We have presented and evaluated a set of spatial ordering/encoding methods to transform spatial relations into a one-dimensional ordering and encoding, which preserves spatial
patterns as much possible. Such ordering and encoding can then be used in a variety of spatial or spatio-temporal data mining tasks. We designed a comprehensive set of measures to evaluate different orderings/encodings. The results revealed a number of important characteristics and unique behaviors for each ordering/encoding method. We showed that the optimal ordering/encoding with the complete-linkage clustering consistently gives the best overall performance, with various data distributions tested. We also briefly introduced two possible applications (out of many) that make use of the spatial ordering and encoding methods we describe.
Evaluation results with various data distributions show that the optimal ordering/encoding based on the complete-linkage clustering gives the best overall performance, surpassing well-known space-filling curves, in preserving spatial patterns. It can preserve spatial locality in both directions, i.e., on one hand spatial neighbors are close in the ordering and on the other hand neighbors in the ordering are also spatially close. The spatial ordering and encoding can then help in a variety of geographic data mining problems.
Although the optimal ordering strategy generally produces a better ordering/encoding than the simple ordering strategy for a given hierarchical clustering method, the primary factor that controls the ordering/encoding quality is the clustering method. For example, the single-linkage clustering gives very poor results in all tests, no matter which ordering strategy is used. The two space-filling curves (i.e., the Hilbert curve and the Morton curve) have very different characteristics. They generally work better with random data than clustered data but neither of them gives a good overall performance—they are good on one type of measures and perform badly on the other type of measures. Another advantage of the cluster-based ordering methods is that they can work with non-Euclidean data spaces.
Lastly, we would like to briefly compare the computational scalability of each ordering method. Space-filling curves are of O(nlogn) complexity and thus can process very large data sets. For each clustering-based method, the computational complexity involves two parts: the clustering procedure and the ordering procedure. The simple ordering strategy is of O(n) complexity while the optimal ordering strategy is of O(n3). The single-linkage clustering is of O(n2logn) complexity and the complete-linkage clustering is of O(n3). Therefore, the CLO_OPT and SNN_CLO_OPT ordering methods are the most time-consuming ones among all clustering-based methods. To derive an ordering of the 3128 US cities, the CLO_OPT method takes about 5 minutes on a desktop computer with 2.0GB of RAM memory and a 3.60GHz Pentium 4 CPU. Where efficiency is an issue, perhaps due to dataset size or time criticality of the application, then the simple ordering strategy might provide a more viable alternative.
Acknowledgement
This research was partially supported by grant CA95949 from the National Cancer Institute.
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