Stability of the surface generated from distributed points of uncertain location

Transcription

1 INT. J. GEOGRAPHICAL INFORMATION SCIENCE, 2003 VOL. 17, NO. 2, Research Article Stability of the surface generated from distributed points of uncertain location YUKIO SADAHIRO Department of Urban Engineering, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo , Japan; (Received 18 June 2001; accepted 29 April 2002) Abstract. Conversion of a point distribution into a surface is one of the spatial operations used in GIS. This supports the visual analysis of point patterns, which is usually followed by more sophisticated statistical and mathematical analysis. If the location of points is uncertain, however, the surface obtained becomes unstable and consequently the results of analysis may be unreliable. Though unavoidable in spatial data, locational uncertainty has been rather neglected in the context of spatial analysis. To fill this gap, this paper proposes a method for representing and analysing the stability of the surface generated from an uncertain point distribution. The surface stability is represented by a scalar function called the slope stability function. Its definition, calculation procedure, visualization method, and summary indices are proposed. The method is evaluated through an empirical study, and some findings are shown which help us understand the effect of the smoothing parameter, locational uncertainty, and spatial pattern of points on the stability of a surface. 1. Introduction In point pattern analysis, the first step of exploration is to visualize the distribution of points (Slocum 1999). The map visually reveals the spatial structure of a point distribution, and may suggest underlying factors to an investigator. Visualization of a point distribution greatly helps its further analysis such as statistical analysis and mathematical modelling. If we have only a small number of points, they are represented and visualized as point objects in a GIS; it is easy to understand their spatial structure by looking at their spatial distribution directly. However, if a large number of points are visualized, say, a thousand or a million points, it becomes quite difficult to find spatial patterns in the point distribution. The points are then converted into a surface by the kernel method (Silverman 1986, Scott, 1992), quadrat method (Upton and Fingleton 1985, Cressie 1993), or other methods (Bailey 1994, Bonham-Carter 1994, Bailey and Gatrell 1995). The kernel method, for instance, has been applied to the distributions of crimes (Goldsmith et al. 2000), volcanic craters (Bailey and Gatrell 1995), redwoods (Fotheringham et al. 2000), population (Bracken and Martin 1989, Martin and Bracken 1991, Bracken 1993) and retail stores (Sadahiro 2001). The quadrat International Journal of Geographical Information Science ISSN print/issn online 2003 Taylor & Francis Ltd DOI: /

2 140 Y. Sadahiro method has also been frequently used in ecology (Pielou 1977), criminology (Brantingham and Brantingham 1981) and epidemiology (Alexander and Boyle 1996). Both methods are used for not only the visualization of point distributions but also in statistical analysis and modelling. In visualization, however, the kernel method is preferred because, unlike the quadrat method, it generates a smooth surface from a point distribution. One important view, that has been rather neglected in spatial analysis including the conversion of a point distribution into a surface, is locational uncertainty in spatial data. If the location of points is uncertain, the surface generated from the points is inevitably uncertain and unstable. Spatial data are usually inaccurate to some extent (Goodchild and Gopal 1989, Burrough and Frank 1996, Heuvelink 1998), and thus uncertainty is unavoidable in spatial analysis. Though this fact is widely recognized in the GIS community, it is often implicitly assumed that spatial data are accurate and the result of analysis is reliable. This is usually not the case. Recently, effort has been made to analyse the effect of uncertainty in spatial operations. Leung and Yan (1997), for instance, discussed how the locational uncertainty affects the point-in-polygon operation. Error propagation in spatial operations such as spatial interpolation, spatial smoothing, overlay and buffer operations has been studied by statistical models and empirical studies ( Veregin 1994, Davis and Keller 1997, Hunter and Goodchild 1997, Heuvelink 1998). Jacquez (1996) proposed statistical tests for uncertain spatio-temporal point distributions. Sadahiro (2002) analyzed cluster detection in uncertain point distributions. However, conversion of a point distribution into a surface has not yet been discussed in terms of locational uncertainty in the point data. To fill the gap in the research, this paper proposes a method for representing and analysing the stability of the surface generated from a distribution of uncertainly located points. We focus on the kernel smoothing method because it is suitable for visual analysis of point distributions, the first step of exploratory spatial analysis. In 2, we outline kernel smoothing and propose a scalar function called the slope function that describes the spatial structure of the surface generated from the distribution of certain points. In 3, extending the method proposed in 2, we propose a method for representing and analysing the stability of the surface generated from the distribution of uncertain points. To represent the surface stability, we introduce a scalar function called the slope stability function. The section describes its calculation and visualization, and summary indices. To test the validity of the method, we perform an empirical study in 4. Section 5 provides a concluding discussion. 2. Surface under certainty 2.1. Kernel smoothing Suppose there are n points in a region S. The ith point and its locational vector are denoted by P i and z i, respectively. The surface value at a location x given by the kernel smoothing is f (x)= k( x z ) i i P k( y z )dy i j yµs (1)

