Clustering analysis identifies samples as groups based on either their mutual closeness or homogeneity. In order to detect clusters in arbitrary shapes, a novel and generic solution based on boundary erosion is proposed. The clusters are assumed to be separated by relatively sparse regions. The samples are eroded sequentially according to their dynamic boundary densities. The erosion starts from low density regions, invading inwards, until all the samples are eroded out. By this manner, boundaries between different clusters become more and more apparent. It therefore offers a natural and powerful way to separate the clusters when the boundaries between them are hard to be drawn at once. With the sequential order of being eroded, the sequential boundary levels are produced, following which the clusters in arbitrary shapes are automatically reconstructed. As demonstrated across various clustering tasks, it is able to outperform most of the state-of-the-art algorithms and its performance is nearly perfect in some scenarios.
Since the proposal of -means, a variety of clustering algorithms have been proposed in the past three decades, which are in general categorized into seven groups @cite_45 . Namely, they are agglomerative @cite_28 , divisive, partitioning @cite_33 , density based @cite_1 @cite_20 @cite_19 @cite_32 @cite_4 @cite_26 , graph based @cite_10 and neural network based algorithms. In the literature, it is also seen the ensemble of several existing algorithms to boost the performance @cite_11 @cite_38 . For the comprehensive surveys, readers are referred to @cite_45 @cite_17 . In this section, our focus will be on the review of several typical algorithms that are able to identify clusters in arbitrary shapes. Namely, the density based and graph based algorithms are discussed mainly.
@cite_38 @cite_26 @cite_4 @cite_33 @cite_28 @cite_1 @cite_32 @cite_17 @cite_19 @cite_45 @cite_10 @cite_20 @cite_11
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Classifying a set of objects into clusters can be done in numerous ways, producing different results. They can be visually compared using contingency tables [27], mosaicplots [13], fluctuation diagrams [15], tableplots [20] , (modified) parallel coordinates plots [28], Parallel Sets plots [18] or circos diagrams [19]. Unfortunately the interpretability of all these graphical displays decreases rapidly with the numbers of categories and clusterings. In his famous book A Semiology of Graphics [5] Bertin writes “the discovery of an ordered concept appears as the ultimate point in logical simplification since it permits reducing to a single instant the assimilation of series which previously required many instants of study”. Or in more everyday language, if you use good orderings you can see results immediately that with other orderings might take a lot of effort. This is also related to the idea of effect ordering [12], that data should be organised to reflect the effect you want to observe. This paper presents an efficient algorithm based on Bertin's idea and concepts related to Kendall's t [17], which finds informative joint orders for two or more nominal classification variables. We also show how these orderings improve the various displays and how groups of corresponding categories can be detected using a top-down partitioning algorithm. Different clusterings based on data on the environmental performance of cars sold in Germany are used for illustration. All presented methods are available in the R package extracat which is used to compute the optimized orderings for the example dataset. A prominent tool in many problems involving metric spaces is a notion of randomized low-diameter decomposition. Loosely speaking, ( )-decomposition refers to a probability distribution over partitions of the metric into sets of low diameter, such that nearby points (parameterized by ( >0 )) are likely to be “clustered” together. Applying this notion to the shortest-path metric in edge-weighted graphs, it is known that n-vertex graphs admit an (O( n) )-padded decomposition (Bartal, 37th annual symposium on foundations of computer science. IEEE, pp 184–193, 1996), and that excluded-minor graphs admit O(1)-padded decomposition (, 25th annual ACM symposium on theory of computing, pp 682–690, 1993; Fakcharoenphol and Talwar, J Comput Syst Sci 69(3), 485–497, 2004; , Proceedings of the 46th annual ACM symposium on theory of computing. STOC ’14, pp 79–88. ACM, New York, NY, USA, 2014). We design decompositions to the family of p-path-separable graphs, which was defined by Abraham and Gavoille (Proceedings of the twenty-fifth annual acm symposium on principles of distributed computing, PODC ’06, pp 188–197, 2006) and refers to graphs that admit vertex-separators consisting of at most p shortest paths in the graph. Our main result is that every p-path-separable n-vertex graph admits an (O( (p n)) )-decomposition, which refines the (O( n) ) bound for general graphs, and provides new bounds for families like bounded-treewidth graphs. Technically, our clustering process differs from previous ones by working in (the shortest-path metric of) carefully chosen subgraphs. Abstract Clustering based on grid and density for multi-density datasets plays a key role in data mining. In this work, a clustering method that consists of a grid ranking strategy based on local density and priority-based anchor expansion is proposed. In the proposed method, grid cells are ranked first according to local grid properties so the dataset is transformed into a ranked grid. An adjusted shifting grid is then introduced to calculate grid cell density. A cell expansion strategy that simulates the growth of bacterial colony is used to improve the completeness of each cluster. An adaptive technique is finally adopted to handle noisy cells to ensure accurate clustering. The accuracy, parameter sensitivity and computation cost of the proposed algorithm are analysed. The performance of the proposed algorithm is then compared to other clustering methods using four two-dimensional datasets, and the applicability of the proposed method to high-dimensional, large-scale dataset is discussed. Experimental results demonstrate that the proposed algorithm shows good performance in terms of accuracy, de-noising capability, robustness (parameters sensitivity) and computational efficiency. In addition, the results show that the proposed algorithm can handle effectively the problem of multi-density clustering. We present a new method for clustering based on compression. The method does not use subject-specific features or background knowledge, and works as follows: First, we determine a parameter-free, universal, similarity distance, the normalized compression distance or NCD, computed from the lengths of compressed data files (singly and in pairwise concatenation). Second, we apply a hierarchical clustering method. The NCD is not restricted to a specific application area, and works across application area boundaries. A theoretical precursor, the normalized information distance, co-developed by one of the authors, is provably optimal. However, the optimality comes at the price of using the noncomputable notion of Kolmogorov complexity. We propose axioms to capture the real-world setting, and show that the NCD approximates optimality. To extract a hierarchy of clusters from the distance matrix, we determine a dendrogram (ternary tree) by a new quartet method and a fast heuristic to implement it. The method is implemented and available as public software, and is robust under choice of different compressors. To substantiate our claims of universality and robustness, we report evidence of successful application in areas as diverse as genomics, virology, languages, literature, music, handwritten digits, astronomy, and combinations of objects from completely different domains, using statistical, dictionary, and block sorting compressors. In genomics, we presented new evidence for major questions in Mammalian evolution, based on whole-mitochondrial genomic analysis: the Eutherian orders and the Marsupionta hypothesis against the Theria hypothesis.