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Orthonormal-Basis Partitioning and Time-Frequency Representation of Cardiac Rhythm Dynamics Benhur Aysin, Luis F. Chaparro, Senior Member, IEEE,, Ilan Gravé, and Vladimir Shusterman*
Abstract—Although a number of time-frequency representations have been proposed for the estimation of time-dependent spectra, the time-frequency analysis of multicomponent physiological signals, such as beat-to-beat variations of cardiac rhythm or heart rate variability (HRV), is difficult. We thus propose a simple method for 1) detecting both abrupt and slow changes in the structure of the HRV signal, 2) segmenting the nonstationary signal into the less nonstationary portions, and 3) exposing characteristic patterns of the changes in the time-frequency plane. The method, referred to as orthonormal-basis partitioning and time-frequency representation (OPTR), is validated using simulated signals and actual HRV data. Here we show that OPTR can be applied to long multicomponent ambulatory signals to obtain the signal representation along with its time-varying spectrum. Index Terms—Cardiac rhythm dynamics, segmentation, timefrequency analysis, time series analysis.
I. INTRODUCTION
B
EAT-TO-BEAT variations of cardiac rhythm, referred to as heart rate variability (HRV), provide a widely used noninvasive probe of the autonomic nervous system activity [1]. Pharmacological tests have shown that the high frequency component of an HRV signal is modulated by the parasympathetic branch of the autonomic nervous system (ANS), whereas the low-frequency component is modulated by combined sympathetic and parasympathetic effects [2]. Time-dependent spectral analysis of HRV signals represents a major challenge, because the structure of the signal includes multiple periodic, pseudo-periodic, and a-periodic components [3]. To be able to claim stationarity in the analysis, the recordings are performed during short periods of controlled posture and respiration [1]. Alternatively, 24-hour recordings are used to accumulate multiple cycles of the studied periodicities and to provide average spectral estimates [4]. These conditions, however, do not apply to physiological or pharmacological tests, where the precise time Manuscript received March 25, 2003; revised October 19, 2004. This work was supported in part by the American Heart Association under Scientist Development Grant 0030248N, in part by the National Institutes of Health (NIH) under SCOR Grant P50 HL52338. The work of V. Shusterman was supported in part by a grant from Competitive Medical Research Fund of the University of Pittsburgh. Asterisk indicates corresponding author. B. Aysin was with the University of Pittsburgh, Pittsburgh, PA. He is now with the Ansar Group Inc., Philadelphia, PA 19107 USA (e-mail:
[email protected]). L. F. Chaparro is with the Department of Electrical Engineering, University of Pittsburgh, Pittsburgh, PA 15213 USA (e-mail:
[email protected]). I. Gravé was with the University of Pittsburgh, Pittsburgh, PA. He is now with the Department of Physics and Engineering, Elizabethtown College, PA 17022 USA (e-mail:
[email protected]). *V. Shusterman is with the Cardiovascular Institute, University of Pittsburgh, Pittsburgh, PA 15213 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TBME.2005.845228
and structure of the changes need to be investigated. In this setting, short-time Fourier transform of the nonstationary segments would produce erroneous frequency components, and their averaging would decrease the time resolution and smear the spectrogram. To obviate this problem, adaptive time windowing, and subsequent time-frequency analysis as well as an orthogonal linear decomposition have been proposed [2], [3]. A number of time-frequency representations (TFR), including the Wigner-Ville [5] and the Choi-Williams distributions [6], have been applied for the estimation of time-dependent spectrum of nonstationary signals such as HRV. However, the resolution of generalized TFRs with respect to abrupt changes that often arise during physiological and pharmacological tests is limited. Application of these TFRs does not allow the precise time and structure of a change to be detected. We propose a simple method for: 1) detecting both abrupt and slow changes in the structure of the HRV signal; 2) segmenting the nonstationary HRV signal into less nonstationary portions; 3) exposing characteristic patterns of the changes in the timefrequency plane. The proposed algorithm provides the signal representation along with its time-varying spectrum. II. ORTHONORMAL-BASIS PARTITIONING Accurate analysis of nonstationary signals typically requires segmentation [7]–[9]. In this section, we consider methods for partitioning nonstationary signals into less nonstationary segments. We introduce two orthogonal-basis partitioning techniques: 1) a computationally efficient low-resolution partitioning for long signals; 2) a high-resolution, computationally intensive partitioning for short signals or segments of particular interest in the long signal. The low-resolution partitioning is based on the projection of consecutive time segments onto a small set of global basis vectors, compressing the information into a few projection coefficients [3]. The basis vectors are the eigenvectors of the signal covariance matrix. Since these basis vectors are fixed for all the segments, the time series formed by the projection coefficients represents the gross structure of the signal. Changes in this series could be used for tracking major changes in the signal and for selecting the segments of interest [3], [11], which can be subsequently analyzed in-depth using high-resolution partitioning and time-frequency representation. In the high-resolution partitioning, the basis vectors are segment-specific or local. Changes in the signal structure occurring between two consecutive time segments are detected by comparing the number of eigenvalues and the local energy in two overlapping windows of different lengths. If the segmentation windows were nonoverlapping, the signal excerpts in the
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[11]. In this section, we summarize our approach to the partitioning of such a signal into arbitrarily small segments by obtaining an orthogonal linear representation and projecting each segment onto a fixed set of basis vectors. Then we use a much more parsimonious time series, formed by the coefficients of the orthogonal representation, to characterize the gross structure of the signal [3]. Changes in this time series relate to transients, which can be marked for further study. The Karhunen-Loeve (KL) orthogonal representation has long been used in pattern recognition applications for feature selection and ordering [10]. An extended version of the KL representation, which we call the Global Karhunen–Loeve (GKL) expansion, can be used as an efficient way of finding where transients occur. The GKL expansion is applied to an array of random vectors
Fig. 1. The Malvar windows (a), (b) used for the segmentation. Boundary optimization is performed when the second criterion (c) or the first criterion (d) is satisfied in the boundary detection algorithm (see Section II-B-1. for details).
