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TBME-00919-2007.R1 1

     Heart Rate Turbulence Denoising

     Using Support Vector Machines

     JL Rojo-Álvarez, Member, IEEE, O Barquero-Pérez, I Mora-Jiménez,

     E Everss, A.B. Rodríguez-González, A García-Alberola

     AbstractHeart Rate Turbulence (HRT) is the transient ac-

     celeration and subsequent deceleration of the heart rate after a

     premature ventricular complex (PVC), and it has been shown to

     be a strong risk stratification criterion in patients with cardiac

     disease. In order to reduce the noise level of the HRT signal, con-

     ventional measurements of HRT use a patient-averaged template

     of post-PVC tachograms (PPT), hence providing with long-term

     HRT indices. We hypothesize that the reduction of the noise

     level at each isolated PPT, using signal processing techniques,

     will allow to estimate short-term HRT indices. Accordingly, its

     application could be extended to patients with reduced number of

     available PPT. In this paper, several HRT denoising procedures

     are proposed and tested, with special attention to Support Vector

     Machine (SVM) estimation, as this is a robust algorithm that

     allows us to deal with few available time samples in the PPT. Pac-

     ing stimulated HRT during electrophysiological study are used

     as a low noise gold-standard. Measurements in a 24 hour-Holter

     patient database reveal a significant reduction in the the bias and

     in the variance of HRT measurements. We conclude that SVM

     denoising yields short-term HRT measurements and improves

     the signal to noise level of long-term HRT measurements.

     Index TermsHeart Rate Turbulence, Denoising, Short-Term,

     Support Vector Machine, Bootstrap Resampling, "-Huber cost,

     Turbulence Slope.


     Heart Rate Turbulence (HRT) has been defined as the

     behavior of the Heart Rate (HR) after a Premature Ventricular

     Complex (PVC). Under normal and healthy conditions, HRT

     consists of a brief increase in HR after the PVC, that is

     immediately followed by a slower decrease in HR. Measure-

     ments on HRT characteristics in long-term (24 hour) Holter

     recordings have shown a high predictive power for identifying

     patients with high-risk of cardiac disease [1], [2]. The two

     parameters that have been mostly used for measuring HRT

     are the Turbulence Onset (TO) and the Turbulence Slope (TS),

     though other parameters have been also proposed [2], [3]. The

     TO measures the amount of sinus acceleration that follows

     immediately a PVC, and it is defined as the shortening of the

     interval average for the two sinus beats (normal-to-normal,

     NN) after the compensatory pause. The TO is calculated as the

     percentage of the two NN interval average preceding the PVC.

     The TS is an indicator of the amount of sinus deceleration

     JLRA, OBP, IMJ, EE, and ABRG, are with Department of Signal Theory

     and Communications, University Rey Juan Carlos (Spain). AGA is with Ar-

     rhythmia Unit, Hospital Virgen de la Arrixaca of Murcia (Spain). Address for

     correspondence: JL Rojo-Álvarez (joseluis.rojo@urjc.es), B-004, Universidad

     Rey Juan Carlos, Camino del Molino s/n, Fuenlabrada-28943, Madrid (Spain).

     This work has been partially supported by Research Project TEC2007-68096-

     C02/TCM from Spanish Government and by Research Grant from Medtronic.

     following the PVC, and it is usually calculated by first fitting

     the linear regression for each 5 consecutive NN intervals in the

     post-PVC tachogram (PPT) during 15 beats, and then selecting

     the maximum slope among all the regression lines fitted along

     the PPT. An important requirement for obtaining reliable HRT

     measurements is the adequate selection of the PPT that are

     useful, and a set of conditions for inclusion in the analysis

     have been clearly established in the literature [2]. Aiming to

     reduce the strongly present physiological noise of different

     kinds in each PPT, a PPT template is first build in each patient

     by averaging the set of available single PPT, and then, TS

     is calculated on this template. Given that the PPT template

     is usually averaged for 24-hour Holter recordings, indices

     obtained from this template are a long-term measurement of

     the global state of the patient during a day, and this processing

     has been shown to be a powerful risk stratifier not only for

     acute myocardial infarction [1], but also for other diseases

     such as Chagas [4] or heart failure [5].

