Detection of weak chaos in infant respiration - Systems ... - IEEE Xplore

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 31, NO. 4, AUGUST 2001

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Correspondence________________________________________________________________________ Detection of Weak Chaos in Infant Respiration Joydeep Bhattacharya

Abstract—This paper concerns the application of newly developed methods for decomposition of an infant respiratory signal into locally stable nonsinusoidal periodic components. Each estimated component has dynamical variation in its three periodicity attributes, i.e., periodicity, scaling factors, and the waveform or pattern associated with the successive segments. Earlier, it has been reported with the application of conventional surrogate analysis and with the cylindrical basis function modeling that the underlying system is distinctly different from linearly filtered Gaussian process, and most probably the human respiratory system behaves as a nonlinear periodic oscillator with two or three degrees of freedom being driven by a high-dimensional noise source. Here, the surrogate analysis is extended and four new types of nonlinear surrogates have been proposed, which are produced by randomizing one or multiple periodicity attributes while preserving certain individual relationships. In this way, a new type of dissection of dynamics is possible, which can lead to a proper understanding of couplings between different controlling parameters. Index Terms—Chaos, infant respiration, nonlinearity, periodicity, surrogate data, SVD.

I. INTRODUCTION This paper presents a study of infant breathing using new empirical nonlinear modeling techniques. Traditional models incorporating an oscillatory driving signal, a group of neurons, or a cerebral control center was first introduced by van der Pol [1] and then later generalized by Fitzhugh [2]. Delay-differential general equations [3] have been shown to exhibit oscillatory features qualitatively similar to respiration. A variant of this model including the activity of the brain respiratory center produces certain pathological periodic breathing such as Cheyne–Stokes rhythm [4]. Classically, any model for the control of respiration should account for two important behaviors: 1) rhythm and 2) pattern generation. Primarily, a linearly coupled model can be assumed where one assumes the existence of discrete pacemaker cells with intrinsic activity that drives other respiratory neurons which are organized by a pattern generator. A more complex model [5] implies that networks of cells with oscillatory behavior may self-organize or be guided by a pattern generator. All these models are based on equations governing various physical processes. However, the respiratory system, its neuronal control, and the interaction of many internal and external forces make the scenario more complex than any of these models. On the advent of chaos theory [6], [7], to analyze complex looking irregular phenomena by a system with low degrees of freedom, there is a plethora of publications on the applications [8]–[11] of dynamical systems theory in general, and correlation dimension specifically, to physiological systems. Recent studies [12]–[14] suggest that respiration is chaotic. Most of these studies are dependent on estimation of the absolute nonlinear invariant measures such as correlation dimension [15] and Lyapunov exponent [16]. Compared to the controlled synthetic data series, physiological signals pose several difficulties, such as presence Manuscript received August 19, 1999; revised March 5, 2001. This paper was recommended by Associate Editor P. Willett. The author was with the Max Planck Institut für Physik Komplexer Systeme, Dresden, Germany. He is now with the Commission for Scientific Visualization, Austrian Academy of Sciences, Vienna, Austria (e-mail: [email protected]). Publisher Item Identifier S 1083-4419(01)05983-0.

