# Photoplethysmogram Recording Length: Defining Minimal Length Requirement from Dynamical Characteristics

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## Abstract

**:**

## 1. Introduction

## 2. Data

#### 2.1. Photoplethysmogram

#### 2.2. Experimental Data

#### 2.3. Simulated Data

_{i}using the following equation: ${\tilde{x}}_{i}={x}_{i}+\theta {\gamma}_{i}$, where $\theta $ is the noise scaling coefficient, which varies from 0 (i.e., no noise) to 0.5 in increments of 0.05; ${\gamma}_{i}$ is the ith component of a fixed uniform random noise vector; and ${\tilde{x}}_{i}$ is the input of the next iteration. For computational efficiency, the resulting time series x was subsampled using a factor of five. Examples of the resulting time series obtained using the original ($\theta $ = 0) and noisy ($\theta $ = 0.25) x time series are shown in Figure 3, where the addition of dynamical noise affected the amplitude of the time series and created additional fluctuations compared to the original time series.

## 3. Analysis

#### 3.1. Time Delay Reconstruction

#### 3.2. Recurrence Plot

_{i}is located within the sphere centered in X

_{j}with radius ε, then the (i, j)-pixel is included in the RP. In this study, ε was equal to 10% of the reconstructed attractor size [40] defined by the maximum distance between attractor points: $\underset{i,j=1,\text{}\dots ,\text{}N}{\mathrm{max}}\text{}\Vert {X}_{i}-{X}_{j}\Vert $.

#### 3.3. Recurrence Quantification Analysis (RQA)

**Determinism.**Determinism (DET) is one of the most important properties of a dynamical system and defines whether the process can be expressed in the form of a system of equations. Determinism can be estimated as:

_{min}is a given minimum length. Ideally, for deterministic time series, DET is equal to unity, whereas it is lower than one when a limited number of samples are available, when the signal includes noise, etc. Practically, values of DET > 0.9 can be considered as a sign of determinism [53].

**Trajectory divergence.**In RQA, exponential trajectory divergence, which is an important measure of the chaotic time series, can be defined as the inverse of the length of the longest diagonal line (${L}_{max}$) in the RP, expressed as:

**Predictability.**Short-term predictability—that, is the possibility to predict future states of the signal based on past observations for a short time window—is an extremely important property both theoretically and practically. The mean prediction time of the dynamical system can be estimated by computing the average diagonal line length (L), defined as

**Complexity.**Entropy (ENTR) is frequently used in applied studies of RQA. ENTR estimates the complexity of the RP with respect to the diagonal lines and can be calculated as:

#### 3.4. Error Estimation

## 4. Results

#### 4.1. Rössler System

_{max}(b), L (c), and ENTR (d) as a function of noise level (θ) and time series length, as averaged over 100 segments. Table 1 shows four calculated RQA measures reference values for three levels of noise. Table 2 shows the time series length corresponding to relative errors equal to 5% and 1% for three different noise levels. A relative error lower than or equal to 1% is considered acceptable. Therefore, the corresponding length can be used as a benchmark for the minimum length of the time series. Figure 6 shows detailed information of the length of the time series associated to relative error below 5% and 1% for the four RQA features here computed.

#### 4.2. Photoplethysmogram

## 5. Discussion

_{max}, L, and ENTR shown in Figure 5a–d indicate specific patterns associated with the different RQA features. As seen in Figure 5a, accurate estimates of determinism can be achieved even for very short time series (e.g., 1500 points). The lowest acceptable length of the time series increases with increasing noise, as shown in Figure 5a and Figure 6 and Table 2, but DET remains the feature requiring the shortest time series length among the RQA features addressed in this study. It is of note that with high levels of added noise, the time series cannot be considered deterministic, as DET is significantly below 0.9 (Table 1).