twowindows could have the same energy but the signal structure could have changed from one window to the other. In such a case, the algorithm would fail to detect the change. To avoid this, we use two overlapping windows: a short window and a two times longer one (Fig. 1). Obviously, the signal within the short window and in the overlapping part of the long window will be the same. On the other hand, the signal excerpt in the nonoverlapping part of the long window may be different. In this case, the signal in the long window will be more complex than the one in the short window requiring more basis vectors to represent. Since the amplitude of an eigenvalue is related to the contribution of the associated eigenvector to overall energy of the signal, the number of eigenvalues indicates the number of basis vectors required to approximate the signal with a certain error. Although the signal excerpts in the short and the long windows have different lengths, this has no effect on our method, since the number of basis vectors approximating the signal is independent of the signal length. This technique is computationally demanding but has a low sensitivity to artifacts, which is desirable for accurate partitioning. Analysis of ambulatory HRV recordings is made more difficult by their typical long length. To detect changes in such a long nonstationary signal, we propose, first to obtain a parsimonious time series that characterizes the gross structure of the signal using an arbitrary segmentation (low-resolution partitioning) [3], [11]. Changes in this time series help to detect segments where transients may occur. Next, we consider each of these selected segments for an in-depth analysis. Despite being short, these segments can still be nonstationary. We thus proceed to partition these segments into the less nonstationary sub-segments using the high-resolution partitioning. A. Low-Resolution Partitioning Detection of transients in a long nonstationary signal is a challenging problem, which we have addressed in detail before [3],
obtained from dividing a long nonstationary signal into short nonoverlapping segments of equal length . Then the GKL expansion of a vector is given by (1) are the GKL coefficients and are the global basis where vectors or the eigenvectors [10] of the correlation matrix (2) covariance matrix, and one set of The existence of one the basis vectors used for representing explain the “global” nature of the GKL expansion. The GKL coefficients are unique for each time segment and given by (3) , , and where is an matrix whose columns are the eigenvectors of the covariance matrix . Thus, changes in the signal that occur from one segment to another are reflected in these coefficients. Constructing time series of the most significant GKL coefficients (for example, the time series of the first ) and GKL coefficients given by tracking the changes in their amplitudes and local variances allows detection of the segments in which the transients occur [3], [11]. In such segments, we perform further in-depth analysis as described in Section II-B. B. High-Resolution Partitioning , , is one of the nonstaSuppose that tionary segments identified by the global processing and needs to be partitioned into less nonstationary segments. Initial partiis performed using Malvar windows, two short tioning of [see windows of length and a long window of length Fig. 1(a), (b)]. Malvar windows are used to avoid the blocking
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effects that nonoverlapping adjacent short windows could produce [12], [13]. We seek to adapt the initial partition using the number of eigenvalues (of the covariance matrix of a windowed signal) required to represent a fixed proportion of the signal energy. The connection between the signal representation, eigenvalues and local energy is presented in [14]. In short, the win, and its KL exdowed signal is given by pansion is (4) where are uncorrelated random KL coefficients, are orthonormal basis obtained from the covariance matrix of , and is the length of the short window (Fig. 1). Due to , its covariance mathe assumption of nonstationarity of trix has entries depending rather than on their difference, which would on both and were stationary. For nonstationary signals, be the case if these entries can be evaluated more accurately (than the outer vector product used in the low-resolution partitioning) by means of “evolutionary” estimator [15] given by
(5) where if if and
are the time-varying windows defined as where the s are orthonormal functions. The basis vectors are the eigenvectors of the and since the signal is nonstationary, covariance matrix will be different for every windowed signal . We do not use the above evolutionary estimator in the low-resolution partitioning, because, at that stage, the signal is long and we are seeking an approximate, computationally efficient solution. The improvement in accuracy that such evolutionary estimator could provide compared to other methods is not significant at that stage, whereas the time required for computing such an estimator is several orders of magnitude longerthanthatrequiredfortheouter vectorproductcomputation. In contrast, at the high-resolution partitioning stage, we are seeking a highly accurate estimation of the autocorrelation and, therefore, the evolutionary estimator needs to be applied. We use the number and the magnitude of eigenvalues to detect changes in the signal structure as described next. 1) Local Segmentation Algorithm: In this section, we describe the algorithm for high-resolution partitioning. Partitioning is performed by using two short windows and of equal length and a long window of length , where is the window number, and superscripts , stand for short and long window, respectively [Fig. 1(a), (b)]. From preliminary experiments, we found that provide reliable the 120-s-long initial length for the partitioning for the HRV signals under study.