of patches which cause nonstationarity, contamination with measurement noise and dynamical noise, etc., for the successful estimation of these invariant measures [17]. Moreover, it is now well recognized [18], [19] that certain stochastic processes with power law power-spectra can also produce a finite correlation dimension which can be erroneously attributed to low-dimensional chaos. Thus, when these standard nonlinear measures are directly applied to physiological data without considering these problems, spurious results are the rule rather than the exception [17], [20]. Under this constraint, a more realistic and reliable approach is to establish initially the inherent nonlinearity of the underlying physiological systems since nonlinearity is a necessary although not sufficient condition for chaos. The concept of surrogate data has been introduced by Theiler et al. [21], where the experimental time series competes with its linear stochastic counterpart. In other words, the method of surrogate data provides a regime to test specific null hypotheses based on which the surrogates are generated. On generation of suitable surrogates, both the original and the surrogate data are characterized by some suitable nonlinear test statistic. If the result for the original data is explicitly different from the outcome of the surrogate data, the associated null hypothesis of a linearly filtered Gaussian process could be rejected with statistical significance. Through this approach, certain premature claims for chaos can be correctly identified [22], [23]. Most of the available surrogate generators are largely nonparametric and concerned with a linear null hypothesis, i.e., a given data set is the outcome of a linear system. However, physiological systems are complex and nonlinear by definition [24]. Therefore, cancelling only the hypothesis of linearity is too wide to characterize the underlying systems. Parametric- and nonlinear-type surrogate generators have been introduced [25], [26] recently in the analysis of infant respiratory data. Here, the data is modeled using a radial basis modeling algorithm [25], [27], and surrogates are generated based on the null hypothesis of a noise driven nonlinear systems. There are very few attempts [28]–[30] where the surrogates are generated by randomizing some structures of the signal in ad hoc fashions. Recently, a more superior method of surrogate generation is proposed [31] in the context of dynamical systems, where a Markov model is fitted to coarse grained dynamics obtained by quantizing the two-dimensional (2-D) delay vector distribution from the data. The method tests whether certain number of unstable periodic orbits is generated by dynamical structure or not. To describe a complex dynamical system as nonlinear or chaotic may not be fully helpful if the constituent components producing the overall irregular structure cannot be identified. It is more important to classify these components rather than to broadly classify the system as nonlinear. In a nonlinear system, the interactions between different subsystems is often quite complex so proper identification of the coupling can characterize the underlying dynamics in a new perspective. We therefore investigate the structure of the infant respiratory data in greater detail. A new method [32], [33] is used here to decompose a complex looking irregular oscillatory time series into constituent nearly periodic time series. Any periodic or nearly periodic time series can be precisely defined in terms of three basic features or periodicity attributes, namely 1) the periodicity (or period length); 2) the pattern (or waveform) over the successive repetitive segments; 3) the scaling factors (or the amplitudes) associated with the successive pattern segments.

1083–4419/01$10.00 © 2001 IEEE

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The adopted method estimates these three periodicity attributes or parameters from the signal on an adaptive basis. Here, a sliding windowbased estimation is performed to accommodate the nonstationarity of the respiratory data. Shuffling in the individual parameter domain creates nonlinear surrogates which retain the overall noisy limit cycle behavior of the original data. These surrogates are intrinsically nonlinear because limit cycle-type behavior cannot be produced by either linear systems nor by the measurement process. Four classes of new surrogates of this kind have been produced and their correlation dimensions and predictability are statistically compared with the estimated series. It has been found that the infant breathing is clearly inconsistent with a linear Gaussian process and has significant dynamical structure spanning more than one cycle. The nature of hidden mutual coupling between these three periodicity attributes which constitute the irregular signal are also discussed. II. DECOMPOSITION OF THE IRREGULAR SERIES In the study of rhythmicity in physiological systems, three critical questions must be addressed: 1) whether the process under investigation is dominantly cyclic, and if so, 2) what is the best estimate of the period of the oscillation, and 3) what does the associated component look like? Fourier spectral estimates and to a lesser extent, the autocorrelation function are the primary tools to detect the hidden periodicities in time series. However, these methods are not suitable for a nonlinear process with dynamical characteristics [17]. In particular, when the pattern is nonsinusoidal, Fourier-based methods are not appropriate from the practical point of view [34]. A reduced autoregressive model based on an information theoretic criterion has been recently proposed [35] to detect latent periodicity in a time series. However, all these methods for periodicity detection are unable to answer the three important questions raised above in a unique and composite fashion. Here, we employ a recently developed algorithm for the nonsinusoidal dominant periodicity detection which is followed by the estimation of the associated component. The detailed implementation procedure and its robustness against noise has been presented elsewhere [32]. A very brief outline is sketched here for the sake of continuity. Consider a scalar time series fx(k)g; k = 1; . . . ; L. A matrix n is formed as follows:

A

An =

x(1) x(2) 1 1 1 x(n) x(n + 1) x(n + 2) 1 1 1 x(2n) : .. .. .. .. . . . . x (n(m 0 1) + 1) x (n(m 0 1) + 2) 1 1 1 x(nm) (1)

A

The singular values [36] i ; i = 1; . . . ; r; r = min(m; n) of n are computed and the ratio = 1 =2 or = 12 = pi=2 i2 is evaluated.1 The resultant spectrum of versus n can be termed as the p-spectrum [32]. Any dominant periodicity of period N will produce large peaks around n = N and its higher integer multiples since at these values of n, the matrices n are more close to rank one than for other values of n. Once the periodicity (e.g., N ) is identified, we find the singular value decomposition of the matrix N as follows:

A

A

An = U

V T:

(2)

1To reduce the computational load, one can use the former index where only two singular values are required. In this paper, the latter index is used since it has more direct physical interpretation.

