, in the the observed service areabest value and how of ω accessiblefor the HACG the is middle 0.02, while schools it is 0.015 are within for the theirRACG. service Figure areas. 11 illustrates the service area delineation results of the adaptive crystal growth Voronoi diagrams using the raster-based and hexagon-based weighted planes with their respective best observed values. It can be seen from the figure that the results are generally different. Further, the RMSE and Max. Error of the differences between the proportion of enrollment capacity and the proportion of the population in the service areas, as well as the maximum, mean, and standard deviation of the to a middle school from any location in its service, are compared in Table 2 with the observed best results of both methods. As the table shows, the RMSE, Max. Error, Max. , mean and S.D. are all smaller when using the hexagon-based method compared with their values obtained when using the raster-based method, which also indicates that the HACG performs better than the RACG considering both the commensurate the population in each service area is with the enrollment capacity of the corresponding middle school and the accessibility of each middle school within its service area.

Table 2. Comparison of the differences between the proportion of enrollment capacity of each middle school and the proportion of the population in each service area of the 34 middle schools in the observed best delimitation results (ω = 0.015 for RACG; ω = 0.02 for HACG).

RMSE Max. Error Max. Mean S.D. RACG 1.2048% 3.0379% 25.6759 9.2379 5.0803 HACG 1.1779% 2.9833% 21.7985 8.2283 4.3100 Note: RACG represents the raster-based adaptive crystal growth Voronoi diagram; HACG represents the hexagon-based adaptive crystal growth Voronoi diagram.

ISPRS Int. J. Geo-Inf. 2018, 7, 257 15 of 20

Thus, the observed best value of ω for the HACG is 0.02, while it is 0.015 for the RACG. Figure 11 illustrates the service area delineation results of the adaptive crystal growth Voronoi diagrams using the raster-based and hexagon-based weighted planes with their respective best observed ω values. It can be seen from the ﬁgure that the results are generally different. Further, the RMSE and Max. Error of the differences between the proportion of enrollment capacity and the proportion of the population in the service areas, as well as the maximum, mean, and standard deviation of the LTT to a middle school from any location in its service, are compared in Table2 with the observed best results of both methods. As the table shows, the RMSE, Max. Error, Max. LTT, mean LTT and S.D. LTT are all smaller when using the hexagon-based method compared with their values obtained when using the raster-based method, which also indicates that the HACG performs better than the RACG considering both the commensurate the population in each service area is with the enrollment capacity of the corresponding middle school and the accessibility of each middle school within its service area.

RMSE Max. Error Max. LTT Mean LTT S.D. LTT RACG 1.2048% 3.0379% 25.6759 9.2379 5.0803 HACG 1.1779% 2.9833% 21.7985 8.2283 4.3100 Note: RACG represents the raster-based adaptive crystal growth Voronoi diagram; HACG represents the hexagon-based adaptive crystal growth Voronoi diagram.

ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 15 of 20

(a)

(b)

Figure 11. The delineation result of middle school service areas by using raster-based (a) and hexagon- Figure 11.basedThe (b) delineationadaptive crystal result growth of Voronoi middle diagrams. school service areas by using raster-based (a) and hexagon-based (b) adaptive crystal growth Voronoi diagrams. Considering calculation efficiency, Figure 12 illustrates the number of cells for the zoning with different spatial resolutions and their computation time for both raster-based and hexagon-based adaptive crystal growth Voronoi diagrams. It can be seen from the figure that the number of cells increases (from 0.7 to 2.8 million) with finer resolution (from 200 to 50 m2), and the computation time for both methods rises. The computation time for RACG increases from 0.09 to 0.3 min while that for HACG increases from 0.35 to 2.34 min. With the longest computation time of about 2 min with a personal computer, the calculation efficiency for HACG is acceptable and feasible to be utilized in service area delineation problems.

ISPRS Int. J. Geo-Inf. 2018, 7, 257 16 of 20

Considering calculation efficiency, Figure 12 illustrates the number of cells for the zoning with different spatial resolutions and their computation time for both raster-based and hexagon-based adaptive crystal growth Voronoi diagrams. It can be seen from the figure that the number of cells increases (from 0.7 to 2.8 million) with finer resolution (from 200 to 50 m2), and the computation time for both methods rises. The computation time for RACG increases from 0.09 to 0.3 min while that for HACG increases from 0.35 to 2.34 min. With the longest computation time of about 2 min with a personal computer, the calculation efficiency for HACG is acceptable and feasible to be utilized in service area delineation problems. ISPRS Int. J. Geo-Inf. 2018, 7, x FOR PEER REVIEW 16 of 20

