Quantifying Redundancy and Information Content of Lines in Recurrence Plots Using the Theory of Framework Rigidity

We address redundancy in the information content of unthresholded recurrence plots (URPs). The theory of framework rigidity is employed to explain and analyze this redundancy geometrically. First we show that the domain of a URP can be restricted to just a finite number of vertical or horizontal lines without loss of information. Then we construct a globally rigid framework to demonstrate a similar property for diagonal lines. This result gives theoretical support to recurrence quantification analysis (RQA), which analyzes and extracts features from an RP along such lines. Third, we construct a finite set of curves, one of which is a contour line, for which it again holds that the URP contains all information along them. This links the information content of lossy (thresholded) recurrence plots to that of URPs. This study is also a starting point in employing redundancy to improve existing recurrence plots based methods and algorithms, and to develop new ones. Several examples clarify the methods and an application from EEG artifact detection shows some of their practical potential.