Adaptive Ar Modeling of Nonstationary Time Series by ... - IEEE Xplore

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 45, NO. 5, MAY 1998

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Adaptive AR Modeling of Nonstationary Time Series by Means of Kalman Filtering Matthias Arnold,* Wolfgang H. R. Miltner, Herbert Witte, Member, IEEE, Reinhard Bauer, and Christoph Braun Abstract—An adaptive on-line procedure is presented for autoregressive (AR) modeling of nonstationary multivariate time series by means of Kalman filtering. The parameters of the estimated time-varying model can be used to calculate instantaneous measures of linear dependence. The usefulness of the procedures in the analysis of physiological signals is discussed in two examples: First, in the analysis of respiratory movement, heart rate fluctuation, and blood pressure, and second, in the analysis of multichannel electroencephalogram (EEG) signals. It was shown for the first time that in intact animals the transition from a normoxic to a hypoxic state requires tremendous short-term readjustment of the autonomic cardiac-respiratory control. An application with experimental EEG data supported observations that the development of coherences among cell assemblies of the brain is a basic element of associative learning or conditioning. Index Terms— Associative learning, blood pressure, cardiorespirography, (partial) coherence, conditional somato-sensoric stimuli, EEG, heart rate, Kalman filter, linear dependence, respiration, synchrony, time-varying multivariate autoregression.

I. INTRODUCTION

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ENERALLY, physiological systems can be modeled by means of multi-input–multi-output systems where the interconnections between input and output signals are nonlinear, time-varying and contaminated by noise [1]. Often, a linear approach can be accepted because within defined ranges the working characteristic is linear. Therefore, multivariate and time-varying (linear) analysis methods can be seen as efficient tools for functional identification of physiological systems and phenomena. This is also true for such systems which consist of accessible subsystems (i.e., for additional internal signals) and for multichannel signals [electroencephalogram (EEG), electrocardiogram (ECG)]. Multivariate spectral analysis of physiological signals is performed predominantly by means of fast Fourier transform Manuscript received August 26, 1996; revised November 19, 1997. This work was supported by the German Federal Ministry of Research and Technology under Project: “Clinic-oriented neurosciences,” 01 ZZ 9602, and by the Deutsche Forschungsgemeinschaft, Schwerpunktprogramm “Mechanismen assoziativen Lernens,” Mi 265/3-1. Asterisk indicates corresponding author. *M. Arnold is with the Institute of Medical Statistics, Computer Sciences and Documentation, Friedrich Schiller University Jena, D-07740 Jena, Germany (e-mail: [email protected]). W. H. R. Miltner is with the Institute of Psychology, Friedrich Schiller University Jena, D-07740 Jena, Germany. H. Witte is with the Institute of Medical Statistics, Computer Sciences and Documentation, Friedrich Schiller University Jena, D-07740 Jena, Germany. R. Bauer is with the Institute of Pathophysiology, Friedrich Schiller University Jena, D-07740 Jena, Germany. C. Braun is with the Institute of Medical Psychology and Behavioral Neuroscience, Eberhard Karls University Tuebingen, D-72070 Tuebingen, Germany. Publisher Item Identifier S 0018-9294(98)02873-0.

(FFT). This is mainly due to the numerical efficiency of the algorithm. However, this method is interval-related, demands stationarity of the segments studied, and suffers from an algorithm-specific discrepancy between time and frequency (where denotes the interval resolution since only length) frequency points are taken into consideration. To keep the benefits of the FFT also in the analysis of nonstationary signals with temporarily changing spectra it is common to makes use of the short-time Fourier transformation (STFT) where the FFT is applied to short overlapping sequences which are assumed to be stationary. However, this approach increases the problem of limited frequency resolution. Because such disadvantages can be overcome by parametric spectral analysis based on linear models, the parametric approach is very attractive for processing data from various areas, e.g., EEG and cardiorespiratory data. Furthermore, linear modeling is altogether a versatile tool for signal analysis, as it includes model selection, prediction, and description of signals by a few (model) parameters. Because of their general nature vector autoregressive (VAR) models (1) are very popular among the linear models. Compared with methods based on FFT the use of such models makes it possible to calculate spectral measures with a high-frequency resolution: the spectral density can be calculated at each frequency point using the model parameters (c.f. Section IIA). On the other hand these models again are only suited to analyzing stationary signals or signals with particular types of nonstationarity [2]. One solution is to fit separate models to consecutive segments of a fixed sufficiently small length or to segments of variable length which result from suitable segmentation algorithms for the detection of change points [3]. Taking into account that the signals are samples from time-series with varying spectral properties, it is also natural to assume the coefficients to be time dependent (2) Several algorithms for the estimation of scalar autoregressive (AR) and VAR models with either constant or timedependent coefficients (random coefficients [4], stochastic [5], [6], or deterministic [7] evolution of coefficients) are currently available. Since there are more parameters than data ordinary least squares (LS) and maximum likelihood methods cannot