The proposed algorithm first compares the number of eigenvalues (of the covariance matrix), corresponding to a fixed proportion of the signal energy, in the first and the second short window with those in the long window and then analyzes energy changes between the two consecutive short windows. The basis vectors are the eigenvectors of the covariance matrix estimated using (5). If the eigenvectors are ordered according to the amplitudes of the corresponding eigenvalues, so that the first eigenvector corresponds to the largest eigenvalue, the second eigenvector corresponds to the second largest eigenvalue, etc., the most of the information contained in the signal can be represented by the first several (most significant) basis vectors. The number of eigenvalues of the covariance matrix required for approximating a fixed proportion of the signal energy reflects its dimensionality (i.e. number of orthogonal eigenvectors required to span the corresponding subspace). If a change in the signal structure (dimensionality) occurs between two consecutive short windows, then the long window would require more eigenvalues (and corresponding eigenvectors) to represent a fixed proportion of energy than the first or the second short window. Thus dimensionality or complexity of the signal in the long window would be greater than that and ). The algorithm in each short windows ( compares the number of eigenvalues in the long window with the number of eigenvalues in the short windows and . To avoid repetitive statements in the algorithm, we describe the comparison between the long window and the only. Comparison between the long first short window can be done window and the second short window in a similar way. At the next step, the energies and of the windowed signals corresponding to and are computed and compared. Because the sum of eigenvalues is equal to the expected value of the energy of the signal and because each eigenvector has unit energy, energy changes between the short windows could be detected by comparing the sums of the eigenvalues in each segment [14]. The segmentation algorithm consists of two parts: 1) boundary detection [Fig. 2(a)]; 2) boundary optimization [Fig. 2(b), (c)]. In the first part, the location of the boundary is determined approximately. In the second part, this location is adjusted to obtain an optimal boundary. Boundary detection [Fig. 2(a)]: Step 1) Find the covariance matrix of the signal in the short window [see Fig. 1(a)] by using (5), calculate the eigenand order them values of the covariance matrix . Next, deso that that corretermine the number of eigenvalues spond to 90% of the total energy of . (Details regarding the choice of the energy threshold are proof vided in Section II-B-2). Find the total energy . the signal Step 2) Repeat step 1 for the signal in the long window , [see Fig. 1(b)] to find the number of eigenvalues required for representing 90% of the energy of the . Next, find the total energy of the signal
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Fig. 2. Flow diagram of the segmentation method, which includes boundary detection (a) and boundary optimization (b) and (c). T denotes the threshold value defined in Section II-B2.
signal
, .
The eigenvectors in the short and in the long windows might be different, however, dimensionality
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of the signal subspace (the number of the eigenvectors required to span the subspace) is invariant under any rotational transformation (mapping) of the orthogonal set of basis vectors. Therefore, even if the sets of eigenvectors for two segments are different, the number of eigenvalues (eigenvectors) required to represent a fixed proportion of the energy is independent of the choice of the eigenvectors and depends only on the signal dimensionality. The number of eigenvalues (as opposed to their magnitudes) is also independent of the segment length, because the number of orthogonal basis vectors required to span the signal space does not depend on the length of the signal. Because the long window always overlaps the short one, the dimensionality of the signal in the long window can become greater than that in the short window if and only if the number of orthogonal components increases from the short window to the long one. This, in turn can happen if a change occurs in the second half of the longer window, which is not included in the shorter window. Step 3) Compare the number of eigenvalues and the energy according to the following criteria: , then there is a change a) If in the structure (dimensionality) of the signal, do the boundary optimization, otherwise check condition 2, or b) If then there is a change in the energy of the signal, do the boundary optimization. Constants , are determined a-priori from the statistics (variances) of the corresponding distributions.1 If none of the above criteria is met, then and are approximately the same, and there is no need for partitioning. In this case, move all segmentation windows by the length of the short window and repeat the first two steps. If either criand are different, in other words, a terion is met, change in the structure (criterion 1) and/or energy (criterion 2) and (see Fig. 1). Then, we perform occurred between the boundary optimization. Boundary optimization: By using boundary detection we and have already determined that a change from has occurred at some point between and . Now, we are searching for the location of this point of change as described below: a) If the first criterion in boundary detection is met by re[Fig. 2(b)]: Shrink the long window ducing its length point-by-point consecutively as shown in Fig. 1(d). Compare the number of eigenvalues in the long and short windows using the conditions described in the boundary detection. b) If the second criterion in boundary detection is met [Fig. 2(c)]: Shift the second short window backward point-by-point consecutively as shown in 1In
our study of human HRV, = 1, = :75.
Fig. 3. (a) Simulated signal comprised of 3 sections using sines of length/frequency: 1) 70 points/.12 Hz, 2) 40 points/.12 and .35 Hz, 3) 55 points/.35 Hz. Segments were obtained using the following thresholds (the proportion of the signal energy represented by the eigenvalues used in the segmentation): 90% (b), 75% (c), and 100% (d). Panel e shows normalized eigenvalues for the section A marked by dashed lines in panel (a). Note, that the first 5 eigenvalues represent 92% of the signal energy; these 5 eigenvalues will be selected with the 90% threshold. Only three first eigenvalues (81%) will be selected with the 75% threshold, whereas with the 100% threshold, all 32 eigenvalues will be used.
