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`Human photoplethysmogram: New insight into chaotic characteristics
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`Article  in  Chaos Solitons & Fractals · August 2015
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`DOI: 10.1016/j.chaos.2015.05.005
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`Chaos, Solitons and Fractals 77 (2015) 53–63
`
`Contents lists available at ScienceDirect
`
`Chaos, Solitons and Fractals
`Nonlinear Science, and Nonequilibrium and Complex Phenomena
`
`journal homepage: www.elsevier.com/locate/chaos
`
`Human photoplethysmogram: new insight into chaotic
`characteristics
`Nina Sviridova a, Kenshi Sakai b,∗
`
`a United Graduate School of Agriculture, Tokyo University of Agriculture and Technology, 3-5-8 Saiwai-cho, Fuchu-shi, Tokyo 183-8509, Japan
`b Environmental and Agricultural Engineering Department, Tokyo University of Agriculture and Technology 3-5-8 Saiwai-cho, Fuchu-shi,
`Tokyo 183-8509. Japan
`
`a r t i c l e
`
`i n f o
`
`a b s t r a c t
`
`Article history:
`Received 3 March 2015
`Accepted 3 May 2015
`Available online 19 May 2015
`
`The photoplethysmogram is widely used in medical settings and sports equipment to measure
`biological signals. The photoplethysmogram, which is measured noninvasively, can provide
`valuable information about cardiovascular system performance. The present study sought to
`investigate the underlying dynamics of photoplethysmographic signals from healthy young
`human subjects. In previous studies the photoplethysmogram was claimed to be driven by
`deterministic chaos [Tsuda 1992, Sumida 2000]; however, the methods applied for chaos de-
`tection were noise sensitive and inconclusive. Therefore, to reach a consistent conclusion it
`is important to employ additional nonlinear time series analysis tools that can test different
`features of the signal’s underlying dynamics. In this paper, methods of nonlinear time series
`analysis, including time delay embedding, largest Lyapunov exponent, deterministic nonlinear
`prediction, Poincaré section, the Wayland test and method of surrogate data were applied to
`photoplethysmogram time series to identify the unique characteristics of the photoplethys-
`mogram as a dynamical system. Results demonstrated that photoplethysmogram dynamics is
`consistent with the definition of chaotic movement, and its chaotic properties showed some
`similarity to Rossler’s single band chaos with induced dynamical noise. Additionally it was
`found that deterministic nonlinear prediction, Poincaré section and the Wayland test can re-
`veal important characteristics of photoplethysmographic signals that will be important tools
`for theoretical and applied studies on the photoplethysmogram.
`© 2015 The Authors. Published by Elsevier Ltd.
`This is an open access article under the CC BY-NC-ND license
`(http://creativecommons.org/licenses/by-nc-nd/4.0/).
`
`1. Introduction
`
`Physiological signals derived from the cardiovascular sys-
`tem show an extreme intricacy that arises from the interac-
`tion of many processes, structure units and feedback loops in
`humans. Attempts to improve our understanding of physio-
`logical complexity and develop new tools for promising ap-
`plications for human mental and physical health monitoring
`
`∗
`
`Corresponding author: Tel.: +81 423675755.
`E-mail addresses: nina_svr@mail.ru (N. Sviridova), ken@cc.tuat.ac.jp (K.
`Sakai).
`
`have made physiological signals such as electrocardiogram
`(ECG), electro-encephalogram (EEG), blood pressure, heart
`rate variability (HRV) and photoplethysmograph (PPG) the
`subject of recent studies [1–9]. Due to an increase in the suc-
`cessful use of nonlinear time series analysis (NTSA) methods
`in many scientific disciplines to quantify the complexity of
`signals [4,9], methods of nonlinear dynamics analysis have
`become a new and powerful tool for physiological signal in-
`vestigation. NTSA allows one not only to quantify, but to qual-
`ify data.
