Phase Space Reconstruction from a Biological Time Series: A Photoplethysmographic Signal Case Study

Abstract

In the analysis of biological time series, the state space is comprised of a framework for the study of systems with presumably deterministic and stationary properties. However, a physiological experiment typically captures an observable that characterizes the temporal response of the physiological system under study; the dynamic variables that make up the state of the system at any time are not available. Only from the acquired observations should state vectors be reconstructed to emulate the different states of the underlying system. This is what is known as the reconstruction of the state space, called the phase space in real-world signals, in many cases satisfactorily resolved using the method of delays. Each state vector consists of m components, extracted from successive observations delayed a time τ . The morphology of the geometric structure described by the state vectors, as well as their properties depends on the chosen parameters τ and m. The real dynamics of the system under study is subject to the correct determination of the parameters τ and m. Only in this way can be deduced features have true physical meaning, revealing aspects that reliably identify the dynamic complexity of the physiological system. The biological signal presented in this work, as a case study, is the photoplethysmographic (PPG) signal. We find that m is five for all the subjects analyzed and that τ depends on the time interval in which it is evaluated. The Hénon map and the Lorenz flow are used to facilitate a more intuitive understanding of the applied techniques.