3 Stability of surface generated from uncertain points 141 where k( ) is the kernel function. A typical choice for k( ) is the bivariate normal distribution k(u)= a exp( a u 2) (2) p where a is the smoothing parameter (Silverman 1986). A small value of a generates a smooth surface, while a large a yields a rough surface which emphasizes large local variations in point density. In the following we use the bivariate normal distribution as the kernel function. Figure 1 shows an example of kernel smoothing using the bivariate normal distribution Representation of surface under certainty In point pattern analysis, we are interested in the spatial structure of points characterized by point clusters, spatial variation of point density and so forth. When a point distribution is converted into a surface, the structure of the original distribution is transferred to the resultant surface. Therefore, it is useful to examine the surface generated from a point distribution as well as to analyse the point distribution directly, in order to understand its spatial structure. To this end, we propose a binary function representing a surface with emphasis on its spatial structure. Let v(h) be the unit vector of angle h measured counterclockwise from the x-axis (0 h<2p). The gradient of the surface function f (x) atxin direction h is given by the directed differentiation of f (x): f (x, h)=(f (x) v(h) (3) Using f (x, h), we define a binary function s(x, h; f )by if f (x, h)0 s(x, h; f )= G1 (4) 0 otherwise and call it the slope function. This function indicates the surface gradient as a binary variable: if the surface ascends at a location x in direction h, the slope function Figure 1. Kernel smoothing. (a) A point distribution, (b) a surface generated by the kernel smoothing (a=0.001).

4 142 Y. Sadahiro becomes one; otherwise, it becomes zero. Figure 2 shows the slope function of the surface in figure 1(b). The slope function has two values of h where it discretely changes between one and zero, which agrees with the direction of contour lines of the surface. Therefore, the slope function also shows the distribution of the surface aspect, and consequently, gives us the outline of the spatial structure of points. For instance, figure 2 roughly tells us the location of point clusters and spatial variation of point density. The slope function is a summary of the original surface function, emphasizing its spatial structure. 3. Surface under uncertainty If the location of points is certain, the surface generated by the kernel smoothing can be either visualized as it is or outlined by the slope function proposed in the previous section. If their location is uncertain, however, the surface obtained becomes unstable and can be represented by neither the simple surface function nor the slope function. To solve this problem, this section extends the slope function so that it represents the stability of a surface instead of the surface itself Representation of locational uncertainty of points Representation of locational uncertainty in spatial data is a controversial research topic in GIS (Goodchild 1989, Burrough and Frank 1996, Burrough and McDonnell 1998); at least two methods exist for representing locational uncertainty in terms of spatial analysis. One uses a continuous function such as the fuzzy membership function (Burrough and McDonnell 1998) and the probability density function (Shi 1998). The other defines the region of a deterministic boundary within which a spatial object is possibly located (Perkal 1966, Chrisman 1982a, 1982b, Dunn et al. 1990, Sadahiro 2002). This paper follows the latter representation of locational uncertainty; we are certain that a spatial object is located in a bounded region called the error zone but we do not know its exact location. This is a key to calculating the surface stability; it gives us high tractability in computation of the slope function and its extension as shown later. The deterministic boundary approach is reasonable Figure 2. The slope function of the surface shown in figure 1(b).

