# American Institute of Mathematical Sciences

doi: 10.3934/fods.2021009

## Wave-shape oscillatory model for nonstationary periodic time series analysis

 1 Department of Anesthesiology, Taipei Veterans General Hospital, 201, Section 2, Shih-Pai Road, Taipei, Taiwan 11217 2 School of Medicine, National Yang Ming Chiao Tung University, 155, Sec. 2, Linong St., Taipei, Taiwan 112304 3 Department of Mathematics, Duke University, 120 Science Drive, Durham, NC 27708, USA 4 Department of Mathematics and Department of Statistical Science, Duke University, 120 Science Drive, Durham, NC 27708, USA

* Corresponding author: Hau-Tieng Wu

Received  December 2020 Revised  March 2021 Published  April 2021

Fund Project: 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

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.

Citation: Yu-Ting Lin, John Malik, Hau-Tieng Wu. Wave-shape oscillatory model for nonstationary periodic time series analysis. Foundations of Data Science, doi: 10.3934/fods.2021009
##### References:
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##### References:
 [1] S. Alagapan, H. W. Shin, F. Fröhlich and H.-T. Wu, Diffusion geometry approach to efficiently remove electrical stimulation artifacts in intracranial electroencephalography, J. Neural Engrg., 16 (2019). doi: 10.1088/1741-2552/aaf2ba.  Google Scholar [2] A. P. Avolio, L. M. Van Bortel, P. Boutouyrie, J. R. Cockcroft and C. M. McEniery, et al., Role of pulse pressure amplification in arterial hypertension: Experts' opinion and review of the data, Hypertension, 54 (2009), 375-383. doi: 10.1161/HYPERTENSIONAHA.109.134379.  Google Scholar [3] M. S. Baker, A. K. Gehi, J. P. Hummel and J. P. Mounsey, Atrial fibrillation: Rate versus rhythm, in Netter's Cardiology, Netter Clinical Science, Elsevier, 2018, 257–261. Google Scholar [4] R. Barbieri, E. C. Matten, A. A. Alabi and E. N. Brown, A point-process model of human heartbeat intervals: New definitions of heart rate and heart rate variability, Am. J. Physiol. Heart Circ. Physiol., 288 (2005), 424-435.  doi: 10.1152/ajpheart.00482.2003.  Google Scholar [5] P. M. Barrett, R. Komatireddy, S. Haaser, S. Topol, J. Sheard, J. Encinas, A. J. Fought and E. J. Topol, Comparison of 24-hour Holter monitoring with 14-day novel adhesive patch electrocardiographic monitoring, Amer. J. Medicine, 127 (2014). doi: 10.1016/j.amjmed.2013.10.003.  Google Scholar [6] J. Bates, The embedding dimension of Laplacian eigenfunction maps, Appl. Comput. Harmon. Anal., 37 (2014), 516-530.  doi: 10.1016/j.acha.2014.03.002.  Google Scholar [7] P. H. Bérard, Spectral Geometry: Direct and Inverse Problems, Lecture Notes in Mathematics, 1207, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0076330.  Google Scholar [8] P. Bérard, G. Besson and S. Gallot, Embedding Riemannian manifolds by their heat kernel, Geom. Funct. Anal., 4 (1994), 373-398.  doi: 10.1007/BF01896401.  Google Scholar [9] M. Bikkina, M. G. Larson and D. 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Wu, How nonlinear-type time-frequency analysis can help in sensing instantaneous heart rate and instantaneous respiratory rate from photoplethysmography in a reliable way, Frontiers in Physiology, 8 (2017). doi: 10.3389/fphys.2017.00701.  Google Scholar [14] G. D. Clifford, F. Azuaje and P. McSharry, Advanced Methods and Tools for ECG Data Analysis, Artech House, Inc., Norwood, MA, 2006. Google Scholar [15] R. R. Coifman and S. Lafon, Diffusion maps, Appl. Comput. Harmon. Anal., 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.  Google Scholar [16] P. de Chazal, M. O'Dwyer and R. B. Reilly, Automatic classification of heartbeats using ECG morphology and heartbeat interval features, IEEE Transactions on Biomedical Engineering, 51 (2004), 1196-1206.  doi: 10.1109/TBME.2004.827359.  Google Scholar [17] A. M. De Livera, R. J. Hyndman and R. D. Snyder, Forecasting time series with complex seasonal patterns using exponential smoothing, J. Amer. Statist. 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Statist., 44 (2016), 346-372.  