0018–9294/98$10.00  1998 IEEE

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be used to estimate the time course of the AR coefficients. An established approach to handling this problem is to express the time-varying coefficients as a linear combination of a set of known base functions (e.g., orthogonal polynomials [7]). Consequently, only a small number of weights have to be estimated. However, this approach presumes a deterministic evolution of the coefficients which, from a theoretical point of view, seems to be inadequate in applications where a pure stochastic system is assumed. Furthermore, known modeling procedures are interval-related and consequently not capable of on-line processing. Up to now, only little attention has been focused on VAR models with time-varying coefficients where the evolution law is assumed to be stochastic (i.e., random walk), whereas for scalar models this approach has been extensively investigated [6]. In this study, an established method for time-varying (adaptive) AR modeling of a scalar time series based on Kalman filtering ([8], [9]) is adapted to the multivariate case. Furthermore, procedures for the subsequent estimation of spectral density and derived spectral parameters—coherence and partial coherence—are described. The time-varying modeling makes it possible to calculate these parameters instantaneously. An adaptive frequency selective measurement of linear dependences between different signals can, thereby, be implemented. The reliability of the proposed algorithm was tested on simulated data and compared with standard methods. The usefulness of the derived instantaneous measures of coherence and partial coherence for the description of physiological phenomena were demonstrated by means of two examples. A trivariate time-varying AR model was used to investigate instantaneously the dependencies between the cardiorespiratory parameters: heart rate fluctuations (HRF), respiratory movement (RM), and arterial blood pressure (aBP) fluctuation. This analysis was carried out during the transition from normoxia to normobaric hypoxic hypoxia and the early period of hypoxia in an animal experiment. The application in EEG analysis was drawn from a study on associative learning. The association was of a differential nature, i.e., different visual stimuli were associated with a painful stimulus and no pain respectively. Associations between cell assemblies of the brain were investigated by means of time-varying coherence analysis of the EEG. Consequently, the present EEG study aimed at disclosing stimulus-induced synchronization of larger brain areas that became activated coherently during the association. The concept of binding by synchronous activation was suggested by Hebb [10] and has already been supported by several studies (cf. [11]). II. METHODS A. Adaptive AR Model Fitting A scalar AR process of order

is given by (3)

where is assumed to be a sequence of independent and normal distributed random variables with zero expectation and (i.d. ). This variable can be a variance of interpreted as the uncertainty of the prediction of the next signal value by regressing the previous observations with the AR coefficients (prediction error). If the scalar values and are replaced by vectors and matrices respectively, the VAR process (1) results. A basic condition for the use of the Kalman algorithm is that the signal model has a representation in state-space form. Such a model consists of two joined linear equations: the state equation and the observations equation. Generally, the Kalman algorithm can be used with the following state-space representation: (4) (5) and depend only on and where the system matrices past observations . Let denote the covariance matrix . If the terms and are of a process and i.d. , assumed to be i.d. respectively, then the state of the system can be recursively estimated according to (6) The gain matrix

is obtained by the Kalman recursion [6] (7) (8) (9)

is the pseudo-inverse of matrix . where In the univariate case, the use of Kalman filters to fit regression models is common ([6], [8], [9], [12]). A simple state-space representation for the univariate AR processes in (3) that is suitable for a recursive estimation of the AR coefficients can be given by (10) (11) with (12) (13) Obviously, this representation matches (4) and (5). Thus, the AR coefficients can be recursively estimated from the samples , where is the signal length. From the state equation (10) it is obvious that the AR coefficient vector is assumed to be unchanged during the course of the signal analyzed. This is disadvantageous with regard to the adaptability of the algorithm to changes in signal properties: as soon as the estimate of the coefficients converges with the true values, the algorithm will no longer be able to perform further modifications of the estimates because the Kalman gain simultaneously tends irreversibly toward zero.