5 Stability of surface generated from uncertain points 143 when the point location is represented by a higher level of address system such as blocks and cities, and it gives a good approximation to continuous functions we can determine a threshold of uncertainty or probability to define the boundary of error zones. In this paper we assume that error zones are represented as polygons, which also approximate spatial objects with curved boundaries. Let Q i be the error zone of point P i in S. The set of error zones {Q 1, Q 2,..., Q n } is denoted by V Representation of surface under uncertainty Uncertainty in the location of points yields instability of the surface generated by the kernel smoothing; the surface fluctuates according to the location of points. Its global structure, however, is usually robust against uncertainty. Remember the point distribution and its surface representation shown in figure 1. We intuitively believe that the global structure of the surface would not drastically change even if a little uncertainty exists in the location of points; there are two peaks in the region, one around the centre and the other in the south of the region. Though it depends on the degree of smoothing determined by the parameter a (equation (2)), there often exists a stable structure in an unstable surface and we may detect it by visual analysis. This intuitive discussion can be described by the behaviour of the slope function under uncertainty. If there is only a small locational uncertainty the slope function is likely to be stable almost everywhere. On the other hand, if the location of points is quite uncertain, the slope function may become unstable. To treat the surface stability formally, we introduce the slope stability function defined by l(x, h; V)= if YP G1 i µq, s(x, h; f )=1 i 0 otherwise (5) If the surface ascends at x in direction h for any possible location of points, the slope stability function l(x, h; V) becomes one. Otherwise, if the surface can descend according to the location of points, the function becomes zero. The slope stability function shows the possible surface gradient as a binary variable, and consequently, the stability of a surface. Figure 3 shows an example of the slope stability function generated on the assumption that a certain amount of uncertainty exists in the locations of points shown in figure 1. The fan-shaped figures indicate the direction in which the surface ascends for any possible location of points. Around the largest point cluster (see also figure 1) the figures face in the direction to the centre of the cluster. This indicates that the surface tends to ascend to the cluster centre, even if the location of points is uncertain. The size of fan-shaped figures indicates the stability of a surface. Therefore, figure 3 shows that the surface is not stable around the centre of the cluster, while it is stable on the periphery, facing in the direction to the cluster centre. Comparing figures 2 and 3, we notice that the slope function and the slope stability function are quite similar. It is reasonable because the latter is an extension of the former. For instance, if the point location is certain, the slope stability function is equivalent to the slope function; the range of h where the slope stability function is equal to one reduces with an increase of uncertainty. In addition, both functions outline the spatial structure of points. Figures 2 and 3 roughly indicate the location

6 144 Y. Sadahiro Figure 3. An example of the slope stability function. of point clusters, the spatial variation of point density, and so forth. We can understand the spatial structure of points from the slope stability function even if the point location is uncertain. From this we can say that the slope stability function shows the structural stability of a surface, and consequently, that of its original point distribution. This is one advantage of the surface stability function. The slope stability function indicates the aspects that the surface can take according to the location of points. Therefore, we can evaluate the surface stability at x by integrating the slope stability function by h. For instance, if the point location is certain, then the aspect of the surface is fixed, which results in P 2p l(x, h; V)dh=p (6) 0 If the point location is so uncertain that the surface can take any aspect at x according to the location of points, then we have P 2p l(x, h; V)dh=0 (7) Computational procedure for calculating the slope stability function Evaluation of the slope stability function is based on the calculation of the minimum gradient of the surface at every location. To this end, we consider the kernel functions of individual points separately. The kernel function for the point P is i k (x)=exp( a x z ) (8) i i where z µq. For the sake of simplicity, we omit the coefficient a/p in equation (2) i i without the loss of generality. The gradient of the kernel function at x in direction h is given by k (x, h)=(k (x) v(h) (9) i i (recall equation (3)). Let k (x, h) be the minimum of k (x, h): i,min i k i,min (x, h)= min P i µq i k i (x, h) (10)

