A New Measure for Analyzing and Fusing Sequences of Objects

This work is related to the combinatorial data analysis problem of seriation used for data visualization and exploratory analysis. Seriation re-sequences the data, so that more similar samples or objects appear closer together, whereas dissimilar ones are further apart. Despite the large number of current algorithms to realize such re-sequencing, there has not been a systematic way for analyzing the resulting sequences, comparing them, or fusing them to obtain a single unifying one. We propose a new positional proximity measure that evaluates the similarity of two arbitrary sequences based on their agreement on pairwise positional information of the sequenced objects. Furthermore, we present various statistical properties of this measure as well as its normalized version modeled as an instance of the generalized correlation coefficient. Based on this measure, we define a new procedure for consensus seriation that fuses multiple arbitrary sequences based on a quadratic assignment problem formulation and an efficient way of approximating its solution. We also derive theoretical links with other permutation distance functions and present their associated combinatorial optimization forms for consensus tasks. The utility of the proposed contributions is demonstrated through the comparison and fusion of multiple seriation algorithms we have implemented, using many real-world datasets from different application domains.