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Wave-shape oscillatory model for nonstationary periodic time series analysis

  • * Corresponding author: Hau-Tieng Wu

    * Corresponding author: Hau-Tieng Wu
The first author is supported by National Science and Technology Development Fund (MOST 107-2115-M-075-001) and the LEAP@Duke program of the Ministry of Science and Technology (MOST), Taipei, Taiwan
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  • The oscillations observed in many time series, particularly in \biomedicine, exhibit morphological variations over time. These morphological variations are caused by intrinsic or extrinsic changes to the state of the generating system, henceforth referred to as dynamics. To model these time series (including and specifically pathophysiological ones) and estimate the underlying dynamics, we provide a novel wave-shape oscillatory model. In this model, time-dependent variations in cycle shape occur along a manifold called the wave-shape manifold. To estimate the wave-shape manifold associated with an oscillatory time series, study the dynamics, and visualize the time-dependent changes along the wave-shape manifold, we propose a novel algorithm coined Dynamic Diffusion map (DDmap) by applying the well-established diffusion maps (DM) algorithm to the set of all observed oscillations. We provide a theoretical guarantee on the dynamical information recovered by the DDmap algorithm under the proposed model. Applying the proposed model and algorithm to arterial blood pressure (ABP) signals recorded during general anesthesia leads to the extraction of nociception information. Applying the wave-shape oscillatory model and the DDmap algorithm to cardiac cycles in the electrocardiogram (ECG) leads to ectopy detection and a new ECG-derived respiratory signal, even when the subject has atrial fibrillation.

    Mathematics Subject Classification: Primary: 37M10; Secondary: 92-08.


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  • Figure 1.  An arterial blood pressure (ABP) signal obtained from an anesthetized patient undergoing the endotracheal intubation procedure. This oscillatory physiological time series changes in amplitude and frequency over time in response to the noxious stimulation associated with intubation. In addition, the shape of its non-sinusoidal oscillatory cycles (pulses) is also time-varying. This distinct type of modulation has strong clinical importance, and we refer to it as wave-shape modulation

    Figure 2.  An illustration of a contrived wave-shape manifold. We build a collection of oscillatory cycles by dilating and scaling a template. We show a three dimensional embedding of this collection obtained by principle component analysis. Points in the embedding are colored according to their associated dilation factor

    Figure 3.  An illustration of the process generating the parameters of $ f $, a time series adhering to the wave-shape oscillatory model. On the surface of the wave-shape manifold $ \mathcal{M} $, we show the sequence of wave-shape functions $ \{s_j\}_{j\geq 1} $ observed in the time series

    Figure 4.  We show an ECG signal featuring atrial fibrillation. This recording also features premature ventricular contractions (PVCs). This time series is difficult to model because the ventricular contractions appear irregularly. Moreover, the shape of each PVC is significantly different than the shape of its preceding normal beat

    Figure 5.  The time series $ \mathbf{U} $, representing transitions between connected components of the wave-shape manifold, corresponds to coarse changes in heartbeat morphology

    Figure 6.  A low-rank approximation of $ \mathbf{X} $ through the singular value decomposition yields a visualization of the set of cardiac cycles in the ECG signal. Heartbeats belonging to the set $ C_1 $ are colored green, and those belonging to the set $ C_2 $ are colored purple. The set $ C_2 $ contains 2113 heartbeats, and the set $ C_1 $ contains $ 563 $ heartbeats

    Figure 7.  The time series $ \widehat{\mathbf{V}} $ (blue) correlates well with the simulataneously recorded respiratory signal (bottom). The traditionally acquired EDR signal (based on measuring the RS heights of non-PVC beats) shown in orange appears to provide less reliable respiratory information. For reference, we show the time series $ \mathbf{U} $ (top) whose peaks indicate the presence of ectopic heartbeats in the ECG signal

    Figure 8.  We show a two-dimensional embedding of the pulse wave-shapes extracted from a $ 360 $-second arterial blood pressure (ABP) signal. The signal was recorded from a patient undergoing general anesthesia before and after an endotracheal intubation event. The ABP signal is shown in Figure 1. The intubation event takes place at the 30th second, indicated by the black arrow on the colorbar

    Figure 9.  We superimpose the normalized pulse wave-shapes from the set $ \mathcal{X}_{\mathbf{A}} $. After normalization, only frequency (width) and wave-shape modulation effects are perceivable. After the stimulus event at the 30-second mark, the pulses become less wide, and a prominent peak protrudes between the systolic peak and the dicrotic notch. These effects then continue to subtlely evolve throughout the recording

    Figure 10.  We show the same two-dimensional embedding depicted in Figure 8 with a different coloring scheme. We color the $ i $-th pulse according to the quantity $ {}^{60f_s}\!\!\diagup\!\!{}_{a_i - a_{i-1}}\; $, known as the instantaneous pulse rate (in beats per minute). The pulse rate surges in response to the stimulus and then descends below the original resting rate

    Figure 11.  The instantaneous amplitude of an ECG signal gives surrogate respiratory information. Top: we plot a respiratory flow signal (nasal thermistor recording temperature differential) in blue; bottom: we plot the simultaneously-recorded ECG signal and illustrate its amplitude modulation in a traditional way using the dotted pink lines

    Figure 12.  The instantaneous frequency of an ECG signal is known as the instantaneous heart rate. In red, we plot the instantaneous heart rate signal corresponding to the ECG signal in black

    Figure 13.  Landmarks on the enigmatic ECG template correspond to distinct stages of a healthy heart contraction. The P wave corresponds to the depolarization of the atria, the QRS complex corresponds to the depolarization of the ventricles, the T wave corresponds to the repolarization of the ventricles, and the U wave corresponds to the repolarization of the papillary muscles

    Figure 14.  Atrial fibrillation is characterized by a lack of P waves, irregularly-occurring ventricular contractions, and a rapid component called the fibrillatory wave

    Figure 15.  ECG signals featuring premature ventricular contractions are difficult to model because the wave-shape is not slowly varying. Premature ventricular contractions are morphologically distinct from normal beats and are not triggered by the SA node

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