`Many studies have investigated ECG, EEG and HRV sig-
`nals obtained from healthy human subjects, as well as from
`
`http://dx.doi.org/10.1016/j.chaos.2015.05.005
`0960-0779/© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
`(http://creativecommons.org/licenses/by-nc-nd/4.0/).
`
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`N. Sviridova, K. Sakai / Chaos, Solitons and Fractals 77 (2015) 53–63
`
`patients with mental or heart illnesses [1,3–9]. In early stud-
`ies PPG as well as ECG and HRV were claimed to be chaotic
`mostly based on results of time-delay reconstructed trajec-
`tory, correlation dimension and largest Lyapunov exponent.
`Later, with the development of methods of nonlinear time
`series analysis for real world data, evidence for the chaotic
`nature of many biological signals was questioned [8,9]. Many
`tools that were previously thought to provide explicit evi-
`dence of chaotic motion were discovered to be noise sen-
`sitive and could produce misleading results. Thus it is still
`quite controversial whether these signals’ dynamics involve
`chaotic motion or not [8,9]. Signals like HRV and ECG that
`were believed to be driven by deterministic chaos have been
`subjected to detailed reinvestigation [8,9]. Similarly PPG sig-
`nal was claimed to be chaotic in the early 90s, however
`nowadays past evidence of chaos was found to be neces-
`sary rather than sufficient, especially in biological studies
`[8,9]. Therefore it is also still not known whether the PPG
`signal, which is commonly measured by commercially avail-
`able medical and sport equipment devices to obtain HRV
`data, oxygen saturation and blood pressure, is chaotic or
`not. In addition, many of its characteristics are not yet well
`studied.
`The microcirculation of the skin is rather complex dy-
`namic system which is important for skin metabolism and
`temperature regulation and plays an important role in an
`organism’s defense system. As the skin surface is directly ac-
`cessible, it has become a valuable organ for many studies.
`Photoplethysmography (PPG) is one of the widely-used tech-
`niques that allows registration of pulsatile changes in the
`dermal vasculature [10,11].
`Photoplethysmography is a simple and low-cost optical
`technique that can be used to detect blood volume changes
`in the microvascular bed of tissue. The PPG wave form com-
`prises a pulsatile physiological waveform attributed to car-
`diac synchronous changes in the blood volume with each
`heart beat and is superimposed on a slowly varying base-
`line with various lower frequency components attributed to
`respiration, sympathetic nervous system activity and ther-
`moregulation [10–12]. Even though pulsation in a finger’s
`capillary vessels (i.e. PPG obtained from finger) in normal sub-
`jects was claimed to be chaotic [13,14], only classical methods
`such as time-delay reconstructed trajectory, power spectrum,
`correlation dimension (CD) and Lyapunov exponent (LE) and
`surrogation have been applied to characterize PPG time se-
`ries [5,13,14]. Positive Lyapunov exponent was believed to
`provide strong evidence of chaotic behavior. However, these
`tests (CD and LE) are inconclusive since as it was found in
`recent studies that they may indicate chaos even in systems
`that are not driven by chaos [9,15]. Therefore, a clear answer
`regarding the nature of the PPG signal dynamics cannot be
`obtained by only applying these types of classical measure-
`ments, although they may provide useful results for medical
`applications.
`In this study we applied the time delay embedding
`method, calculated the power spectrum, largest Lyapunov
`exponent (LLE), deterministic nonlinear prediction’s (DNP)
`correlation coefficient (CC) and relative route mean square er-
`ror (RRMSE), Poincaré section, Wayland test translation error
`and applied method of surrogate data to investigate whether
`the underlying dynamics of the PPG signal involve motion on
`
`a strange attractor and to study the chaotic motion charac-
`teristics of the PPG signal. This expanded toolkit is designed
`to cover most of the important characteristics of chaotic mo-
`tion and is expected to help us investigate a wider range of
`PPG signal characteristics, compared with previous studies,
`and thus to extract its underlying properties. Additionally,
`in an effort to analyze the PPG signal not only quantitatively,
`but qualitatively we conducted a comparative analysis of PPG
`signal and Rosser’s single band chaos.