Fig. 1(c). Compare the energies in the two short windows using the conditions described in boundary detection. c) When either condition is satisfied for a -point shift, then . the location of the boundary is Once the optimal boundary is located, the procedure is repeated starting from that point. 2) Energy Threshold for Comparison of the Numbers of Eigenvalues in Short and Long Segmentation Windows: The total number of the basis vectors (eigenvectors) required for accurate reconstruction of a signal, which has a full-rank covariance matrix, is equal to the signal length. Therefore, a 100% threshold would provide a perfect representation, since in this case the number of the eigenvalues (and corresponding eigenvectors) would be equal to the signal length. Since the length of the short and the long windows are always different, the number of eigenvalues in the two windows would also be different all the time, and the algorithm would partition the signal into small segments regardless of the signal properties (Fig. 3). On the other hand, a few of the most significant eigenvectors that correspond to the largest eigenvalues contain most of the information, whereas the rest of the eigenvalues are very close to zero and do not contribute significantly to the description of the signal structure. Thus, we were seeking a threshold that includes only the most significant eigenvalues
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Fig. 4. Segmentation of the simulated signal in Fig. 3 delayed by 32 samples (a), this signal advanced by 8 samples (b), and by 16 samples (c). Boundary localization error at each (1–16 sample) shift (d). The length of the initial segmentation window used was 16 samples. Segmentation of an HRV signal obtained during head-up tilt (e), signal advanced by 14 samples (f), and by 29 samples (g). Boundary localization error at each (1–30) sample shift (h). The length of the initial segmentation window used was 30 samples. Note, that the localization error in (d) and (h) were identical for all segment boundaries.
and excludes the near-zero ones. The 90% threshold has been found empirically to provide this result for the HRV signals. Representing 90% of the signal energy, by definition, corresponds to a 10% representation error. Since the eigenvalues are selected in a decreasing order (from largest to smallest), this representation includes the contributions of the most significant basis functions (eigenvectors) and excludes the least significant ones. Choosing a lower threshold ( 90%) would result in a poorer representation of the local signal and, therefore, the algorithm could become inaccurate and overlook the changes in the signal structure. 3) Analysis of Time-Shift Invariance: Time-shift invariance has been examined using previously described methodology for both simulated and human HRV signals [16], [17]. To examine the effect of time shifts on the accuracy of segmentation, we first partitioned the signal without a shift and determined the original boundaries. Then, the signal was shifted point-by-point consecutively (Fig. 4), and partitioned at each shift to obtain new boundaries. If the partitioning is time-shift invariant, new boundary locations should follow the signal shift every time. The error calculated as a difference between the new and the original boundaries (shifted with the signal) is shown in Fig. 4(d), (h). The error was zero at each shift indicating that the partitioning algorithm is time-shift invariant. Because point-by-point shifting produced every possible phase shift in
the signal, the boundaries of the segmentation windows usually did not match the exact boundaries of the signal segments. Nevertheless, the algorithm accurately determined location of the boundaries in both simulated and human HRV signals. 4) Comparison With Other Techniques: The local segmentation algorithm (Algorithm I) has been compared with another segmentation algorithm (Algorithm II), which is described by Andersson [9]. Algorithm II builds AR or ARMA models assuming that the model parameters are piecewise constant over time; it splits the data record into segments over which the model remains constant. The model order and the noise variance need to be chosen a-priori. Performance of these two segmentation algorithms is demonstrated using a simulated signal (Fig. 5). The signal consists of concatenated sinusoids with different amplitudes and frequen. Solid vertical cies embedded in white noise lines correspond to the actual segments. Fig. 5(a) shows the segments (vertical dotted lines) obtained using the algorithm I. Fig. 5(b) shows the results obtained from algorithm II. Algorithm I performed reasonably well, giving the segments that were close to the original ones. Although the algorithm II produced similar results [Fig. 5(b)], it could not detect a change around 225 samples. Fig. 5(c) shows that the RMS segmentation error over a wide range of SNR was smaller for the Algorithm I compared to the Algorithm II.
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’s are the expansion coefficients, and correwhere sponds to the length of the th segment. In (7), the basis are derived from the KL basis functions functions in each segment. The expansion coefficients are given by ; , . The functions are the product of s, extendefined on the th sion of the orthonormal KL bases , and the window shown in Fig. 1. time interval Properties of the window and the extension of the orthonormal are given in [12], [18]. Then functions
The proof of orthonormality of the basis functions be found in [12].
(8) can
B. Time-Frequency Representation of the Partitioned Signal and of the Orthogonal Basis
Fig. 5. Performance of the two segmentation algorithms. Algorithm I in Panel (a) with the segmentation window length = 16 and Algorithm II in Panel (b) with the model order of 7 are compared by using a simulated signal (SNR = 3:1 dB). The solid vertical lines indicate the actual segments; the dotted lines represent the boundaries obtained by the corresponding algorithms. The RMS error of the boundary detection vs. SNR for the Algorithm I (solid line) and Algorithm II (dotted line) is shown in Panel (c).
