J. Geogr. Sci. 2010, 20(5): 787-798 DOI: 10.1007/s11442-010-0811-8 © 2010
Science China Press
Springer-Verlag
General multidimensional cloud model and its application on spatial clustering in Zhanjiang, Guangdong DENG Yu1,4, *LIU Shenghe1, ZHANG Wenting2, WANG Li3,4, WANG Jianghao1,4 1. Institute of Geographic Sciences and Natural Resources Research, CAS, Beijing 100101, China; 2. School of Resources and Environment Science, Wuhan University, Wuhan 430079, China; 3. Institute of Policy and Management, CAS, Beijing 100080, China; 4. Graduate University of Chinese Academy of Sciences, Beijing 100049, China
Abstract: Traditional spatial clustering methods have the disadvantage of “hardware division”, and can not describe the physical characteristics of spatial entity effectively. In view of the above, this paper sets forth a general multi-dimensional cloud model, which describes the characteristics of spatial objects more reasonably according to the idea of non-homogeneous and non-symmetry. Based on infrastructures’ classification and demarcation in Zhanjiang, a detailed interpretation of clustering results is made from the spatial distribution of membership degree of clustering, the comparative study of Fuzzy C-means and a coupled analysis of residential land prices. General multi-dimensional cloud model reflects the integrated characteristics of spatial objects better, reveals the spatial distribution of potential information, and realizes spatial division more accurately in complex circumstances. However, due to the complexity of spatial interactions between geographical entities, the generation of cloud model is a specific and challenging task. Keywords: multi-dimensional cloud; spatial clustering; data mining; membership degree; Zhanjiang
1
Introduction
With the rapid development of modern science and technology, the capacity of accessing data has been greatly improved. However, the complexity of massive data and the timeliness of data processing has prevented the effective use of data, we get into a contradiction in “rich data, meager knowledge” (Liu, 2007). In order to search for more valuable knowledge, Data Mining and Knowledge Discovery emerges, which has become the focus of international research and applications (Macqueen, 1967). Spatial clustering is one of the important methods applied to spatial data mining and knowledge discovery. There are many methods of spatial clustering including partitioning method (Kaufman and Rousseeuw, 1990), hierReceived: 2010-02-06 Accepted: 2010-04-16 Foundation: National Natural Science Foundation of China, No.40971102; Knowledge Innovation Project of the Chinese Academy of Sciences, No. KZCX2-YW-322; Special Grant for Postgraduates’ Scientific Innovation and Social Practice in 2008 Author: Deng Yu (1985–), Ph.D Candidate, specialized in urban development and land use. E-mail:
[email protected] * Corresponding author: Liu Shenghe, Professor, E-mail:
[email protected]
www.scichina.com
www.springerlink.com
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archy method (Berkhin, 2000; Zhang et al., 1996; Karypic and Han1999), method based on network (Wang et al., 1997; Sheikholeslami et al., 1998), and method based on density (Ester et al., 1996; Ankerst et al., 1999). Traditional methods of spatial clustering can not overcome the defect of hardware division effectively, as well as a reasonable expression of dynamic change. Therefore, it is particularly urgent to look for new methods of spatial clustering. The cloud model, which was introduced to China by Li Deyi, is a qualitative and quantitative uncertainty conversion model which was built on the basis of traditional fuzzy set theory, probability and statistics. It organically combines fuzziness and randomness of uncertain concept and realizes the conversion between uncertain language value and quantitative value (Li et al., 1995). Subsequently, Li Deyi discussed the universal nature of normal cloud model and broadened the scope of its applications (Li and Liu, 2004). In order to raise the awareness of cloud model and its application level, Liu Changyu analyzed the statistical significance and parameters characteristics of the normal cloud model (Liu, 2005). On this basis, the cloud model is widely used in spatial generalized knowledge and association rules mining, founded knowledge expression, continuous data discretization, spatial database uncertainty query and inference, remote sensing image interpretation and identification (Li et al., 1997; Li et al., 1998; Di et al., 1999; Liu et al., 2004). As the application deepens, cloud theory system is increasingly mature, such as cloud model, virtual cloud, cloud computing, cloud transform, uncertainty reasoning and so on (Li and Liu, 2004; Li, 