`
`2. Methods and materials
`
`2.1. Photoplethysmography
`
`PPG can be defined as the continuous recording of the light
`intensity scattered from a given source by the tissues and col-
`lected by a suitable photodetector [10]. Modern PPG sensors
`usually utilize low cost semiconductor technology with LED
`and matched photodetector devices working at the near in-
`frared (NIR) wavelengths (NIR band 0.8–1 μm), which allows
`measurement of deep-tissue blood flow [11,12]. The light
`from the LED is absorbed by hemoglobin, and the backscat-
`tered radiation is then detected and recorded. The backscat-
`tered light depends on the amount of hemoglobin in the skin,
`and the obtained result therefore reflects the cutaneous blood
`flow [10,11].
`The microcirculation of the skin is rather complex and
`dynamic system which is important for skin metabolism and
`temperature regulation and is an important part of the organ-
`ism’s defense system against invaders. The cutaneous blood
`supply is carried out into microcirculatory bed composed of
`three segments—arterioles, arterial and venous capillaries,
`and venules; most of this microvasculature is contained in
`the papillary dermis 1–2 μm below the epidermal surface
`[10–12]. PPG waveform reflects heartbeat synchronized cu-
`taneous blood flow pulsatile changes in the dermal vascula-
`ture. The shape of the waveform is related to anacrotic and
`catacrotic phases. The anacrotic phase corresponds to the ris-
`ing edge of the pulse, and the catacrotic phase to the falling
`edge of the pulse curve. The first one is primarily connected
`with contraction of the heart and therefore with the systolic
`phase of cardiac cycle, while the second corresponds to dias-
`tolic phase (Fig. 1) and wave reflection from the periphery. A
`dicrotic notch, connected with wave reflected from the pe-
`riphery, usually can be seen in the catacrotic phase of subjects
`with healthy arteries [6,11].
`The pulsatile component of the PPG waveform that relates
`to cardiac pulsation is usually called the ‘AC’ (alternative cur-
`rent) component and its fundamental frequency depends on
`the heart rate and typically varies around 1–1.4 Hz. The AC
`component is superimposed onto a large ‘DC’ (direct current)
`component that depends on the structure of the tissue and
`the average blood volume of both arterial and venous blood.
`The DC component varies slowly due to respiration, vaso-
`motor activity, vasoconstrictor waves, thermoregulation and
`other slow circulatory changes [10–12].
`Although the origins of the components of the PPG signal
`are not fully understood, it is generally accepted that PPG can
`provide valuable clinical information about the cardiovascu-
`lar system [11].
`
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`55
`
`Fig. 1. Components of PPG signal waveform for healthy young subjects.
`
`Fig. 2. Example of 30-second long portion of the healthy young subject PPG signal (9th subject’s 2nd measurement).
`
`2.2. Data collection
`
`The PPG signal was recorded using a finger PPG recorder
`by detecting the near infrared light reflected by vascular
`tissue following illumination with a LED. Data were collected
`from nine healthy 19–27-year old volunteers among Tokyo
`University of Agriculture and Technology (TUAT) students.
`Experimental data collection was approved by TUAT author-
`ities. Written informed consent was obtained from partici-
`pants prior to the experiment. At the time of the study all
`subjects were healthy non-smokers, physically active to sim-
`
`ilar levels, were not taking any medication, and none declared
`a history of heart disease.
`For each subject five measurement repeats were done. The
`measured period was 5 min with 5 ms sampling steps. For all
`data collection sessions, a BACS (Computer convenience, Inc.)
`transmission-mode PPG sensor was located on the right fore-
`finger. Every measurement was preceded by a blood pressure
`check. Since physical and mental activity, as well as exter-
`nal effects such as temperature, noise etc. can considerably
`affect cutaneous blood flow [10,11] all measurements were
`done with the subject in a relaxed sitting position in a room
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`N. Sviridova, K. Sakai / Chaos, Solitons and Fractals 77 (2015) 53–63
`
`Fig. 3. Example of typical spectra obtained by Fourier analysis, where LF is low frequency and PF is predominant frequency component.
