Matrix comparison, Part 1: Motivation and important issues for measuring the resemblance between proximity measures or ordination results

Abstract

The present two-part article introduces matrix comparison as a formal means of evaluation in informetric studies such as cocitation analysis. In this first part, the motivation behind introducing matrix comparison to informetric studies, as well as two important issues influencing such comparisons, are introduced and discussed. The motivation is spurred by the recent debate on choice of proximity measures and their potential influence upon clustering and ordination results. The two important issues discussed here are matrix generation and the composition of proximity measures. The approach to matrix generation is demonstrated for the same data set, i.e., how data is represented and transformed in a matrix, evidently determines the behavior of proximity measures. Two different matrix generation approaches, in all probability, will lead to different proximity rankings of objects, which further lead to different ordination and clustering results for the same set of objects. Further, a resemblance in the composition of formulas indicates whether two proximity measures may produce similar ordination and clustering results. However, as shown in the case of the angular correlation and cosine measures, a small deviation in otherwise similar formulas can lead to different rankings depending on the contour of the data matrix transformed. Eventually, the behavior of proximity measures, that is whether they produce similar rankings of objects, is more or less dataspecific. Consequently, the authors recommend the use of empirical matrix comparison techniques for individual studies to investigate the degree of resemblance between proximity measures or their ordination results. In part two of the article, the authors introduce and demonstrate two related statistical matrix comparison techniques the Mantel test and Procrustes analysis, respectively. These techniques can compare and evaluate the degree of monotonicity between different proximity measures or their ordination results. As such, the Mantel test and Procrustes analysis can be used as statistical validation tools in informetric studies and thus help choosing suitable proximity measures.

On the normalization and visualization of author co‐citation data: Salton's Cosine versus the Jaccard index

“…Directed graphs can be visualized using Waldo Tobler's Flow Mapper, available at http://www.csiss.org/clearinghouse/ FlowMapper/ 3 Pajek is a software package for social network analysis and visualization which is freely available for academic usage at http://vlado.fmf. uni-lj.si/pub/networks/pajek/ matrix, supports the analytical conclusions given earlier about the expected monotonicity between these two measures (Schneider & Borlund, 2007). There are, however, some differences in the values which matter for the visualization.…”

“…In many cases, one can expect the Jaccard and the cosine measures to be monotonic to each other (Schneider & Borlund, 2007); however, the cosine metric measures the similarity between two vectors (by using the angle between them) whereas the Jaccard index focuses only on the relative size of the intersection between the two sets when compared to their union. Furthermore, one can normalize differently using the margin totals in the asymmetrical occurrence or the symmetrical co-occurrence matrix.…”

“…2 Table 2 provides the Spearman rank-order correlations among the lower triangles of the various similarity matrices under discussion. Spearman's r is used instead of Pearson's r because objects in proximity matrices are based on dyadic relationships (Kenny, Kashy, & Cook, 2006); the assumption of independence required for parametric significance tests is violated (Schneider & Borlund, 2007).…”

Training-Free, Generic Object Detection Using Locally Adaptive Regression Kernels

“…We compare the feature matrices F T i and F Q from the ith patch of T and Q to look for matches. Inspired in part by the many studies [19], [20], [21], [22], [23], [24] which took advantage of cosine similarity over the conventional euclidean distance, we employ and justify the use of "Matrix Cosine Similarity" as a similarity measure which generalizes the cosine similarity between two vectors [25], [26], [27] to the matrix case. We illustrate the optimality properties of the proposed approach using a naive Bayes framework, which leads to the use of the Matrix Cosine Similarity (MCS) measure.…”

“…In several earlier papers, including [25], [26], it has been shown that Pearson correlation is less discriminating than the cosine similarity due to the fact that centered values are less informative than the original values and the computation of centered values is sensitive to zero or small values in the vectors. Since the discriminative power is critical in our detection framework, we focus on the cosine similarity.…”