u v u v

The time series configured from the matrix 1 1 1T ( 1 ; 1 are the first column of , and , respectively, and 1 is the first element of ) represents the associated most dominant periodic component of periodicity N while fulfilling the least squares criterion of having maximum energy of the set of all possible combinations of time series of periodicity N that can be extracted from the time series fx(k)g. Here, 1 represents the pattern or shape of the waveform and the series 1 1 will give the amplitudes or multiplicative scaling factors for the successive periodic segments. This method is completely data adaptive (no prior information of the pattern is assumed unlike Fourier-based approaches) and extracts an amplitude modulated component. Since the physiological signal, like a respiratory signal, is quite nonstationary, all three periodicity attributes may vary with time. In order to capture local variations, a sliding window based periodicity detection and subsequent extraction of components is used. The necessary details of data segmentation have been relegated to [32]. The size of the data window is not set to have a constant number of data points; it depends on the most recently detected dominant periodicity. Generally speaking, if Ni is the periodicity in the ith data window then the length of the (i + 1)th data window is mNi ; two successive data-windows overlap over (m 0 1)Ni data points.2 In each window, the dominant periodicity is detected through the p-spectrum and the associated periodic component is extracted. Only the first period is separated, and the rest is forwarded to the next window. Finally, we adjoin successive unit periodic segments from the successive data windows to form the resultant component. This is actually composed of different segments having different periodicities, different patterns, and different scaling factors over the consecutive segments. Therefore, this component possesses the dynamics interwoven with frequency and amplitude modulation. This kind of decomposition is of help here since it is well recognized [37] that respiration in quiet sleep exhibits cyclic amplitude modulation indicative of substantial structure in breath-to-breath variability.

U

V

v

u

III. ANALYSIS WITH SURROGATE DATA It has been already mentioned that it is very difficult to tell if a dynamical behavior is chaotic; especially for the physiological signals which are notoriously nonstationary and contains considerable amount of noise, proving of chaos is almost an insurmountable task [20]. Therefore, a much less ambitious task than proving the presence of low-dimensional chaotic attractors in respiratory data signal is to detect the inherent nonlinearity. The introduction of surrogate data helps us to discriminate linear dynamics and weakly nonlinear signatures—strong nonlinearity is more or less easily detectable [38]. The two most popular surrogate generators, the unwindowed Fourier transform surrogate (UWFT or FT surrogate) and amplitude adjusted Fourier transform surrogate (AAFT surrogate) described by Theiler et al. [21], test the null hypothesis of linear transformation of linear filtered noise, and of monotonic transformation of linearly filtered noise, respectively. Rejection of these two hypotheses should not necessarily be attributed to the inherent nonlinearity [26]. If any linearity system is subjected to nonmonotonic transformation or noninvertible observable, the inherent linearity can also not be truly identified. Schreiber and Schmitz [39] proposed a modified and iterative version of AAFT (which can be called IAAFT) such that the surrogate and the original data will have identical probability distributions and Fourier spectra (thus autocorrelation, too) in an asymptotic sense.

m= N

2In this paper, 5 is used, which is not a limitation. Ideally speaking, if is the dominant periodicity present in the data, then the window length should be at least 2 long to produce the first fundamental peak in the p-spectrum and 4 points are needed to obtain the second peak which is required for confirmation of the presence of dominant periodicity.

N

N


Fig. 1. Uppermost figure on the left shows abdominal movement for an infant in quiet sleep. The vertical axis is in arbitrary units (proportional to cross-sectional area measured by inductance plethysmography). The component estimated from the above data series through sliding window-based periodic decomposition is depicted in the second row. Three conventional linear surrogates (UWFT, AAFT, and IAAFT) and their phase space plots [x(k ) versus x(k + 2)] (on their right side) are also shown. It is clearly explicit that the linear surrogates are unable to preserve the noisy limit cycle structure of the original signal.