`
`with temperature, noise and vibration control. Each test sub-
`ject was asked to rest for 5 min under quiet conditions in the
`laboratory room in a sitting position in which the record-
`ings were obtained, and with the test site uncovered. An
`example of a 30-second long portion of the obtained PPG
`signal is shown in Fig. 2.
`
`applied a complex of nonlinear time series analysis tools. Or-
`bital instability was tested with LLE, determinism with DNP,
`WTE, recurrence of motion with time-delay embedding and
`Poincaré section; additionally we applied phase randomized
`surrogation.
`
`3. Results
`
`Various definitions of chaos can be found in the literature;
`Thompson et al. [20] provide quite broad, so called “positive”
`definition of chaos as “recurrent motion in simple systems or
`low-dimensional behavior that has some random aspects as
`well as certain order”, which covers wide range of systems
`that produce chaos and yet have significantly different prop-
`erties, as for example chaotic Lorenz and Rossler systems.
`In this paper to analyze data sets obtained in the experi-
`ments described in Section 2.2 and to investigate whether the
`PPG signal is consistent with the above definition by Thomp-
`son et al. and study its chaotic characteristics in details, we
`
`3.1. Spectral analysis
`
`An example of typical plot of the Fourier spectrum in the
`studied time series is shown in Fig. 3. In Fig. 3, small fluc-
`tuations, which indicate environmental noise, can be distin-
`guished around the predominant component (PF) whose pe-
`riod is approximately equal to the heart cycle period. Lower
`frequency (LF) components correspond to respiration and
`other effects, such as thermoregulation and nervous system
`activity. Table 1 shows values of amplitude and frequency
`corresponding to the predominant component obtained by
`Fourier transform. As seen from Table 1, all predominant fre-
`quencies (PF) are in the range 1.02–1.52 Hz, which is the
`range of normal heart beat frequencies.
`
`Table 1
`Amplitude (|FT|) and frequency (PF) of PPG predominant component obtained by Fourier analysis.
`
`Subject
`
`Repeat
`
`1
`
`|FT|
`
`5339.0
`1877.5
`6198.9
`1057.7
`7843.4
`4382.0
`7085.3
`3028.2
`5324.7
`
`2
`
`|FT|
`
`5608.5
`2063.1
`6960.6
`3475.7
`6379.4
`5915.2
`6758.1
`2557.6
`7062.0
`
`PF
`
`1.12
`1.21
`1.15
`1.28
`1.09
`1.19
`1.12
`1.10
`1.47
`
`3
`
`|FT|
`
`5057.4
`2906.8
`5438.9
`2384.0
`7048.2
`7298.9
`5449.0
`2451.6
`8937.0
`
`PF
`
`1.09
`1.21
`1.11
`1.24
`1.05
`1.12
`1.15
`1.11
`1.43
`
`4
`
`|FT|
`
`4185.4
`2276.4
`5765.3
`4018.1
`4945.1
`5763.3
`4378.1
`3563.0
`5444.2
`
`PF
`
`1.04
`1.12
`1.03
`1.29
`1.03
`1.14
`1.07
`1.12
`1.41
`
`5
`
`|FT|
`
`1814.5
`1839.4
`2947.3
`3792.7
`2784.6
`4100.3
`4711.1
`3006.1
`3658.1
`
`PF
`
`1.02
`1.22
`1.19
`1.34
`1.05
`1.15
`1.13
`1.06
`1.39
`
`PF
`
`1.20
`1.24
`1.16
`1.40
`1.07
`1.17
`1.17
`1.09
`1.52
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`
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`
`57
`
`Fig. 4. Trajectory reconstructed in 4 dimensional phase space by time-delay embedding.
`
`3.2. Time delay embedding
`
`By using the time delay embedding technique the possi-
`ble dynamics of time series can be reconstructed in phase
`space. The structure of reconstructed trajectory is considered
`to be an important characteristic of time series; the obtained
`geometric pattern of the trajectory may provide valuable
`information about the PPG signal properties, for example,
`it can reflect the level of physical or mental activity or the
`level of maturity [13,14].
