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Analysis of Physiological Time Series UsingWavelet Transformsime-series analysis is the basis for char-Tacterization and modeling of an ob-

served system. The aim of this analysis isthe prediction of the short-term evolutionof a dynamic system and the developmentof a model that capture features of itslong-term behavior. In this article, wepresent a a method of time-series analysisthat uses wavelet transforms to analyzeheart rhythm, chest volume, and blood-oxygen saturation data from a patient suf-fering from sleep apnea.

Time-Series TechniquesThe start of modern time-series tech-niques might be set at1927,when Yule in-

vented the autoregressive method. Hismethod predicted the next value as aweighted sum of previous values. Around1980, an important development oc-curred: Takens[11proved that it was pos-sible to build a dynamic system with aunique variable and state-space recon-struction with time-delay techniques. Hedrew on ideas from differential topologyand provided a technique for recognizingwhen a time series can be generated by de-terministic equations.

From these techniques, it is possible tocalculate the Lyapunov exponents [2],which are ameasure of the speed of loss ofsystem information and are directly re-lated to the possibility of predicting if asystem has any exponent greater thanzero. From this analysis, it can be con-cluded that the dynamic system is chaoticand, hence, will loss information in time.In other words, the Lyapunov exponentcan evaluate how many future events canbe predicted from the current information.In spite of Takens theorems, this tech-nique is only valid for series that are with-out noise and are stationary in time andthus is not suitable for the type of seriesdiscussed in this work.We present another point of the viewthat is based in the signal time-frequencycharacteristics. The principal idea is tobuild a tool that can evaluate the order ofany system or the loss of this order as afunction of its dynamic changes. The mostpopular way of performing frequency

Aleiandra Figliola and Eduardo Serrano2lnstituto de Calculo,

Departamento de Motematica,Ciudad Universitaria, Buenos Aires

analysis has been the Fourier transform(FT) using the fast Fourier transform(FFT) algorithm. However, the FFTmethod assumes the signal to be station-ary and is thereby insensitive to its vary-ing features.

Time-localization can be achieved byfirst sliding a window along the signal andthen taking theFTover each interval. TheGabor transform (GT), which uses aGaussian window function, is often ap-plied in signal processing.

The principal problem of this methodis that its window width is fixed. Thus,Gabor analysis is badly adapted to signalswhere pattems with different scales ap-pear, and it poorly resolves short-timephenomena associated with high frequen-cies [3].The wavelet transform (WT), analo-gous to the GT, provides a similar time-scale description and can decompose anysignal into frequency bands. The WT useswavelet functions that have time-widthsadapted to each frequency as windows:window width is very narrow for high fre-

quencies and wider for low frequencies.As a result, the WT is better than the GT(or windowing FT) for expanding zoneswhere the series has very high frequen-cies, such as transients, and it can analyzeevents that occur at different scales. Anextensive bibliography on this subject canbe foundinthe literature [3-61.

We apply the discrete wavelet trans-form (DWT), using asplineasthe waveletfunction. From the wavelet coefficients,we calculate the energy distribution on thefrequency spectrum. Also, we calculatean information cost function (ICF), whichis the information associated with the en-ergy distribution of the system[7].The data analyzed consist of a multi-channel physiological time series of a pa-tient who suffers from sleep apnea(periods in which he takes a few quickbreaths and then stops breathing) [l ,21.These series are nonstationary becausetheir statistical moments (means and vari-ances) are not constant over time. Further-more, overlapping transient waveformswith unpredictable arrival times often ap-pear. A common problem is to extract theparameters of interest when they involvejoint variations of time and frequency.The ICF is an entropy-like function thatgives estimate of the disorder of thesystem[7,10].Our work presents ameth-odology to obtain a characterization for

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Heart Rate

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-40u2 3 5 7 8Minutes(b )I o3 Oxygen S aturation3.4r Ic.33.23 00 2 3 5 7 8Minutes(C)1. Temporal series of (a) heart rhythm,(b) lung volume variation, and (c) oxy-gen saturation for pre-apnea and peri-odicbreathing.