ARNOLD et al.: ADAPTIVE AR MODELING OF NONSTATIONARY TIME SERIES

A solution to this problem is the assumption that the vector of coefficients performs a random walk ([8], [9]), i.e., that the coefficients vary according to a simple Markov process (14) With this modification of (10), a state-space representation of an AR process with stochastic (time-varying) coefficients is given. Using this model a high adaptation speed can be achieved. On the other hand, since the state of system (5) is conditionally [6], the estimates can be Gaussian with covariance of high variance. Therefore, a smoothing procedure for the estimates should be involved. To maintain the adaptation speed a nonlinear recursive filter of the form

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Again, an estimation of the state vector is an output of the Kalman filter. By inverting the vec-operator we get the estimation of coefficient matrices .1 As in the univariate case, the smoothing procedure provided by (15) and (16) is advised. An instantaneous estimation of the spectral density, which in the multivariate case is a matrix valued function of frequency, can be given in terms of the AR coefficient matrices (23) where (24)

(15) (16) is suggested for each component of the estimated coefficient vector , where is a suitable positive constant. Since , (15) defines a lowpass filter for each , where the cutoff frequency increases with the value of . Such filters cause an exponential smoothing of the signal where the degree of smoothing reduces (decrease of memory) as nears one. In our application, this procedure was used to perform an adaptive estimation of the mean of the estimated AR coefficients. During stable periods (only little variation in the estimates), a good smoothing is performed, whereas throughout sudden strong and directed deviations in the estimates a quick adaptation can be achieved. An instantaneous estimation of the spectral density (updated for each new sample point) can be given in terms of the estimated AR coefficients [13] (17) To make use of the Kalman algorithm in the multivariate case too, it is necessary to develop a state-space representation of the model (1). This can be achieved by rearranging the elements of the matrices of coefficients in vector form using the vec-operator, which stacks the columns of a matrix on top of each other. Then, with the following notation: (18) (19) (20) (where denotes the Kronecker-product of matrices, the dimension of the vector process, and the identity matrix of dimension ) an appropriate state-space representation of the multivariate AR model with stochastic coefficients can be given by (21) (22)

denotes the conjugate complex of . and In the application of the Kalman algorithm the problem arises that the variance and covariance matrices respectively of the measurement noise are not known a priori. Thus, the corresponding terms in (17) and (23) as well as in the Kalman algorithm itself must be replaced by estimates of these quantities. The following recursive adaptive procedures can be used to estimate the variance and covariance matrix respectively of the measurement noise: (25) (26) (27) (28) determines the speed The adaptation constant of adaptation. These procedures have already been used in previous applications [15]. Each iteration of the Kalman algorithm involves multiplications of matrices of maximal dimension and and matrix inversion of dimension . The computational effort of such matrix inversion and multiplication is and essential floating point operations respectively. Consequently, the effort of one Kalman iteration . Since each iteration of the recursive least squares is (RLS) algorithm with forgetting factor [16] involves multiplication of matrices by vectors with maximal dimension , its . The effort of conventional LS estimation [6] effort is of a data segment of length is given by operations. Consequently, the effort of the procedure based on Kalman filtering is higher then the RLS algorithm. If the LS algorithm is applied to data segments according to a sliding window the effort of both algorithms is lower, then for LS estimation for sufficient large . The performances of these algorithms are compared in Section III-A. 1 A similar approach can be found in [14], but the authors did not make use of the random walk of the coefficients. This, however, is disadvantageous for the adaptability of the model.

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B. Measurement of Linear Dependence In [17], measures of linear dependence and feedback between components of multiple time series and their decomposition by frequency were introduced. In the case of twodimensional (2-D) signals, the measure of linear dependence can be given in terms of the coherence function. Generally, coherence of the components and of a multivariate time , is defined as series (29) where denotes the element in the th row and the th column of the spectral density matrix of the process . The idea that a high level of coherence between two components of a multiple time series could be due to the influence of any of the remaining components on both led to the concept of partial coherence [13]. Let be an index vector. The partial coherof and , with the regression on ence removed, can be computed according to (30) (31) (32) (33) By means of (23) and (28), the spectral density matrices can be estimated instantaneously (time-varying). Consequently, (29) and (33) can now be used to calculate the parameters coherence and partial coherence instantaneously (cf. [18]). Thus, it is possible to track the time course of gradation of coupling between components of multiple time series within desired frequency ranges. C. Subjects and Material 1) Analysis of the Cardiorespiratory System: a) Subjects: Recordings were derived from a group of seven male newborn piglets (two days old, body weight around 1819 146 g). b) Experimental protocol: Anesthesia was introduced with an initial gas mixture of 1.5% halothane in 70% nitrous oxide and 30% oxygen, maintained with a dose of 0.5–0.7% halothane, and applied by a mask while the animal spontaneously breathed throughout the whole experiment. A control period of 20 min was recorded under normoxia (PaO between 102 and 142 mmHg). Moderate normobaric hypoxic hypoxia was then applied for 1 h (FiO of 0.12 in the ventilated gas mixture to get an arterial O -saturation of between 45–50% and a PaO of 29–35 mmHg). Then, reoxygenation with the prehypoxic ventilatory gas mixture was performed. After 30 min reoxygenation, ECG, RM, and aBP were recorded over a period of 20 min. For cholinergic blockade a total of 0.5-mg/kg body weight intravenous atropine was injected and again a period of 20 min was recorded.