_{max}showed that accurate estimates of this feature—that is, the inverse of the divergence—cannot be achieved for short time series (Figure 5b and Figure 6), as L

_{max}requires long time series, i.e., longer than 48,500 points, for obtaining errors below 1%. It is of note that the average L

_{max}value for original data (i.e., 3763.9 points, Table 1) is significantly shorter than the used maximal time series length; nevertheless, the resulting lower limit for time series length is compatible with the maximal time series length used. Figure 5b and Figure 6 and Table 2 also clearly demonstrated that in the absence of noise, the lowest acceptable length of the time series is higher than that observed with noisy data due to the fact that the presence of noise significantly shortens the length of the maximal diagonal lines (Table 1).

_{max}, longer time series are required to reach error below 1%, specifically 43,000 (131.15 average cycles or 104.92 s) and 49,120 (149.75 average cycles or 119.8 s), respectively. However, the L

_{max}lower limit of the time series almost reaches the maximal time series length, and the actual L

_{max}value—unlike the results of the simulated data—is comparable with the maximal time series length, thus making it inapplicable for short recordings of the investigated PPG type.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Cardiovascular Diseases (CVDs). Available online: https://www.who.int/news-room/fact-sheets/detail/cardiovascular-diseases-(cvds) (accessed on 5 June 2022).
- Allen, J. Photoplethysmography and Its Application in Clinical Physiological Measurement. Physiol. Meas.
**2007**, 28, R1–R39. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Elgendi, M. PPG Signal Analysis: An Introduction Using MATLAB; CRC Press: Boca Raton, FL, USA, 2020; ISBN 978-0-429-44958-1. [Google Scholar]
- Kyriacou, P.A. Introduction to Photoplethysmography. In Photoplethysmography; Allen, J., Kyriacou, P., Eds.; Academic Press: Cambridge, MA, USA, 2022; pp. 1–16. ISBN 978-0-12-823374-0. [Google Scholar]
- Tamura, T. Current Progress of Photoplethysmography and SPO
_{2}for Health Monitoring. Biomed. Eng. Lett.**2019**, 9, 21–36. [Google Scholar] [CrossRef] [PubMed] - Celka, P.; Charlton, P.H.; Farukh, B.; Chowienczyk, P.; Alastruey, J. Influence of Mental Stress on the Pulse Wave Features of Photoplethysmograms. Healthc. Technol. Lett.
**2019**, 7, 7–12. [Google Scholar] [CrossRef] [PubMed] - Charlton, P.H.; Celka, P.; Farukh, B.; Chowienczyk, P.; Alastruey, J. Assessing Mental Stress from the Photoplethysmogram: A Numerical Study. Physiol. Meas.
**2018**, 39, 054001. [Google Scholar] [CrossRef] [PubMed] - Correia, B.; Dias, N.; Costa, P.; Pêgo, J.M. Validation of a Wireless Bluetooth Photoplethysmography Sensor Used on the Earlobe for Monitoring Heart Rate Variability Features during a Stress-Inducing Mental Task in Healthy Individuals. Sensors
**2020**, 20, 3905. [Google Scholar] [CrossRef] [PubMed] - Tsuda, I.; Tahara, T.; Iwanaga, H. Chaotic Pulsation in Human Capillary Vessels and Its Dependence on Mental and Physical Conditions. Int. J. Bifurc. Chaos
**1992**, 2, 313–324. [Google