III. PARTITIONED TIME-FREQUENCY REPRESENTATION In this section, we consider the time-frequency representation of nonstationary segments of interest, which were detected by means of the global, low-resolution partitioning and then locally partitioned into less nonstationary components via the high-resolution partitioning. A. Orthonormal Expansion of the Partitioned Signal , and it is partiAssume we have a nonstationary signal by using the segtioned into overlapping segments mentation method introduced in the previous section
Although partitioning facilitates the analysis of time-dependent variations in the structure of multicomponent signals, widerange changes often produce smeared TFRs. Therefore, two approaches to the time-frequency analysis of multicomponent signals are proposed. For signals that exhibit changes in a few frequency components, a direct time-frequency representation is applied. For signals with complex, multicomponent changes, partitioning and exposure of the most significant changes in each time segment is needed to avoid the smearing caused by direct time-frequency representation. This is achieved by: 1) extracting the most significant eigenvectors in each time segment; 2) representing each eigenvector in each time segment in the time-frequency plane (decomposed time-frequency representation); 3) constructing the time series of the corresponding representations by concatenating all segments. Examples of “direct” and “decomposed” time-frequency representations are shown later in the Results section. Contrary to other orthogonal representations, the basis funcused in our representation do not have a stantions dard form and are signal dependent. Furthermore, they are ordered not according to the frequency content but according to the weights of the corresponding eigenvalues. As a result, the exdo not carry any information about spepansion coefficients cific frequencies of their basis vectors. To obtain the frequency information, one can use the following Fourier bases. Representing in terms of exponentials, or , where , and substituting it in (8), we get
(9) (6) If
is represented as a linear combination of basis functions , i. e., , then (6) becomes
Then, replacing tation of :
in (7), results in the following represen-
(7)
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(10)
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TABLE I COMPARISON OF SYMPTOMATIC AND ASYMPTOMATIC SUBJECTS UNDERGOING HEAD-UP TILT TEST
Fig. 6. (a) A simulated signal obtained by concatenating segments with different properties, and its time-frequency spectra obtained using (b) OPTR (segmentation window length of 30 samples), (c) the short-time Fourier transform using a 64-point Hanning window, (d) the Wigner-Ville, (e) the Smoothed Pseudo Wigner-Ville, and (f) the Choi-Williams ( = 1) distributions. All spectra are shown on a logarithmic scale.
which is similar to the evolutionary spectral representation [19], and as such, the time-frequency kernel is defined as (11) is given by Thus, the evolutionary spectrum of where corresponds to time and corresponds to frequency. Detailed theoretical analysis of this TFR can be found in [19]. In summary, we used the KL expansion, first, for segmentation and then for time-frequency representation of the most significant eigenvectors, since this expansion allows selection of the best (in a least-squares sense) eigenvector set. The Fourier expansion was applied to obtain the frequency information, which could not be derived from the KL basis. IV. VALIDATION OF ORTHONORMAL BASIS PARTITIONING AND TIME-FREQUENCY REPRESENTATION (OPTR) A. Simulations ToverifytheperformanceofOPTR,weapplyittosimulatedsignals composed of sinusoidal and linear FM signals and sinusoids of different frequencies [Fig. 6(a)]. The time-frequency spectrum obtained using OPTR [Fig. 6(b)] is compared with the one obtained using the short-time Fourier transform(a 64-point Hanning window), the Wigner-Ville, the smoothed pseudo Wigner-Ville, and the Choi-Williams distributions [Fig. 6(c)–(f)]. Notice that the structure of this signal is relatively simple, and its time-frequency representation is sufficiently clear, obviating the need for time-frequency analysis of the most signifi-
cant basis vectors. OPTR produces sharp inter-segment boundaries that coincide with the actual boundaries of the time segments, whereas in the other methods, the boundaries between adjacent time segments are blurred. Note also that the dominant frequency content of each time segment in the OPTR is “sharper” than in the other TFR. When two simultaneous sinusoidal components are present (Fig. 6, from 200 to 280 time samples), OPTR does not suffer from the cross terms in contrast to the Wigner-Ville and the Choi-Williams TFR. On the other hand, in the case of the linear or sinusoidal FM, OPTR produces artificial boundaries, which is not the case with the other methods. However, as indicated in [19], improvements in the time-frequency spectrum can be obtained by estimating the instantaneous frequency. B. Heart Rate Variability Measurement OPTR was tested on heart rate variability (HRV) signals obtained from human subjects undergoing tilt test or 24-h ambulatory monitoring. Electrocardiographic signals were recorded continuously using ambulatory (Holter) recorders. ECG data were sampled at 400 Hz, and QRS complexes were detected using a commercial scanning system and custom software and verified by a cardiologist. The RR-intervals between normal QRS complexes were extracted, and a regularly spaced time series was sampled at 1 Hz using a boxcar low-pass filter [20]. Gaps in the time series resulting from noise or ectopic beats were filled in with linear splines, which can cause a small reduction in high-frequency power but do not affect other components of the power spectrum [4]. Series of RR-intervals were high-pass filtered (FIR Remez filter, cutoff frequency 0.005 Hz) to remove components below 0.005 Hz [21]. C. Head-Up Tilt Protocol The experiment consisted of three phases: baseline in a supine position during 10 min, passive, head-up 70 tilt during 45 min followed by 10-min rest in a supine position. The protocol was approvedbytheInstitutionalReviewBoardoftheUniversityofPittsburgh. Each subject was asked to sign an informed consent prior to thestudy.OPTRwereappliedtotheHRVsignalsobtainedfrom20 subjects undergoing the test (Table I). The subjects were divided
