Quantitative Methods and Applications in GIS - Chapter 3

3 Spatial Spatial Smoothing and Interpolation This chapter covers two more generic tasks in GIS-based spatial analysis: spatial smoothing and spatial interpolation. Spatial smoothing and spatial interpolation are closely related and are both useful to visualize spatial patterns and highlight spatial trends. Some methods (e.g., kernel estimation) can be used in either spatial smooth-ing or interpolation. There are varieties of spatial smoothing and spatial interpolation methods. This chapter only covers those most commonly used. Conceptually similar to moving averages (e.g., smoothing over a longer time interval), spatial smoothing computes the averages using a larger spatial window. Section 3.1 discusses the concepts and methods for spatial smoothing, followed by case study 3A using spatial smoothing methods to examine Tai place-names in southern China in Section 3.2. Spatial interpolation uses known values at some locations to estimate unknown values at other locations. Section 3.3 covers point-based spatial interpolation, and Section 3.4 uses case study 3B to illustrate some common point-based interpolation methods. Case study 3B uses the same data and further extends the work in case study 3A. Section 3.5 discusses area-based spatial interpolation, which estimates data for one set of (generally larger) areal units with data for a different set of (generally smaller) areal units. Area-based interpolation is useful for data aggregation and integration of data based on different areal units. Section 3.6 presents case study 3C to illustrate two simple area-based interpolation methods. The chapter is concluded with a brief summary in Section 3.7. 3.1 SPATIAL SMOOTHING Like moving averages that are calculated over a longer time interval (e.g., 5-day moving-average temperatures), spatial smoothing computes the value at a location as the average of its nearby locations (defined in a spatial window) to reduce spatial variability. Spatial smoothing is a useful method for many applications. One is to address the small numbers problem, which will be explored in detail in Chapter 8. The problem occurs for areas with small populations, where the rates of rare events such as cancer or homicide are unreliable due to random error associated with small numbers. The occurrence of one case can give rise to unusually high rates in some areas, whereas the absence of cases leads to a zero rate in many areas. Another application is for examining spatial patterns of point data by converting discrete point data to a continuous density map, as illustrated in Section 3.2. This section discusses two common spatial smoothing methods (floating catchment area method and kernel estimation), and Appendix 3 introduces the empirical Bayes estimation. 35 © 2006 by Taylor & Francis Group, LLC 36 Quantitative Methods and Applications in GIS 11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28 31 32 33 34 35 36 37 38 41 42 43 44 45 46 47 48 51 52 53 54 55 56 57 58 61 62 63 64 65 66 67 68 72 73 75 76 77 78 71 74 81 82 83 84 85 86 87 88 91 92 93 94 95 96 97 98 FIGURE 3.1 The FCA method for spatial smoothing. 3.1.1 FLOATING CATCHMENT AREA METHOD The floating catchment area (FCA) method draws a circle or square around a location to define a filtering window and uses the average value (or density of events) within the window to represent the value at the location. The window moves across the study area until averages at all locations are obtained. The average values have less variability and are thus spatially smoothed values. The FCA method may be also used for other purposes, such as accessibility measures (see Section 5.2). Figure 3.1 shows part of a study area with 72 grid-shaped tracts. The circle around tract 53 defines the window containing 33 tracts (a tract is included if its centroid falls within the circle), and therefore the average value of these 33 tracts represents the spatially smoothed value for tract 53. The circle centers around each tract centroid and moves across the whole study area until smoothed values for all tracts are obtained. A circle of the same size around tract 56 includes another set of 33 tracts that defines a new window for tract 56. Note that windows near the borders of a study area do not include as many tracts and cause a lesser degree of smoothing. Such an effect is referred to as edge effect. The choice of window size is very important and should be made carefully. A larger window leads to stronger spatial smoothing, and thus better reveals regional than local patterns; a smaller window generates reverse effects. One needs to exper-iment with different sizes and choose one with balanced effects. Implementing the FCA in ArcGIS is demonstrated in case study 3A in detail. We first compute the distances (e.g., Euclidean distances) between all objects, and then distances less than or equal to the threshold distance are extracted.1 In ArcGIS, we then summarize the extracted distance table by computing average values of © 2006 by Taylor & Francis Group, LLC Spatial Smoothing and Spatial Interpolation 37 Kernel function K( ) Xi Data point Bandwidth Grid FIGURE 3.2 Kernel estimation. attributes by origins. Since the table only contains distances within the threshold, only those objects (destinations) within the window are included and form the catchment area in the summarization operation. This eliminates the need of pro-gramming that implements iterations of drawing a circle and searching for objects within the circle. 