doi: 10.1214/14-AOS1275.  Google Scholar [23] M. Elgendi, Optimal signal quality index for photoplethysmogram signals, Bioengineering, 3 (2016). doi: 10.3390/bioengineering3040021.  Google Scholar [24] M. Elgendi, I. Norton, M. Brearley, D. Abbott and D. Schuurmans, Systolic peak detection in acceleration photoplethysmograms measured from emergency responders in tropical conditions, PLoS One, 8 (2013). doi: 10.1371/journal.pone.0076585.  Google Scholar [25] D. Escalona-Vargas, H.-T. Wu, M. G. Frasch and H. Eswaran, A comparison of five algorithms for fetal magnetocardiography signal extraction, Cardiovascular Engineering and Technology, 9 (2018), 483-487.  doi: 10.1007/s13239-018-0351-4.  Google Scholar [26] C. L. Feldman, P. G. Amazeen, M. D. Klein and B. Lown, Computer detection of ventricular ectopic beats, Computers and Biomedical Research, 3 (1970), 666-674.  doi: 10.1016/0010-4809(70)90034-0.  Google Scholar [27] P. Flandrin, Time-Frequency/Time-Scale Analysis, Wavelet Analysis and its Applications, 10, Academic Press, Inc., San Diego, CA, 1999.  Google Scholar [28] N. García Trillos, M. Gerlach, M. Hein and D. Slepčev, Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace–Beltrami operator, Found. Comput. Math., 20 (2020), 827-887.  doi: 10.1007/s10208-019-09436-w.  Google Scholar [29] I. M. Gel'fand and N. Y. Vilenkin, Generalized functions. Vol. 4: Applications of Harmonic Analysis, Academic Press, New York - London, 1964.   Google Scholar [30] T. Hasan, Complex demodulation: Some theory and applications, Time Series in the Frequency Domain, Handbook of Statist., 3, North-Holland, Amsterdam, 1983,125–156. doi: 10.1016/S0169-7161(83)03009-6.  Google Scholar [31] E. Helfenbein, R. Firoozabadi, S. Chien, E. Carlson and S. Babaeizadeh, Development of three methods for extracting respiration from the surface ECG: A review, J. Electrocardiology, 47 (2014), 819-825.  doi: 10.1016/j.jelectrocard.2014.07.020.  Google Scholar [32] T. Y. Hou and Z. Shi, Extracting a shape function for a signal with intra-wave frequency modulation, Philos. Trans. Roy. Soc. A, 374 (2016), 17pp. doi: 10.1098/rsta.2015.0194.  Google Scholar [33] R. Latchamsetty and F. Bogun, Premature ventricular complex-induced cardiomyopathy, JACC: Clinical Electrophysiology, 5 (2019), 537-550.  doi: 10.1016/j.jacep.2019.03.013.  Google Scholar [34] C.-Y. Lin, L. Su and H.-T. Wu, Wave-shape function analysis: When cepstrum meets time-frequency analysis, J. Fourier Anal. Appl., 24 (2018), 451-505.  doi: 10.1007/s00041-017-9523-0.  Google Scholar [35] Y. Lu, H.-T. Wu and J. Malik, Recycling cardiogenic artifacts in impedance pneumography, Biomedical Signal Processing and Control, 51 (2019) doi: 10.1016/j.bspc.2019.02.027.  Google Scholar [36] J. 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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
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
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
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
The time series $\mathbf{U}$, representing transitions between connected components of the wave-shape manifold, corresponds to coarse changes in heartbeat morphology
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
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
. The intubation event takes place at the 30th second, indicated by the black arrow on the colorbar">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
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
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 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
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
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
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
Atrial fibrillation is characterized by a lack of P waves, irregularly-occurring ventricular contractions, and a rapid component called the fibrillatory wave
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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