A general method for the analysis of observations

Abstract

A general method for constructing the science of a complex system from observational data has been developed from the view point of mathematical epistemology. A complete description of an observed system is achieved by establishing a large number of addresses under which all of the data are systematically arranged and by adopting an embedding dimension (number of variables to describe the system) appropriately for the complexity of the system. The variables are then normalized, and descriptive principal-component analyses (DESPCA) are performed to describe the system. Then, the addition of time derivatives (or variables of describing law) to the set of PCA's of the previous DESPCA provides an extended data set to be applied to a dynamical principal-component analysis (DYNPCA) which follows. The advantage of DYNPCA lies, among others, in a systematic improvement of the system used for analysis and in a quantitative estimation of the uncertainty of the differential equation of the dynamical system (or of the law) determined from the minimum eigen-value of the DYNPCA. As a simple application of the DYNPCA, the distance determination of mass-losing super-giants considered in a previous study is re-discussed from the point of view of methodology. The traditional use of classification in empirical sciences is found to be well adaptable in cooperation with the DYNPCA.