The Mathematical Gnostics (Advanced Data Analysis) - Kovanic, Pavel (Information Processing and Management of Uncertainty in Knowledge-Based Systems, 2016)
A brief survey of mathematical gnostics is presented. Mathematical gnostics is a tool of advanced data analysis, consisting oftheory of individual uncertain data and small samples,algorithms to implement the theory,applications of the algorithms. The axioms and definitions of the theory are inspired by the Laws of Nature dealt with by physics and the investigation of data uncertainty follows the methods of analysis of physical processes. The first axiom is a reformulation of the measurement theory which mathematically formalizes the empirical cognitive activity of physics. This axiom enables the curvature of the data space to be revealed and quantified. The natural affinity between uncertain data and relativistic mechanics is also shown. Probability, informational entropy and information of individual uncertain data item are inferred from non-statistical Clausius’ thermodynamical entropy. The quantitative cognitive activity is modeled as a closed cycle of quantification and estimation, which is proved to be irreversible and maximizes the result’s information. A proper estimation of the space’s curvature ensure a reliable robustness of the algorithms successfully proven in many applications. Gnostic formulae of data weights and errors, probability and information, which has been proved as valid for small samples of strongly uncertain data converge to statistical ones when uncertainty becomes weak. From this point of view, the mathematical gnostics can be considered as an extension of statistics useful under heavy-duty conditions.