Nonlinear Analysis of Physiological Time Series

Chinese Journal of Biomedical Engineering( English Edition) Volume 16 Number 4 , December 2007Nonlinear Analysis of Physiological Time SeriesMENG Qing-fang'，PENG Yu-hua' ,XUE Yu-i? ,HAN Min'1 School of Information Science and Engineering , Shandong Unirersity, Jinan 250100 ,China2 Shandong Youth Administrative Cadres College ,Jinan 250103 ,ChinaAbstract. The heart rate variability could be explained by a low. dimensional gover-ning mechanism. There has been increasing interest in verifying and understanding thecoupling between the respiration and the heart rate. In this paper we use the nonlinear de-tection method to detect the nonlinear deterministic component in the physiological timeseries by a single variable series and two variables series respectively, and use the condi-tional information entropy to analyze the correlation between the heart rate, the resprationand the blood oxygen concentration. The conclusions are that there is the nonlinear deter-ministic component in the heart rate data and respiration data, and the heart rate and therespiration are two variables originating from the same underlying dynamics.Key words : Nonlinear time series analysis ; nonlinear detection ; conditional informa-tion entropy ; heart rate variabilityINTRODUCTIONNonlinear time series analysis that are suit to analyze complex phenomena have been extensivelystudied in the past two decades'; " .6]1 . Various methods have been proposed to extract nonlinear informa-tion from real time series. Important problems in this field are that if time series are indicative of under-lying nonlinear dynamics , and arise from a low dimensional attractor, etc. Detecting the nonlinear dy-namics in real time series and determining the nature of time series are very important for natural da-tal7141A different strategy, widely accepted at present , is to test the original data conformance with cer-tain regularity patterns ，often through a statistical comparison with randomly generated surrogate da-t[-10]or to infer the presence of nonlinear determinism in time series on the basis of its nonlinear short-term predictabilityHowever, most of them require rather long time series and are not robust to中國煤化工Heart rate variability has attracted much attention froMYHCNMHG1980s. It hasCLC number:R 318. 04 Document code:A Article ID; 1004-0552( 2007 )04-0163-07Crant sponsor:Scientifc Research Foundation for the Retumed Overseas Chinese Scholars of China ;Grant number :2004. 176. 4;Grant sponsor: Natural Sei-ence Foundation of Shandong Province;Grant rumber:72004C0LReceivea 27 Tuly 20:eie 2 Novembter 209@ou.co--163-CHINESE J. BIOMED. ENG. VOL.16 NO.4, DEC.2007been understood that a metronomic heart rate is pathological, and that the healthy heart is influenced bymultiple neural and hormonal inputs that result in variations in interbeat ( RR ) intervals , at time scalesranging from less than a second to 24 h. Even after 20 years of study , new analytic techniques continueto reveal properties of the time series of RR intervals. Many researches in this area aim to discover or ex-plain how changes in variability can be related to specific pathologies.There is growing evidence that observed variations in the heart rate ,it might be related to a low-di-mensional governing mechanism, so to understand this mechanism is obviously important. The heart rateand the respiration are potentially interacting variables. A correlation between breathing and the heartrate, called respiratory sinus arrhythmia, is almost always observed. There has been increasing interestin verifying and understanding the coupling between the respiration and the heart rate.In this paper we use the nonlinear detection method to detect the nonlinear deterministic componentin the physiological time series by a single variable series and two variables series respectively, and usethe conditional information entropy, which is minimized at zero relative shift of the two signals if the sig-nals have their origin in same underlying dynamics, so as to analyze the correlation between the heartrate, the respiration and the blood oxygen concentration.ANALYSIS PHYSIOLOGICAL TIME SERIES USING NONLINEAR METHODIn the following we briefly review the method of detecting nonlinear dynamics in time series4 ,which is suit to short noisy time series. For time series, a closed-loop version of the dynamics in whichthe output feeds back as a delayed input was proposed. Within this framework, we analyze time series byusing a discrete Volterra autoregressive series of degree, and memory as a model to calculate the predic-ted time series yin :yn =0o +a.yn-1 +a2Yn-2 +... +arYn-k +ag+1Yn-1+M-14+2+-1-+...+.-1y%-h=. Samm(n)(1)Where the functional basis {zm (n)} is composed of all the distinct combinations of the embeddingspace coordinates (yn-1ny--,ynm-3,",yxn_k)up to degree d, with a total dimension M= Ck+d=(h+d)! /(k! d!). Thus, each model is parameterized by k and d which correspond to the embedding di-mension and the degree of the nonlinearity of the model (i.e. d=1 for linear model andd > 1 for nonlin-ear model). The coefficients am are recursively estimated through a Gram- Schmidt procedure from linearand nonlinear autocorrelations of time series itself.The goodness of fit of a model ( linear vs. nonlinear) is measured by the normalized residual sum ofsquare errors :s [y%a'-y,]2ε(h,d)2==(2)中國煤化工1CNMHGwhereF=N-N.,yn and e( m,d)2 is, in fact ,on a normallzed vanance o1 tne error residuals. The op-timal model {roor ,dor } is the model that minimizes the Akaike information criterion:-164-Chinese Journal of Bionedical Engineering( English Edition) Volume 16 Number 4 , December 2007C(r) =loge(r) +r/N(3)where r∈[1 ,M]is the number of polynomial terms of the truncated Volterra expansion from a certainpair {k,d}.The numerical procedure is as follows : for each time