Modelling long-term heart rate variability: an ARFIMA approach

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1 Biome Tech 006; 51: by Walter e Gruyter Berlin New York. DOI /BMT Moelling long-term heart rate variability: an ARFIMA approach Argentina S. Leite 1 3, *, Ana Paula Rocha 1,3, M. Euara Silva 1,4 an Ovíio Costa 5 1 Departamento e Matemática Aplicaa, Universiae o Porto, Porto, Portugal Departamento e Matemática, Universiae e Trásos-Montes e Alto Douro, Vila Real, Portugal 3 Centro e Matemática a Universiae o Porto, Porto, Portugal 4 Uniae e Investigacão Matemática e Aplicacões, Universiae e Aveiro, Aveiro, Portugal 5 Faculae e Meicina, Universiae o Porto, Porto, Portugal Abstract Long-term heart rate variability (HRV) series can be escribe by time-variant autoregressive moelling. HRV recorings show epenence between istant observations that is not negligible, suggesting the existence of long-range correlations. In this work, selective aaptive segmentation combine with fractionally integrate autoregressive moving-average moels is use to capture long memory in HRV recorings. This approach leas to an improve escription of the low- an high-frequency components in HRV spectral analysis. Moreover, it is foun that in the 4-h recoring of a case report, the long-memory parameter presents a circaian variation, with ifferent regimes for ay an night perios. Keywors: long-range correlations; selective aaptive segmentation; spectral analysis. Introuction Cariovascular variables such as heart rate, arterial bloo pressure an the shape of the QRS complexes in the electrocariogram are almost perioic, showing some variability on a beat-to-beat basis. This variability reflects the interaction between perturbations to the cariovascular variables an the corresponing response of cariovascular regulatory systems. Therefore, both time an frequency analysis of such variability can provie a quantitative an non-invasive metho to assess the integrity of the cariovascular system. The iscrete series of successive RR intervals (the tachogram) is the simplest signal that can be use to characterise heart rate variability *Corresponing author: Argentina Soeima Leite, Departamento e Matemática Aplicaa, Faculae e Ciências, Universiae o Porto, Rua o Campo Alegre 687, Porto, Portugal Phone: q Fax: q amsleite@fc.up.pt (HRV) an has been applie in various clinical situations w1, 10x. Ambulatory long-term HRV series typically correspon to beats in a 4-h recoring an exhibit nonstationary characteristics. These long-term HRV series can be escribe by time-variant autoregressive (AR) moelling. The AR moels are calle short-memory moels, since their autocorrelations (ACFs) ecay to zero exponentially, meaning that observations far apart are not correlate. However, the sample autocorrelations (SACFs) of HRV series show a ifferent type of behaviour: the ecay is very slow, inicating that the epenence between istant observations is not negligible. This is illustrate in Figure 1 by a tachogram for a normal subect, its SACF an the ACF of the fitte AR moel. Correlations exhibiting this type of behaviour are calle long-range correlations an the processes are enote as long-memory or persistent processes w4x. This long-memory property is reveale in the frequency omain by a spectral ensity function that follows a power law (1/f) in the very low frequencies an has been the subect of several stuies using etrene fluctuation analysis an coarse graining spectral analysis (w7x an references therein). An alternative approach to long-memory escription in HRV ata is to use fractionally integrate autoregressive moving average (ARFIMA) moels, which are an extension of the well-known autoregressive moving average (ARMA) moels. ARFIMA moels allow escription of both the short- an long-term correlation structure an have been use in numerous application areas. In this work, selective aaptive segmentation combine with ARFIMA moels is use in the escription of long-term HRV recorings. Materials an methos ARFIMA moelling of HRV A stationary process µx t tgz is sai to have long-range epenence if there exists a real number ag x0,1w an a constant c r )0 such that rž k.;crzz k -a, k `, covž X t,xtqk. where rž k. s is the ACF. Alternatively, a sta- varž X t. tionary process µx t tgz is sai to have long-range epenence if there exists a real number bg x0,1w an a constant c f )0 such that fž v.;c Zv Z -b f, v 0, where f(ø) is the spectral ensity function. 006/5140