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into asymptomatic (Group 1) and symptomatic (Group 2); none of the subjects had structural heart disease. D. Statistical Analysis To analyze energy distribution in time-frequency plane, we used the low-order conditional time-frequency moments, which effectively compress the information by tracking the changes in important physical properties of the signal [22]. The mean and the lower frequency, the upper frequency bounds, where 70% of the energy was contained, were used to quantify changes in the time-frequency distributions. In preliminary studies, these frequency bounds were found to discriminate subjects who were symptomatic during tilt testing from asymptomatic ones. Comparisons between the groups shown in Table I were performed using nonparametric Mann–Whitney U-test. V. RESULTS A. Head-Up Tilt Fast change from supine to vertical body position causes an increase in blood volume below the diaphragm and a decrease in the blood flow in the organs above the level of the heart [2]. Because steady blood flow is critical for normal brain functioning, changes in body position cause an immediate increase in the sympathetic nervous system activity, which maintains normal level of blood flow to the brain by contracting peripheral blood vessels and increasing cardiac output. A typical response to head-up tilt includes an increase in heart rate and a decrease in the high frequency spectral power of HRV. Note that abrupt or multicomponent changes in the structure of the HRV signal could be difficult to discern using “direct” time-frequency representation (Fig. 7). For such signals, the time-frequency representation of the most significant basis vectors provides effective filtering of the information from the least significant eigenvectors to reveal the dominant changes in the signal properties (Fig. 8). At the beginning of the tilt, RR-intervals decrease in most patients and remain short until return to the supine position (Fig. 7(a)). In asymptomatic patients (Fig. 8, right column), the spectral energy is stable during the test and concentrated near 0.1 Hz, the frequency of the sympathetically modulated vasomotor tone. It has long been known, that efficient adjustment to the vertical body position is accompanied by an increase in vascular activity [2]. OPTR of the eigenvectors reveals the underlying patterns and shows that symptomatic subjects have unstable and widely spread energy distribution (Fig. 8, left column). The differences between the energy distributions in the groups of symptomatic and asymptomatic subjects were confirmed statistically with respect to the upper and lower frequency bounds and the mean frequency of the most significant basis vectors (38%, 73%, and 45% lower in Group 2 than in Group 1, Table I). Note that changes in the traditional HRV indices (the ratio of the low (0.04–0.15 Hz)/high (0.15–0.4 Hz) frequency power and the high frequency power of the spectrum) during the tilt allowed separation of this patient group according to the presence or absence of the structural heart disease [23].
Fig. 7. A representative example of the HRV signal obtained from asymptomatic subject during head-up tilt (a), the high-pass filtered signal with a Remez filter, cutoff frequency 0.005 Hz (b), and time-frequency representations of the filtered signal obtained using OPTR (c), the short-time Fourier transform with a 256-point Hanning window (d), the Smoothed Pseudo Wigner-Wille with a Kaiser window (e), and the Choi-Williams distribution ( = 1) with a Kaiser window (f). Dashed lines mark the onset and offset times of the tilt.
However, broader energy distribution, which includes very low frequency power, is required to find the HRV differences between the symptomatic and asymptomatic subjects [24]. VI. CONCLUSION We described a new approach to the analysis of nonstationary, multicomponent HRV signals that allows detection of abrupt changes and changes in the structure. The proposed technique provides both a representation and a time-varying spectrum of a signal. The performance of the method has been compared with the short-time Fourier transform, the Wigner-Ville, the smoothed pseudo Wigner-Ville, and the Choi-Williams TFR. The OPTR had a relatively poor time resolution in quantifying gradually changing signals, such as a chirp, but accurately represented abrupt or multicomponent changes (for example, two simultaneous sinusoidal components between 200 and 280 time samples in Fig. 6) in the simulated and real-life HRV signals (shown in Fig. 8). Time-frequency representations have long been used to demonstrate HRV changes during controlled physiological and pharmacological experiments. In these controlled conditions,
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Fig. 8. Time-frequency representations of the 3 most significant HRV eigenvectors during head-up tilt in two symptomatic subjects [left column, (a), (e)] and two asymptomatic subjects [right column, (i), (m)]. The signal shown in (i) is the same as the signal shown in Fig. 7(a). Dashed lines in (a), (e), (i), and (m) correspond to the tilt start and end times.