     Nevertheless, relevant information could be masked by the

     long-term averaging in this calculation procedure, both from

     a clinical and from a signal analysis points of view. First,

     relevant short-time fluctuations in the TS along the day [6]

     could be hidden by the 24-hour template averaging. Second,

     several influences of the physiological state can affect the

     HRT, such as the described effect of HR level that precedes

     to the PVC on the HRT oscillation amplitude [3], [7]. More

     specifically, the vegetative tone is probably controlling both the

     HR level and the HRT oscillation amplitude, but nevertheless,

     averaging along the different states during the day could result

     in a reduction of the true magnitude of the HRT fluctuation

     and in a smoothing not only in noise level, but also in signal

     level [6], [8]. And third, averaging precludes the comparison

     of HRT in a given moment to other fluctuating physiological

     variables. For instance, comparison of long-term Heart Rate

     Variability (HRV) to long term HRT has been reported [9], but

     the short-term regulation of the autonomous nervous system

     on HR can not be studied jointly with the HRT.

     Therefore, our hypothesis is that efficient cancelation of

     physiological noise from each isolated PPT will allow the

     short-term quantification of TS. This would allow us also to

     measure the HRT in a higher number of patients, beyond the

     current limits given by the exclusion criteria for TS averaging

     with a minimum number of available PPT. Accordingly, a

     signal processing method capable of canceling the noise in

     a single PPT will be a valuable tool to evaluate the short-term

     HRT, and the development of such method is the purpose of

     this paper. Two main technical issues appear when addressing

     Copyright (c) 2008 IEEE. Personal use of this material is permitted. However,

    permission to use this material for any

     other purposes must be obtained from the IEEE by sending an email to pubs-permissions@ieee.org.

    TBME-00919-2007.R1 2

     the signal denoising of a single PPT. First, the PPT has a very

     short duration (current recommendation is 15 NN samples),

     and hence, a very robust signal processing method will be

     required. Second, the HRT is usually measured from 24-hour

     Holter recordings, which are surely influenced by a variety of

     noise sources, including the daily activity and the changes in

     the physiological state of the patient. But a clear gold-standard

     for HRT behavior and measurement will be needed if we want

     to benchmark and compare the performance of any proposed

     denoising algorithm.

     To overcome the first issue, we propose the use of Support

     Vector Machines (SVM), in particular, the SVM regressor [10].

     The SVM framework has been shown specially advantageous

     in problems with few samples available, due to their excellent

     generalization performance. We will show that the SVM

     regression based on a robust cost function (the "-Huber cost),

     together with the use of bootstrap resampling techniques for

     tuning the free parameters of the algorithm [11], [12], can

     provide us with an efficient HRT denoising technique.

     We also propose to study the performance of the denoising

     procedure in a gold standard given by HRT induced with car-

     diac electrical stimulation (pacing) during electrophysiological

     study (EPS), which can be considered an almost noise-free

     environment, because the patient is maintained at rest. Pacing-

     induced HRT has started to receive increasing interest, and

     conditions for its measurement have been established [13],

     [14], [15], [16], [17]. This gold standard will allow us to

     quantify the HRT shape in the temporal and spectral domain

     in an almost noise-free environment, an then to compare the

     performance of the signal processing algorithms used for HRT

     denoising in Holter recordings.

     The scheme of the paper is as follows. In the next section,

     the SVM approach to HRT denoising is presented, and other

     algorithms used for comparison are also described. Then, the

     algorithms are benchmarked in two scenarios: (1) analysis of

     induced HRT in a patient database during EPS; and (2) anal-

     ysis of 24-hour Holter HRT recordings. Finally, conclusions

     are drawn and future research is suggested.


