We propose a fully Bayesian approach to wideband, or broadband, direction-of-arrival (DoA) estimation and signal detection. Unlike previous works in wideband DoA estimation and detection, where the signals were modeled in the time-frequency domain, we directly model the time-domain representation and treat the non-causal part of the source signal as latent variables. Furthermore, our Bayesian model allows for closed-form marginalization of the latent source signals by leveraging conjugacy. To further speed up computation, we exploit the sparse ``stripe matrix structure'' of the considered system, which stems from the circulant matrix representation of linear time-invariant (LTI) systems. This drastically reduces the time complexity of computing the likelihood from $\mathcal{O}(N^3 k^3)$ to $\mathcal{O}(N k^3)$, where $N$ is the number of samples received by the array and $k$ is the number of sources. These computational improvements allow for efficient posterior inference through reversible jump Markov chain Monte Carlo (RJMCMC). We use the non-reversible extension of RJMCMC (NRJMCMC), which often achieves lower autocorrelation and faster convergence than the conventional reversible variant. Detection, estimation, and reconstruction of the latent source signals can then all be performed in a fully Bayesian manner through the samples drawn using NRJMCMC. We evaluate the detection performance of the procedure by comparing against generalized likelihood ratio testing (GLRT) and information criteria.
The usefulness of the information contained in biomedical data relies heavily on the reliability and accuracy of the methods used for its extraction. The conventional assumptions of stationarity and autonomicity break down in the case of living systems because they are thermodynamically open, and thus constantly interacting with their environments. This leads to an inherent time-variability and results in highly nonlinear, time-dependent dynamics. The aim of signal analysis usually is to gain insight into the behavior of the system from which the signal originated. Here, a range of signal analysis methods is presented and applied to extract information about time-varying oscillatory modes and their interactions. Methods are discussed for the characterization of signals and their underlying nonautonomous dynamics, including time-frequency analysis, decomposition, coherence analysis and dynamical Bayesian inference to study interactions and coupling functions. They are illustrated by being applied to cardiovascular and EEG data. The recent introduction of chronotaxic systems provides a theoretical framework within which dynamical systems can have amplitudes and frequencies which are time-varying, yet remain stable, matching well the characteristics of life. We demonstrate that, when applied in the context of chronotaxic systems, the methods presented facilitate the accurate extraction of the system dynamics over many scales of time and space.
The effect of the drugs atropine (a parasympathetic blocker) and propranolol (a sympathetic blocker) is investigated. In the experiment, the subjects were measured under an experimental protocol that used saline controls, with both spontaneous and paced breathing as well as apnea. The recorded data included an electrocardiogram, end tidal CO 2 , blood pressure and a direct measurement of the muscle sympathetic nerve activity. The signals were analysed using time-frequency methods and an information theory approach, revealing information about the change in coupling and coherence that has not previously been studied. The results show that atropine strongly reduces the power in the signals and also removes the coupling and coherence between cardiovascular oscillations. The effects occur across a wide range of frequencies and provide insight into the neurophysiological mechanisms involved in the regulation of the cardiovascular system.
Time series analysis is commonly applied to both chaotic and stochastic systems, which are collectively described as turbulence. However, explicitly time-dependent non-autonomous systems can also generate turbulent dynamics, which makes them useful for describing many physical phenomena. Nevertheless, many of the methods used to analyse turbulence are based around autonomous systems. In this paper, time series from the chaotic, stochastic and non-autonomous Duffing system are analysed using these methods to gauge their suitability to non-autonomous systems. It is found that time-dependent representations are vitally important in the study of this class of systems. Moreover, when time-dependence is neglected in the representation a completely deterministic non-autonomous system is often indistinguishable from a stochastic system.
Nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies have often been treated as stochastic, inappropriately. We therefore formulate them as a new class and discuss how they generate complex behavior. We show how to extract the underlying dynamics, and we demonstrate that it is simple and deterministic, thus paving the way for a diversity of new systems to be recognized as deterministic. They include complex and nonautonomous oscillatory systems in nature, both individually and in ensembles and networks.
