Recurrence-based reconstruction of dynamic pricing attractors

Abstract

Dynamic pricing depends on the understanding of uncertain demand. We ask the question whether a stochastic system is sufficient to model this uncertainty. We propose a novel paradigm based on statistical analysis of recurrence quantification measures. The paradigm fits nonlinear dynamics by simultaneously optimizing both the determinism and the trapping time in recurrence plots and identifies an optimal time delay embedding. We firstly apply the paradigm on well-known deterministic and stochastic systems including Duffing systems and multi-fractional Gaussian noise. We then apply the paradigm to optimize the sampling of empirical point process data from RideAustin, a company providing ride share service in the city of Austin, Texas, the USA, thus reconstructing a period-7 attractor. Results show that in deterministic systems, an optimal embedding exists under which recurrence plots exhibit robust diagonal or vertical lines. However, in stochastic systems, an optimal embedding often does not exist, evidenced by the inability to shrink the standard deviation of either the determinism or the trapping time. By means of surrogate testing, we also show that a Poisson process or a stochastic system with periodic trend is insufficient to model uncertainty contained in empirical data. By contrast, the period-7 attractor dominates and well models nonlinear dynamics of empirical data via irregularly switching of the slow and the fast dynamics. Findings highlight the importance of fitting and recreating nonlinear dynamics of data in modeling practical problems.