Similarity in Computational Sciences

The advent of fast computer systems has enabled scientists to visualize and analyze complex phenomena (such as explosions of stars and expressions of genes) [2][3][7][8][9][10][12][13]. These complex phenomena (whether simulated or observed) generate large-scale data sets. For instance, simulations of supernovae easily produce terabytes of data [1]. Given such massive amounts of data, it is not surprising that clustering algorithms are quite popular in computational sciences. In particular, non-projected clustering algorithms (such as k-means, k-mediods, hierarchical, and smooth clustering) are widely used [4][5][6]. A crucial input to any of these clustering algorithms is the similarity function that assigns data objects to specific groups. Depending on the purpose of clustering and the characteristics of data objects, the task of selecting an appropriate similarity function can be nontrivial. In computational sciences, data objects are represented by n-dimensional vectors in space and time [10][11]. In particular, each element of an ndimensional data object can be either a scalar quantity (such as density) or a vector quantity (such as velocity). Due to the presence of both scalar and vector quantities within the data objects, it is important that a similarity function encodes both magnitude and direction. For instance, the popular Euclidean distance can only measure dissimilarities in scalar quantities and magnitude of vector quantities. That is, two vectors β α r r and are considered identical when β α β α r r r r ∠ ≠ ∠ ≡ and ! In contrast, the Pearson’s correlation coefficient can only measure similarities in directions of vector quantities. That is, two vectors β α r r and are considered identical when β α β α r r r r ∠ ≡ ∠ ≠ and ! In both of these examples, a similarity function that captures both magnitude and direction will not consider β α r r and as identical!