changes in the signal structure usually involve a few, distinct frequency components, which could be readily detected by visual inspection and quantified by the spectral energy integration over the range of interest [21]. In uncontrolled, real-life conditions, however, simultaneous or abrupt changes in multiple, overlapping frequency components can be difficult to discern using “direct” time-frequency representations (Fig. 7). To obviate this problem, we propose a generalized approach, referred to as the orthonormal basis partitioning and time-frequency representation (OPTR). Our method is based
on the number of the eigenvalues required to represent a fixed proportion of signal energy. This number represents signal dimensionality (i.e. the number of orthogonal basis vectors required to span the signal space) [8]. Because dimensionality represents a fundamental characteristic of the signal structure, the eigenvalues (dimensionality)-based approach is less sensitive to noises and artifacts than AR-model based or spectral segmentations. Changes in signal dimensionality indicate pronounced structural changes; therefore, analysis of eigenvalues provides a reliable segmentation of multidimensional signals
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 5, MAY 2005
[8]. Of note, analysis of the eigenvalues can be also used to quantify the degree of signal nonstationarity. First, we partition the signal into the segments with distinct properties and then represent each segment in the time-frequency plane using the basis functions that are derived from the signal itself. The idea of using eigenvalues for segmentation of a nonstationary, multidimensional signal is not new. Basseville et al. [25] analyzed changes in the eigenvalues and eigenvectors of the state transition matrix to detect small changes in the characteristics of a vibrating mechanical system. OPTR can be considered a simplified implementation of a more general eigenstructure analysis that allows partitioning of nonstationary, multidimensional signals into the less nonstationary segments. A combined application of a long-term and a short-term window, corresponding to the periods before and after a possible change, has been described in [8]. The authors also described the choice of the window size as a tradeoff between the estimation precision and the mean time between changes actually present in the processed signal. They concluded that this implementation unavoidably introduces a limitation of the resulting algorithm with respect to the presence of frequent changes or equivalently short segments. The same problem of choice of a window length, obviously, applies to OPTR as well. In our studies, we used a-priori knowledge about the structure of HRV signals and extensive experimentation to select the initial window length that was slightly longer then the longest period of the studied oscillations. Importantly, the proposed segmentation was time-shift invariant for simulated and HRV signals with various relationships between the borders of segmentation window and the location of structural changes. Various applications of this approach for different types of HRV signals are possible. For hours-long signals with multicomponent structure, first, an efficient low-resolution partitioning could be applied to select the transients and short segments of interest. For short segments, a more computationally demanding high-resolution partitioning could be used with subsequent time-frequency representation. Finally, for such complex multicomponent signals as the one shown in Fig. 7, we used the time-frequency representation of the most significant basis vectors, which provided an effective compression of the information and reveal the underlying dominant pattern (Fig. 8).
REFERENCES [1] S. Akselrod, D. Gordon, F. A. Ubel, D. C. Shannon, A. C. Barger, and R. J. Cohen, “Power spectrum analysis of heart rate fluctuation: a quantitative probe of beat-to-beat cardiovascular control,” Sci., vol. 213, pp. 220–222, 1981. [2] L. Keselbrener and S. Akselrod, “Selective discrete Fourier transform algorithm for time-frequency analysis: method and application on simulated and cardiovascular signals,” IEEE Trans. Biomed. Eng., vol. 43, no. 8, pp. 789–803, Aug. 1996. [3] V. Shusterman, B. Aysin, K. P. Anderson, and A. Beigel, “Multidimensional rhythm disturbances as a precursor of sustained ventricular tachyarrhythmias,” Circ. Res., vol. 88, pp. 705–712, 2001. [4] V. Shusterman, B. Aysin, R. Weiss, S. Fahrig, S. Brode, V. Gottipaty, D. Schwartzman, and K. P. Anderson, “Autonomic nervous system activity and spontaneous initiation of ventricular tachycardia,” J. Am. Coll. Cardiol., vol. 32, pp. 1891–1899, 1998. [5] T. A. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis,” Philips J. Res., vol. 35, pp. 276–300, 1980.
[6] H. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 6, pp. 862–871, Jun. 1989. [7] B. Lovell and B. Boashash, “Segmentation of nonstationary signals with applications,” in Proc. IEEE Int. Conf Acoustic, Speech and Signal Processing, 1988, pp. 2685–2688. [8] M. Basseville and N. Nikiforov, The Detection of Abrupt Changes—Theory and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1993. [9] P. Andersson, “Adaptive forgetting in recursive identification through multiple models,” Int. J. Control, vol. 42, pp. 1175–1193, 1985. [10] J. T. Tou and R. C. Gonzales, Pattern Recognition Principles. Reading, MA: Addison-Wesley, 1979. [11] B. Aysin, L. F. Chaparro, I. Grave, and V. Shusterman, “Detection of transient changes in heart rate variability signals using a time-varying Karhunen–Loeve expansion,” in Proc. IEEE Int. Conf. Acoustic, Speech and Signal Processing, 2000, pp. 3586–3589. [12] X. G. Xia, B. W. Suter, and M. E. Oxley, “Malvar wavelets with asymmetrically overlapped windows,” IEEE Trans. Signal Process., vol. 44, no. 3, pp. 723–728, Mar. 1996. [13] H. S. Malvar, “Lapped transforms for efficient transform/subband coding,” IEEE Trans. Signal Process., vol. 38, no. 6, pp. 969–978, Jun. 1990. [14] B. Aysin, “Orthonormal-basis partitioning and time-frequency representation of non-stationary signals,” Ph.D. dissertation, Univ. Pittsburgh, Pittsburgh, PA, 2002. Available: