Finding the "natural" vector bases for multidimensional reference values

The concept of reference values can be extended to multidimensional results. A probability function describes the relative density of the observations in the multivariate space. When the density of a given point is measured relative to all other points, we get an estimate of the density rank of a given point. If the rank of a point is lower than 95 per cent of all points, the multidimensional result is outside the multidimensional reference range. The single-dimensional case is a special case of this general concept. Many observations are needed to define multidimensional distributions. However, less points are needed if the dimensionality of the data matrix is reduced by statistical methods such as principal component analysis (PCA). Also other vector bases than the orthogonal solution produced by PCA are possible, and all of them compress data equally well. So the choice must be based on other criteria than compression. We propose using a vector basis that consists of positive numbers. The positive vectors can be found by direct methods such as Alternating Regression (AR) or they can be modified from the results of the PCA. Positive vectors resemble the spectra that are familiar in chemistry and physics. They are a "natural" way to describe multidimensional results. It is easier to name the positive vectors than the purely statistical vectors of PCA. To obtain a unique positive solution, additional constraints besides positivity are needed.