`For time delay reconstruction, the time delay lag τ needs
`to be sufficiently large to ensure that the resulting individual
`coordinates are relatively independent; however, it should
`not be too large to make it completely independent statis-
`tically [16]. In this study we have defined the time lag for
`further calculations as a quarter of the period of the predom-
`inant component [17] obtained by Fourier analysis (Table 1).
`Following previous studies [7,13,14] where it was shown
`that at least 4 dimensions need to be used for the PPG sig-
`nal, we performed a time-delay embedding technique with
`4 dimensions to obtain the reconstructed trajectory in phase
`space. An example (corresponding to the 1st subject’s 1st
`measurement) of our typical data for a time-delay recon-
`struction is shown in Fig. 4, where the fourth dimension is
`represented by color. As seen in Fig. 4, the reconstructed tra-
`jectory has clear, screwing structure similar to Möbius band.
`
`3.3. Largest Lyapunov exponent
`
`Sensitive dependence to initial conditions is one of the
`most important properties of the chaotic system. Any chaotic
`system should have at least one positive Lyapunov exponent,
`with the magnitude reflecting the time scale on which sys-
`tem dynamics become unpredictable [18,19]. Lyapunov ex-
`ponent, which provides a qualitative and quantitative char-
`acterization of dynamical behavior, is a useful dynamical di-
`
`Table 2
`Largest Lyapunov exponents for all collected PPG time series
`(calculated by Wolf’s method).
`
`Subject
`
`Repeat
`
`1
`
`2
`
`3
`
`4
`
`5
`
`1.041
`0.731
`1.143
`1.100
`1.279
`1.206
`1.250
`1.079
`1.014
`
`0.961
`0.810
`1.565
`0.968
`1.020
`1.692
`1.010
`1.150
`0.938
`
`1.145
`1.099
`1.398
`0.861
`1.209
`0.937
`1.121
`1.194
`1.182
`
`0.744
`0.902
`1.303
`1.019
`1.167
`1.621
`1.157
`1.040
`1.119
`
`0.787
`0.754
`0.968
`1.274
`0.766
`1.242
`0.929
`1.000
`1.008
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`
`agnostic for chaotic systems since it gives a measure of the
`rate of divergence of neighboring trajectories [18,19].
`In this study LLE was calculated with Wolf’s method [19],
`which allows the estimation of LLE from an experimental
`time series. As many studies have mentioned [8,9,15] most
`computational methods for estimating LLE have limitations
`and can produce positive LLE even for non-chaotic systems
`[9] or can overestimate its value [15].
`LLE was calculated for all measured PPG data, and in all
`cases resulted in a positive LLE as shown in Table 2.
`
`3.4. Deterministic nonlinear prediction
`
`As it is widely known, chaotic systems show predictability
`in the short term that should decay rapidly, however different
`systems demonstrate different predictabilities, like for exam-
`ple prediction performance for the Lorenz system in chaotic
`regime and Rossler’s single band chaos differs (as shown in
`Fig. 5).
`
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`N. Sviridova, K. Sakai / Chaos, Solitons and Fractals 77 (2015) 53–63
`
`Fig. 5. (a) Correlation coefficient (CC) and (b) relative root mean squared error (RRMSE) curves for nonlinear deterministic prediction of Rossler’s single band
`chaos, Rossler’s single band chaos data with 7% additive noise and PPG.
`
`Rossler’s system is described by the following equations
`[20]:
`
`⎧⎪⎨
`⎪⎩
`
`˙x = −y − z,
`˙y = x + ay,
`˙z = b + z (x − c);
`
`and Lorenz system by equations[20]:
`
`⎧⎪⎨
`⎪⎩
`
`˙x = −σ (x − y) ,
`˙y = ρx − y − xz,
`˙z = xy − βz.