2.5 3 HACG RACG No. of cells 2.5 2

2 1.5 1.5 1 1

0.5 0.5 No. of cells (millions) No. of cells

Computation time (minutes) 0 0 R1 R2 R3 R4 Grid cell resolution

Figure 12. The computation efficiency of the raster-based (blue) and hexagon-based (orange) adaptive Figure 12. The computation efficiency of the raster-based (blue) and hexagon-based (orange) adaptive crystal growth Voronoi diagrams for different spatial resolutions. (The left vertical axis is the crystalcomputation growth Voronoi time in diagrams minutes, while for different the right spatialvertical resolutions.axis is the number (The of left cells vertical for the weighted axis is the computationplanes timewith indifferent minutes, resolution.) while the right vertical axis is the number of cells for the weighted planes with different resolution.) 4. Discussion 4. DiscussionThe best delineation results for HACG and RACG can be obtained by iterating and calibrating Thethe best parameter delineation ω. According results to for the HACG calibration and results RACG of can the beω settings, obtained the by best iterating observed and delimitation calibrating results were selected based on how commensurate the population in each service area is with the the parameter ω. According to the calibration results of the ω settings, the best observed delimitation enrollment capacity of the middle school in the service area and how accessible the middle schools results were selected based on how commensurate the population in each service area is with the are within their service areas. The delimitation results with observed best ω values are illustrated in enrollmentFigures capacity 11a (RACG) of the and middle 11b (HACG). school in In the these service figure areas, the and polygons how accessible with distinct the colors middle are schools the are withinservice their areas service of different areas. Themiddle delimitation schools. Becaus resultse the with road observed network, best naturalω values barriers, are illustratedand the in Figurepopulation 11a (RACG) distribution and Figure were taken 11b into (HACG). account In th theserough figures,two weighted the polygons planes in the with adaptive distinct crystal colors are thegrowth service Voronoi areas of diagrams, different different middle schools schools. have Because differently the road sized network, service areas; natural generally barriers, speaking, andthe populationthe service distribution area of were a middle taken school intoaccount is smaller through when the two road weighted network planes is sparser, in the the adaptive population crystal growthdensity Voronoi higher, diagrams, and more different natural schools barriers have exist differentlyin its surrounding sized servicearea. Not areas; surprisingly, generally the speaking, shape the serviceof some area middle of a middle schools’ school service is areas smaller is elongated when the along road the network transport is sparser,network. the This population is because these density middle schools are located in sparsely populated areas where the crystal growth is not constrained higher, and more natural barriers exist in its surrounding area. Not surprisingly, the shape of some and the crystal growth speed along transport network cells is faster than in other cells due to the high middle schools’ service areas is elongated along the transport network. This is because these middle accessibility. When comparing the delimitation results of the raster-based and hexagon-based schoolsadaptive are located crystal in growth sparsely Voronoi populated diagrams, areas Figure where 11 highlights the crystal the growthfact that isthese not two constrained service area and the crystaldelimitation growth methods speed along lead to transport different results network with cells the same is faster crystal than growth in other rules. cells due to the high accessibility.Regarding When comparing the ω value, the which delimitation is the only results parameter of the in raster-basedthe method and and represents hexagon-based how restrictive adaptive crystalthe growth growth Voronoi constraint diagrams, rule is, the Figure maximum 11 highlights value of theω for fact calibration that these could two be service defined area by the delimitation specific methodsquestion lead to of different how great results a percentage with theof service same over crystal its capacity growth for rules. each facility is allowed and tolerable. RegardingFor instance, the consideringω value, which the middle is the onlyschool parameter service capacity, in the methodwe require and that represents the service how population restrictive the growthshould constraint not go over rule its is, service the maximum capacity by value 10%. of Otherwise,ω for calibration the school could would be not defined be able by to the provide specific quality education for students. Based on that, the maximum value of ω could be set as 0.1. Therefore, question of how great a percentage of service over its capacity for each facility is allowed and tolerable. the value can be calibrated in the range from 0 to 0.1. Further, the process to find the optimal ω value For instance, considering the middle school service capacity, we require that the service population is problem-dependent. It depends on the particular service facility in question, and tolerance of errors shouldof not the go two over measurements its service capacityfor the specific by 10%. questions Otherwise, of (1) how the commensurate school would the not population be able to in provide each service area is with the enrollment capacity of the corresponding middle school (RMSE or Max. Error); and (2) how accessible each middle school is within its service area (Max. or Mean LLT). If the tolerance of errors for how commensurate the population in each service area is with the enrollment capacity of the corresponding middle school is 10%, and the target is to find the optimal results with the least mean LLT, then the calibration procedure is to find the optimal ω value with smallest mean LLT while the Max. Error is no larger than 10%. If the target is to find the best fit of the service

