Decomposition using maximum autocorrelation factors

This paper presents methods for the analysis and decomposition of multivariate data sets where a given ordering/structure of the observations or the variables exists. Examples of such data sets are found in remote sensing imagery, where observations (pixels) each consisting of a reflectance spectrum are organized 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 co‐ordinates 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. For the case with observation ordering, the maximum autocorrelation factor (MAF) transform was proposed for multivariate imagery (Switzer P. In Computer Science and Statistics, Billard L (ed.). Elsevier: Amsterdam, 1985; 13–16). This corresponds to an R‐mode analysis of the data matrix. We propose to extend this concept to situations with variable ordering. This corresponds to a Q‐mode analysis of the data matrix. We call this method the Q‐MAF decomposition. It turns out that in many situations the new variables resulting from MAF and Q‐MAF analyses can be interpreted as a decomposition of (spatial) frequency. However, contrary to Fourier decomposition, these new variables are located in frequency as well as location (space, time, wavelength, etc.).