`
`For calculation of Rossler’s data, the system coefficients
`chosen were a = 0.398, b = 2, c = 4 which corresponds to
`single-band chaos [20–22], and for the Lorenz system and
`σ = 10, ρ = 28, β = 8/3; numerical simulation was done
`by the 4th order Runge–Kutta method with time step 0.01.
`Information about predictability can be used to distinguish
`different behaviors. Additionally, the presence of short-term
`prediction indicates the determinism of the system under
`investigation [9].
`To examine whether PPG time series are predictable in the
`short-term and investigate how forecasting quality changes
`with increasing prediction time, we conducted direct deter-
`ministic nonlinear prediction (DNP) in this study. Fig. 5(a)
`shows an example of typical data of CC and Fig. 5(b) RRMSE
`
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`
`59
`
`Fig. 6. Performance of deterministic nonlinear prediction. Original signal vs. 160 steps (0.8 s) forward prediction.
`
`between real and predicted signals with increasing prediction
`time.
`While short-time predictability can be easily identi-
`fied, conclusions regarding long-term prediction are not as
`straightforward, and in many cases are rather empirical and
`might be based on researcher experience. For example, pre-
`diction performance of well-known chaotic Lorenz data ex-
`hibits rapid and apparent decay of CC, which is not always
`observed for other well-known systems generating chaos.
`For example, in Fig. 5 Rossler’s single band chaos shows high
`CC (higher than 0.96) over more than 400 steps, while CC cor-
`responding to Lorenz reaches zero. It is important to notice
`in Fig. 5(a) and (b) that CC and RRMSE curves corresponding
`to chaotic Rossler band initially demonstrate decay, as would
`be expected, and later both curves stabilize at a high CC value
`(CC > 0.96) and a low RRMSE value (RRMSE < 0.28) over a
`long period, however, it is known that chaotic systems do not
`have long-term prediction and therefore despite the stable
`high CC value and low value of RRMSE these results should
`not be recognized as a sign of long-term prediction.
`As discussed later in Fig. 9, PPG trajectory have some topo-
`logical similarity to Rossler’s single band chaos; therefore, to
`illustrate changes in the PPG’s CC and RRMSE curves it was
`compared with the CC and RRMSE curves of the correspond-
`ing Rossler’s data. In addition, since the PPG data were ob-
`tained experimentally, it inevitably contains noise, so we also
`compared PPG prediction results with the same Rossler’s sys-
`tem with 7% additive (dynamical) noise; results are shown in
`Fig. 5(a) and (b).
`Fig. 6 shows example of actual vs. 0.8 s (160 time steps)
`predicted PPG, even after 160 steps forecast still resemble
`original time series with sufficiently high quality to reproduce
`not only the general trend of the PPG waveform, but also
`smaller details such as the dicrotic notch.
`As seen from Figs. 5 and 6, time series clearly demon-
`strated the presence of short-term prediction, which is in-
`dicative of underlying determinism. The chosen coefficients
`for the Rossler system correspond to a chaotic regime, and
`
`therefore the obtained Rossler’s data should not have long-
`term prediction. Thus comparison between CC and RRMSE
`curves corresponding to PPG, Rossler and noise induced
`Rossler data in Fig. 5 demonstrate the absence of long-term
`forecasting in the PPG signal, which is typical for chaotic mo-
`tion. Besides finding that PPG’s CC and RRMSE curves showed
`similar trends with CC and RRMSE corresponding to 7% noise
`induced Rossler’s data, these results have demonstrated that
`it might be misleading to compare results obtained from
`PPG with chaotic Lorenz data, which played a role of one
`of the classical examples of chaos over decades, but actu-
`ally demonstrates only one of many other possible chaotic
`behaviors.