c) Data recording: The ECG was digitized with a sample rate of 2048 Hz. RM and aBP were sampled with a sampling rate of 128 Hz. For representation of HRF the instantaneous heart rate (HR) was used, where the reciprocal of the interval duration between subsequent R waves is held within the corresponding interval (used in this study) or the subsequent interval (output signal of cardiotachographs). This type of representation is visually informative, most frequently used in clinical practice, and accessible for automatic analysis [19]. The resulting data were checked for artifacts and trends by continuous playback, and disturbed intervals were rejected. The resulting traces were resampled at a rate of 128 Hz. 2) EEG Analysis a) Subjects: Sixteen healthy, right-handed student volunteers (nine female) participated in the experiment. b) Experimental protocol: The Subjects’ task was to learn that a visual stimulus (e.g., of red color) was always terminated by a painful stimulus applied to the left midfinger, whereas no such painful stimulus was given at the end of another visual stimulus (e.g., of green color). For painful stimulation somatosensory intracutaneous electrical stimuli were used (for details see [20]). All stimuli were bipolar rectangular pulses of 10-ms duration. Prior to the conditioning experiment, the pain threshold was determined for each subject by the method of limits. Stimuli during conditioning were set to an intensity of 40% above each individual’s pain threshold. Differential visual stimuli consisted of red and green light that illuminated the whole experimental chamber. Duration of visual stimuli was 3000 ms. During acquisition of differential conditioning, one of both visual stimuli (stimulus condition A) became established as a cue for the succeeding painful stimulus applied within the last 10 ms of light presentation. The other visual stimulus (condition B) was never terminated by a painful stimulus. Both visual stimuli were applied in random order for a total number of 120 trials with 60 stimuli representing condition A and 60 stimuli representing condition B. Intertrial interval was 4 s. Stimulus type for A or B was matched across subjects. Conditioning was followed by an extinction phase consisting of 40 CS /40 CS trials with no reinforcement of the association between visual and painful stimuli. The experiment was carried out in a sound-proofed electrically shielded room. Throughout the experiment, subjects sat in a reclining chair with their hands relaxed on an armrest and were requested to keep their eyes open. c) Data recording: EEG data were recorded from 31 electrodes (Ag/AgCl electrodes) mounted on the subject’s scalp according to the international 10–20 system, with additional electrodes interspaced between standard electrodes and at left and right earlobes, as outlined in Fig. 7. Cz was used as the common reference. Vertical and horizontal electrooculographical (EOG) data were recorded for eye blinks and eye movements. EEG and EOG data were collected for a total period of 4250 ms, starting at 250 ms prior to the beginning of the visual stimulus presentation. Data were recorded using a high-pass filter of 70 Hz and a time constant of 10 s. All electrophysiological recordings were sampled with 200 Hz.


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TABLE I PAIRS OF ELECTRODES USED FOR COHERENCE ANALYSIS BETWEEN OCCIPITAL AND LEFT, RIGHT, OR MIDLINE AREAS OF THE BRAIN

(a)

D. Data Analysis 1) Analysis of the Cardiorespiratory System: In a preprocessing step, a reduction of the sampling rate to 8 Hz and a trend elimination were performed for RM, HR, and aBP. For both procedures FIR filters were used. The trend-eliminated HR is called heart rate fluctuation (HRF). Using the procedures described in Section II-A, a trivariate AR model of order 22, with time-varying coefficients, was fitted to the simultaneously measured signals. Investigations regarding the model order were performed by means of Akaike’s information criterion (cf. [6]). From the estimated AR coefficients the spectral parameters instantaneous coherence and instantaneous partial coherence were computed for all possible pairs of HRF, aBP, and RM (see Fig. 6). 2) EEG Analysis: EEG data were corrected for eye movements and eye blinks [21]. Additionally, a current source density analysis was performed in order to maximize the topographical distribution of sources and sinks and to obtain reference-free measures for each electrode ([22]). Resulting waveforms were then submitted to the coherence procedure described, before using pairs of electrode sites according to Table I (cf. Fig. 7). Pairs of electrodes were arranged so that they covered the occipital lobe, the primary and secondary somatosensory projection areas of fingers on the homunculus of both hemispheres, and the midline of the brain. An order of 16 was chosen for the fitted bivariate AR models (cf. Section II-D). According to studies by Singer et al. on synchronous activities among pairs of single cells (e.g., [23]), coherence analysis with a bandwidth of 37 to 43 Hz was performed separately for these pairs of electrodes addressing a time window from 2750 to 3000 ms. Coherence values were computed and averaged within this frequency band and time window, whereas the time-varying AR model was fitted for each entire trial. This time window covered the last 250 ms prior to the presentation of the painful stimulus during condition A, which contained the maximal brain electrical negativity that developed in anticipation of the painful stimulus [24]. An equivalent time window was used for trials of condition B, where no painful stimulus was applied. Separate analyses were performed for each subject and for each trial of the acquisition phase. Coherence measures