7 Stability of surface generated from uncertain points 145 Since the surface function is the sum of the individual kernel functions (equation (1)), the minimum gradient of the surface is given by the sum of the minimum gradient of kernel functions, that is, S k (x, h). i i,min The slope stability function becomes one if the surface ascends at a location x in direction h for any possible location of points. Therefore, equation (5) becomes l(x, h; V)= if S G1 i k (x, h)0 i,min (11) 0 otherwise To calculate equation (11), we consider two points at x and x+dx denoted by A and A, respectively. Let ds be the distance between the points, that is, ds= dx. Then we have dx=(ds cosh, ds sinh) (12) Substituting equation (12) into equation (10), we obtain k (x+dx) k (x) k (x, h)= min lim i i (13) i,min P i µq i ds0 ds For the sake of simplicity, we translate and rotate A, A, and Q so that A is at i the origin of the coordinate system and A is on the positive side of the x-axis (figure 4). New coordinates of the points A and A are denoted by (0, 0) and (ds, 0), respectively. The coordinate of P is given by (l cosq, l sinq), where the angle Q is i measured counterclockwise from the x-axis (0 Q<2p). Equation (13) then becomes k i,min (x, h)=min l,q lim ds0 exp( a{(l cosq ds)2+l2 sin2q}) exp( al2) ds exp( ads2)exp(2aldscosq) 1 =min exp( al2) lim (14) l,q ds0 ds To solve the right side of equation (14), we first fix the variable l and minimize k (x, h) with respect to Q, which is equivalent to minimizing k (x, h) with respect to i i Figure 4. The coordinate system for calculation of the slope stability function.

8 146 Y. Sadahiro cosq. Let D(Q; l) be the domain of the variable Q for a given l. We can obtain it by overlaying a regular polygon on Q i that approximates the circle of radius l centred at the origin and calculate their intersections (figure 5). If pµd(q; l), then k i (x, h) is minimized when Q=p. Otherwise, k i (x, h) is minimized when Q is located exactly on one of the boundaries of D(Q; l). We then minimize k i (x, h) with respect to l. Let D(l) be the domain of l given by the minimum and maximum distances from the origin to Q i. Gradually increasing l from its minimum in D(l), we obtain k i,min (x, h) and finally the slope stability function l(x, h; V) using equation (11) Unstable regions In analysis of surfaces, critical points, that is, peaks, bottoms, and cols (saddle points) play a crucial role (Warntz 1966, Warntz and Waters 1975, Okabe and Masuda 1984, Iri et al. 2000, Sadahiro 2001). They are defined by the locational vector x satisfying d f (x) =0 (15) dx and describe the spatial structure of a surface in a simple but efficient way. Similar concepts can be defined for uncertain surfaces. As mentioned earlier, there may be regions in S where the surface can take any aspect according to the location of points (equation (7)). We call them unstable regions. Stable regions, on the other hand, are the regions in which every location has at least one aspect not taken by the surface. Unstable regions are further classified into three types: peak regions, bottom regions, and neutral regions. If the surface always ascends in a certain direction into Figure 5. Calculation of the domain D(Q; l).