2.Temporal series for regular stateof(a) heart r hythm, (b) lung volume varia-tion, and (c)oxygen saturation.

nonstationary noisy series. It is thus possi-ble to use the ICF to characterize a systemeven when it is impossible to calculate itsLyapunov exponent.

The Data SetThe data set is a multivariable physio-logical time series, consisting of 4 hoursand 43 minutes of simultaneous heart rate,lung volume change, blood-oxygen satu-ration, and electroencephalogram(EEG)

state. The data were recorded over some5hours from a 49-year-old male in theSleep Laboratoryof Boston's Beth IsraelHospital [S,91.The signals were recordedwith a multichannel instrumentation re-corder and were subsequently playedback and digitized at250Hz.

This patient suffered from extremedaytime drowsiness, a result of sleep ap-nea [ l l , 121. When he starts to fall asleep,he stops breathing; his blood-oxygen con-

centration decreases and carbon dioxideincreases. This increase in carbon dioxidein blood that perfuses certain centers inthe brain results in areflex arousal, and thepatient takes some short, quick breathsand awakens. This process is repeated,which is a pattern called Cheyne-Stokesbreathing.

We will briefly describe the series re-corded: A single channel electrocardio-gram (ECG) was recorded andpreprocessed to determine the times atwhich theR wave (electrical depolariza-tion of the ventricles) appeared. TheR-R' interval was interpolated and sam-pled every 0.5seconds to synthesize aninstantaneous heart rate signal, where thedata correspond to the inverse of theR-R' interval, a measure of cardiacrhythm [9].The second series recordedwas the flux of air flow through the nose,a linear function of lung volume varia-tion. Finally, arterial blood-oxygen satu-ration was measured by pulse oximetry,which is based on the colorof the hemo-globin, which varies with the quantity ofbound oxygen. Figures 1 and 2 show theseries at different states.

Wavelet AnalysisAs discussed above, the FT and GTpresent difficulties for the analysis of non-stationary series or when there is a combi-nation of short- and long-timephenomena. Wavelet analysis (WA)givesus a powerful tool to confront verydiverse problems in applied sciences orpure mathemathics. The wavelet is asmooth and quickly vanishing oscillatingfunction with good localization both infrequency and in time. It can be inter-preted as single signals, or atoms, of shorttimes with an oscillating structure. Figure3 shows a typical wavelet function. Awavelet family, is a set of waveletsgenerated for dilations and translationsofan unique mother wavelet,v(t):

where a is the scale parameter, b is thetranslation parameter, and ti s the time. Asa increases, the wavelet becomes morenarrow.Thus,we have a unique analytic pattemand its replicas at different scales and withvariable localization in time. Given afunction, f , the different correlations,


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WT synthesizes the numerical informa-tion thus obtained. From a different view-point, the wavelets of a family play therole of elemental functions, representingthe function as a superposition of thewavelets correlated with the function fordifferent scales (different as). Thismakes it possible to organize the informa-tion in some particular structure-to dis-tinguish, for example, trends or the shapeassociated with long scales of the local de-tails from correspondingshoi-tscales.The CWT off(t) is defined as the corre-lation between the function (t) with thefamily waveletUT,,[>for eacha [3]:

Pre-apneaPeriodic BreathingRegular State

dt=V.,,(t)> (2)For special selections of the function,andadiscrete netofparameters,uj =2-/

andb;,k =2-/k, with , k E and the scale2-J ,giveus theshift parameter. The sub-family

(3)constitutes an orthonormal basis (ONB)of L2(R)hat is the space of the functionswith finite energy or the functions of inte-grable square. In this way, we can obtaindiscrete transformations, and it is possibleto expand the function in series ofwavelets. Then, we can join the advan-tagesof the WT with the atomic decom-posingoff(t).