For decades the role of autonomic regulation and the baroreflex in the generation of the respiratory sinus arrhythmia (RSA) - modulation of heart rate by the frequency of breathing - has been under dispute. We hypothesized that by using autonomic blockers we can reveal which oscillations and their interactions are suppressed, elucidating their involvement in RSA as well as in cardiovascular regulation more generally. R-R intervals, end tidal CO 2 , finger arterial pressure, and muscle sympathetic nerve activity (MSNA) were measured simultaneously in 7 subjects during saline, atropine and propranolol infusion. The measurements were repeated during spontaneous and fixed-frequency breathing, and apnea. The power spectra, phase coherence and couplings were calculated to characterise the variability and interactions within the cardiovascular system. Atropine reduced R-R interval variability ( p < 0.05) in all three breathing conditions, reduced MSNA power during apnea and removed much of the significant coherence and couplings. Propranolol had smaller effect on the power of oscillations and did not change the number of significant interactions. Most notably, atropine reduced R-R interval power in the 0.145–0.6 Hz interval during apnea, which supports the hypothesis that the RSA is modulated by a mechanism other than the baroreflex. Atropine also reduced or made negative the phase shift between the systolic and diastolic pressure, indicating the cessation of baroreflex-dependent blood pressure variability. This result suggests that coherent respiratory oscillations in the blood pressure can be used for the non-invasive assessment of autonomic regulation.
Following the development of a new class of self-sustained oscillators with a time-varying but stable frequency, the inverse approach to these systems is now formulated. We show how observed data arranged in a single-variable time series can be used to recognize such systems. This approach makes use of time-frequency domain information using the wavelet transform as well as the recently developed method of Bayesian-based inference. In addition, a set of methods, named phase fluctuation analysis, is introduced to detect the defining properties of the new class of systems by directly analyzing the statistics of the observed perturbations.We apply these methods to numerical examples but also elaborate further on the cardiac system.
Until recently, deterministic nonautonomous oscillatory systems with stable amplitudes and time-varying frequencies were not recognized as such and have often been mistreated as stochastic. These systems, named chronotaxic, were introduced in Phys. Rev. Lett. 111, 024101 (2013). In contrast to conventional limit cycle models of self-sustained oscillators, these systems posses a time-dependent point attractor or steady state. This allows oscillations with time-varying frequencies to resist perturbations, a phenomenon which is ubiquitous in living systems. In this work a detailed theory of chronotaxic systems is presented, specifically in the case of separable amplitude and phase dynamics. The theory is extended by the introduction of chronotaxic amplitude dynamics. The wide applicability of chronotaxic systems to a range of fields from biological and condensed matter systems to robotics and control theory is discussed.
Structure and function go hand in hand. However, while a complex structure can be relatively safely broken down into the minutest parts, and technology is now delving into nanoscales, the function of complex systems requires a completely different approach. Here the complexity clearly arises from nonlinear interactions, which prevents us from obtaining a realistic description of a system by dissecting it into its structural component parts. At best, the result of such investigations does not substantially add to our understanding or at worst it can even be misleading. Not surprisingly, the dynamics of complex systems, facilitated by increasing computational efficiency, is now readily tackled in the case of measured time series. Moreover, time series can now be collected in practically every branch of science and in any structural scale—from protein dynamics in a living cell to data collected in astrophysics or even via social networks. In searching for deterministic patterns in such data we are limited by the fact that no complex system in the real world is autonomous. Hence, as an alternative to the stochastic approach that is predominantly applied to data from inherently non-autonomous complex systems, theory and methods specifically tailored to non-autonomous systems are needed. Indeed, in the last decade we have faced a huge advance in mathematical methods, including the introduction of pullback attractors, as well as time series methods that cope with the most important characteristic of non-autonomous systems—their time-dependent behaviour. Here we review current methods for the analysis of non-autonomous dynamics including those for extracting properties of interactions and the direction of couplings. We illustrate each method by applying it to three sets of systems typical for chaotic, stochastic and non-autonomous behaviour. For the chaotic class we select the Lorenz system, for the stochastic the noise-forced Duffing system and for the non-autonomous the Poincaré oscillator with quasi-periodic forcing. In this way we not only discuss and review each method, but also present properties which help to clearly distinguish the three classes of systems when analysed in an inverse approach—from measured, or numerically generated data. In particular, this review provides a framework to tackle inverse problems in these areas and clearly distinguish non-autonomous dynamics from chaos or stochasticity.
'%3e%3cpath fill='%23012169' d='M0 0v30h60V0z'/%3e%3cpath stroke='%23fff' stroke-width='6' d='m0 0 60 30m0-30L0 30'/%3e%3cpath stroke='%23C8102E' stroke-width='4' d='m0 0 60 30m0-30L0 30' clip-path='url(%23b)'/%3e%3cpath stroke='%23fff' stroke-width='10' d='M30 0v30M0 15h60'/%3e%3cpath stroke='%23C8102E' stroke-width='6' d='M30 0v30M0 15h60'/%3e%3c/g%3e%3c/svg%3e)