9 Stability of surface generated from uncertain points 147 an unstable region everywhere on its boundary, the region contains at least one peak independently of the location of points. We thus call it a peak region. Ifthe surface always ascends out of an unstable region, the region contains at least one bottom and we call it a bottom region. Unstable regions classified into neither of them are called neutral regions. Stable regions do not contain either peaks or bottoms. Figure 6 shows an example of unstable regions. Unstable regions, as well as the slope stability function, show the structure of an uncertain surface and its stability against the locational uncertainty of points. They are in a sense substitutes for critical points in surface analysis V isualization and evaluation of surface stability Having obtained the slope stability function and unstable regions, we visualize them to understand the structural stability of a surface, and consequently, that of its original point distribution. A simple but useful visualizing method is to use the fanshaped figures shown in figure 6. Using the centroid of error zones, we can also calculate an average surface function which helps our interpretation of the slope stability function. Figure 7 shows an example of the visualization of the stability of the surface generated from an uncertain point distribution. In figure 7(c) we notice that the peak at the centre of the region (figure 7(b)) is stable against data uncertainty; there exists at least one peak in that region independently of the location of points in error zones. The peak in the south, on the other hand, is in the neutral region, which implies that the peak may disappear according to the location of points. To evaluate the overall stability of a surface quantitatively, we propose two indices: the stable region ratio (SRR) and the stability ratio (SR). Let d(x; V) be a binary function representing stable regions: if d(x; V)=G1 P 2p l(x, h; V)dh>0 0 0 otherwise (16) The SRR is the ratio of the area of stable regions to that of the region S: P d(x; V)dx SRR= xµs dx P xµs The SR is an extension of the SRR weighted by the slope stability function: P P 2p l(x, h; V)dhdx SR= xµs 0 p P xµsdx (17) (18) Either index shows a large value if the surface is stable against locational uncer-

10 148 Y. Sadahiro Figure 6. Unstable regions. Figure 7. Visualization of the stability of the surface generated from an uncertain point distribution. (a) Error zones, (b) a surface generated by the kernel smoothing (a= 0.001, the points are located at the centroid of error zones), (c) the slope stability function and unstable regions. tainty of points. A small value indicates that the surface is unstable and attention should be paid to its visual analysis. From the definitions they satisfy 0 SRR, SR 1 (19) The SR and SRR can be used to evaluate not only the stability of the surface but also the smoothing parameter a in terms of the surface stability. If the surface

11 Stability of surface generated from uncertain points 149 is extremely unstable, the slope stability function gives us little information about the spatial structure of points. Therefore, to perform a meaningful analysis of a point distribution, we should apply a values that keep the surface stability above a certain level. Whether an a value is acceptable can be judged by calculating the SR or SRR value and comparing it with its desirable level determined by analysts. 4. Empirical study This section empirically analyses the stability of the surface generated from a point distribution, in order to test the validity of the method proposed in the previous section. We also discuss the effect of the smoothing parameter, locational uncertainty of points, and spatial pattern of points on the surface stability in a practical setting. Point data used in the analysis are based on the list of retail stores and restaurants in the NT T telephone directory published monthly. Among 1670 categories we chose coffee shops, clothing shops, and convenience stores in the Shinjuku area in Tokyo, Japan (figure 8). The numbers of shops are 689, 555, and 145, respectively. We extracted their addresses in September 1999 in ASCII format, and converted them into point data by geocoding. The locational data obtained by geocoding are highly accurate; their locational error is usually kept below 10 m, especially in the central area of Tokyo. We thus considered hypothetical error zones of points represented by the circle of radius r, which was set to 50, 100, and 150 m, successively. Figure 9 shows the error zones of radius 100 m assumed for the distribution of coffee shops. We then calculated the slope stability function and unstable regions, following the procedure simply illustrated in figure 10. We overlaid a lattice of side 10 m over the study region, and calculated the slope stability function at each lattice node and the middle point of each edge by the procedure shown in the previous section. The slope stability function calculated on edges was used for detection of unstable regions. The smoothing parameter a (equation (2)), which reflects the scale of spatial analysis, was set to 0.001, 0.01, and 0.1, successively. The results are shown in table 1 and figure 11. In table 1 we notice that the surface stability decreases as uncertainty of point location increases; both the SRR Figure 8. Shinjuku area in Tokyo, Japan.