     A. HRT and PVC Signals

     HRT represents a biphasic chronotropic response of sinus

     rhythm to a single VPC [3], [18], and it is given by an early HR

     acceleration followed by a deceleration. Its pathophysiological

     background has been investigated, aiming to understand the

     underlying mechanisms in order to give an explanation of HRT

     power as independent postinfarction risk stratifier. It has been

     hypothesized that HRT could be triggered by a transient vagal

     inhibition in response to the missed baroreflex afferent input

     due to hemodynamically inefficient PVC induced ventricular

     contraction. Late deceleration of HR also increases systolic

     blood pressure due to vagal recruitment, and hence, this should

     be consistent with the baroreflex mechanism involved. Phys-

     iological mechanisms of HRT and of systolic blood pressure

     dynamics in the late HRT phase have been studied [18]. HRT

     could also be influenced by the underlying HR, as far as it

     defines the hemodynamic setting and the autonomic milieu in

     which the PVC happens [3].

     From a digital signal point of view, HRT signal is a

     short-length sequence of beat-to-beat time intervals (or R-R

     intervals), comprising three periods: First, previous 4 or 5 RR

     intervals represent the basal state in which the PVC occurs;

     Second, the PVC yields a much shorter interval, followed

     by a much longer interval due to the compensatory pause;

     And Third, the turbulence signal consists of the fast initial

     acceleration, followed by an oscillation in R-R intervals, which

     usually lasts no longer than about 15-20 beats. Therefore,

     this is an extremely short signal duration for conventional

     denoising or filtering techniques.

     B. SVM Denoising Algorithm

     According to the preceding description of HRT signal, the

     signal model considered here uses the NN intervals from an

     ECG or EGM recording. A PVC has happened at discrete time

     instant n = ?1, which is then followed by a compensatory

     pause at n = 0, so that the following NN intervals (from n = 1

     to n = 20) represent the PPT under study. Assume that the

     observed PPT is given by fyn; n = 1; : : : ; 20g contains two

     contributions: one is the actual HRT as the metabolic response

     to the PVC perturbation, given by fxn; n = 1; : : : ; 20g, and

     the other one consists of noise contributions from different

     sources, and is given by fen; n = 1; : : : ; 20g. The HRT model

     is then:

     yn = xn + en (1)

     for n = 1; : : : ; 20. A first approach for denoising fyng and

     obtaining an estimate of HRT, denoted by fx^ng, is to use linear

     filtering. For instance, a Qth order moving-average filter can

     be used, which will be given by the following signal model:

     x^n =



     bqyn?q+1 (2)

     where bq are fixed as the coefficients for an adequate filter in

     the frequency domain. Independently of the kind of filter used,

     this denoising scheme relies on the assumption of HRT being

     a band-limited process. Alternatively, a Qth median filter can

     be used, given by

     x^n = medianfyn?bQ2 c; : : : ; yn+dQ2 eg (3)

     where b?c and d?e denote the rounding up and down to zero,

     respectively. This denoising scheme is known to be more

     appropriated whenever impulse noise can be present.

     Note that these basic filtering schemes rely on the distri-

     bution of noise being known to some extent, which is an a

     priori information that we do not have yet. According to this

     fact, we propose to use a SVM modeling approach. The SVM

     regressor can be seen as a nonparametric procedure, in the

     sense that it does not rely on any specified form of the HRT.

     Also, we propose to consider the "-Huber cost [19], which

     represents a cost function that can adapt itself to the noise

     distribution. Finally, given that we can not split the extremely

     short PPT signal into a training and a validation subset, we

     propose to use nonparametric bootstrap resampling, which has

     been previously used in SVM classifiers for the same purpose

     of previously tuning of the SVM free parameters [12]. TBME-00919-2007.R1 3

     The SVM model for HRT denoising can be described as

     follows. The nonlinear regression model is given by

     yn = xn + en = hw; `(n)i+ b+ en (4)

     where `(n) is a nonlinear application of n to a possibly high-

     dimensional (say P -dimensional) feature space F, where a

     linear approximation is built by the dot product with vector

     w 2 F. This model can be seen as a nonlinear interpolation.