2000). Moreover, theoretical research has instructive significance on practical applications. Cloud model wakened new interest in the application of clustering because it applies fuzziness and randomness of cloud drops to retain the uncertainty membership of spatial information. It can overcome many problems such as avoiding the defect of hardware division effectively, as well as expressing the process of dynamic change of spatial objects reasonably (Tang, 1986; Chen, 1998; Hou et al., 2008), and really implement “soft division”, which traditional spatial clustering methods can not achieve. Qin Kun (2006) applied cloud model in image classification and clustering, which was the first attempt of practical application (Qin, 2006). Wang classified the spatial object successfully using cloud model (Wang, 2007a). Wang did clustering research on the spatial object after “potential transform” (Wang, 2007b), which made some progress as well. However, all of them did not fully reflect the multi-dimensional characteristics of spatial data. These methods which achieve the purpose of dimensionality reduction only relying on data integration still have many defects, while weighting ascertainment is too subjective. Moreover, the error produced by normal atomic cloud fitting can not be controlled effectively when the leaf nodes in pan concept tree generates, and this method can not express the complexity of the characteristics of spatial objects accurately. In view of the above, this paper sets forth a general multi-dimensional cloud model, this model describes the characteristics of spatial objects more reasonably according to the idea of non-homogeneous and non-symmetry, therefore the new method is more accordant to practice. On this basis, this paper analyzes the space object with clusters by applying general multi-dimensional cloud model. The method avoided the issue of index weight ascertainment (Li and Zheng, 2004; Zhang et al., 2004), and its clustering results embodied the integrated characteristics of space objects, reflected the spatial distribution of the potential in-
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formation, and realized spatial division more accurately in complex circumstances. Therefore, general multidimensional cloud model can be widely applied to the research on spatial data mining and knowledge discovery.
2 2.1
Cloud model and general multidimensional cloud model Cloud model
Cloud model takes expectation value (Ex), entropy (En) and super-entropy (He) as a token of qualitative concept. It combines fuzziness and randomness together during qualitative transform. Ex reflects the cloud center of the cloud drops. En reveals the fuzziness of the range of concept numerical. The value of He reflects the discrete degree of cloud drops. Digital characteristics of cloud are shown as Figure 1.
Figure 1
Digital characteristics of cloud
It has been demonstrated that atomic cloud is universally adaptable (Li and Liu, 2004). However, in complex real world, this homogeneity and symmetry are difficult to meet. In order to portray the objective things more accurately, general multidimensional cloud model emerged. Figure 2 is a comparison diagram of a standard one-dimensional and a general multidimensional cloud model.
Figure 2
Comparison diagrams of 1D normal cloud and general cloud
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General multidimensional cloud model
General cloud overcomes the shortcomings of unreasonable spatial division, it has several non-equilibrium and non-symmetric forms; on the other hand, the heterogeneous characteristics of general multi-dimensional cloud model has great superiority in simulating complex phenomena. For example, the delimitation of influence radius of business service center usually spread along the traffic route, or spread to residential areas in a certain direction unsymmetrical. In order to deal with such problems effectively, the presentation of general cloud model seems particularly important. When necessary, using the relative position to the “cloud cores” to describe the direction range, and selecting appropriate normal cloud function can meet the requirements of complex situations. General cloud model is substantially piecewise, and its basic formula is shown in Figure 3 (taking two-dimensional cloud model as an example): 1 ( x1 Ex1)2 ( x 2 Ex 2 )2 e 2 ( En11)2 ( En 21)2 x1 Ex1 and x 2 Ex 2 (1) i 2 2 1 ( x1 Ex12) ( x 2 Ex 22) ( En 22 ) e 2 ( En12 ) others where i is degree of membership, xi is abscissa value of random dot, and x2 is ordinate value of random dot. (Ex1, Ex2) is expectation pair of two-dimensional cloud model. (En11, En21) is entropy pair of one direction, and (En12, En22) is entropy pair of the other direction.
