American Institute of Mathematical Sciences

Advanced
June  2021, 3(2): 99-131. doi: 10.3934/fods.2021009

Wave-shape oscillatory model for nonstationary periodic time series analysis

Yu-Ting Lin 1,2, , John Malik 3, and  Hau-Tieng Wu 4,,

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  June 2021 Early access  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.

Keywords: Wave-shape manifold, wave-shape oscillatory model, dynamic diffusion maps, time series analysis.
Mathematics Subject Classification: Primary: 37M10; Secondary: 92-08.
Citation: Yu-Ting Lin, John Malik, Hau-Tieng Wu. Wave-shape oscillatory model for nonstationary periodic time series analysis. Foundations of Data Science, 2021, 3 (2) : 99-131. doi: 10.3934/fods.2021009
References:
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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.

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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.

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N. El Karoui and H.-T. Wu, Graph connection Laplacian methods can be made robust to noise, Ann. Statist., 44 (2016), 346-372.  doi: 10.1214/14-AOS1275.

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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.

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show all references

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.

[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.

[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.

[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.

[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.

[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.

[7]

P. H. Bérard, Spectral Geometry: Direct and Inverse Problems, Lecture Notes in Mathematics, 1207, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0076330.

[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.

[9]

M. Bikkina, M. G. Larson and D. Levy, Prognostic implications of asymptomatic ventricular arrhythmias: The framingham heart study, Ann. Internal Medicine, 117 (1992), 990-996.  doi: 10.7326/0003-4819-117-12-990.

[10]

S. Bordignon, M. C. Corti and C. Bilato, Atrial fibrillation associated with heart failure, stroke and mortality, J. Atrial Fibrillation, 5 (2012).

[11]

P. J. Brockwell and R. A. Davis, Introduction to Time Series and Forecasting, Springer Texts in Statistics, Springer-Verlag, New York, 2002. doi: 10.1007/b97391.

[12]

Y.-C. Chen, M.-Y. Cheng and H.-T. Wu, Non-parametric and adaptive modeling of dynamic periodicity and trend with heteroscedastic and dependent errors, J. R. Stat. Soc. Ser. B. Stat. Methodol., 76 (2014), 651-682.  doi: 10.1111/rssb.12039.

[13]

A. Cicone and H.-T. 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.

[14]

G. D. Clifford, F. Azuaje and P. McSharry, Advanced Methods and Tools for ECG Data Analysis, Artech House, Inc., Norwood, MA, 2006.

[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.

[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.

[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. Assoc., 106 (2011), 1513-1527.  doi: 10.1198/jasa.2011.tm09771.

[18]

X. Ding and H.-T. Wu, Phase transition of graph Laplacian of high dimensional noisy random point cloud, preprint, arXiv: 2011.10725.

[19]

J. W. Dukes, T. A. Dewland, E. Vittinghoff, M. C. Mandyam and S. R. Heckbert, et al., Ventricular ectopy as a predictor of heart failure and death, J. Amer. College of Cardiology, 66 (2015), 101-109. doi: 10.1016/j.jacc.2015.04.062.

[20]

D. B. Dunson, H.-T. Wu and N. Wu, Spectral convergence of graph Laplacian and heat kernel reconstruction in $L^\infty$ from random samples, preprint, arXiv: 1912.05680.

[21]

N. El Karoui, On information plus noise kernel random matrices, Ann. Statist., 38 (2010), 3191-3216.  doi: 10.1214/10-AOS801.

[22]

N. El Karoui and H.-T. Wu, Graph connection Laplacian methods can be made robust to noise, Ann. Statist., 44 (2016), 346-372.  doi: 10.1214/14-AOS1275.

[23]

M. Elgendi, Optimal signal quality index for photoplethysmogram signals, Bioengineering, 3 (2016). doi: 10.3390/bioengineering3040021.

[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.

[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.

[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.

[27]

P. Flandrin, Time-Frequency/Time-Scale Analysis, Wavelet Analysis and its Applications, 10, Academic Press, Inc., San Diego, CA, 1999.

[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.

[29] I. M. Gel'fand and N. Y. Vilenkin, Generalized functions. Vol. 4: Applications of Harmonic Analysis, Academic Press, New York - London, 1964. 
[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.

[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.

[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.

[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.

[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.

[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.

[36]

J. Malik, A Geometric Approach to Biomedical Time Series Analysis, Ph.D thesis, Duke University, 2020. 162–170.

[37]

J. Malik, N. Reed, C.-L. Wang and H.-T. Wu, Single-lead f-wave extraction using diffusion geometry, Physiological Measurement, 38 (2017), 1310-1334.  doi: 10.1088/1361-6579/aa707c.

[38]

J. Malik, E. Z. Soliman and H.-T. Wu, An adaptive QRS detection algorithm for ultra-long-term ECG recordings, J. Electrocardiology, 60 (2020), 165-171.  doi: 10.1016/j.jelectrocard.2020.02.016.

[39]

M. Malik, Problems of heart rate correction in assessment of drug-induced QT interval prolongation, J. Cardiovascular Electrophysiology, 12 (2001), 411-420.  doi: 10.1046/j.1540-8167.2001.00411.x.

[40]

G. V. Naccarelli, H. Varker, J. Lin and K. L. Schulman, Increasing prevalence of atrial fibrillation and flutter in the United States, Amer. J. Cardiology, 104 (2009), 1534-1539.  doi: 10.1016/j.amjcard.2009.07.022.

[41] W. W. Nichols, M. F. O'Rourke and C. Vlachopoulos, McDonald's Blood Flow in Arteries – Theoretical, Experimental and Clinical Principals, CRC Press, London, 2011.  doi: 10.1201/b13568.
[42]

M. O'Rourke and A. Adji, An updated clinical primer on large artery mechanics: Implications of pulse waveform analysis and arterial tonometry, Current Opinion in Cardiology, 20 (2005), 275-281.  doi: 10.1097/01.hco.0000166595.44711.6f.

[43]

C.-K. Peng, S. Havlin, H. E. Stanley and A. L. Goldberger, Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series, Chaos: An Interdisciplinary J. Nonlinear Science, 5 (1995), 82-87.  doi: 10.1063/1.166141.

[44]

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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
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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
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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
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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
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Figure 5.  The time series $ \mathbf{U} $, representing transitions between connected components of the wave-shape manifold, corresponds to coarse changes in heartbeat morphology
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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
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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
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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
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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
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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
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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
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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
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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
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Figure 14.  Atrial fibrillation is characterized by a lack of P waves, irregularly-occurring ventricular contractions, and a rapid component called the fibrillatory wave
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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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