Scholar] [CrossRef] - Pham, T.D.; Thang, T.C.; Oyama-Higa, M.; Sugiyama, M. Mental-Disorder Detection Using Chaos and Nonlinear Dynamical Analysis of Photoplethysmographic Signals. Chaos Solitons Fractals
**2013**, 51, 64–74. [Google Scholar] [CrossRef] - Sumida, T.; Arimitu, Y.; Tahara, T.; Iwanaga, H. Mental Conditions Reflected by the Chaos of Pulsation in Capillary Vessels. Int. J. Bifurc. Chaos
**2000**, 10, 2245–2255. [Google Scholar] [CrossRef] - Depression. Available online: https://www.who.int/news-room/fact-sheets/detail/depression (accessed on 6 June 2022).
- Sviridova, N.; Sakai, K. Application of Photoplethysmogram for Detecting Physiological Effects of Tractor Noise. Eng. Agric. Environ. Food
**2015**, 8, 313–317. [Google Scholar] [CrossRef] - Hwang, S.; Seo, J.; Jebelli, H.; Lee, S. Feasibility Analysis of Heart Rate Monitoring of Construction Workers Using a Photoplethysmography (PPG) Sensor Embedded in a Wristband-Type Activity Tracker. Autom. Constr.
**2016**, 71, 372–381. [Google Scholar] [CrossRef] - Bradke, B.S.; Miller, T.A.; Everman, B. Photoplethysmography behind the Ear Outperforms Electrocardiogram for Cardiovascular Monitoring in Dynamic Environments. Sensors
**2021**, 21, 4543. [Google Scholar] [CrossRef] [PubMed] - Elgendi, M. On the Analysis of Fingertip Photoplethysmogram Signals. Curr. Cardiol. Rev.
**2012**, 8, 14–25. [Google Scholar] [CrossRef] [PubMed] - Millasseau, S.C.; Ritter, J.M.; Takazawa, K.; Chowienczyk, P.J. Contour Analysis of the Photoplethysmographic Pulse Measured at the Finger. J. Hypertens.
**2006**, 24, 1449–1456. [Google Scholar] [CrossRef] [PubMed] - Bhat, S.; Adam, M.; Hagiwara, Y.; Ng, E.Y.K. The Biophysical Parameter Measurements from Ppg Signal. J. Mech. Med. Biol.
**2017**, 17, 1740005. [Google Scholar] [CrossRef] - Kumar, A.K.; Ritam, M.; Han, L.; Guo, S.; Chandra, R. Deep Learning for Predicting Respiratory Rate from Biosignals. Comput. Biol. Med.
**2022**, 144, 105338. [Google Scholar] [CrossRef] - Rong, M.; Li, K. A Multi-Type Features Fusion Neural Network for Blood Pressure Prediction Based on Photoplethysmography. Biomed. Signal Process. Control.
**2021**, 68, 102772. [Google Scholar] [CrossRef] - Kim, J.W.; Choi, S.-W. Normalization of Photoplethysmography Using Deep Neural Networks for Individual and Group Comparison. Sci. Rep.
**2022**, 12, 3133. [Google Scholar] [CrossRef] - Small, M.; Judd, K.; Lowe, M.; Stick, S. Is Breathing in Infants Chaotic? Dimension Estimates for Respiratory Patterns during Quiet Sleep. J. Appl. Physiol.
**1999**, 86, 359–376. [Google Scholar] [CrossRef] [Green Version] - Poon, C.-S.; Merrill, C.K. Decrease of Cardiac Chaos in Congestive Heart Failure. Nature
**1997**, 389, 492–495. [Google Scholar] [CrossRef] - Shelhamer, M. Nonlinear Dynamics in Physiology: A State-Space Approach; World Scientific: Singapore, 2006; ISBN 978-981-270-029-2. [Google Scholar]
- Sviridova, N.; Sakai, K. Human Photoplethysmogram: New Insight into Chaotic Characteristics. Chaos Solitons Fractals