12 150 Y. Sadahiro Figure 9. Error zones of radius 100 m assumed for the distribution of coffee shops. Figure 10. Calculation of the slope stability function and unstable regions. (a) The slope stability function at lattice points, (b) the slope stability function calculated on edges (an arrow indicates that the surface always ascends in its direction), (c) unstable regions detected by the result shown in (b). and SR monotonically decrease with an increase of r. Similarly, the stability decreases with an increase of the smoothing parameter a. When a=0.001, the surface is quite smooth and the SRR shows large values over 98%. As a increases, however, the stability monotonically decreases in all cases. This implies that enlargement of the scale of analysis, say, transition from global to local analysis, reduces the reliability of the result. From this we obtain one obvious solution to uncertainty in spatial analysis; we should analyse spatial phenomena at a scale small enough to keep the reliability of the result at a certain level. The SRR and SR seem to be closely related with each other. In fact, Pearson s correlation coefficient of these indices is 0.954, a large value indicating that they are highly correlated with each other. Figure 11 shows that the unstable region increase mainly at the expense of the

13 Stability of surface generated from uncertain points 151 Table 1. The stable region ratio (SRR) and stability ratio (SR) (percentage). The variable r is the radius of the circles that represent the hypothetical error zones, and a is the smoothing parameter used in equation (2). r: 50m r: 100 m r: 150 m SRR SR SRR SR SRR SR (a) Coffee shop a: a: a: (b) Clothing shop a: a: a: (c) Convenience store a: a: a: neutral region. When a=0.1, for instance, the neutral region accounts for at least four-fifths of the unstable region in all cases. This implies that an increase of a and r reduces the relative amount of information that we obtain from the unstable region. Among three types of unstable regions, the peak and bottom regions assure the existence of at least one peak and bottom, respectively. On the other hand, the neutral region does not provide any information about the existence of critical points, and thus is less informative than the peak and bottom regions. Let us then discuss the effect of point pattern on the surface stability, comparing the results of three categories of shops. They have different spatial patterns. Coffee shops are strongly clustered around Shinjuku station. Clothing shops also show a clustered pattern, but they have two clusters: one around Shinjuku station, and the other near Harajuku station. Unlike these categories, convenience stores are widely dispersed in the study region as shown in figure 11(c). This is because convenience stores are quite similar to each other so that they receive little additional profit by agglomeration and travel distances are generally low for non-luxury items. Table 1 shows that the results are quite similar among the three categories when a= A difference arises with an increase of a. When a=0.01, the clothing shops have the largest SRR and SR values among the three categories, which implies that the surface generated from the distribution of clothing shops is the most robust against locational uncertainty. On the other hand, the convenience stores yield the worst result the smallest SRR and SR values. When a=0.1, however, the convenience stores become the best and the coffee shop the worst. The order of the three categories in terms of the surface stability is independent of the radius of error zones. The order of these categories is closely related to the spatial pattern of shops, and it can be explained by the separability of points as follows. Consider two uncertain points in a one-dimensional region represented by error zones (figure 12(a)). In this case there are three unstable regions: two exactly coincide with the error zones, and one is in the middle of the error zones (figure 12(b)). If the points are closer as shown in figure 12(c), the middle unstable region expands (figure 12(d)).

14 152 Y. Sadahiro Figure 11. Results of analysis. (a) The coffee shop, (b) clothing shop, (c) convenience store. The first column shows the centroid of error zones. The second column is the surface generated from the points located at the centroid of error zones. The third, fourth, and fifth columns show the slope stability function (the fan-shaped figure) and unstable regions (the grey area indicates the neutral region, and the areas enclosed by the solid and broken lines indicate the peak and bottom regions, respectively).

15 Stability of surface generated from uncertain points 153 Figure 11(c). However, if the points get much closer (figure 12(e)), the unstable regions are merged into smaller one (figure 12( f )). To sum up, when two uncertain points get closer, the surface stability first decreases and then increases. The above observation indicates that the surface stability is greatly affected by the separability of points determined by the distance between error zones and the shape of the kernel function. The unstable region is small when either error zones are almost separable with respect to the kernel function (figure 12(b)) or they are too close to distinguish (figure 12( f )). Instability arises when error zones are neither separable nor indistinguishable. In our example, it seems that coffee shops become indistinguishable when a decreases from 0.1 to Convenience shops, on the other hand, experience it when a decreases from 0.01 to This is understandable because the convenience stores are more dispersed than the coffee shops as mentioned earlier. The above discussion suggests that, given a point distribution, a parameter value of a exists that minimizes SRR and SR, and consequently, generates the most unstable surface from the distribution. In our empirical study, however, we did not see such a case where the stability reaches its minimum in terms of SRR and SR. It seems to appear when a>0.1, but such a steep kernel function is not meaningful in point pattern analysis because the surface obtained is almost equivalent to the distribution of error zones itself. 5. Concluding discussion In this paper we have developed a method for representing and analysing the surface generated from an uncertain point distribution. We proposed the slope stability function to evaluate the surface stability quantitatively. It also enables us to understand the spatial structure of a surface, and consequently, that of its original point distribution, even if the point location is uncertain. Visualization of the slope stability function and unstable regions outlines the structural stability of an uncertain