Figure 3
3
Comparison diagram of 2D normal cloud and general cloud
Study of clustering based on general multidimensional cloud model
The basic idea in spatial clustering method based on general multidimensional cloud model is as follows: First of all, determine reasonable multidimensional cloud model parameters according to the radiation range and attribute dimension as well as related characteristics of a spatial object, and then generate atomic clouds, that is, leaf node in pan-concept-tree. Secondly, raise the lower conceptual fineness to a higher level according to synthetic operator of a multidimensional cloud model. Stop this step when the number of concepts equals to the number of classification grades. Finally, get the membership degree of each spatial objects from a higher-level concept. The concept with the highest membership degree to an object is the related concept of that object.
DENG Yu et al.: General multidimensional cloud model and its application on spatial clustering in Zhanjiang
3.1
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Generation of atomic cloud
Spatial objects are represented by concept, while atomic cloud is the smallest concept particle. The single object can be considered as atomic cloud, and thus how to determine the relevant parameters is especially important according to the complexity and comprehensiveness of spatial objects. Expectation value (Ex) reflects the horizontal location of the atomic cloud, a reflection of the concept of “core”. The membership degree of the location where expectation stands is 1, and it decreases gradually with the distance. Entropy (En) has an explicit geographic meaning, it is a metric of spatial radiation range according to the attribute value of spatial objects. R max R min 1 Ai A min ( Ai A min) (2) Eni 3 A max A min b Ai A min where A is the attribute value of spatial object, Amin, Amax are the minimum and maximum of attribute values; Rmin, Rmax are the minimum and maximum of radiation distance; b is a un1 1 determined constant; constant is obtained by the formula en＝ R. 3 3 Based on the non-homogeneous and non-symmetry attributes of spatial objects, entropy also has piecewise features, that is, anisotropy. Thus we should introduce a correction , which can depict the complex geographic phenomenon more accurately. The expression is as follows: Eni En＝ (3) (1 + ) Eni
The normal cloud is a normal distributed cloud whose deviation degree from the normal distribution is measured by super entropy He. If He = 0, cloud model degenerate into ordinary normal distribution. In order to show the dynamic changes of radiation range, and control the fuzzy degree of “Both This and That” characteristic, the setting of He is very important. Take all factors into consideration, the setting of He is 0.1 (Qin et al., 2006). After parameter setting, the atomic cloud--the generation of leaf nodes of pan concept tree terminated. This method takes consideration of both the accurate expression of central concept and the characteristics of edge dynamic changes. 3.2
Climbing of universal conceptual number--Cloud computing
Atomic cloud is raised from extended multidimensional cloud model to a higher level conceptual fineness gradually by comprehensive operations. The conceptual tree built by cloud model has property of uncertainty, and the boundary between concepts is vague. Concept fineness in lower level can climb to a high level to generate the needed leaf nodes in pan concept tree. The number of types decides the number of root nodes. Piecewise and multi-formation characteristic of atomic cloud have raised a higher demand for cloud operation. Superposition of different atomic clouds can be treated flexibly, and atomic clouds in the same membership interval are described as follows: C (E x1,E x2,E n1,E n2, H e1,H e2) , A 2 (E x21, E x22,E n21, E n22,H e21,H e22) , ...,
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A m (E xm1 , E xm2 ,E nm1,E nm2 ,H em1,H em2 ) Applying “Soft-Or” method, cloud synthetic algorithm can be ameliorated and represented as follows (Jiang et al., 2000): When the distance between A1and A2 is the minimum, that is Dmin= A1,A2 : Ex1= ( Ex11 + Ex21 ) / 2 + ( En21 En11 ) / 4 ;
(4)
Ex2= ( Ex12 + Ex22 ) / 2 + ( En22 En12 ) / 4 ;
(5)
En1 = ( Ex21 Ex11 ) / 4 +( En11 + En21 ) / 2 ;
(6)
En2 = ( Ex22 Ex12 ) / 4 +( En12 + En22 ) / 2 ;
(7)
He1 = max ( He11 , He21 ) ;