2 16 A.S. Leite et al.: Moelling long-term HRV: an ARFIMA approach Figure 1 (A) Tachogram of a normal subect. (B) SACF an ACF of the fitte AR moel, AR(8) with orer inicate by AIC criterion. A class of processes with this property are the ARFIMA processes. These processes were introuce by Hosking w5x an have special interest for applications because of their capability of moelling both short- an long-term behaviour of a time series. A stochastic process µxt tgzis ARFIMA(p,,q), p,qg N µ0 an gr, if it satisfies the equation Ž. Ž. f B = Xtsu B t, where µ t tgz is Gaussian white noise WN(0,s ), fž z. s p q i i an Ž. is1 s1 1-8fz u z s1-8uz are polynomials such that fž z./0 an už z./0 for ZZ z F1, B is the backwar-shift operator, BXtsX t-1, = is the fractional ifference operator efine by ` GŽ -. = sž 1-B. s1q8 B, GŽ q1. GŽ -. s1 an G(Ø) is the gamma function. The parameter etermines the long-term behaviour, whereas p, q an the corresponing parameters in f(b) an u(b) allow the moelling of short-range properties. For , ARFIMA(p,,q) is stationary an invertible. Moreover, for , the process has long memory an for s0 ARFIMA(p,0,q) reuces to the usual short-memory ARMA(p,q) moel. In this work, we consier ARFI- MA(p,,0) moels, since they are a natural extension of the classic AR(p) moels. The spectral ensity function of µx t is then given by: fž v. sf v Z1-e -ivz - SMŽ., -pfvfp (1) s where fsmž v. s is the spectral ensity of the -iv pfe Z Ž. Z corresponing short memory, ARFIMA(p,0,0) war(p)x process. Given a HRV series, X 1,«,Xn, estimation of the parameters of the ARFIMA(p,,0) moels is as follows w3, 4x: estimate using the semi-parametric local Whittle estimator (LWE) an estimate the AR parameters in the filtere ata Uts= Xt. Robinson w8x an Velasco w11x prove that the LWE is consistent for The AR parameters are estimate from the Yule-Walker equa- Figure (A) Tachogram of a normal subect uring a sleeping perio. (B) Spectrum using AR moelling. Spectrum using (C) ARFIMA moelling an (D) the corresponing f SM spectrum. Panel (D) also shows the HF component resulting from the automatic ecomposition of f SM. The frequency axis is normalise by the mean RR.

3 A.S. Leite et al.: Moelling long-term HRV: an ARFIMA approach 17 Table 1 Parameter estimates for AR(p) an ARFIMA(p,,0) moels of the tachogram in Figure A. ˆ p ˆf 1 ˆf ˆf 3 ˆf 4 ˆf 5 ˆf 6 ˆf 7 ˆf 8 ˆf 9 ˆf 10 ˆf 11 ŝ AR ARFIMA Figure 3 Tachogram of a normal subect. Holter recoring (4 h), with localisation of the break points obtaine using ARFIMA moelling combine with selective aaptive segmentation. The arrows inicate the sleeping an waking times. tions an the Levinson algorithm, with the orer p etermine by the Akaike information criterion (AIC) w3x. To illustrate the use of ARFIMA compare to AR moels in HRV, the tachogram represente in Figure A was moelle using both methos. The results are summarise in Table 1. Note that the estimate value for, ˆs0.30 inicates that the recor has long memory. Moreover, in ARFIMA(4,0.30,0), all the parameters are statistically significant, whereas in AR(11) there are only four statistically significant parameters, namely f ˆ, ˆ 1 f, fˆ an ˆ 4 f11. The corresponing spectra, represente in Figure B,C, contain similar global information. However, the spectrum obtaine after removing the long-range correlations, f SM, shows more clearly the components associate with the low (LF, 0.04 to 0.15 Hz) an high (HF, 0.15 to 0.4 Hz) frequencies (Figure D). Furthermore, the usual automatic ecomposition may be applie to the f SM spectrum, which is an AR spectrum, as illustrate in Figure D for the HF component. These results inicate that ARFIMA moels are aequate in HRV, allowing more parsimonious moelling an a better escription of the spectral components. Similar results were obtaine in other recorings. ARFIMA moelling of long-term ata Long-term HRV series can be escribe by AR moelling combine with aaptive segmentation w6x: the long recor is ecompose into short recors of variable length L i an the break points, which mark the en of consecutive shorts recors, are etermine using the AIC criterion for AR moels. In view of the long-memory properties of the ata, the aaptive segmentation here is base on ARFIMA moelling. The break points are obtaine using the AIC criterion for ARFIMA(p,,0) moels wx: Ž. Ž. AIC p snlogsˆ q pq. Figure 4 (A) Tachogram of Figure 3 an (B) evolution over 4 h of ˆ. The otte lines inicate the sleeping (:0 h) an waking (08:45 h) times. Specifically, let N be the minimum allowable segment length an L i initialise with N samples, L i GN. Each segment S i starts at the last break point etermine an has length L i qn. A new break point is selecte when D aic saic 0 -AIC 1 G0, where AIC 0 is the value of the AIC in

4 18 A.S. Leite et al.: Moelling long-term HRV: an ARFIMA approach Figure 5 Three-imensional spectra (pseuo-spectra where appropriate) of the long-term HRV series in Figure 3: (A) f(ø) an (B) f SM (Ø) for the selecte AIC moel orers in (C). (D) Evolution over time of the HF an HFc parameters etermine automatically from f SM (Ø). Spectra were obtaine with interpolation an resampling of the iniviual spectra to ensure correct matching of the frequency axis.

