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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 |
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.
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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.
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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. |
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On information plus noise kernel random matrices, Ann. Statist., 38 (2010), 3191-3216.
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Graph connection Laplacian methods can be made robust to noise, Ann. Statist., 44 (2016), 346-372.
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M. Elgendi, Optimal signal quality index for photoplethysmogram signals, Bioengineering, 3 (2016).
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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.
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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. |
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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.
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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.
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Premature ventricular complex-induced cardiomyopathy, JACC: Clinical Electrophysiology, 5 (2019), 537-550.
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C.-Y. Lin, L. Su and H.-T. Wu,
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J. Malik, N. Reed, C.-L. Wang and H.-T. Wu,
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J. Malik, E. Z. Soliman and H.-T. Wu,
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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] |
J. W. Portegies,
Embeddings of Riemannian manifolds with heat kernels and eigenfunctions, Comm. Pure Appl. Math., 69 (2016), 478-518.
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