`In this study DNP became an important method for testing
`the predictability properties of the PPG signal. In several stud-
`ies it was claimed that PPG time series of young healthy sub-
`jects are deterministic chaos, based mostly on results of the
`Lyapunov exponent, CD and attractor reconstruction, which
`as it is nowadays known, may produce misleading results
`for noise contaminated real-world signals. Results of DNP
`(Figs. 5 and 6) and its comparison with chaotic Rossler’s data
`provided not only substantial support for the claim of the
`chaotic nature of PPG, but more importantly, it provides us
`with quantitative (CC and RRMSE) as well as qualitative (sim-
`ilarity of prediction performance with noise induced chaotic
`Rossler’s data) characteristics of PPG dynamics.
`
`3.5. Poincaré section
`
`The Poincaré section is one of the most powerful tools for
`qualitative exploration of the dynamics of a system [9], as
`it enables a demonstration of the process generating chaos
`in the phase space. The Poincaré section was obtained from
`a three-dimensional time-delay reconstructed attractor by
`a clock-wise rotating slicing two-dimensional plane. Fig. 7
`demonstrates areas on trajectory sliced by the plane; in Fig. 8,
`an example of Poincaré sections for the 1st subject’s 1st mea-
`surement is shown.
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`Fig. 7. The areas sliced by the rotating plane on the reconstructed trajectory of PPG signal.
`
`Fig. 8. Poincaré sections for the PPG signal, where Y(t) =
`
`(cid:6)
`x(t)2 + x(t + τ )2.
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`61
`
`As seen from Fig. 8 dense trajectories (α = 75°) tend to
`expand (α = 150°), bend (α = 225°) and stretch (α = 300°) and
`then fold back (α = 0° and α = 75°) along the attractor. This
`stretch-and-fold behavior generates sensitive dependence on
`the initial conditions, which is recognized as an important
`property of chaotic dynamics.
`
`3.6. Wayland test
`
`In order to explain the variability observed in complex
`time series and distinguish whether it is due to external
`stochastic noise, internal deterministic dynamics, or a combi-
`nation of both, we applied the Wayland test, which is a com-
`putationally simple variation of the Kaplan–Glass method
`and uses the phase space continuity observed in time series
`to measure determinism [23].
`We calculated Wayland test translation error (WTE),
`which can quantify smoothness of the flow reconstructed
`in the phase space. WTE performs well in high levels of un-
`correlated noise and provides a robust measure of the de-
`terminism in the trajectory reconstructed in the phase space
`[23]. WTE is insensitive to an overall scaling of the original
`time series. If the time series is deterministic, then the WTE
`will be small [23]. For instance, for chaotic Rossler time se-
`ries (dataset containing 60,000 points) with parameters as
`described in Section 3.4, the WTE is 0.0002, for Rossler with
`10% induced noise the WTE is 0.0045, and for white noise is
`1.08 (60,000 points). Table 3 shows the results of WTE calcu-
`lations for PPG (time series size is 60,000 points).
`According to Table 3, most of the WTE values are small,
`which is an indication of determinism in the PPG data.
`
`3.7. Surrogation
`
`The surrogation approach is to specify a defined null hy-
`pothesis and generate a set of surrogate signals that embody
`
`Table 3
`Results of Wayland test’s translation error (WTE) calculation
`for all collected PPG time series.
`
`Subject
`
`Repeat
`
`1
`
`2
`
`3
`
`4
`
`5
`
`0.007
`0.041
`0.023
`0.016
`0.092
`0.015
`0.012
`0.031
`0.018
`
`0.006
`0.041
`0.012
`0.024
`0.024
`0.010
`0.017
`0.030
`0.019
`
`0.005
`0.045
`0.013
`0.020
`0.059
`0.019
`0.016
`0.015
`0.027
`
`0.010
`0.023
`0.015
`0.034
`0.027
`0.019
`0.026
`0.039
`0.045
`
`0.024
`0.048
`0.050
`0.085
`0.017
`0.037
`0.035
`0.048
`0.024
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`
`a hypothesis about the time series. Then by determining the
`distribution of the index under investigation obtained from
`surrogates, empirical statistical boundaries can be found. By
`comparing the index of the original time series with the dis-
`tribution of the index from surrogates, the null hypothesis
`under which surrogates were generated can either be re-
`jected if the value from the original signal does not overlap
`with the distribution from surrogates, or fail to be rejected if
`the original index is within distribution with high significance
`[9,24].