ISPRS Int. J. Geo-Inf. 2018, 7, 257 17 of 20 quality education for students. Based on that, the maximum value of ω could be set as 0.1. Therefore, the value can be calibrated in the range from 0 to 0.1. Further, the process to find the optimal ω value is problem-dependent. It depends on the particular service facility in question, and tolerance of errors of the two measurements for the specific questions of (1) how commensurate the population in each service area is with the enrollment capacity of the corresponding middle school (RMSE or Max. Error); and (2) how accessible each middle school is within its service area (Max. or Mean LLT). If the tolerance of errors for how commensurate the population in each service area is with the enrollment capacity of the corresponding middle school is 10%, and the target is to find the optimal results with the least mean LLT, then the calibration procedure is to find the optimal ω value with smallest mean LLT while the Max. Error is no larger than 10%. If the target is to find the best fit of the service population with the enrollment capacity and the tolerance of travel time in each service area is 25 min, then the calibration procedure is to find the optimal ω value with smallest RMSE while the Max. LLT is no larger than 25 min. The results prove that HACG generates better delineation results than does RACG in many ways. Considering how commensurate the population in each service area is with the enrollment capacity of the corresponding middle school, Figures6 and7 show that the proportion of the population in different service areas based on the results of the hexagon-based method is more commensurate with the proportion of middle schools’ enrollment capacity. This may suggest that the hexagon grid may suffer less from orientation bias and sampling bias and thus can generate service areas that better match the service capacities of the middle schools in the study area. Regarding the accessibility of each middle school within its service area, Figures8–10 indicate that the LTT of the service areas delimited by the hexagon-based method is smaller than the LTT of the service areas delimited by the raster-based method. These results indicate that the hexagon-based adaptive crystal growth Voronoi diagrams produced results superior to those obtained with the raster-based diagrams in addressing both the socioeconomic context and accessibility (travel time based on the road network and natural barriers) when delimiting service areas of public schools. Regarding the spatial resolution of the grid cells, the results indicate that the finer the resolution, the better the delineation results for both RACG and HACG. However, with the same resolution, HACG always performs better than RACG. Although the proposed method extends the adaptive crystal growth Voronoi diagrams and performs better than conventional raster-based methods, there are some limitations which should be addressed in future research. First, a considerable amount of inner-city travel is made by public transit (e.g., bus, subway), which is not considered in this research. It should be integrated into the accessibility-weighted plane to better evaluate the accessibility to public facilities. In addition, all areas are considered as walkable area except buildings, rivers, and lakes, and this is not accurate. Routes that are unsuitable for walking were ignored, and pedestrian sidewalks and pedestrian cut-throughs should be considered in future studies. Further, the socioeconomic weighted plane in this study only considered population distribution; weighted planes of other socioeconomic attributes and new crystal growth rules based on multiple socioeconomic weighted planes are also topics for future research. Lastly, this study is a pilot exploration of the method on school service area delineation to justify the advantages of HACG compared with RACG. However, more studies in different locations with different context settings are needed to investigate the merits and shortcomings of HACG further. Moreover, the method should be further examined in a broader set of test problems in order to be generalizable to the service area delineation of other public facilities.

5. Conclusions The hexagon-based adaptive crystal growth Voronoi diagram approach for service area delineation presented in this paper represents the most recent and advanced stage in the development of the method by incorporating the newest features, which extends the conventional adaptive crystal growth Voronoi diagrams by utilizing a hexagon-based spatial framework instead of the raster grid structure to represent the demand for service and the environmental contexts. Also, this study found that ISPRS Int. J. Geo-Inf. 2018, 7, 257 18 of 20 using a different representation of space (i.e., continuous space covered with hexagonal cells) led to an improvement of the service area delineation results. The results of the middle school service area delineation reveal the advantage of the hexagon-based adaptive crystal growth Voronoi diagrams when compared with the raster-based method, considering both how commensurate the population in each service area is with the enrollment capacity of the corresponding middle school, and how accessible the middle school is within its service area. In the hexagon-based method, the estimated continuous socioeconomic weighted plane mitigates the MAUP, and the hexagon grid accessibility-weighted plane suffers less from orientation bias and sampling bias from edge effects, which may help to delimit service areas that better match the service capacities of public facilities and improve the accessibility of public facilities in their service areas. In the future, other case studies on different kinds of public facilities with varying settings of context will be further explored to see if the conclusion that HACG works better than RACG for service area delineation is generalizable. The proposed method could provide decision support for policymakers in city management and urban planning (e.g., educational administration) based on robust quantitative analysis results. Although this study implemented hexagon-based adaptive crystal growth Voronoi diagrams for delineating middle school service areas that consider the distribution of population, the method may also be used for delimiting the service area of other public facilities (e.g., ﬁre stations, police stations, and emergency response centers) and may be extended through considering other socioeconomic factors. The hexagon-based adaptive crystal growth Voronoi diagrams and their implementation may help city managers to serve their citizens better and allocate public service resources more efﬁciently.

Author Contributions: J.W. and M.-P.K. conceived the idea; J.W. designed and performed the experiments and analyzed the data; J.W. wrote the paper; M.-P.K. contributed to refining and revising the paper. Funding: This research received no external funding. Acknowledgments: The authors would like to thank the anonymous reviewers for their valuable comments on the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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