Bivariate phase-rectified signal averaging

📝 Abstract

Phase-Rectified Signal Averaging (PRSA) was shown to be a powerful tool for the study of quasi-periodic oscillations and nonlinear effects in non-stationary signals. Here we present a bivariate PRSA technique for the study of the inter-relationship between two simultaneous data recordings. Its performance is compared with traditional cross-correlation analysis, which, however, does not work well for non-stationary data and cannot distinguish the coupling directions in complex nonlinear situations. We show that bivariate PRSA allows the analysis of events in one signal at times where the other signal is in a certain phase or state; it is stable in the presence of noise and impassible to non-stationarities.

💡 Analysis

Phase-Rectified Signal Averaging (PRSA) was shown to be a powerful tool for the study of quasi-periodic oscillations and nonlinear effects in non-stationary signals. Here we present a bivariate PRSA technique for the study of the inter-relationship between two simultaneous data recordings. Its performance is compared with traditional cross-correlation analysis, which, however, does not work well for non-stationary data and cannot distinguish the coupling directions in complex nonlinear situations. We show that bivariate PRSA allows the analysis of events in one signal at times where the other signal is in a certain phase or state; it is stable in the presence of noise and impassible to non-stationarities.

📄 Content

arXiv:0804.2798v1 [physics.data-an] 17 Apr 2008 Bivariate phase-rectified signal averaging Aicko Y. Schumann a, Jan W. Kantelhardt a, Axel Bauer b and Georg Schmidt b aInstitut f¨ur Physik, Martin-Luther-Universit¨at, Halle, Germany bMedizinische Klinik und Deutsches Herzzentrum der Technischen Universit¨at M¨unchen, Germany Abstract Phase-Rectified Signal Averaging (PRSA) was shown to be a powerful tool for the study of quasi-periodic oscillations and nonlinear effects in non-stationary signals. Here we present a bivariate PRSA technique for the study of the inter-relationship between two simultaneous data recordings. Its performance is compared with tra- ditional cross-correlation analysis, which, however, does not work well for non- stationary data and cannot distinguish the coupling directions in complex nonlinear situations. We show that bivariate PRSA allows the analysis of events in one signal at times where the other signal is in a certain phase or state; it is stable in the presence of noise and impassible to non-stationarities. Key words: Time-series analysis; Quasi-periodicities; Non-stationaritiy behavior; Cross-correlation analysis; Phase-rectified signal averaging PACS: 05.40.−a; 05.45.Tp; 02.50.Sk; 87.19.Hh 1 Introduction Many natural systems generate periodicities on different time scales because some of their components form closed regulation loops in addition to causal linear control chains. In biology and physiology, cardio-respiratory rhythms, rhythmic motions of limbs in walking, rhythms underlying the release of hor- mones and gene expression, membrane potential oscillations, oscillations in neuronal signals, and circadian rhythms are just a few examples (see, e.g., [1,2]). Oscillations also occur in geophysical data, e.g., for the El-Ni˜no phe- nomenon, sunspot numbers, and ice age periods [3]. In many cases several signals from different components of the complex system can be recorded si- multaneously. For understanding the control chains and loops in the system, Preprint submitted to Elsevier 23 October 2018 we want to know how periodicities in the signals are generated by (possibly di- rected and/or nonlinear) interactions between its components. Consequently, there is a need for identifying periodicities in one recorded signal together with the direction of causal relations to periodicities in other signals. Cross-correlation analysis and transfer function analysis are traditional tools for this type of analysis. However, there are three major drawbacks of these methods: (i) only rather stationary data can be studied, (ii) a linear rela- tionship between the signals is usually assumed, and (iii) the identification of causalities is hindered by the fact that the exchange of the two signals under study is identical with time inversion. We thus propose a method which helps to overcome these problems. Non-stationarities are a major problem in the analysis of signals recorded from complex systems over a prolonged period of time [4,5,6,7,8]. Many internal and external perturbations are continuously influencing the system and causing interruptions of the periodic behavior. The interruptions often ’reset’ the reg- ulatory mechanisms resulting in phase de-synchronization of the oscillations. The signals thus become quasi-periodic, consisting of many periodic patches as well as noise and trends. Cross-correlation and transfer function techniques are thus problematic. In addition, there might be causal inter-relations between two signals that cannot be revealed by these methods. For illustration, let us assume that a large increase and a large decrease in signal X (trigger signal) cause the same specific effect in signal Y (target signal), while there is no such effect in Y if X remains unchanged. In this situation with an essentially nonlinear coupling between the signals, both, cross-correlation analysis and spectral analysis cannot reveal the effect. They show the superposition of the two branches of the interaction with opposite signs, i.e., no effect. Even if the effects on signal Y were different for increases and decreases of signal X, one could see some relation but could not distinguish the two effects. Hence, one needs a method that can separately study effects in signal Y which might oc- cur in response to different causes in signal X, and vice versa. A separation of effects with different typical duration or frequency scale seems also appropri- ate for distinguishing frequency-band selective inter-relations between signals X and Y . Our approach for extracting inter-relations between two or more simultaneous data recordings from a complex system is based on the phase-rectified signal averaging technique (PRSA) [9,10], which was shown to be a powerful tool for the study of quasi-periodic oscillations in noisy, non-stationary signals. The original method extracts the features in one signal before and after in- creases in the same signal (or, alternatively, decreases). This way, information on charac

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