16 154 Y. Sadahiro Figure 12. Relationship between the point distribution and the surface stability. (a)(c)(e)two points in a one-dimensional region (solid and broken lines represent the error zones of the points), (b) (d) ( f ) kernel functions (solid and broken lines) and unstable regions ( black rectangles).

17 Stability of surface generated from uncertain points 155 point distribution. The overall stability of a surface is quantitatively evaluated by two indices, the SRR and SR. They also serve as criteria for evaluating the smoothing parameter a in terms of the surface stability. To test the validity of the method, we analysed the stability of surfaces generated from the point data of retail stores and restaurants in Tokyo, Japan. The empirical study yielded some interesting findings that help us understand the nature of surface stability such as the effect of the smoothing parameter, locational uncertainty of points, and spatial pattern of points. We finally discuss some limitations of our method for further research. First, we have assumed discrete representation of locational uncertainty, which may be called the epsilon band approach (Perkal 1966, Chrisman 1982a, 1982b). One of the advantages of this approach is high tractability in computation of the slope stability function. However, this representation is not the only way to treat uncertainty; representation of locational uncertainty in spatial data is a research topic in progress as mentioned in 3. If the locational uncertainty of points is represented by a continuous function, one practical solution is to determine a threshold of uncertainty or probability and use it to define the boundary of error zones. This enables us to calculate the slope stability function and to evaluate the stability of the surface generated from the points. Otherwise, we may have to calculate the joint probability distribution of the surface function directly from the probability distributions of points. Though this is an important and challenging research topic, we leave it for future research. Second, we have used the bivariate normal distribution as the kernel function throughout this paper. This is because the bivariate normal distribution is most widely used in GIS and spatial analysis. However, there are many other kernel functions such as the Epanechnikov kernel, triangular kernel, and so forth (Silverman 1986). When a kernel other than the bivariate normal distribution is used, calculation of the slope stability function requires a different computational procedure because equation (14) takes a different form. A new computational procedure with wider applicability should be developed to treat a variety of kernel functions. Third, we have discussed the effect of locational uncertainty in a rather limited context: conversion of a point distribution into a surface. As discussed in the introductory section, however, uncertainty affects the results of a wide range of spatial operations and analysis including spatial statistics, spatial optimization, and spatial decision making. It is necessary to discuss the effect of uncertainty in a broader context, and to establish a common methodology of handling uncertainty in these spatial procedures. References ALEXANDER, F. E., and BOYLE, P., 1996, Methods for Investigating L ocalized Clustering of Disease (Lyon, France: International Agency for Research on Cancer). BAILEY, T. C., 1994, A review of statistical spatial analysis in geographical information systems. In Spatial Analysis and GIS, edited by S. Fotheringham and P. Rogerson (London: Taylor & Francis), BAILEY, T. C., and GATRELL, A. C., 1995, Interactive Spatial Data Analysis (London: Taylor & Francis). BONHAM-CARTER, G. F., 1994, Geographic Information Systems for Geoscientists (Oxford: Pergamon). BRACKEN, I., 1993, An extensive surface model database for population-related information: concept and application. Environment and Planning B, 20, BRACKEN, I., and MARTIN, D., 1989, The generation of spatial population distributions from Census centroid data. Environment and Planning A, 21, BRANTINGHAM, P. J., and BRANTINGHAM, P. L., 1981, Environmental Criminology (Prospect Heights, Illinois: Waveland Press).

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