A Unifying Tool for Linear Multivariate Statistical Methods: The RV- Coefficient

Abstract

Consider two data matrices on the same sample of n individuals, X(p x n), Y(q x n). From these matrices, geometrical representations of the sample are obtained as two configurations of n points, in BfP and Bf·. It is shown that the RV-coefficient (Escoufier, 1970(Escoufier, ,1973 can be used as a measure of similarity of the two configurations, taking into account the possibly distinct metrics to be used on them to measure the distances between points. The purpose of this paper is to show that most classical methods of linear multivariate statistical analysis can be interpreted as the search for optimal linear transformations or, equivalently, the search for optimal metrics to apply on two data matrices on the same sample; the optimality is defined in terms of the similarity of the corresponding configurations of points, which, in turn, calls for the maximization of the associated R V-coefficient. The methods studied are principal components, principal components of instrumental variables, multivariate regression, canonical variables, discriminant analysis; they are differentiated by the possible relationships existing between the two data matrices involved and by additional constraints under whichthe maximum of R Visto be obtained. It is also shown that the R V-coefficient can be used as a measure of goodness of a solution to the problem of discarding variables.

Structure of endemic fish assemblages in the upper Yangtze River Basin

“…Then the CIA was used to summarize the relationships between species lists and land-cover variables, in which a random Monte Carlo test with 1000 permutations was performed to reveal the significance of the co-structure of this CIA. The RV-coefficient is calculated to measure the overall similarity (Robert and Escoufier, 1976) and this has a range 0 to 1, where a high RV-coefficient indicates a high degree of co-structure.…”

Sequences Classification Based on Group Technology for Flexible Manufacturing Cell Design

“…To compare the 3 classifications P1, P2 and P3, we are going to use the index of vectorial correlation RV [18]. This index is formulated as follows [19]:…”

Modern pollen rain and fungal spore assemblages from pasture woodlands around Lake Saint-Point (France)

“…Contrary to asymmetric constrained ordination methods, such as CCA (Canonical Correspondence Analysis) or RDA (Redundancy Analysis), MFA does not assume a priori any causal relationship among variables, and can be applied to datasets where the number of objects is very low as compared to the number of descriptors in each group (Dray et al, 2003). The similarity between the geometrical representations derived from each group of variables is measured by the RV-coefficient, ranging from 0 to 1 (Escoufier, 1973;Robert and Escoufier, 1976). RV-coefficients can be tested by permutations .…”