The DWT associated withu/ is simplyseen asarestriction of the CWT at the pa-rameters aj ,b;, k. he wavelet coefficients(WC) are:

yJ J , k =2J 2yJ (2J t-k)

,,k w(,, b,,k) = (4)The functionf can decompose as:

Heart Rhythm Chest Volume Oxygen Saturation0 ,83 0 ,63 0, 720,72 0,44 0,700,72 0,47 0, 70

The analytic properties of the DWT areanalogous to the CWT. From the WC, thesum E = ,


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1 -0.8 fI I I I I I I I I I I0.0 0. 2 0.4 0.6 0.8 1 o 1.2Samples(4

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0 I I I I I0. 0 0.1 0.2 0.3 0.4 0.5Frequency (Hz)

(b )

3.Wavelet spline functiony~(a) and the amplitude and phase of the Fourier trans-form of the wavelet@ (b)used in this work.

where the sum is interpreted as zero forany q =0. The ICF evaluates the infor-mation associated to the energy distribu-tion of the system. The ICF is anentropy-like function that is easy to calcu-late and gives a good estimate of the de-greeof disorder of a system. To test theICF function, we take as a temporal series,one variable of the simplest possiblemodel for forced nonlinear oscilla-tions-the Duffing oscillator [131. For pa-rameters where the system has chaoticbehavior, we compare the maximumLyapunov exponent,h,with the ICF func-tion for the Duffing system. Figure 4shows the maximum Lyapunov exponentas a function of the ICF function. We havefitted the pairs of points with quadraticfunctions at different chaotic states of theDuffing oscillator (different routes tochaos) [7]. In all cases, the best fit is by aquadratic function. In this case the rela-tion is:

ZCF =16.39x hz-1.74 x h+6.16 (9)with a correlation coefficientR2=0.992.We should say that these are experimentalnumeric results.

Application s to the SignalsFrom the total multichannel data se-

ries, we select two different conditions:one corresponding to the transition fromapnea to periodic breathing; the other nor-mal behavior when the patient is awake.Figures 1and2 show the data set of theseconditions for the three variables: cardiacrhythm, variation of lung volume, andblood-oxygen saturation. We computenearly 2-5 wavelet coefficients, CJ ,k, foreachj=-1, ...,- logfl1,whenNis thetotaldataof the section series that we want toanalyze. We use the Mallat algorithm fortheCJ ,kcalculations[14].

The coefficients achieve local infor-mation and details of the signal and giveus a time-scale representation. Figure 5displays the temporal evolution of thewavelet coefficient of the series for thehigh-frequency band. From Eq. (6), wecalculate the energy distribution as a func-tion of the frequency level, , or both phe-nomena. Tables 1, 2, and 3 give acomparison of the percentage of energydistribution for the different frequencybands for the three series, correspondingto pre-apnea, periodic breathing, and theregular state.Note that the pre-apnea series has anenergy spectrum function that is wideband. When apnea is present, the major

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part of the energy is carried by one or twoprincipal frequency bands (depending ofthe series). At the start of the apnea, it ap-pears that the energy is localized aroundthe same principal frequency bands. Forthe regular state, the major part of the en-ergy is carried in the frequency band cor-responding to j =-2, for typical timesbetween2 and4 sec. Table2shows thatthe principal bands at pre-apnea and peri-odic breathing is j = -3, which corre-spond at typical times between4 and 8sec. Also, in the regular state, the mediumand low frequencies (less than or equal toj= -4) are negligible.The cardiac spectrum also changes itsprincipal frequency bands from pre-apneaorapnea to the regular state. Table1showsthat principal frequencies for the pre-apneastate at =-3 andj =-6, and for the peri-odic breathing at j=-3 and j=-7. Theregular state shows the principal peaks tobe at low frequencies. The energy distribu-tion spectrum of the oxygen saturationvariable is similar to the cardiac distribu-tion, but also contains low frequencies.Inother words, the level corresponding toj =-3 disappears (see Table3).