**2015**, 77, 53–63. [Google Scholar] [CrossRef] [Green Version] - Sviridova, N.; Zhao, T.; Aihara, K.; Nakamura, K.; Nakano, A. Photoplethysmogram at Green Light: Where Does Chaos Arise From? Chaos Solitons Fractals
**2018**, 116, 157–165. [Google Scholar] [CrossRef] - Kaneko, K.; Tsuda, I. Complex Systems: Chaos and Beyond; Springer: Berlin/Heidelberg, Germany, 2001; ISBN 978-3-642-63132. [Google Scholar]
- Liang, Y.; Elgendi, M.; Chen, Z.; Ward, R. An Optimal Filter for Short Photoplethysmogram Signals. Sci. Data
**2018**, 5, 180076. [Google Scholar] [CrossRef] [PubMed] - Ram, M.R.; Madhav, K.V.; Krishna, E.H.; Komalla, N.R.; Reddy, K.A. A Novel Approach for Motion Artifact Reduction in PPG Signals Based on AS-LMS Adaptive Filter. IEEE Trans. Instrum. Meas.
**2012**, 61, 1445–1457. [Google Scholar] [CrossRef] - Peng, F.; Zhang, Z.; Gou, X.; Liu, H.; Wang, W. Motion Artifact Removal from Photoplethysmographic Signals by Combining Temporally Constrained Independent Component Analysis and Adaptive Filter. BioMed. Eng. Online
**2014**, 13, 50. [Google Scholar] [CrossRef] [Green Version] - Waugh, W.; Allen, J.; Wightman, J.; Sims, A.J.; Beale, T.A.W. Novel Signal Noise Reduction Method through Cluster Analysis, Applied to Photoplethysmography. Comput. Math. Methods Med.
**2018**, 2018, e6812404. [Google Scholar] [CrossRef] [Green Version] - Yan, Y.; Poon, C.C.; Zhang, Y. Reduction of Motion Artifact in Pulse Oximetry by Smoothed Pseudo Wigner-Ville Distribution. J. NeuroEng. Rehabil.
**2005**, 2, 3. [Google Scholar] [CrossRef] [Green Version] - Sviridova, N. Detection of Preprocessing-Induced Changes in Chaotic Characteristics of Human Photoplethysmogram. In Proceedings of the International Symposium on Nonlinear Theory and its Applications, Kuala Lumpur, Malaysia, 2–6 December 2019; pp. 540–543. [Google Scholar]
- Badii, R.; Broggi, G.; Derighetti, B.; Ravani, M.; Ciliberto, S.; Politi, A.; Rubio, M.A. Dimension Increase in Filtered Chaotic Signals. Phys. Rev. Lett.
**1988**, 60, 979–982. [Google Scholar] [CrossRef] - Liang, Y.; Chen, Z.; Liu, G.; Elgendi, M. A New, Short-Recorded Photoplethysmogram Dataset for Blood Pressure Monitoring in China. Sci. Data
**2018**, 5, 180020. [Google Scholar] [CrossRef] [Green Version] - Sviridova, N.; Ikeguchi, T. Application of Recurrence Quantification Analysis to Hypertension Photoplethysmograms. In Proceedings of the International Symposium on Nonlinear Theory and Its Applications 2020, Virtual Online Conference, 16–19 November 2020; pp. 56–58. [Google Scholar]
- Sato, S.; Miao, T.; Oyama-Higa, M. Studies on Five Senses Treatment. In Knowledge-Based Systems in Biomedicine and Computational Life Science; Pham, T.D., Jain, L.C., Eds.; Studies in Computational Intelligence; Springer: Berlin/Heidelberg, Germany, 2013; pp. 155–175. ISBN 978-3-642-33015-5. [Google Scholar]
- Webber, C.L.; Zbilut, J.P. Dynamical Assessment of Physiological Systems and States Using Recurrence Plot Strategies. J. Appl. Physiol.