`One of the typical applications of surrogate data is to check
`for determinism and whether data are a realization of a spe-
`cific random process. In this study the null hypothesis is that
`the signal is a realization of a linear Gaussian stochastic pro-
`cess. Surrogate time series were created by phase random-
`ization of Fourier transforms for the original PPG data and
`following inverse Fourier transform. As a result, the obtained
`surrogates are stochastic, but have the same power spectrum
`as the original data. Applying this type of surrogates allows
`one not only to place empirical boundaries, but also to test
`for nonlinearity of PPG.
`
`Fig. 9. Deterministic nonlinear prediction correlation coefficient for PPG (red line) and 30 surrogate datasets (blue lines). For interpretation of the references to
`colour in this figure legend, the reader is referred to the web version of this article
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`
`DNP were applied to 30 surrogate datasets. Fig. 9 demon-
`strates the results of DNP for surrogate time series generated
`under the null hypothesis from the original data. As seen from
`the CC curve, the original signal is clearly distinct and does
`not overlap with curves corresponding to surrogates. Since
`the PPG data and surrogates are clearly distinguishable, the
`null hypothesis of a linear Gaussian stochastic process should
`be rejected. Additionally, we may suggest that there is no
`significant effect of Gaussian noise, which PPG may contain,
`based on results of DNP.
`
`4. Discussion
`
`The objective of this study was to provide evidence that
`the PPG signal, obtained from healthy human subjects, is con-
`sistent with the definition of chaotic motion given by Thomp-
`son et al. [20] and conduct a comprehensive study of the
`chaotic characteristics of PPG dynamics. Produced by sophis-
`ticated mechanisms of the cardiovascular system, the PPG
`signal is still not fully understood. In the in early 90s PPG was
`claimed to be chaotic based on time-delay reconstruction, LLE
`and CD results [13,14], which were previously believed to be
`sufficient to identify chaos. However, many studies subse-
`quently showed that these measures can be misleading and
`provide false evidence of chaos in the case of, for example,
`noise contaminated experimental data [8,9,15]. In addition, it
`is well known that physiological data are inevitably contam-
`inated by environmental noise and movement artifacts. And
`the same measures were used in application studies inves-
`tigating the dependence of PPG characteristics from human
`subject’s mental and physiological conditions [5,13,14].
`In an attempt to obtain reliable results, we have applied
`various methods of NTSA, including classical and widely used
`ones—power spectrum, time delay embedding, and largest
`Lyapunov exponent and have conducted deterministic non-
`linear prediction, Poincaré section, Wayland test and sur-
`rogation, results of which are not only considered to be
`
`more reliable in the case of noise-contaminated data, but can
`also provide new insights into the chaotic properties of PPG
`dynamics.
`Additionally, to study in more detail trajectories’ evolving
`process, we have directly investigated trajectories spreading
`in phase space. An arbitrary chosen fixed point on an attractor
`and its nearest neighbor were selected as the starting points
`of two trajectories. Fig. 10 shows the separation of these two
`trajectories based on the example of 140 and 300-step long
`segment for PPG time series (left side) and for Rossler’s single
`band chaos with 7% additive noise (right side), in both cases
`the trajectories evolve clock-wise. Star and circle markers are
`corresponding to trajectories originating from an arbitrarily
`chosen fixed point on an attractor and its neighboring tra-
`jectory, respectively, where red color indicates the starting
`and black color the ending points of the segment. The dis-
`tance between trajectories increases immediately with in-
`creasing evolution time step, however later trajectories are
`getting closer and remain in the bounded region. Obtained
`PPG trajectories are continuously bending due to that fold-
`ing is realized. In this sense PPG’s attractor seems to belong
`to the same folded band topological group as Rossler’s single
`band chaotic attractor, rather than the Lorenz group, in which
`folding is produced by splitting and layering [20].
`The investigated data primarily demonstrated