Procrustes-based distances for exploring between-matrices similarity

Abstract

The statistical shape analysis called Procrustes analysis minimizes the Frobenius distance between matrices by similarity transformations. The method returns a set of optimal orthogonal matrices, which project each matrix into a common space. This manuscript presents two types of distances derived from Procrustes analysis for exploring between-matrices similarity. The first one focuses on the residuals from the Procrustes analysis, i.e., the residual-based distance metric. In contrast, the second one exploits the fitted orthogonal matrices, i.e., the rotational-based distance metric. Thanks to these distances, similarity-based techniques such as the multidimensional scaling method can be applied to visualize and explore patterns and similarities among observations. The proposed distances result in being helpful in functional magnetic resonance imaging (fMRI) data analysis. The brain activation measured over space and time can be represented by a matrix. The proposed distances applied to a sample of subjects—i.e., matrices—revealed groups of individuals sharing patterns of neural brain activation. Finally, the proposed method is useful in several contexts when the aim is to analyze the similarity between high-dimensional matrices affected by functional misalignment.

Acoustic tunnel lining cavity detection using cepstral coefficients with optimized filter bank

“…From the above figures, it can be seen that EFCC exhibits better discrimination between cavity and dense acoustic signals compared to MFCC in terms of low-dimensional features. In addition, a matrix Frobenius parametric similarity measurement is performed on the features extracted by both methods, which is presented below [24]: where A and B are matrices and the distance between them needs to be evaluated. The values of the matrix Frobenius norm for both MFCC and EFCC algorithms are 31.9 and 37.3, respectively.…”