5 A.S. Leite et al.: Moelling long-term HRV: an ARFIMA approach 19 segment S i an AIC 1 is the sum of AIC in the first L i samples with AIC in the last N samples of the segment S i. The values of the AIC are obtaine consiering a fixe value for p. In this methoology, when a break point is ientifie, N samples are necessary to start the search for the next break point. Therefore, no break points are ientifie in segments of length N. This limitation may be overcome by selecting the most significant break points (maximum D aic ) from a set of caniates in the next N samples. This segmentation is referre to as selective aaptive segmentation. The short recors thus obtaine are subsequently moelle using ARFIMA moels. The selective segmentation leas to segments of increase length an consequently to a better estimation of the memory parameter,. Results an iscussion The methoology presente above is illustrate in a case report. The ata are an experimental 4-h Holter recoring of a 19-year-ol normal subect, obtaine with a Mortara H-Scribe 1-lea ECG monitor (Mortara Instrument Inc., Milwaukee, WI, USA). The sleeping an waking times were registere by the subect in the Holter iary. Figure 3 shows the break points thus obtaine (marke with otte vertical lines) with Ns51 an ps1 (a value reporte in the literature w9x). ARFIMA moelling combine with selective aaptive segmentation leas to aequate localisation of rhythm ifferences an transient perios, as the sleeping an waking times, marke with arrows. As illustrate in Figure 4B, ˆ evolves over time, presenting a circaian variation. Most of the values estimate uring the night (sleeping perio) range from 0 to 0.5, whereas uring the ay (waking perio) they range from 0.5 to 1. Hence, uring the ay the maority the HRV recorings are non-stationary an Equation (1) is then sai to be a pseuo-spectral ensity w11x. Figure 5A represents the three-imensional parametric spectra (pseuo-spectra where appropriate), f(.), of the long-term HRV series an Figure 5B represents the corresponing parametric f SM (Ø), compute from AR moels with orers selecte by AIC an represente in Figure 5C. During the night, a significant HF component is observe, corresponing to ominance of the parasympathetic nervous system activity an to the regularity of respiratory activity. However, these features are emphasise in f SM (Ø) spectra (Figure 5B). Furthermore, the use of f SM (Ø) allows the application of current approaches to the analysis of HRV. Specifically, Figure 5D represents the evolution over time of the HF component an its typical central frequency, HFc, etermine automatically from the parametric moelling. Similar results were obtaine from the analysis of other Holter recorings of normal subects. Conclusion HRV recorings are long-memory or persistent processes, i.e., they show spectra that follow a power law in the very low frequencies, hinering the analysis of LF an HF frequencies. ARFIMA moels are use to capture an remove long memory in HRV recorings, leaing to an improve escription of low- an high-frequency components in the spectra. ARFIMA moelling combine with selective aaptive segmentation of 4-h HRV ata shows that the memory parameter has a circaian variation with two regimes: sleeping an waking perios. The ARFIMA approach escribe in this work allows the use of current techniques in the analysis of HRV. References w1x Appel ML, Berger RD, Saul JP, Smith JM, Cohen RJ. Beat to beat variability in cariovascular variables: noise or music? J Am Coll Cariol 1989; 14: wx Beran J, Bhansali RJ, Ocker D. On unifie moel selection for stationary an nonstationary short an long memory autoregressive processes. Biometrika 1998; 85: w3x Brockwell PJ, Davis RA. Time series: theory an methos. New York: Springer-Verlag w4x Doukhan P, Oppenheim G, Taqqu MS. Theory an applications of long-range epenence. Boston: Birkhäuser 003. w5x Hosking JRM. Fractional ifferencing. Biometrika 1981; 68: w6x Niezwiecki M. Ientification of time-varying processes. Somerset, NJ: John Wiley 000. w7x Rienzo MD, Mancia G, Parati G, Peotti A, Zanchetti A. Frontiers of bloo pressure an heart rate analysis. Amsteram: IOS press w8x Robinson PM. Gaussian semiparametric estimation of long range epenence. The Annals of Statistics 1995; 3: w9x Rocha AP, Leite A, Gouveia S, Lago P, Costa O, Freitas F. Spectral characterization of long-term ambulatory heart rate variability signals. In: Digest of the 5th IMA Conference on Mathematics in Signal Processing, Warwick, UK, 000. Essex: IMA e.: 1 4. w10x Task Force of the ESC an NASPE. Heart rate variability: stanars of measurement, physiological interpretation an clinical use. Eur Heart J 1996; 17: w11x Velasco C. Gaussian semiparametric estimation of nonstationary time series. Journal of Time Series Analysis 1999; 0: