The extended fuzzy C-means algorithm for hotspots in spatio-temporal GIS

In spatial analysis buffer impact areas are called hotspots and are determined by means of density clustering methods. In a previous work, we found these hotspots in the context of a Geographic Information System (GIS) by using the extended fuzzy C-means (EFCM). Here we show how the spatial distribution of the hotspots can evolve temporally and like applicational example, we present the spatial-temporal evolution in the period 2000–2006 of the fire point-events data of the Santa Fè district (NM) (downloaded from URL: www.fs.fed.us/r3/gis/sfe_gis.shtml). 2011 Elsevier Ltd. All rights reserved. 1. Clustering algorithms and GIS It is well known that the primitive operations in a Geographic Information System (GIS) concern points, lines and polygons. Thus the geographic location, for instance, of a criminal event is represented from a point, which is center of a circle considered as buffer influence area of that event and its radius is usually called the buffer distance. The buffer area is called hotspot (Chainey, Reid, & Stuart, 2002; Grubesic & Murray, 2001). Another typical example of hotspot is the circle having center in the epicenter of a earthquake. When the user faces with many point-events data, the buffer distance can also be a constant value for all these points (cf. Fig. 1) and the related circles are merged for creating new buffer areas (cf. Fig. 2). If one must manipulate a huge number of pointevents, the classical density clustering methods are not adequate in the determination of hotspots. Based fuzzy C-means algorithms (FCM) (Bezdek, 1981) seem more adequate: indeed it has been used in the determination of hotspots with high number of crimes (e.g., Chainey et al., 2002; Harries, 1999; McGuire & Williamson, 1999; Murray, McGuffog, Western, & Mullins, 2001). As in our previous papers (Di Martino, Loia, & Sessa, 2007; Di Martino & Sessa, 2009) we continue to propose the usage of the extended fuzzy C-means (EFCM) (Kaymak, Babuska, Setnes, Verbruggen, & van Nauta Lemke, 1997; Kaymak & Setnes, 2002) algorithm for three advantages with respect to the FCM’s: robustness to noise and outliers, linear computational complexity and automatic determination of the optimal number of clusters. After the determination of the hotspots, we analyze their temporal evolution between two successive years. In GIS literature the spatio-temporal evolution of some topics is already known (cf., e.g., Liu, Zhang, Cai, & Tong, 2010). In Section 2 we give an overview of the EFCM algorithm. In Section 3 we give an application of the EFCM algorithm in the specific problem of fire prevention of a forest area, located in New Mexico: indeed we construct hotspots which represent dangerous areas of fire point-events. In Section 4 we study their spatio-temporal evolution in the period 2000–2006, by making a comparison of the hotspots for every pair of successive years. Section 5 concludes the paper. 2. EFCM algorithm: an overview For sake of completeness, we recall the main steps of the EFCM (Kaymak & Setnes, 2002). Let X = {x1, . . . ,xN} R, xj = (x1j,x2j, . . . ,xnj) 2 R, be the dataset, where xij is the jth component (feature) of the vector xi, j = 1, . . . ,N. In the classical FCM algorithm the objective function to be minimized is the following: