Decomposition of Multivariate Datasets with Structure/ordering of Observations or Variables Using Maximum Autocorrelation Factors

This article presents methods for the analysis and decomposition of multivariate datasets where a given ordering/structure of the observations or the variables exist. Examples of such data sets are remote sensing imagery where observations (pixels) each consisting of a reflectance spectrum are organised in a two-dimensional grid. Another example is biological shape analysis. Here each observation (e.g. human bone, cerebral ventricle) is represented by a number of landmarks the coordinates of which are the variables. Here we do not have an ordering of the observations (individuals). However, normally we have an ordering of landmarks (variables) along the contour of the objects. In this context a landmark is a point with anatomical or geometrical meaning across observations. A further example is reflectance spectra from samples, where the samples do not exhibit any order but the variables do. For the case with observation ordering the maximum autocorrelation factor (MAF) transform was proposed for multivariate imagery in [1]. this corresponds to a R-mode analyse of the data matrix. We propose to extend this concept to situations with variable ordering. This corresponds to a Q-mode analysis of the datamatrix. We denote this methods Q-MAF decomposition. It turns out that in many situations the new variables resulting from the MAF and the Q-MAF analyses can be interpreted as a frequency analysis. However, contrary to Fourier decomposition these new variables are located in frequency as well as location (space, time, wavelength etc).