Strong oscillation [Fig. 1] is visually explicit in the respiratory signal. The problems with the available surrogate data for periodic processes (or systems with long coherence time) are discussed in [40] and [41]. Ideally, the surrogate of a periodic time series should be another periodic time series [23]. In other words, if the original time series produces a limit cycle behavior in phase-space, surrogate data should also correspond to a limit cycle after phase randomization. Time series producing noisy limit cycles cannot be described by a linear stochastic process, even observed through a static monotonic nonlinear transformation. A more realistic approach to the surrogate generation of oscillatory time series is introduced in [29], and it tests the hypothesis of temporal correlation between cycles; here the surrogates are generated by shuffling the complete cycles, but the separation of individual cycles remains a difficult task. In Section II, the method to estimate an oscillatory time series from the original time series is outlined. This oscillatory time series has three controlling parameters known as periodicity attributes, and these are constantly varying. The resultant time series is the outcome of the complex interaction between these dynamical parameters. Let us hypothetically assume that these three parameters (periodicity, scaling, and block pattern) are actually generated by three individually different sub-systems P , S , and BP , respectively. In this parametric regime, we propose the following four nonlinear surrogates using a Monte Carlo approach in the individual parameter domain. Surrogate-1: Here, we shuffle only the amplitude or multiplicative scaling factor sequence while all other information remains intact. This surrogate tests for the determinism in the amplitude sequence. Here,

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the null hypothesis is that there is no deterministic contribution by the system-S to the overall deterministic structure. Surrogate-2: It is created by shuffling the temporal sequence of block normalized pattern while maintaining the pattern-period relationship. Scaling sequences are undisturbed. Here, we are concerned with the determinism in the temporal sequencing of the pattern and thus test the null hypothesis that information is contained in the deterministic coupling between systems P and BP whereas S remains undisturbed. Surrogate-3: This is the same as Surrogate-2 but with shuffling in the scaling sequences too, thus destroying any temporal information of scaling factor sequence. Here, the associated null hypothesis is that the dynamics are dependent only on the mutual coupling between P and BP , and S has no contributions to the overall dynamics. Surrogate-4: Shuffle all the attributes while preserving their mutual interconnectivity. Here, we randomize the complete cycle but preserving the total coupling between systems P , S , BP . This surrogate is very close to Theiler’s proposed cycle shuffled surrogates [29], but here, one has greater control over individual cycles. Once the surrogates are generated, the next step is to compare these surrogates with the original data series through a statistic Q. A suitable statistic should provide a nontrivial measure of the underlying dynamics and should be independent of the way the surrogates are generated. The associated null hypothesis is rejected if the observed value of Qd for the data is statistically different from the set fQs g for the set of surrogates. The level of significance (S ) is computed as p S = jQd 0 Qs j= & , where Qs and & are the mean and variance of fQs g, respectively. Assuming that fQs g has normal distribution, the associated null hypothesis can be rejected for two-sided testing with 99% level of confidence when S > 2:33 [21]. The correlation dimension (dc ) [15] and simple nonlinear prediction [42] are employed here as test statistics. First, the phase space is reconstructed by the time delay embedding technique [43]. From the scalar time series fx(k)g, a vector time series is constructed via x(k) ! xn := [x(k); x(k 0 ); x(k 0 2 ), . . . ; x(k 0 (d 0 1) ], where is the time lag, and d is the embedding dimension. The most widely used method [15] estimates correlation dimension by calculating the density of the reconstructed attractor in the phase space. The correlation integral is defined as

( ) = L1 (L21 0 1)

C r

where

L

L

=1 j =i+w

2 (r 0 kx 0 x k) i

j

(3)

i

number of phase space vectors (=L 0 (d 0 1) ); heaviside function [i.e., 2(r) = 0; r < 0; 2(r)=1; r > 0); w Theiler’s window [44], aimed at excluding from the computation those nearby state vectors which are temporally correlated due to oversampling. It is assumed that the correlation dimension (dc ) is related to the correlation integral by C (r) / rd . It should be stressed here that we are not giving the absolute importance on the dimension value; rather, the value obtained from the original signal is statistically compared with the values obtained from the surrogates. For this reason, a less stringent criterion of scaling has been imposed in the dimension estimation. The time lag ( ) is set to the first zero-crossing of the autocorrelation function [17]; w is set to two times the correlation time of the signal [23], and the embedding dimension (d) is set to 10, since this algorithm is mostly suitable for a low-dimensional system (dc < 5) [45]. Prediction is a sine qua non to determinism. The method of simple local nonlinear prediction [42] is implemented here. In the reconstructed phase space, for any reference vector xn , a number M of mutual neighbors are considered. Let rn; j ; j = 1; . . . ; M denote L1

2

640


Fig. 2. Global variation of amplitudes or scaling factors, period and the associated pattern or wave-shape, respectively. The set of patterns is normalized to have same length and it is clear that the waveshapes are not sinusoidal in nature.