(8)
He2 = max ( He12 , He22 ) ;
(9)
Concept promoting is to get the difference between concept fineness in the same level, the most common of which is Euclidean distance, and to combine the concept with minimal expectation difference. Using multi-dimensional cloud synthetic operator mentioned above, the level of nodes in pan-concept-tree can be raised stage by stage. It is worth emphasizing that we should employ the union of different scope, when facing concepts in the same level with different direction. 3.3
Determining class--X-condition generators
When the number of father conceptual clouds in the highest level reaches the number of classification categories, pan-concept-tree based on cloud model has been built, and the cloud synthesis finishes. Then, the membership degree of each ordered set to its relevant concept in a higher level is gained on the basis of an X-condition normal cloud model. Among all classes, the one with the highest membership degree is defined as the membership analysis result of its relevant object. The specific algorithm can be described in the following steps: Step 1: Estimate the positional membership between (x1, x2) and (Ex1, Ex2), find out the corresponding cloud function Step 2: Compute formula ( P1i, P 2i ) R1( Ex1, Ex 2, En1, En 2) , and get ( P1i, P 2i ) as a random number under normal distribution with En as its expectation value and He as its standardization difference. Step 3: Compute formula i e
1 ( x1 Ex1)2 ( x 2 Ex 2 ) 2 2 ( P1i )2 ( P 2 i )2
(10)
Step 4: Compute Max i, and the object that is being studied falls into the relevant Class i.
4
Application on a case study in Zhanjiang
According to the General Multidimensional Cloud Model and its basic ideas in the Spatial Clustering Research, this paper uses the clustering of high schools in Zhanjiang city as a study case. Zhanjiang lies in the west of Guangdong Province, which is one of the first batch of open coastal cities. Owing to its predominant geological location, Zhanjiang has had a rapid development of society and economy. The author has participated in the multiple-index land price evaluation of Zhanjiang and mastered the basic data and service conditions of
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various infrastructures including high schools. Therefore, the related measurement results and spatial modification information in the base land value evaluation report of Zhanjiang can be used to ascertain the conceptional parameter of the general cloud model. Furthermore, comparative analysis of the clustering results and residential standard land price distribution map can be used to further embody the value and advantage of general cloud model. Divide the research area into 120×120 grids and each of the grids is 250×250 m2. Figure 4 shows the distribution map of the general cloud model defined by the basic data of the high schools in the research area. We can tentatively divide spatial pattern of high schools’ service zone into four areas. Then every school may belong to a certain zone according to the classification. Figure 5 shows the clustering results operated by the cloud synthetic operator, and the number of the classes can be different with different situations. We can see from Figure 5 that the result of the clustering cloud is obviously anisotropic, and the division of the spatial classes can be reflected by the spatial coverage range of the classes. It is noteworthy that none of the spatial entities totally belong to a certain class, that is to say, the division of the classes is “Both This and That”. However, each entity has the maximum membership degree according to itself.
Figure 4
Distribution of cloud model
Figure 5
Results of classification
Just as what Figure 6 shows, there are four cluster districts of membership degree in space, the maximum membership degree isoclines attenuate from the center to the peripheral area
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of each class. Due to the correction of the calculation of high schools’ service radius, the membership degree changes sharply in the direction of northwest, while attenuating smoothly in other directions. This results in the noncontinuous distribution of the membership degree in the direction of north and west. We can classify these high schools according to the maximum membership degree principle, and finish the spatial clustering work.
