Principal component analysis of measures, with special emphasis on grain-size curves

A new exploratory method for analyzing a set {} fenced(F¹, ..., Fⁿ) of distribution functions is proposed. It consists in analyzing an approximation of the table of distances between the densities {} fenced(d F¹ / d μ, ..., d Fⁿ / d μ), where μ is a fixed reference probability. This probability makes it possible to take into account changes of scale, or methodological options linked with the phenomenon investigated. It is shown that this analysis is (in some sense) invariant under change of scale, once the reference probabilistic framework has been fixed. It practically consists in conducting an ordinary principal components analysis on raw data (i.e. cumulative curves), using a metrics depending on the reference probability space. It is noteworthy that this method is immediately extensible to any set of absolutely continuous bounded signed measures supported by a fixed bounded interval. This method is applied to surface sediments from the Berre lagoon (Southern France), since grain-size curves have the same characteristics as distribution functions. Two reference probabilities were used: the first one stems from the physics of sediment transport, the second one is classical in sedimentology. Both types of analyses evidenced the main features of the lagoon, with some differences due to distinct reference probabilities.