their time indices. Then, using a zeroth- order approximation locally, the k -step ahead prediction will be

+k

x ^n

=

1 M

M

j

=1

x(rn; j

+ k):

(4)

The forecast error normalized to the variance of the data is studied here. To calculate this statistic, one can choose directly the same reconstruction parameters as used in the estimation of correlation dimension. However, in this context, it was argued [46] to be advantageous to choose parameters in order to obtain a small spread of prediction error, even if at these parameters, the embedding is not strictly valid. For the reconstruction procedure in this calculation, we choose an embedding dimension of four, and time delay of three time samples. The value of M is set to 30, and the prediction horizon is equal to 30 time samples. Due to the computational burden, we cannot explore all the possible combinations, but we hope that the chosen parameters are not far from the optimal.

Fig. 3. Time series of four nonlinear surrogates generated by shuffling individual periodicity attributes. For details, see Section III. Note the obvious similarities between all these nonlinear surrogates and the estimated component (Fig. 1). Two-dimensional x(k ) versus x(k + 2) phase space diagrams are also plotted where the limit cycle behavior has been produced by all four surrogates. TABLE I LINEAR AND NONLINEAR SURROGATE ANALYSIS OF ESTIMATED COMPONENT. VALUES OF S -STATISTIC HAVE BEEN LISTED FOR DIFFERENT SURROGATES

IV. DATA With the help of noninvasive inductance plethysmography, measurements proportional to the cross-sectional area of the abdomen of infants during natural quiet sleep were collected. The infants were all healthy, and their parents were informed of the procedures and its purpose and had given full consent. Five cases are reported here. Data were initially sampled at 50 Hz using a 12-bit A/D-converter and later downsampled again by four to ease the computational load. Sleep stages were determined using conventional polysmnogrophic criteria [47]. All other details about the recording and preprocessing of this experimental data are described in [48]. V. RESULTS AND DISCUSSIONS Fig. 1 shows a section of the time series obtained from one healthy infant in sleeping condition. An initial window length of 100 data samples, which corresponds to time span of 8 s, has been chosen for the estimation procedure. As described in Section II, successive windows are not of equal length; rather the length is completely adaptive based on the most recent periodicity detected. The resultant estimated component has been displayed in Fig. 1. The associated three time varying periodicity attributes are depicted in Fig. 2. Although the original time series looks quite innocent at first sight, the detailed variation in each individual parameter shows irregular dynamics acting at a finer level. Fig. 1 shows the state space plot in two dimensions. It is clear that the estimated component captures the dynamics of the raw data macroscopically. The time series of three conventional surrogates (i.e., UWFT, AAFT, and IAAFT) of the estimated data are displayed in Fig. 1 along with their phase space plots. From direct visual inspection, it is quite obvious that these surrogate series possess possibly different dynamics from the raw data. All these surrogates have beating phenomena, which

is distinctly absent in the raw data. Typically, this is a problem of the application of the FFT algorithm used for the surrogate generators especially for cyclical time series [40]. This fact is more prominent in their phase space plots; the original (noisy) limit cycle3 of the data is completely destroyed by these types of surrogates. The series of four nonlinear surrogates produced by shuffling different periodicity attributes are shown in Fig. 3 All these four surrogates resemble the data in the time domain as well as in the phase space domain, but smearing of data points is seen in the state space plot. These new surrogates are free from the waxing and waning phenomena, unlike conventional linear surrogates. For each data set ten surrogates according to each of the seven surrogate generators (three linear surrogates and four nonlinear surrogates) are generated. Table I lists the values of S for all types of surrogates. For all the linear surrogates, the value of S is found to be much larger than 2.33, thus rejecting the null hypothesis of linear correlated noise subjected to a static and monotonic nonlinear transformation. This is also supported by earlier report [26] on these data. The only difference is that we are comparing the estimated component rather than the raw time series itself. Next, we focus on the outcome of new surrogates 3The original phase space plot is not strictly a limit cycle, rather similar to that. It is clear that periodicity attributes are not constant throughout the process evaluation, thus causing the deviation from strict limit cycle-type behavior.