**1994**, 76, 965–973. [Google Scholar] [CrossRef] - Marwan, N.; Carmen Romano, M.; Thiel, M.; Kurths, J. Recurrence Plots for the Analysis of Complex Systems. Phys. Rep.
**2007**, 438, 237–329. [Google Scholar] [CrossRef] - Marwan, N. How to Avoid Potential Pitfalls in Recurrence Plot Based Data Analysis. Int. J. Bifurc. Chaos
**2011**, 21, 1003–1017. [Google Scholar] [CrossRef] [Green Version] - Hertzman, A.B. Photoelectric Plethysmography of the Fingers and Toes in Man. Proc. Soc. Exp. Biol. Med.
**1937**, 37, 529–534. [Google Scholar] [CrossRef] - Maeda, Y.; Sekine, M.; Tamura, T.; Suzuki, T.; Kameyama, K. Performance Evaluation of Green Photoplethysmography. J. Life Support Eng.
**2007**, 19, 183. [Google Scholar] [CrossRef] - Maeda, Y.; Sekine, M.; Tamura, T. The Advantages of Wearable Green Reflected Photoplethysmography. J. Med. Syst.
**2011**, 35, 829–834. [Google Scholar] [CrossRef] - Gagge, A.P.; Stolwijk, J.A.J.; Hardy, J.D. Comfort and Thermal Sensations and Associated Physiological Responses at Various Ambient Temperatures. Environ. Res.
**1967**, 1, 1–20. [Google Scholar] [CrossRef] - Přibil, J.; Přibilová, A.; Frollo, I. Comparison of Three Prototypes of PPG Sensors for Continual Real-Time Measurement in Weak Magnetic Field. Sensors
**2022**, 22, 3769. [Google Scholar] [CrossRef] - Leitner, J.; Chiang, P.-H.; Dey, S. Personalized Blood Pressure Estimation Using Photoplethysmography: A Transfer Learning Approach. IEEE J. Biomed. Health Inform.
**2022**, 26, 218–228. [Google Scholar] [CrossRef] - Sviridova, N.; Sawada, K.; Ikeguchi, T. Consistency of Determinism Detection in Sparse Photoplethysmogram Recordings. In Proceedings of the BIBE2022: The Sixth International Conference on Biological Information and Biomedical Engineering, Online, 19–20 July 2022; pp. 1–4. [Google Scholar]
- Rössler, O.E. An Equation for Continuous Chaos. Phys. Lett. A
**1976**, 57, 397–398. [Google Scholar] [CrossRef] - Takens, F. Detecting Strange Attractors in Turbulence. In Dynamical Systems and Turbulence, Warwick 1980; Rand, D., Young, L.-S., Eds.; Springer: Berlin/Heidelberg, Germany, 1981; pp. 366–381. [Google Scholar]
- Sauer, T.; Yorke, J.A.; Casdagli, M. Embedology. J. Stat. Phys.
**1991**, 65, 579–616. [Google Scholar] [CrossRef] - Kennel, M.B.; Brown, R.; Abarbanel, H.D.I. Determining Embedding Dimension for Phase-Space Reconstruction Using a Geometrical Construction. Phys. Rev. A
**1992**, 45, 3403–3411. [Google Scholar] [CrossRef] [Green Version] - Albano, A.M.; Muench, J.; Schwartz, C.; Mees, A.I.; Rapp, P.E. Singular-Value Decomposition and the Grassberger-Procaccia Algorithm. Phys. Rev. A
**1988**, 38, 3017–3026. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Thiel, M.; Romano, M.C.; Kurths, J.; Meucci, R.; Allaria, E.; Arecchi, F.T. Influence of Observational Noise on the Recurrence Quantification Analysis. Phys. D Nonlinear Phenom.
**2002**, 171, 138–152. [Google Scholar] [CrossRef] [Green Version] - Anton, H.; Bivens, I.; Davis, S. Calculus, 7th ed.; Anton, H., Bivens, I., Davis, S., Eds.; John Wiley & Sons: New York, NY, USA, 2002; ISBN 978-0-471-43312-5. [Google Scholar]
- Rubenstein, D.A.; Yin, W.; Frame, M.D. Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation; Academic Press: Cambridge, MA, USA, 2021; ISBN 978-0-12-818034-1. [Google Scholar]
- Baselli, G.; Cerutti, S.; Porta, A.; Signorini, M.G. Short and Long Term Non-Linear Analysis of RR Variability Series. Med. Eng. Phys.
**2002**, 24, 21–32. [Google Scholar] [CrossRef]