(b) Fig. 1. Results of coefficient estimation of a simulated bivariate AR process (order eight), with periodic switching between two different sets of coefficients (regimes). The estimated coefficients were averaged over ten trials. (a) Time course of a particular estimated coefficient on the basis of Kalman filtering (bold line), by means of the LS algorithm (batch processing, sliding window of length 200, thin line), and by means of the RLS algorithm with forgetting factor (-.). The true coefficients are given by the disconnected line. (b) Time course of the mean squared estimation errors (all coefficients).

were then transformed using Fisher’s -transformation and collapsed to average coherence measures for each subject, each pair of electrodes, and each condition of visual stimulation, whereby the trial sequence were subdivided into five blocks of 12 trials each. Blocking into groups of 12 trials was carried out to trace the process of development of the coherence between different electrode sites throughout the course of acquisition. After testing for normal distribution, these mean coherences were then submitted to an analysis of variance (ANOVA) for repeated measurements with the within factors blocks (five levels) and experimental condition (condition A or B of visual stimulation). III. RESULTS A. Simulations The reliability of the proposed algorithm was tested on simulated data. First, the coefficients of a simulated bivariate AR process of order eight were estimated, where a switch between two sets of coefficients (regime 1 and 2) occurred periodically. The estimations resulting from the suggested procedure were averaged over ten trajectories of that stochastic process. Additionally, two reference methods were used: conventional LS VAR estimation applied to data segments according to a sliding window of constant length (batch processing) and RLS with forgetting factor [16]. Fig. 1 demonstrates the adaptability of the resulting estimates to changes in the model parameters, and the adaptation speed. Major differences between the three methods occur during transition from regime 2 toward regime 1. Accuracy of the LS procedure is superior

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(a)

(b)

(c) Fig. 2. Components of a 3-D signal. The components are narrow-band oscillations with the band limits given in the plot. Fractions of the filtered component three [plot (c)] were added to the components one [plot (a)] and two [plot (b)]. The band limits of component one change at sample point 2000.

(a)

(d)

(b)

(e)

(c)

(f)

Fig. 4. Estimations of coherence (bold lines) and partial coherence between the components shown in Fig. 2 at two sample points (see Fig. 3) between components 1 and 2 [plots (a) and (d)], 1 and 3 [plots (b) and (e)], and 2 and 3 [plots (c) and (f)].

(a)

(b)

(c)

Fig. 3. Time-frequency plot of the autospectra of the first component of the signal given in Fig. 2. The vertical plot on the left side shows the corresponding data segment.

beyond the sample point after a switch has taken place. In the same situation the adaptation speed of the RLS algorithm is worse. The Kalman filter-based procedure responds most quickly to the model changes. Second, investigations of the performance of the algorithm with respect to the estimation of power spectra and related spectral parameters were performed on simulated narrow band signals with changing spectra. An example is given in Figs. 2, 3, and 4. Fig. 2 shows a segment of a simulated three-dimensional (3-D) signal with an abrupt change in the spectrum of the first component. The three components were

(d) Fig. 5. Estimation of the frequency of a sinusoidal oscillation with changing frequency (a) by searching for the frequency of maximal power of the STFT spectra (b), the adaptive AR-spectrum (c), and by calculation of instantaneous frequency by means of Hilbert transformation (d).

generated by passing independent sequences of white noise through different Chebychev-type narrow-bandpass filters. The corresponding limit frequencies are given in the figures. Additionally, the output of a linear filter applied to component (c) was added to components (a) and (b). A 3-D AR model of order eight was fitted to these data. In Fig. 3, a time-frequency plot of the estimated autospectra of the first component (ob-


tained from the first diagonal element of the spectral density matrices) is given. After a short period of oscillating peak configurations, the power is shifted toward higher frequencies as expected from the signal generation. The power around 0.5 due to the (constant) influence of the third component is preserved. Fig. 4 shows the estimates of coherence and partial coherence at a sample point before (left-hand plots) and 50 sample points after (right-hand plots) the point of alteration. The plots of coherence and partial coherence correspond to the expected shape. Since only the spectral components at about 0.5, which occur in all components, are due to a single (coherent) source [component (c)], coherence between pairs of components nears the value one around this frequency and is lower elsewhere. Accordingly, partial coherence between components (a) and (b), with regression from component (c) removed, is low everywhere. The superiority of adaptive spectrum estimation by means of Kalman filtering, compared to STFT, with respect to tracking of spectral peaks is demonstrated in Fig. 5(b) and (c). The frequency of the sinusoidal oscillation shown in Fig. 5(a) was determined by searching the frequency with maximal power and by calculation of the instantaneous frequency by means of Hilbert transformation Fig. 5(d), (cf. [25]), respectively.

