| The multiplicatively weighted Voronoi diagram is a spatial division method that takes generator’s weights into account. The gravity model and its modified versions are of one kind of spatial interaction model that measures spatial correlation of spatial objects, both of which reflect the discrepancies of the attractiveness of the geographic entity and consider the feature that the attractiveness of the geographic entity becomes lower as the distance among them increases,becoming two common and direct ways to define the attracting scopes of the geographic entity, which are widely applied in various fields, such as research on the space pattern evolvement of the city, the construction of the spatial hierarchies of the cities, the space pattern optimization of rural residential areas, the spatial accessibility analysis of the facitilities, and the facility location optimization of the facitilities. The multiplicatively weighted Voronoi diagram and the gravity model can be used to effectively guide the urban planning, the regional resource allocation and the reasonable layout of the facilities. The gravity model and its modified versions depict interaction of the geographic entity in the form of actuating quantity, while the multiplicatively weighted Voronoi diagram describe tangible boundaries of the geographic entity in the form of figure. There is an intimate connection between these two theories and methods. They can be combined together effectively. The extended breaking point theory model is one of the combined cases. Since proposed, it has been applied in various fields and has gradually becoming a research focus in the context of ―regional and urban integration‖ and ―equalization of public service facilities‖. Limited by the difficulty to create and expand the multiplicatively weighted Voronoi diagram, the combination form of the two methods is usually by taking advantage of the gravity model and its modified versions to confirm the weight form of the geographic entity and using multiplicatively weighted Voronoi diagram to depict the attracting scope of the geographic entity, which lacks deep research and function expansion. The combination theory of the two methods remains to be deeply digged and their combination applications need to be extensively explored. Based on it, we determine the subject of this paper, i.e. ―the combination of the multiplicatively weighted Voronoi diagram and the spatial interaction model and its application and extension‖. Using GIS tools to create the multiplicatively weighted Voronoi diagram and combining it with the gravity model, we propose a set of spatial models and methods that are used to express spatial pattern of the geographic entity. Aiming at the situation of the aging population and responding to the call of national policies, we choose the nursing institutions for the aged in Beijing as the research objects and explore their spatial pattern from the perspective of spatial interaction, spatial hierarchy and spatial accessibility, which aims to provide the foundation for planning of the nursing institutions for the aged in Beijing.The main research contents and results are as follows:(1) We conclude that the gravity model, the Reilly model, the breaking point model and the Huff model conform to the definition of the multiplicatively weighted Voronoi diagram after transformation. A graphic expression of the gravity model and its modified versions is confirmed.(2) Research on the gravity-multiplicatively weighted Voronoi diagram model① We prove that a normal form of the gravity model satisfies the definition of the multiplicatively weighted Voronoi diagram. Based on it, we propose a uniform graphic expression of the gravity model, the Reilly model, the breaking point model and the Huff model, that is, the gravity-multiplicatively weighted Voronoi diagram model. The properties, deductions and relevant spatial pattern evaluation indexes of the proposed model are elaborated.② We compare the four division methods of the attracting scope for the geographical entities, which concludes the buffer method, the Voronoi diagram method, the service area analysis method and the gravity-multiplicatively weighted Voronoi diagram model method. Applicable conditions of the proposed model are highlighted.③ We review and compare the common vector-based and the raster-based algorithms of the multiplicatively weighted Voronoi diagram. Based on the topological union method, we proposed a vector-based algorithm to generate the multiplicatively weighted Voronoi diagram using C# and Arc GIS Engine. The basic idea and implementation steps of the algorithm are elaborated. Experimental results show that the diagram created by our algorthm contains disconnected regions and holes, satisfying the definition and properties of the multiplicatively weighted Voronoi diagram. Using patch shape index to test the ratio of the hole and the perimeter of a circle with equal area with that hole, we prove that the holes are standard circles, that is, the boundaries of the multiplicatively weighted Voronoi diagram created by our algorithm are standard circular arcs, or the result precision is 100%. It takes 24.95 second to create a multiplicatively weighted Voronoi diagram of 200 generators(the scope: 100m×100m), which illustrates that the algorithm efficiency is acceptable when the amount and the scope of the generators is not very large.④ The evaluation indexes of the nursing institutions for the aged are confirmed. The construction method of the attraction index of the nursing institution for the aged is given.⑤ We portray the scope of the nursing institutions for the aged with the gravity-multiplicatively weighted Voronoi diagram model. Using the spatial pattern evaluation indexes, we analyze the service area of the nursing institutions for the aged, the coverage rate of the aging population and the supply and demand ratio. Some conclusions are drawn:(a) The service areas of the nursing institutions for the aged in the six center districts are generally smaller, while those in the outskirts are universally larger.(b) The difference of the attraction of the nursing institutions for the aged in Shunyi is unconspicuous, but obvious in Changping, Mentougou and Yanqing.(c) The bed supply is extraordinary insufficient in the six center districts, while it oversupplies in the ourskirts.(d) The spatial interaction among the nursing institutions for the aged in Beijing is balanced.