     Following the conventional SVM methodology, a regularized

     cost function of the residuals is to be minimized. In [19], the

     following robust cost function of the residuals was proposed,

     L(en) =




     0; jenj • "


     2 (jenj ? ")2; " • jenj • eC

     C(jenj ? ")? 12–C2; jenj ‚ eC


     where eC = " + C; " is the insensitive parameter, and

     and C control the trade-off between the regularization and

     the losses. The "-insensitive zone ignores errors lower than

     "; the quadratic cost zone uses the L2-norm of errors, which

     is appropriate for Gaussian noise; and the linear cost zone

     controls the effect of outliers. The SVM coefficients are

     estimated by minimizing the previous loss function regularized

     with the squared norm of model coefficients,





     w2p +





     (?2n + ??2n ) +C



     (?n + ??n)?






     with respect to wp, f?(?)n g (notation for both f?ng and f??ng),

     and b, and constrained to

     yn ? hw; `(n)i ? b • "+ ?n (7)

     ?yn + hw; `(n)i+ b • "+ ??n (8)

     ?n; ??n ‚ 0 (9)

     for n = 1; ? ? ? ; 20; f?(?)n g are slack variables or losses, which

     are introduced to handle the residuals according to the robust

     cost function; and I1; I2 are the sets of samples for which

     losses have a quadratic or a linear cost, respectively.

     Similar derivations of the dual functional can be found in

     the literature [19], [20]. In brief, by including linear constraints

     (7)-(9) into (6), the primal-dual functional (or Lagrange func-

     tional) is obtained:

     LPD = 12



     w2p +





     (?2n + ??2n ) + C



     (?n + ??n)?





     2 ?



     (fln?n + fl?n??n)? "



     (fin + fi?n)+




     (fin ? fi?n) (yn ? hw; `(n)i ? b? ?n)


     constrained to fi(?)n ; fl(?)n ; ?(?)n ‚ 0. By making zero the gradi-

     ent of LPD with respect to the primal variables [19], we obtain

     fi(?)n = 1 ?


     n (n 2 I1), fi(?)n = C?fl(?)n (n 2 I2), to be fulfilled,

     and if these constrains are included into (10), primal variables

     can be removed. The correlation matrix of input space vectors

     can be identified, and denoted as R(s; t) ? h`(s); `(t)i. The

     dual problem can now be obtained and expressed in matrix

     form, and it corresponds to the maximization of


     ?)T [R+ –I] (fi?fi?)+(fi?fi?)Ty?"1T (fi+fi?)


     constrained to C ‚ fi(?)n ‚ 0, where fi(?) = [fi(?)1 ; ? ? ? ; fi(?)20 ]T ;

     y = [y1; y2; : : : ; y20]T ; and 1 denotes a column vector of ones.

     After obtaining Lagrange multipliers fi(?), the time series

     model for a sample at time instant m is:

     x^m =



     (fin ? fi?n) h`(n); `(m)i+ b (12)

     which is a weighted function of the nonlinearly observed times

     in the feature space. Note that only a reduced subset of the

     Lagrange multipliers is nonzero, which are called the support

     vectors, and the HRT solution is built in terms of them.

     A Mercer’s kernel is a bivariate function that is equivalent

     to calculate a dot product in a possibly infinite dimensional

     feature space [10]. Examples of valid Mercer’s kernels are the

     linear kernel, given by K(s; t) = hs; ti, and the (nonlinear)

     Gaussian kernel, given by

     KG(s; t) = exp

? (s? t)


     2 2



     where is the width of the Gaussian kernel, and it must

     be properly chosen. For a fixed value of , it is fulfilled

     that KG(s; t) = h`(s); `(t)i in some unknown feature space.

     However, we do not need to know explicitly neither the feature

     space nor the nonlinear application, but still the dot products

     in the feature space can be readily calculated by means of the

     kernel. Thus, the final solution of SVM for HRT denoising

     can be expressed simply as

     x^m =



     (fin ? fi?n)KG(n;m) + b (14)

     which is just a linear combination of shifted Gaussian kernels

     of a given width.

     C. Bootstrap Tuning of the Free Parameters

     Note that several free parameters need to be previously

     tuned in the described SVM denoising algorithm, namely,

     width of the Gaussian kernel, and the free parameters of

     the cost function ("; ; C). Cross-validation techniques are

     often used for this purpose in SVM approaches, but in our

     case only 20 observations are available, and splitting them

     involves dramatically reducing the amount of information

     in the training set. We propose to search using bootstrap

     resampling techniques for finding the bootstrap bias-corrected

     error as a function of each free parameter, and then fixing

     the free parameters and training a machine with the whole

     20-samples set of the PPT.