Figure 6
Regional isoline distribution of the largest membership
Further investigate the membership between membership degree and spatial position of each type of objects, as shown in Figure 7, there is a good correlation between them：The correlation coefficients with different zones are: 0.91 (west, F=167, sig=0.00), 0.67 (south, F=39, sig=0.00), 0.98 (north, F=26, sig=0.01), and 0.99 (east, F=2800, sig=0.00). West zone includes 17 objects, and the maximum covering distance is 9807 m. Compared with the mark of the spatial object in Figure 6, objects a1, a2 and a3 in west zone are far away from the center, which is resulted from the extension of the service radius in the direction of southeast. South zone includes 22 objects, the most distant object b1 which class is not in the direction of northwest, in the contrary, the closer object b2 is in this area, and thus membership degree of b2 is smaller. North zone and east zone have fewer objects, thus the fitting results are more satisfactory. Figure 8 reveals the fundamental similar information from the aspect of fitting error. In south zone, the maximal error is up to 62% because of the existence of b2, which is the reflection of the anisotropy of spatial objects’ radiation. Overall, membership degree of objects decreases with the increase of the distance to the class center. The spatial influence among objects causes the unbalanced decrease of the membership degree, which reflects the actual situation more rationally.
DENG Yu et al.: General multidimensional cloud model and its application on spatial clustering in Zhanjiang
Figure 7
Changing trends of membership of different types
Figure 8
Residual of membership of different types
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In order to represent the advantages of clustering method based on general multidimensional cloud, here we analyze the clustering results and FCM comparatively. FCM is a clustering algorithm which determines the subjection degree to a certain class according to membership degree. Membership function describes the sharing level of models between fuzzy classes. FCM was put forward in order to absorb the advantages of traditional C-means clustering, such as its convergent speed, insensitivity to initialization and ability to show the similar information among samples. The fuzzy partition matrix (U) is not only partly explicit but also maintains the fuzziness of samples’ spatial distribution thereby increases the accuracy of classification (Bezdek, 1981). The clustering result based on FCM is shown in Figure 9. Compared with the method in this paper, a1, a2, a3 and b1 are all classified as north zone. Although FCM can ensure the smallest differences among classes, it still could not consider effect characteristics of all the spatial objects in a complex geographic world. Therefore, FCM is hard to meet the requirements of scientific classification under the constraints of comprehensive factors.
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Figure 9
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Clustering results of fuzzy C-means
Figure 10 shows the distribution of residential affecting factors value of Zhanjiang city. The assessment coverage of the spatial distribution does not completely coincide with the city boundaries. Consequently, clustering classes have an obvious aggregation with the four extreme areas of land value distributions. 65% of spatial objects are located in the regions of high land prices, only 8% is distributed in regions of low land prices. The higher the land prices of the region, the more intensive the distribution of the spatial objects. The regions with a small membership degree often have
Figure 10
Relationship of house price and clustering results in 2008
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a low land prices either, especially in the price lowland between classes, just as the spatial object a1, a2, a3 and b1 in Figure 6, which reflect the fierce competition for space objects between class centers. The price lowlands reveal the characteristics of weakly absolute controlling force of all the classes over objects, and the clustering result is mostly influenced by these regions.
5
Conclusions and discussion
(1) Cloud model has dual characteristics of fuzzyness and randomness, and it shows great superiority in the expression of conceptual fineness. One-dimensional cloud model can not accurately reflect multi-attribute characteristics of the real-world, and essential information of spatial objects was lost during the procedure of simple data fusion. Standard two-dimensional cloud model overcomes some shortcomings of one-dimensional cloud, but it still can not meet the needs of simulating the non-homogeneous and non-symmetry characteristics of complex geographical phenomena. This paper puts forward a general multi-dimensional cloud model, and the problems above were resolved effectively and factually. (2) Based on the demonstrative study, a detailed interpretation of clustering results is made from the integrated perspective of the spatial distribution of membership degree of spatial clustering, and the comparative study of FCM and a coupled analysis of residential land prices. General multi-dimensional cloud model could reflect the integrated characteristics of spatial objects better, while the spatial clustering results can reveal the potential information of spatial distribution, and realize spatial division more accurately in complex circumstances. (3) The practical value of general multi-dimensional cloud model in spatial clustering is notable. However, parameter setting, the accuracy and uncertainty of the model are problems to be overcome. The characteristic of the model is that all the parameters have appropriate geographical meanings. This makes the description of geographical phenomena more reasonable. Due to the complexity of spatial interactions among geographical entities, the generation of a cloud model is still a specific and challenging task.
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