(4) Spatial hierarchy construction for the geographic entity based on the gravity-multiplicatively weighted Voronoi diagram model① We propose a spatial level establishment method based on the gravity-multiplicatively weighted Voronoi diagram model. The specific implementation steps are given.② Based on the gravity-multiplicatively weighted Voronoi diagram model and the Voronoi treemap, we propose a spatial hierarchy construction method using gravity-multiplicatively weighted Voronoi treemap. Its definition and relevant deductions are given. The differences between the gravity-multiplicatively weighted Voronoi treemap and the Voronoi treemap are summarized.③ We compare the similarities and the differences between the Voronoi diagram method and the gravity-multiplicatively weighted Voronoi diagram model method, and discover rules as follows:(a) while neglecting friction of distance factor, the local centers and generator belongingness may be different. If we change the weight values of one generator and its neiborhood, the local centers and generator inclusion relation ascertained by Voronoi diagram method will not verify, but may alter for our proposed method.(b) While considering friction of distance factor, if we use our proposed method to construct spatial hierarchy, two cases should be dicussed. The fisrt one is when b approaches zero, only one local center with the largest weight can be ascertained from the generators and only two hierarchies can be constructed form that local center and the generators. The second one is when b approaches ￥. The spatial hierarchy constructed by our method is the same to that created by the Voronoi diagram method. In other words, Voronoi diagram method is a exception of gravity-multiplicatively weighted Voronoi diagram model method while b approaches ￥ and the friction of distance factor is considered.④ We utilize the gravity-multiplicatively weighted Voronoi diagram model to generate the multiplicatively weighted Voronoi diagram of the nursing institutions for the aged. Based on it, their levels and hierarchies are confirmed and the gravity-multiplicatively weighted Voronoi diagram treemap is created. Then, a spatial hierarchy of the nursing institutions for the aged in Beijing with four levels is constructed. Through analyzing it, some conclusions are drawn as follows:(a) the nursing institutions for the aged with large weight may not be a local center;(b) The local centers in the nursing institutions for the aged are concentrated in outskirts;(c) The local centers of the nursing institutions for the aged with higher level in the six center districts contain more local centers with lower level than that in outskirts;(e) The distribution of the local centers of the nursing institutions for the aged is basically consistent with the density of aging population.(5) The spatial accessibility analysis based on the gravity-multiplicatively weighted Voronoi diagram model① We propose a spatial accessibility method based on the gravity-multiplicatively weighted Voronoi diagram model. The basic idea and implementation steps are elaborated.② We utilize two-step floating catchment method(the space segmentation threshold is set to 3300 m, 5000 m, 7000 m, 10000 m, etc) and the gravity-multiplicatively weighted Voronoi diagram model method(the neighbourhood value is set to 0 and 1) to calculate the accessibilities of the nursing institutions for the aged. The maximum, the minimum, the mean value and the standard deviation of the accessibility values, which are calculated by the two-step floating catchment method and our proposed method, are acquired. The spearmen rank correlation coefficient of the accessibility values acquired by these two methods are gotten. Then, we use ―min-max normalization‖ to transfer the accessibility values to interval [0,10] and create several scatter diagrams. Through analyzing above results, several conclusions can be drawn.(a) The variation trend for each statistical parameter acquired by our proposed is nearly as same as that gotten by the two-step floating catchment method as searching area extends.(b) The mean values gotten by the two methods are approximate.(c) The accessibility values have a highest correlation for two methods, while k is set to 0 and 1 for the gravity-multiplicatively weighted Voronoi diagram model method and s0 is set to 7000 m and 16000 m for the two-step floating catchment method.(d) The majority of accessibility values for our proposed are higher than that for the two-step floating catchment method when s0 is set bigger, and are lower when s0 is set smaller. From a statistical standpoint, we conclude the advantanges of our proposed method comparing with the two-step floating catchment method, i.e. the space segmentation coefficient is used, and the difference among supply points and spatial interaction among supply points are considered, and accessibility value will never be zeor.③ We record the accessibility values acquired by the gravity-multiplicatively weighted Voronoi diagram model and the two-step floating catchment method into the data base of the streets(township) of Beijing and rendering these data using Arc GIS. From the rendering maps we can see that accessibility pattern for our proposed method is similar to that for the two-step floating catchment method when k is set to 0 and s0 is set smaller or when k is set to 1 and s0 is set bigger. From the perspective of geography, we conclude the advantanges of our proposed method comparing with the two-step floating catchment method, i.e. the accessibility pattern for our proposed method has a better effect on the transition of the accessibility values among the adjacent districts in Beijing.④ Based on the gravity-multiplicatively weighted Voronoi diagram model, we depicit the accessibility pattern of the nursing institutions for the aged of Beijing in the street scale and the region scale. In the street scale, we distinguish the districts with the highest and the lowest accessibility values. In the region scale, we draw some conclusions as follows:(a) the accessibility values in the north are higher than that in the south;(b) The accessibility values are higher in the intermediate zone of Beijing, but lower in center zone and at the edge of Beijing, that is, the trend of accessibility value from the city center to the fringe appears a ―low-high-low‖ hierarchical structure;(c) There is a spatial mismatch phenomenon for the bed supply and the needs of the aged in Beijing. |
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