series，obtain the best linear model by search-ing for h'n which minimize C(r) with d=1. Repeat with increasing k and d>1 so as to obtain the bestnonlinear model. Likewise obtain the best linear and nonlinear models for surrogate randomized data setswith the same autocorrelation ( and power spectrum) as the original series , which results in four compe-ting models with error standard deviations eεoirg, eorie, eEsier and Eur. The presence of nonlinear determin-ism is indicated ifd. oe > 1. And the relevance of nonlinear predictors is established when the best nonlin-ear model from the original data is sigificantly more predictive than both (1) the best linear model fromthe data series, and (2) the best linear and nonlinear models , obtained from the surrogate series, thatis, e(ing, Emr, Eoni;>eaum. The procedure can be extended to multivariate series. For instance, the pre-dicting model for y variable series by two variables series y and z is as follows:y% =ao+a]Yn-1 +by7n-1 +a2Yn-2 +bz.n-2 +...+arYn-k +bxzn-k +aG:+13n-1 +612-1 +a+2Y-IYn-2 +b2r27-7n-2+..+a.-.Y%- +by_1Z-n(4)The coarse-graining methodology and conditional entropy which was applied to nonlinear time series byLehrman et al5) were reviewed in the following. A symbolic sequence { S, ,S2,S,..| is associated witha time-series {yr ,2,y3,. via a coarse -graining'5s 17 I such that the information concerning the orbit issuitably encoded. This is accomplished by the following partitioning of the phase space.Given a set of m symbols, {ψ,ψ,°,ψm⊥} ,and a symbolic of m +1 critical points, {xy,x,.，xm} the time series {yv,y,y.,..} is converted into a symbolic sequence by the ruleS,=ψ, ifxg1 and eoink eo+lev,e,ewe.So we deduce there is the nonlinear deterministic component in the heart rate data and respiration forcedata.2)0(b02015d=2d=3s.2165Fig.2 lnε(r) for (a) the heart rate data, (b) the, resniration force data中國煤化工k =4 ( original data口, surrogate data .We use two variables series ( the heart rate data and res:FYHCNMHGdofasinglevaria-ble series to build the model respectively for the heart rate series and the respiration force series, and theresult is shown in Fig. 3. From Fig. 3 we can see that the normalized variance of the error residuals of the-166-Chinese Jourmal of Biomedical Engineering( English Edition) Volume 16 Number 4 , December 2007model built using two variables series is less than that of the model built using a single variable series,and the descend trend is more distinct. So we deduce that the heart rate and the respiration are not inde-pendent, and they affect each other.0z間) !6)0d=2d=305Fig.3 In e(r) from model built simultaneously using the heart rate data and respiration force datafor (a) the heart rate data, ( b) the respiration force data, k =4 ( original data口, surrogatedata-)Although the heart rate and the respiration force will be importantly coupled variables in some physi-ological model , it is difficult to decipher the correlation features by merely looking at the time signals. Aproper quantitative analysis is needed to determine the dynamical coupling of the variables. Quantitiessuch as the correlation coefficient or the correlation function do not often provide unequivocal indicationof the dynamical coupling of physiological variables. For example, the correlation cofficient for the heartrate and respiration force is 0. 12. The values of cross correlation function Cxv(no) =E{X(n)Y(n+ng)}, where X and Y are two variables ( presently heart rale and respiration force) being compared，which are low ( Fig.4(a)).355a)0.33b)oou0325sl^0020315-子g 03n3060四03-z602908Fig.4 (a) Correlation function and (b) conditional entropy as a function of the shift no of the respi-ration force series and heart rate seriesWe applied the coarse graining method, and computed E( Re/Hr), where Re and Hr denote therespiration force and heart rate, respectively. The presen中國煤化工at no(Fig. 4(b))clearly demonstrates the dynamical coupling of the two va:YHCNMHGThe blood oxygen concentration affect, and in turn is affected by the heart rate and the respiration.They are potentially interacting variables. These effects could involve delays. The coarse-graining meth-一167-CHINESE J. BIOMED. ENC. VOL.16 NO.4 ,DEC.2007od is suited to this analysis. We look for such an effect by doing the coarse -graining analysis for theblood oxygen concentration and the heart rate, and for the blood oxygen concentration and the respira-tion.(a)b)010516D0009512Fig.5 The conditional entropy for (a) the blood oxygen concentration and heart rate and(b) the blood oxygen concentration and respirationIn Fig. 5(a) it is shown the conditional entropy of the blood oxygen concentration and heart rate( E( Bo/Hr)), where Bo denotes the blood oxygen concentration, which indicates a minimum at no =17,namely that the heart rate affect the blood oxygen concentration in a time delay of8.5 s. In Fig.5(b) itis shown the conditional entropy of the blood oxygen concentration and respiration ( E( Bo/Re)) , whichindicates a minimum at no = 78, namely that the respiration affect the blood oxygen concentration in atime delay of 39 s.CONCLUSIONIn summary, we get conclusions in the fllowing:1) the heart rate was mainly influenced by fourfactors;2 ) there is the nonlinear deterministic cormponent in the heart rate data and respiration force da-ta;3) the heart rate and the respiration are not independent, they take effect each other, and they origi-nate from the same underlying dynamics;4) the heart rate affect the blood oxygen concentration after atime delay of 8.5 s, and the respiration affect the blood oxygen concentration after a time delay of 39 s.REFERENCES[1] Kantz H, Schreiber T. Nonlinear time series analysis[ M ]. London: Cambridge University Press, 1997.[2] Packard NH , Crutchfield JP , Farmers JD,et al. Geometry from a time series[J]. 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