http://etd.library.pitt.edu/ETD/available/etd-12022002-131851/unrestricted/AYSIN_121102.PDF. [15] A. S. Kayhan, A. El-Jaroudi, and L. F. Chaparro, “Data-adaptive evolutionary spectral estimation,” IEEE Trans. Signal Process., vol. 43, no. 1, pp. 204–213, Jan. 1995. [16] I. Cohen, S. Raz, and D. Malah, “Shift invariant wavelet package bases,” in Proc. ICASSP-95, 1995, pp. 1081–1084. [17] P. Carre and C. Fernandez-Maloigne, “A nonuniform, shift-invariant, and optimal algorithm for Malvar’s wavelet decomposition,” in Proc. IEEE-SP Int. Symp. Time-Frequency and Time-Scale Analysis, Pittsburgh, PA, 1998, pp. 45–48. [18] Y. Meyer, Wavelets: Algorithms and Applications. Philadelphia: SIAM, 1993, pp. 75–85. [19] R. Suleesathira, L. F. Chaparro, and A. Akan, “Discrete evolutionary transform for the time-frequency signal analysis,” J. Franklin Inst., vol. 337, pp. 347–364, 2000. [20] R. Berger, S. Akselrod, D. Gordon, and R. J. Cohen, “An efficient algorithm for spectral analysis of heart rate variability,” IEEE Trans. Biomed. Eng., vol. BME-33, pp. 900–904, 1986. [21] L. T. Mainardi, A. M. Bianchi, and S. Cerutti, “Time-frequency and time-varying analysis for assessing the dynamic responses of cardiovascular control,” Crit. Rev. Biomed. Eng., vol. 30, pp. 175–217, 2002. [22] P. Loughlin, F. Cakrak, and L. Cohen, “Conditional moment analysis of transients with application to helicopter fault data,” Mech. Syst. Signal Process., vol. 14, pp. 511–522, 2000. [23] V. Shusterman, A. Beigel, S. I. Shah, B. Aysin, R. Weiss, V. K. Gottipaty, D. Schwartzman, and K. P. Anderson, “Changes in autonomic activity and ventricular repolarization,” J. Electrocardiol., vol. 32, pp. 185–192, 1999. [24] S. I. Shah, V. Shusterman, B. Aysin, S. Flanigan, D. Cavlovich, S. Brode, V. Gottipaty, R. Weiss, and K. P. Anderson, “Susceptibility to syncope at baseline: a novel approach,” in PACE, vol. 22, 1999, p. 757. [25] M. Basseville, A. Benveniste, and G. Moustakides, “Detection and diagnosis of abrupt changes in modal characteristics of nonstationary digital signals,” IEEE Trans. Inf. Theory, vol. IT-32, no. 3, pp. 412–417, May 1986.
Benhur Aysin received the B.S. degree in electronics and telecommunication engineering from Istanbul Technical University, Istanbul, Turkey, in 1991 and the M.S. and Ph.D. degrees in electrical engineering from University of Pittsburgh, Pittsburgh, PA, in 1995 and 2002, respectively. From 1998 to 2003, he was a Research Specialist at the University of Pittsburgh School of Medicine where he worked on developing signal processing algorithms for biomedical signals. His research interests include digital signal processing, biomedical signal processing, time-frequency analysis, signal segmentation, on-line biomedical computing. Since 2003, he is a Product Engineer at Ansar Group Inc., Philadelphia, PA.
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AYSIN et al.: OPTR OF CARDIAC RHYTHM DYNAMICS
Luis F. Chaparro (S’71–M’72–SM’90) was born in Sogamoso, Colombia, in 1947. He received the B.S. degree in electrical engineering from Union College, Schenectady, NY, in 1971, and the M.S. and Ph.D. degrees in electrical and computer science from the University of California, Berkeley, in 1972 and 1980, respctively. Since 1979, he has been with the Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA, where he is currently an Associate Professor. His research work has focused on statistical signal processing and multidimensional systems theory with applications in biomedical engineering, wireless communications and video communications. He is currently interested in the processing of biomedical nonstationary signals via the evolutionary spectral theory, and the application of time-frequency theory and video compression to wireless communications. Prof. Chaparro has been Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING and of the Journal of the Franklin Institute, member of the IEEE Signal Processing technical committee on Statistical Signal and Array Processing, and organization chair of the IEEE International Symposium in Time-Frequency and Time-Scale held in Pittsburgh in 1998.
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Vladimir Shusterman received the B.S. degree in mathematics, the M.D. degree in 1985, and the Ph.D. degree in biomedical engineering in 1991 from the Novosibirsk State Medical Institute and Institute of Medical and Biological Cybernetics, Novosibirsk, Russia. He completed a postdoctoral fellowship in Biomedical Engineering at Tel Aviv University, Tel Aviv, Israel, in 1993–1995 and in cardiac electrophysiology in 1995–1996 at the University of Pittsburgh, Pittsburgh, PA. He is currently an Assistant Professor and Director of Noninvasive Cardiac Electrophysiology Laboratories at the University of Pittsburgh. His research interests include detection and quantification of changes in biological signals, time series analysis and segmentation, development of algorithms for dynamical analysis of physiological systems, and mathematical modeling of cardiovascular electromechanical functions. Dr. Shusterman is a member of the International Society for Computerized Electrocardiology, Cardiac Electrophysiology Society, and the Heart Rhythm Society (NASPE).
Ilan Gravé received the B.Sc. and M.Sc. degrees in physics and the B.Sc. degree in electrical engineering from Tel Aviv University, Tel Aviv, Israel, and a Ph.D. in applied physics (1993) from the California Institute of Technology, Pasadena. He has held positions in high tech industries, in research and academics in Israel, Italy, and the USA. Among them, he managed advanced projects at the Israeli Aircraft Industries (IAI,) was a senior scientist at the Fondazione Ugo Bordoni, at the ministry of Post and Communications in Rome, Italy; he was also with Department of Electrical Engineering at the University of Pittsburgh, and, since 2002, he has been an Associate Professor of Physics and Engineering at Elizabethtown College, Elizabethtown, PA. His interests and contributions have been in various fields, including superconductivity, semiconductors, quantumconfined devices, nonlinear optics, infrared detectors, semiconductor lasers and, more recently, biomedical signal processing.
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