B. Analysis of the Cardiorespiratory System A representative example from a group including seven piglets demonstrates the transition from normoxia to normobaric hypoxic hypoxia and the early period of hypoxia in a newborn piglet (two days old) (see Fig. 6). It is clearly shown that the reduced PaO from 99 mmHg to 37 mmHg and unchanged arterial PaCO led to an increase of about 75% in the respiration rate and about 18% in the heart rate within the first 240 s of normobaric hypoxic hypoxia. The temporal variation of coherence and partial coherence within the frequency range of respiratory sinus arrhythmia (RSA) is considerable. Coherence between RM and HRF is rather high during baseline conditions (normoxia). During the transition from normoxia to hypoxia and the early readjustment of the autonomic nervous system, indicated by the steep increase of respiration and heart rates, the transfer properties changed dramatically: RM/HRF coherence decreased from 0.9 to 0.52 at the sixtieth second of hypoxia. Simultaneously, the partial coherence of RM/HRF (aBP excluded) increased from 0.3 0.13 during normoxia to maximum values of 0.9, but after about 50 s of hypoxia, partial coherence between RM and HRF within the RSA band dropped below prehypoxic values. RM/aBP coherence and HRF/aBP (RM excluded) coherence showed an opposite course. Partial RM/aBP coherence provided the highest values during stationary states with means 0.5, with, however, considerable fluctuations as well. The occurrence of high coherence between RM and HRF at the frequency of respiration is due to the high-frequency component of HRF synchronous with the RM (RSA) and has been frequently reported ([26]–[28]). Our results suggest that under certain conditions the interactions between RM, HRF, and aBP can be extremely time dependent. To our knowledge, it was shown for the first time in

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TABLE II

p-VALUES (GREENHOUSE GEISSER CORRECTED) CONDITION (F WAS CALCULATED WITH A DF

OF FACTOR OF 1/15)

intact animals that the transition from a normoxic to a hypoxic state required a tremendous short-term readjustment of the autonomic respiro-cardiac output. This leads to a short, but very strong, improvement of the transfer properties between respiratory and cardiomotor activity during the initial period of the simultaneous acceleration of respiration rate and heart rate. This kind of transfer improvement is surprising because prevailing ideas suggested that the respiration-related RSA should be predominantly mediated by vagal activity. However, our findings recommend that under certain circumstances, such as hypoxic stimulation of chemoreceptors and during the readjustment of the autonomic nervous system, even sympathetic activity might be responsible for the ameliorated transfer of respiratory rhythmic activity to the HRF. This could be a result of the time-variable ontogenesis of the autonomic control of cardiovascular functions. The sympathetic nervous system develops and becomes functional earlier than the parasympathetic system, which becomes more important near term and thereafter [29]. The possibilities and limitations of RSA coherence analysis on the basis of instantaneous heart rate representation—disadvantages of that representation are described in [30], [31]—where shown by Witte and Rother [32]. C. EEG Analysis The ANOVA revealed no meaningful effect for the main factor blocks. Therefore, the following presentation of results will be restricted to those for the factor condition. Mean coherences between different electrode sites for condition A and condition B for all trials during acquisition were rather small with only slight variation between electrodes. Table II presents the ANOVA results for the comparison between conditions A and B for all 60 trials during acquisition. Due to the exploratory nature of the present experiment the criterion for significant 10%. In all significant cases, the main effects was set to mean coherence of condition A systematically exceeds that of condition B. There is clear evidence that condition A was associated with larger coherences than were seen for condition B between occipital electrode sites and electrode sites on the contralateral somatosensory projection areas of the left midfinger where the processing of the succeeding painful stimulus is associated. No such difference of coherence was found for other pairs of electrode sites, indicating that conditioning during experimental condition A led to a systematic increase in coherence between visual and somatosensory areas affected

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Fig. 6. Representative example of a recording of RM, HRF, and aBP of a newborn piglet and the results of coherence analysis (for further details see text). The inset shows a part of the original trace obtained from the period of hypoxia-induced RM acceleration and heart rate increase, but no relevant changes in the aBP pressure trace.


