Data Mining: The Textbook

16.2. MINING WITH CONTEXTUAL SPATIAL ATTRIBUTES

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is illustrated in Fig. 16.1. The different shades in the chart represent the different sea sur-face temperatures. These correspond to the values of the behavioral attributes at different spatial locations.

Another example is the case of image data, where the intensity of an image is measured in pixels. Such data is often used to capture diagnostic images. Examples of PET scans for a cognitively healthy person and an Alzheimer’s patient are illustrated in Fig. 16.2. In this case, the values of the pixels represent the behavioral attributes, and the spatial locations of these pixels represent the contextual attributes. The behavioral attributes in spatial data may present themselves in a variety of ways, depending on the application domain:

1. 
   For some types of spatial data, such as images, the analysis may be performed on the contour of a specific shape extracted from the data. For example, in Fig. 16.3, the contour of the insect may be extracted and analyzed with respect to other images in the data.
   

2. 
   For other types of spatial data, such as meteorological applications, the behavioral attributes may be abstract quantities such as temperature. Therefore the analysis can be performed in terms of the trends on these abstract quantities. In such cases, the spatial data needs to be treated as a contextual data type with multiple refer-ence points corresponding to spatial coordinates. Such an analysis is generally more complex.
   

The specific choice of data mining methodology often depends on the application at hand. Both these forms of data are often transformed into other data types such as time series or multidimensional data before analysis.

16.2.1 Shape to Time Series Transformation

In many spatial data sets such as images, the data may be dominated by a particular shape. The analysis of such shapes is challenging because of the variations in sizes and orientations. One common technique for analyzing spatial data is to transform it into a different format that is much easier to analyze. In particular, the contours of a shape are often transformed to time series for further analysis. For example, the contours of the insect shapes in Fig. 16.3 are difficult to analyze directly because of their complexity. However, it is possible to create a representation that is friendly to data processing by transforming them into time series.

A common approach is to use the distance from the centroid to the boundary of the object, and compute a sequence of real numbers derived in a clockwise sweep of the bound-ary. This yields a time series of real numbers, and is referred to as the centroid distance signature. This transformation can be used to map the problem of mining shapes to that of mining time series. The latter domain is much easier to analyze. For example, consider the elliptical shape illustrated in Fig. 16.4a. Then, the time series representing the distance from the centroid, using 360 different equally spaced angular samples, is illustrated in Fig. 16.4b. Note that the contextual attribute here is the number of degrees, but one can “pretend” that this represents a timestamp. This facilitates the use of all the powerful data mining techniques available for time series analysis. In this case, the sample points are started at one of the major axes of the ellipse. If the sample point starts at a different position, or if the shape is rotated (with the same angular starting point), then this causes a cyclic translation of the time series. This is quite important because the precise orientation of a shape may not be known in advance. For example, the shapes in Figs. 16.3b and c are rotated from the

534 CHAPTER 16. MINING SPATIAL DATA

Figure 16.1: Contour charts for sea surface temperatures: Image courtesy of the NOAA Satellite and Information Service