TABLE II SURROGATE ANALYSIS FOR THE RESIDUAL ERROR. HERE S -STATISTIC IS SHOWN ONLY FOR IAAFT SURROGATES

which retain the overall phase space structure. For three infants, the null hypotheses associated with all the four surrogates are found to be incompatible with the estimated component. Rejection of the first three new hypotheses would suggest that the deterministic structure present in period/pattern/amplitude is interrelated and not simply a result of determinism in amplitude or in period or in pattern alone. Failure to reject all four hypotheses tests would suggest that it is likely that there is no determinism in any of these components. For the three infants (I1, I3, and I5), all the four new surrogates are rejected with more than 99% statistical level of significance, thus implying that there exist important contributions from all these hypothetical subsystems (B , P , and BP ) and their mutual coupling is not trivial. However, for two infants, (I2 and I4), we find that the null hypotheses associated with first three surrogates cannot be rejected although they possess nontrivial temporal correlation between cycles leading to the rejection of surrogate-4. These discrepancies are rather awkward since the infants reported here are all healthy. This apparently embarrassing fact prompt us to test the error series obtained in the modeling for any interesting structures left out. Table II lists the S values obtained through the comparison of the dimension values of the residual error and their IAAFT surrogates. It should be stressed that no such new type of surrogate can be constructed from the residual series due to the lack of presence of any periodic component(s), which can be detected through the adopted method irrespective of how small the window size may be. It is quite interesting to note that residual error series of only those two infants who accept the first three null hypotheses, clearly show that nonlinear structure is still remaining in it. This fact is noteworthy since the estimated component seemingly captures the entire dynamics which is also reflected in terms of energy and their similarity of phase space plot, but it has been found that nontrivial dynamics is still contained in some of the error series, especially when the newly proposed null hypotheses have been failed to be rejected. For some surrogates, we find that their dimension are even less than that of raw data itself, even though they are created with randomly shuffling the periodicity attributes. This may be because the structure of the respiratory signal depends to some extent on long-range correlations over time or the global coupling between these attributes. The whole study is repeated using the simple nonlinear prediction error as a second discriminating statistic. Since all these new type of surrogates are not completely random and their phase space trajectories repeat within bounded region, some degree of determinism are present in these surrogates. Table III lists the S values by comparing the normalized prediction error of the original signal and the surrogates. We choose the prediction horizon to be 30 since it is greater than the average period; thus, the prediction performances beyond one cycle can be compared. Overall, the results are in accordance with Table II. The profile of S versus k for one infant (I5) has been shown in Fig. 4. It is very interesting to note that for surrogate-1, S drops to almost zero for k = 14, while later on, it is again above the level of significance. The reason for this sudden drop is not very clear and it warrants further investigations.

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TABLE III

S

= 30 IN (1)] OBTAINED BY USING THE NONLINEAR

VALUES [FOR k PREDICTION

ERROR AS A DISCRIMINATING STATISTIC

Fig. 4. Prediction performances in terms of S score for the infant I5. For the estimated component, it is shown for five types of surrogates: IAAFT (line with “ ”), Surr-1 (with “o”), Surr-2 (solid), Surr-3 (dotted), Surr-4 (with “3”), respectively. S values for the residual component (dashed) are computed by comparison with their IAFT surrogates. The horizontal line denotes the level of significance (S ) above which a null hypothesis is rejected. :

+

= 2 33

VI. SUMMARY In this paper, an extension of surrogate generation technique has been suggested where the signal contains dominant periodic components. The regular breathing of infants is studied. Through linear nonparametric surrogate analysis, the underlying nonlinearity is established, and it is quite explicit that linear filtered Gaussian processes with an invertible static nonlinearity are not the appropriate model for the underlying system. The fact is evident even visually which is reflected by the “hole” in the phase space plot of the signal. The nonlinear dynamics is further dissected through proposing new types of nonlinear surrogates. It has been shown that these surrogates do not distort the noisy limit cycle-type structure of the signal (i.e., they preserve the hole in the phase space), and they are generated by destroying one/or more mutual coupling between different periodicity attributes. It can be conjectured that the regular quiet breathing from healthy infants are possibly “very weak” chaos. Using the same data set, it was reported [37] that the dynamics of infant breathing during quiet sleep is similar to a complex system comprised of large-scale low-dimensional and relatively small-scale high-dimensional dynamics. The dimension values of the signal lie between two and three. This system with apparently dominating two degrees of freedom is mainly due to the inspiration-expiration cycle along with the breath-to-breath variation. In all cases, it has been found that there is significant deterministic coupling between the period of breathing, their amplitude, and the breathing pattern. Further, the predictability power of the signal is also higher than these nonlinear surrogates even for higher prediction horizons, which indicates the presence of deterministic structure. This present study seems promising and may provide a clearer insight into the respiratory dynamics although it will

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