Fig. 7. Results of ANOVA for mean coherences of frequency range 37–43 Hz during the time window from 2750 to 3000 ms. Arrows indicate pairs of electrodes with significant increase of coherence for condition A. The marked areas cover the occipital cortex (below) and the fields of somatosensory association (middle) and postcentral projection (middle and right), respectively, for the left hand.

by the presentation of both types of stimuli (see Fig. 7). Again, the differences between coherences of condition A and B were small, but systematic across subjects. These findings are in accordance with previous results [24], where we could demonstrate that toward the end of the visual stimulus associated with the succeeding application of pain there was a significant increase of electrical negativity over primary somatosensory brain areas. This negativity did not develop toward the end of the visual stimulus not associated with the painful stimulus. This result gave ample evidence for the existence of a distinct neural communication between occipital brain areas processing the visual stimulus and primary somatosensory brain areas processing the painful stimulus. Both brain areas seemed to communicate toward the end of the visual stimulus as a result of conditioning. IV. DISCUSSION We presented an efficient method for multivariate spectral analysis of nonstationary signals by means of time-variant AR modeling where the Kalman filter was used to perform an adaptive fitting of the model. The reliability of the procedure was tested on simulated data. Compared with standard methods, the Kalman filter-based estimation responded most quickly to parameter changes. The usefulness of our approach in the analysis of physiological recordings was demonstrated in two examples. Referring to these applications, but also from a general point of view, the use of Kalman filters for the estimation procedure has some advantages compared with other approaches. First, Kalman filters can be constructed for multivariate systems with stochastic variation of the parameters. The properties of the resulting estimates can be described theoretically [6]. It is, in particular, not necessary to search for a suitable set of base functions to model the temporal evolution of coefficients [7] or to implement procedures for the detection of change points [3]. Second, the Kalman algorithm is suitable for implementation on microcomputers. Because of its recursive structure, it allows on-line processing, even of huge data sets. Interval-related computations could be avoided.

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It was shown that the procedure is sensitive enough to disclose even small and/or transient changes of the spectrum. The analysis of short-term, but important, functional processes in the intact organism, e.g., variations in transfer properties of the cardiorespiratory system, is only possible using a multivariate method with high time resolution to characterize the joint variability between incorporated signals. Partial coherence between each pair of three components, with the regression of the third (remaining) component removed as an additional analytical tool, enables an identification of the origin and quantifies the interrelation between two relevant signals, excluding the influence of a third inherent signal of the functional system investigated. In the case of EEG, it must be taken into consideration that the measured activity covers much broader levels of brain areas than single cell observations in microbiological paradigms. The present results provide further evidence that even such small synchronous activities of cell assemblies can be observed with EEG procedures and calculated with an adequate and sensitive procedure for coherence analysis. From a physiological point of view, the effects demonstrated seem to be meaningful and could be shown, to our knowledge, for the first time. REFERENCES [1] P. Z. Marmarelis and V. Z. Marmarelis, Analysis of Physiological Systems. New York: Plenum, 1978. [2] S. K. Ahn and G. C. Reinsel, “Estimation for partially nonstationary multivariate autoregressive models,” J. Amer. Statistical Assoc., vol. 85, no. 411, pp. 813–823, 1990. [3] A. Cohen, F. Flomen, and N. Drori, “EEG sleep staging using vectorial autoregressive models,” in Advances in Processing and Pattern Analysis of Biological Signals, I. Gat and G. F. Inbar, Eds. New York: Plenum, 1996. [4] D. F. Nicholls and B. G. Quinn, “The estimation of multivariate random coefficient autoregressive models,” J. Mult. Anal., II, pp. 544–555, 1981. [5] Y. Grenier, “Time–dependent ARMA modeling of nonstationary signals,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 899–911, Apr. 1983. [6] H. H. Chen and L. Guo, Identification and Stochastic Adaptive Control. Boston, MA: Birkäuser, 1991. [7] W. Gersch, A. Gevins, and G. Kitagawa, “A multivariate time varying autoregressive modeling of nonstationary covariance time series,” in Proc. 22nd IEEE Conf. Decis. Contr., 1983, pp. 579–584. [8] P. J. Harrison and C. F. Stevens, “Bayesian forecasting,” J. Roy. Statist. Soc. Series B. Methodological, vol. 38, pp. 205–228, 1976. [9] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: PrenticeHall, 1986. [10] D. O. Hebb, The Organization of Behavior, New York: Wiley, 1949. [11] W. Singer, “Synchronization of cortical activity and its putative role in information processing and learning,” Annu. Rev. Physiol., vol. 55, pp. 349–374, 1993. [12] M. Kay and S. L. Marple, “Spectrum analysis—A modern perspective,” in Proc. IEEE, vol. 69, pp. 1380–1418, Nov. 1981. [13] L. H. Koopmans, The Spectral Analysis of Time Series. New York: Academic, 1974. [14] T. Cipra and I. Motykovà, “Study on Kalman filter in time series analysis,” Commentations Mathematicae Universitatis Carolinae, vol. 28, no. 3, pp. 549–563, 1987. [15] B. Schack, G. Grießbach, M. Arnold, and J. Bolten, “Dynamic crossspectral analysis of biological signals by means of bivariate ARMA processes with time-dependent coefficients,” Med. Biol. Eng., Comput., vol. 33, no. 5, 1995. [16] M. Campi, “Performance of RLS identification algorithms with forgetting factor: A -mixing approach,” J. Math. Syst., Estimation, Contr., vol. 4, no. 3, pp. 1–25, 1994. [17] J. Geweke, “Measurement of linear dependence and feedback between multiple time series,” J. Amer. Stat. Assoc., vol. 77, no. 378, pp. 304–316, 1982.