3.1.2 KERNEL ESTIMATION The kernel estimation bears some resemblance to the FCA method. Both use a filtering window to define neighboring objects. Within the window, the FCA method does not differentiate far and nearby objects, whereas the kernel estimation weighs nearby objects more than far ones. The method is particularly useful for analyzing and displaying point data. The occurrences of events are shown as a map of scattered (discrete) points, which may be difficult to interpret. The kernel estimation generates a density of the events as a continuous field, and thus highlights the spatial pattern as peaks and valleys. The method may also be used for spatial interpolation. A kernel function looks like a bump centered at each point xi and tapering off to 0 over a bandwidth or window. See Figure 3.2 for illustration. The kernel density at point x at the center of a grid cell is estimated to be the sum of bumps within the bandwidth: f (x) = 1d n K(x − xi ) i=1 where K( ) is the kernel function, h is the bandwidth, n is the number of points within the bandwidth, and d is the data dimensionality. Silverman (1986, p. 43) provides some common kernel functions. For example, when d = 2, a commonly used kernel function is defined as f (x) = n 1π in1 [1− (x − xi)2 + (y − yi)2 ]2 where (x − xi)2 + (y − yi)2 measures the deviation in x-y coordinates between points (xi, yi) and (x, y). © 2006 by Taylor & Francis Group, LLC 38 Quantitative Methods and Applications in GIS Similar to the effect of window size in the FCA method, larger bandwidths tend to highlight regional patterns and smaller bandwidths emphasize local patterns (Fotheringham et al., 2000, p. 46). ArcGIS has a built-in tool for kernel estimation. To access the tool, make sure that the Spatial Analyst extension is turned on by going to the Tools from the main manual bar and selecting Extensions. Click the Spatial Analyst dropdown arrow > Density > choose Kernel for Density Type in the dialog. 3.2 CASE STUDY 3A: ANALYZING TAI PLACE-NAMES IN SOUTHERN CHINA BY SPATIAL SMOOTHING This case study examines the distribution pattern of Tai place-names in southern China. The study is part of an ongoing larger project2 dealing with the historical origins of the Tai in southern China. The Sinification of ethnic minorities, such as the Tai, has been a long and ongoing historical process in China. One indication of historical changes is reflected in geographical place-names over time. Many older Tai names can be recognized because they are named after geographical or other physical features in Tai, such as “rice field,” “village,” “mouth of a river,” “mountain,” etc. On the other hand, many other older Tai place-names have been obliterated or modified in the process of Sinification. The objective of the larger project is to reconstruct all the earlier Tai place-names in order to discover the original extent of Tai settlement areas in southern China before the Han pushed south. This case study is chosen to demonstrate the use of GIS technology in historical-linguistic-cultural studies, a field whose scholars are less exposed to it. We selected Qinzhou Prefecture in Guangxi Autonomous Region, China, as the study area (see the inset in Figure 3.3). Mapping is important for examining spatial patterns, but direct mapping of Tai place-names may not be very informative. Figure 3.3 shows the distribution of Tai and non-Tai place-names, from which we can vaguely see areas with more representations of Tai place-names and others with less. The spatial smoothing techniques help visualize the spatial pattern. The following datasets are provided in the CD for the project: 1. Point coverage qztai for all towns in Qinzhou, with the item TAI identifying whether a place-name is Tai (= 1) or non-Tai (= 0). 2. Shapefile qzcnty defines the study area of six counties. 3.2.1 PART 1: SPATIAL SMOOTHING BY THE FLOATING CATCHMENT AREA METHOD We first test the floating catchment area method. Different window sizes are used to help identify an appropriate window size for an adequate degree of smoothing to highlight general trends but not to block local variability. Within the window around each place, the ratio of Tai place-names among all place-names is computed to represent the concentration of Tai place-names around that place. In implementation, the key step is to utilize a distance matrix between any two places and extract the places that are within a specified search radius from each place. © 2006 by Taylor & Francis Group, LLC Spatial Smoothing and Spatial Interpolation 39 Non-Tai N Tai Guangxi County Qinzhou 0 12.5 25 50 75 100 Kilometers FIGURE 3.3 Tai and non-Tai place-names in Qinzhou. 1. Computing distance matrix between places: Refer to Section 2.3.1 for measuring the Euclidean distances. In ArcToolbox, choose Analysis Tools > Proximity > Point Distance. Enter qztai (point) as both the Input Features and the Near Features and name the output table Dist_50km.dbf. By defining a wide search radius of 50 km, the distance table allows us to experiment with various window sizes £ 50 km. In the distance file Dist_50km.dbf, the INPUT_FID identifies the “from” (origin) place, and the NEAR_FID identifies the “to” (destination) place. 2. Attaching attributes of Tai place-names to distance matrix: Join the attribute table of qztai to the distance table Dist_50km.dbf based on the common keys FID in qztai and NEAR_FID in Dist_50km.dbf. By doing so, each destination place is identified as either a Tai place or non-Tai place by the field point:Tai. 3. Extracting distance matrix within a window: For example, we define the window size with a radius of 10 km. Open the table Dist_50km.dbf > click the tab Options at the right bottom > Select By Attributes > enter the condition Dist_50km.DISTANCE <=10000. For each origin place, only those destination places within 10 km are selected. Click Options > Export, and save the new table as Dist_10km.dbf, which keeps only distances of 10 km. Those records with a distance = 0 (i.e., the origin and destination places are the same) indicate that the search circles are centered around these places. © 2006 by Taylor & Francis Group, LLC ... - tailieumienphi.vn