     Bootstrap resampling techniques are useful for nonpara-

     metric estimation of the pdf of statistical magnitudes, even

     when the observation set is small. A detailed description and

     discussion on bootstrap resampling can be found in [21]. The TBME-00919-2007.R1 4

     procedure used here is described in [12] for SVM classifica-

     tion, its extension to the regression case being straightforward.

     In brief, be = f"; ; Cg is the set of free parameters of

     the SVM for time series yn. The estimated SVM coefficients

     are fi^ = [fi^1 ? fi^1; : : : ; fi^20 ? fi^20] = s(fyng; ), where

     s(; ) denotes the SVM estimation operator. Empirical risk

     R^emp = t(fi^; fyng), where t() is the estimation operator,

     can be defined as the averaged cost in the training set of

     samples. A bootstrap resample is a data subset drawn from

     the training set by following its empirical distribution, and

     accordingly, it consists of sampling with replacement the time

     samples of yn, this is, fyn(b)g = fy1 ; y2 ; : : : ; y20g, and

     the resampling process is repeated for b = 1; :::; B times.

     Note that, for each resample, fyn(b)g contains samples of

     fyng appearing none, one, or several times. A partition of

     fyng set of samples can be done in terms of resample yn(b),

     given by fyng = fyn;in(b)g [ fyn;out(b)g, according to the

     time samples included (in) and excluded (out) in resample b.

     The SVM coefficients from each resample will be given by

     f^i(b) = s(fy1; : : : ; yn(b)ing; ).

     An acceptable approximation to the actual risk (i.e., not

     only empirical, but total risk) can be obtained if we use

     R^act = t(fi^(b); fyn;out(b)g). A bias-corrected estimate of the

     actual risk is obtained by simply taking the replication average.

     Furthermore, this average estimate can be achieved for a grid

     of values of the SVM free parameters, hence allowing us to

     determine their suitable values to train the SVM with the

     whole training set. A good range for B is typically 200 to

     500 resamples. SVM free parameters are not usually mutually

     independent, however, a good heuristic approach is to start

     with an intermediate value of C; , set " = 0, then giving

     an initial guess of the kernel parameter, and then re-estimate

     again each the other parameters, continuing until a stable set

     of parameters is obtained.


     Practical issues for the application of the proposed HRT

     denoising techniques were studied and are next presented. We

     start by analyzing the suitability of bootstrap resampling for

     tuning the free parameters in SVM interpolation algorithm.

     Then, the clinical EPS database that was used as gold standard

     for HRT measurements is described, and application examples

     of denoising are used in order to show the following points:

     (1) pacing-induced HRT during EPS can be considered as

     almost noise-free recordings; (2) The cycle length previous

     to the HRT onset can be physiologically related to the HRT

     oscillation amplitude, which should be taken into account

     when measuring TS parameter; And (3) the spectral domain

     representation of HRT can yield the shape of the denoised

     gold standard HRT. Next, measurements on TS parameter

     are studied in the gold standard EPS patient database. After

     summarizing the clinical data of Holter database, denoising

     examples are considered both in the time and in the frequency

     domains, and poblational measurements of TS are studied

     in this setting. Finally, denoising methods are statistically

     compared in terms of a new parameter, the Turbulence Length,

     which allows us to quantitatively determine the effectiveness

     of the denoising procedures in the time domain.

     A. SVM Free Parameter Selection

     One of the key issues when using SVM algorithms is setting

     appropriate values for the free parameters. In this problem,

     where only 20 discrete-time samples are available, bootstrap

     resampling was used for this purpose. For each PPT, the

     Mean Squared Error (MSE) was estimated with Bootstrap

     resampling on the time series for each tested combination of

     SVM free parameters (C, , ", and ). Bootstrapped MSE (200

     resamplings) was obtained for values of each free parameter in

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