Phase Space Reconstruction and Recurrence Plot Theory

Abstract

As shown in Chaps. 4 and 5 that different hybrid QCM, CMM, CGM, RLM, and SM with meta-heuristic algorithms are applied to select appropriate parameter combination of an SVR-based electric load forecasting model. These forecasting results indicate that all SVR-based hybrid models are superior to other competitive forecasting models. This chapter will introduce a novel approach, hybrid phase space reconstruction (PSR) algorithm and recurrence plot (RP) theory with bi-square kernel (BSK) function, namely PSR-BSK model, to improve the forecasting accuracy. as know that a specific state of the system can be represented by a point in the phase space and time evolution of the system creates a trajectory in the phase space. Where the phase space is a space in which all possible states of the system are represented, with each possible state corresponding to one unique point. Then, the given time series could be a projection of trajectory of the system to one coordinate of phase space. Therefore, based on the theory of time delay and embedding dimension, the phase space reconstruction (PSR) algorithm is employed to reconstruct the phase space of chaotic time series, to extract some valuable features by extending a one-dimensional time series to a high dimensional phase space. On the other hand, recurrence plot (RP) theory is a relatively new technique for the qualitative assessment of time series in a dynamical system. The fundamental assumption of RP is that there exists a realized dynamical process in an observable time series (a sequence of observations) to represent the interaction among the relevant variables over time. It has been proven mathematically that one can recreate a topologically equivalent picture of the original multidimensional system behavior by using the time series of a single observable variable. Therefore, RP reveals all of the times when the phase space trajectory of the dynamical system visits roughly the same area in the phase space, it is can graphically detect hidden patterns and structural changes in data or see similarities in patterns across the time series under study.