As shown in [7], a function of the orderof a system (ICF) can be related to thewavelet coefficients. We calculated theICFs for the multichannel series and com-pared them before and during the apneaperiod. Table 4 summarizes the results.Note that in all cases, the ICF decreasesfrom pre-apnea statetoapnea state, whichis in accord with the wide-band spectrumof the energy for pre-apnea. In otherwords, the apnea state is a more orderedcondition than is the pre-apnea state.

It is important to note that the samestructure of the phenomena appears in allthe series. For the three series, the ICF de-creases from pre-apnea to apnea. For the

breathing series, the ICF is higher in theregular state than in periodic breathing.The other two cases, corresponding to car-diac rhythm and oxygen saturation, pres-ent similarICFs(see Table4).We assumethat the same dynamics would appear inother patients similarly afflicted and thatthis is a proper characterization of thispathological dynamic condition. At anyrate, the aim of this article istopresent analternative methodology for analysis ofnoisy and nonstationary data series.

Results and ConclusionsStudies have shown that cardiac dy-

namics are complex, and if they could bedescribed by a set of differentiable equa-tions, this would beof higher dimension-ality [15-171. To us, the difficulty inapplying this type of analysis is that thephenomenon is nonstationary and con-taminated with noise, which makes mod-eling by dynamic systems theory almostimpossible. We have taken this series notto discover the dynamicsof thepathology,but to present an example of the tools thatthe wavelet transform provides. As hasbeen shown in the tables, the DWT pro-

vides a very powerful techniquetodetectsignal changes by observing changes inthe energy spectra of the series. Also,visualization of WCs changes with timeis useful to synthesize the different dy-namics involved in the phenomenon.

Results from using the ICF can besummarized as follows: in all pre-apneastates, the system has higher ICF valuesthan in the apnea condition. This makes usbelieve that the system acquires more or-der at the start of the apnea state.

For a nonstationary noisy series withunknown dimensionality, Lyapunov ex-ponents are, inpractice, impossible to cal-culate for dynamic characterization. TheICF coefficients give us a relative valuebetween two situations, tellinguswhich isthe most ordered, or which provides moreinformation. From this knowledge, thefirst conclusion is that in all cases the sys-tem has lower ICF coefficients when theapnea crisis appears (in general, duringthe periodic breathing state). It ispossi-ble to believe that the bodys central nerv-ous system has undergone some kind ofadaptation to the new conditions imposedby the patients pathology. We could

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1 I I I I I0.04 0. 06 0.08 0.10 0.12 0.14 0.16Lyapunov Exponent

4.Comparison between the L yapunov exponent and theICF coefficient for theDuffing model, a nonlinear oscillator .i a IEEE ENGINEERING IN MEDICINE AND B IOL OGY M a d J u n e 1997


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5.Wavelet coefficients: (a) heartrhythm, (b) lung volume variation, and(c) oxygen saturation corr esponding atthe same cases presented in Fig. 1(a,band c).think that the system learns from theprevious situation and orders itself, opti-mizing certain variables. Another possi-ble interpretation from these results is thatthe system tries to organize itself, and thisis the origin of the apnea crisis. Still an-other interpretation is that the systemuses the crisis to synchronize itself at its

principal frequency. In any case, such in-terpretation belongs to the physiologists,but even without such specialists, variousconclusions can be reached.

Our method was tested with a typicalnonlinear oscillator. The Duffing oscilla-tor presents a bifurcation route to chaos.We compared the Lyapunov exponentwith the ICF coefficient and found them inagreement. Likewise, comparison of twoICF values should be made between con-ditions close in time. Our method couldalso be applied to other physiological timeseries, but its purpose here was to investi-gate a particular patient suffering with aparticular disorder (sleep apnea).

