Abstract
In practicality, recurrence analyses of dynamical systems can only process short sections of signals that may be infinitely long. By necessity, the recurrence plot and its quantifications are constrained within a truncated triangle that clips the signals at its borders. Recurrence variables defined within these confining borders can be influenced more or less by truncation effects depending upon the system under evaluation. In this study, the question being asked is what if the boundary borders were tilted, what would be the effect on all recurrence variables? This question was prompted by the observation that line entropy values are maximized for highly periodic systems in which the infinitely long line elements are truncated to different unique lengths. However, by redefining the recurrence plot area to a 45-degree tilted box within the triangular area, the diagonal lines would consequently be truncated to identical lengths. Such masking would minimize the line entropy to 0.000 bits/bin. However, what new truncation influences would be imposed on the other recurrence variables? This question is examined by comparing recurrence variables computed with the triangular recurrence area versus boxed recurrence area. Examples include the logistic equation (mathematical series), the Dow Jones Industrial Average over a decade (real-word data), and a square wave pulse (toy series). Good agreement among the variables in terms of timing and amplitude was found for most, but not all variables. These important results are discussed.
Highlights
Because long diagonal lines are necessarily truncated at the borders of the recurrence plot, the question arises how much the truncation effect influences the accuracy of DETERM, diagonal maximum (DMAX) and ENT
Disparity between Recurrence Line Entropy Values and Lyapunov Exponents. When this author [10] was studying the logistic equation by recurrence analysis, it was noted that when the equation was in period 1, period 2, period 4, etc., periodic modes, the line entropy values were maximized when the Lyapunov exponents were low
No mention was made about masking the recurrence matrix to modify the triangular border truncations of diagonal lines
Summary
As an extension of recurrence plots [1], recurrence quantification analysis (RQA) was introduced by Zbilut and Webber [2,3] almost three decades ago and Marwan et al [4]. These quantifications include eight recurrence variables extracted from recurrence plots [5] which have proven to have utility in general-purpose data analyses for linear and nonlinear systems alike [6]. Because long diagonal lines are necessarily truncated at the borders of the recurrence plot, the question arises how much the truncation effect influences the accuracy of DETERM, DMAX and ENT computations, not to mention the remaining recurrence variables. This is the same question formerly asked by Kraemer and Marwan [9]. System studied include a mathematical system, a real-world financial system, and a contrived toy system to clearly illustrate time and amplitude shifts in all compared variables
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