**Figure 3.**Examples of a segment of the original (blue dotted line) and noisy (red solid line) x time series generated using the chaotic Rössler model.

**Figure 4.**Reconstructed attractors from x time series of (

**a**) original (θ = 0) and (

**b**) noisy (θ = 0.25) Rössler systems and the resulting RPs (panels (

**c**) and (

**d**), respectively).

**Figure 5.**Relative error of the RQA features estimates as a function of length and noise level of Rössler time series: (

**a**) determinism, (

**b**) maximal diagonal line length, (

**c**) average diagonal line length, and (

**d**) entropy.

**Figure 6.**Length of the Rössler time series (left: in points, right: in average cycles) as a function of the noise level corresponding to a relative error equal to 5% (solid line) and 1% (dashed line) in the estimation of the determinism (red), maximal diagonal line length (blue), average diagonal line length (magenta), and entropy (green).

**Figure 7.**An example of (

**a**) time-delay-reconstructed attractor and (

**b**) the resulting RP obtained from one of the PPGs recorded in this study.

**Figure 8.**Relative error of the RQA features estimates as a function of the length of the time series.

**Table 1.**Values of determinism, maximal diagonal line length, average diagonal line length, and entropy obtained for long Rössler’s x time series for three different noise levels.

Noise Level, θ | DET | L_{max} | L | ENTR |
---|---|---|---|---|

0 | 0.9992 | 3763.9 | 60.4329 | 4.6079 |

0.245 | 0.4138 | 7.52 | 2.3379 | 0.7544 |

0.5 | 0.0910 | 3.24 | 2.0455 | 0.1765 |

**Table 2.**Time series length corresponding to relative errors equal to 5% and 1% in the estimates of determinism, maximal diagonal line length, average diagonal line length, and entropy for three different noise levels of the Rössler time series. The values highlighted indicate the lower limit of the length of the time series required for accurately estimating the four features.

Noise Level, θ | E_{l} | DET | L_{max} | L | ENTR |
---|---|---|---|---|---|

0 | 5% | 1500 | 46,000 | 4000 | 10,500 |

1% | 1500 | 48,500 | 16,000 | 26,000 | |

0.245 | 5% | 3000 | 36,500 | 2500 | 7500 |

1% | 17,000 | 46,000 | 3500 | 30,000 | |

0.5 | 5% | 5000 | 41,500 | 11,000 | 42,000 |

1% | 7000 | 47,000 | 13,500 | 43,500 |

**Table 3.**Average values of the length of the time series corresponding to relative error below 5% and 1% and the reference RQA values.

Lower Time Series Length Limit, Average Cycles | Reference RQA Values (409.6 Hz) | ||||||
---|---|---|---|---|---|---|---|

409.6 Hz | 204.8 Hz | 102.4 Hz | |||||

E_{l}, 5% | E_{l}, 1% | E_{l}, 5% | E_{l}, 1% | E_{l}, 5% | E_{l}, 1% | ||

DET | 3.05 | 3.05 | 3.05 | 3.05 | 3.05 | 3.05 | 0.998 |

L_{max} | 144.82 | 149.39 | 143.38 | 150.28 | 143.39 | 150.25 | 49,859 |

L | 65.55 | 131.10 | 64.19 | 132.79 | 64.02 | 133.02 | 144.38 |

ENTR | 7.62 | 21.34 | 9.65 | 22.05 | 10.67 | 24.39 | 5.73 |

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**MDPI and ACS Style**

Sviridova, N.; Zhao, T.; Nakano, A.; Ikeguchi, T.
Photoplethysmogram Recording Length: Defining Minimal Length Requirement from Dynamical Characteristics. *Sensors* **2022**, *22*, 5154.
https://doi.org/10.3390/s22145154

**AMA Style**

Sviridova N, Zhao T, Nakano A, Ikeguchi T.
Photoplethysmogram Recording Length: Defining Minimal Length Requirement from Dynamical Characteristics. *Sensors*. 2022; 22(14):5154.
https://doi.org/10.3390/s22145154

**Chicago/Turabian Style**

Sviridova, Nina, Tiejun Zhao, Akimasa Nakano, and Tohru Ikeguchi.
2022. "Photoplethysmogram Recording Length: Defining Minimal Length Requirement from Dynamical Characteristics" *Sensors* 22, no. 14: 5154.
https://doi.org/10.3390/s22145154