AcknowledgmentThis article was partially supported by

CONICET, Argentina.Maria Alejandra F i-gliola was born in Bue-nos Aires, Argentina onNovember 27, 1957.She received the M.S.and Ph.D. degrees inPhysics from the Uni-versity of Buenos Aires,Argentina, in 1983 and1990, respectively. Her research interests

are in the field of nonlinear dynamics,chaotic theory, and, recently, wavelet the-ory with applications in signals process-ing. Dr. Figliola is a member of theNational Research Council (CONICET)of Argentina.Eduardo P . Serranowas born in Buenos Ai-res, Argentina, in 1945.He graduated from Bue-nos Aires University,Department of Mathe-matics, in 1986, and re-ceived the Ph.D. degreein 1996. His researchar -eas include harmonic analysis, waveletstheory, and applications in signal process-ing. He is currently working at Buenos Ai-res University.

A ddress for C orr espondence: MariaAlejandra Figliola, Instituto de Cglculo.Pab. I1 Ciudad Universitaria (1428) Bue-nos Ai res, Argentine. E-mail: [email protected].

References1.TakensF: Detecting Strange Attractors in the

Turbulence, Dynamical Systems and Turbulence,ed. Rand and L .3. Y oung, L ecture Notes inMath., 898,336-381, Springer-V erlag, 1980.2. Gukenheimer J and Holmes P: Nonlinear Os-cillations, Dynamical Systems, and Bifurcationso Vector F ields, Springer-Verlag, 1983.3. Daubechies I: Ten Lectures on Wavelets,SIAM , Philadelphia, 1992.4.M eyer Y : Ondelettes and Operateurs,TomesI,11, 111Paris Hermann, 1992.5. M eyer Y : Wavelets Alghoritms and Applica-tion, SIAM,Philadelphia, 1992.6. Chui Ch K : An Introduction to Wavelets,SanDiego: Academic Press Inc., 19927. Figliola A and Baredes C: An A ltemativeMethod to Evaluate the Order of a System, Insta-bil iti es and Nonequil ibr ium Structures V ,Tirapegui-Zeller Ed. K luwer Academic Pub. (inpress), 1996.8.Gershenfeld N and Weigend A: Times SeriesPrediction: Forecasting the Future and Under-standing the Past - The Future of Time Series:L eaming and Understanding, Proceedings VolXV Santa Fe Institute Studies in the Sciences andComplexity,pp. 1-70, 1994.9. Rigney D, Goldberger A, Ocasio W, Ichi-maru Y , Moody G and Mark R: Times SeriesPrediction: Forecasting the Future and Under-standing the Past - Multi-Channel PhysiologicalData: Description and Analysis, Proceedings VolXV Santa F e Institute Studies in the Sciences andComplexity,pp. 105-129, 1994.10. Coifman RR: A dapted M ultiresolutionAnalysis, Computation, Signal Processing andOperator Theory, Proceedings o the Interna-tional Congress ofklathematici ans, K ioto, Japan,11. Guil leminault V and Partinen M, eds: Ob-structive Sleep Apnea Syndrome. Clinical Re-search and Treatment, New Y ork, Raven Press,1990.12. Edelman NH and Santiago TV: BreathingDisorderso Sleep.New Y ork: Churchill L iving-stone, 1986.13. Wiggins: Introduction to Applied NonlinearDynamical Systems and Chaos,Springer-Verlag,1990.14. M allat S: Multiresolution Representationsand Wavelets, Grasp. Lab. 153, Univ. of Pensyl-vania, Philadelphia, 1988.15.GlassL and Kaplan D: Times Series Predic-tion: Forecasting the Future and Understandingthe Past - Complex Dynamics in Physiology andMedicine,Proceedings Vol XV Santa Fe InstituteStudies in the Sciences od Complexity, pp. 513-527, 1994.16. L efebvre J H, Goodings DA, K amath M Vand Fallen EL : Predictability of Normal HeartRhythms and Deterministic Chaos, Chaos,3, (2),1993.17. Casdagli M and Weigend A: Times SeriesPrediction: Forecasting the Future and Under-standing the Past - Exploring the Continuum Be-tween Deterministic and Stochastic Modelling,Proceedings VolXV Santa Fe Institute Studies inthe Sciences on Complexity,pp. 347-367, 1994.

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