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[18] W. Gersch, “Non-stationary multichannel time series analysis,” in Methods of Analysis of Brain Electrical and Magnetic Signals (Revised Series), vol. 1, A. S. Gevins and A. Rèmond, Eds. Amsterdam, the Netherlands: Elsevier Science B.V., 1987, pp. 261–296. [19] U. Niklasson, U. Winklund, P. Bjerle, and B.-O. Olofsson, “Heart-rate variation: what are we measuring,” Clin. Physiol., vol. 13, pp. 71–79, 1993. [20] B. Bromm and W. Meier, “The intracutaneous stimulus: A new pain model for algesimetric studies,” Meth., Findings Experimental, Clin. Pharmacol., vol. 6, no. 7, pp. 405–410, 1984. [21] G. Gratton, M. G. Coles, and E. Donchin, “A new method for offline removal of ocular artifact,” Electroencephalogr. Clin. Neurophysiol., vol. 55, no. 4, pp. 468–484, Apr. 1983. [22] P. L. Nunez, Electrical Fields of the Brain. New York: Oxford Univ. Press, 1981. [23] C. M. Gray, P. Konig, A. K. Engel, and W. Singer, “Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties,” Nature, vol. 338, no. 6213, pp. 334–337, 1989. [24] H. Waschulewski-Floruß, W. Miltner, S. Brody, and C. Braun, “Classical conditioning of pain responses,” Int.. J. Neurosci., vol. 78, pp. 21–32, 1994. [25] M. Arnold, A. Doering, H. Witte, J. Dörschel, and M. Eiselt, “Use of adaptive Hilbert transformation for EEG segmentation and calculation of instantaneous respiration rate in neonates,” J. Clin. Monit., vol. 12, no. 1, pp. 43–60, Jan. 1996. [26] 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,” Science, vol. 213, pp. 220–222, 1981. [27] B. Pomeranz, R. J. B. Macaulay, M. A. Caudill, I. Kutz, D. Adam, D. Gordon, K. M. Kilborn, A. C. Barger, D. C. Shannon, R. J. Cohen, and H. Benson, “Assessment of autonomic function in human by heart rate spectral analysis,” Amer. J. Physiol. (Heart Circ. Physiol. 25), vol. 256, pp. H151–153, 1985. [28] H. Witte and M. Rother, “High-frequency and low-frequency heartrate fluctuation analysis in newborns—A review of possibilities and limitations,” Basic Res. Cardiol., vol. 87, pp. 193–204, 1992. [29] P. M. Gootman, “Development of central autonomic regulation of cardiovascular function,” in Developmental Neurobiology of the Autonomic System, vol. 6. New York: Humana, 1986, pp. 279–325. [30] R. W. de Boer, “Beat-to beat blood-pressure fluctuations and heart-rate variability in man: Physiological relationships, analysis techniques and a simple model,” Ph.D. dissertation, Univ. Amsterdam, Amsterdam, the Netherlands, Dec. 1985. [31] R. D. Berger, D. Akselrod, S. Gordon, D. Cohen, and R. J. Cohen, “An efficient algorithm for spectral analysis of the heart rate variability,” IEEE Trans. Biomed. Eng., vol. BME-33, p. 900, 1986. [32] H. Witte and M. Rother, “Better quantification of neonatal respiratory sinus arrhythmia—progress by modeling and model-related physiological examinations,” Med. Biol. Eng., Comput., vol. 27, pp. 298–306, 1989.

Matthias Arnold studied mathematics at the Friedrich Schiller University in Jena, Germany. He received the Dipl.-Math. degree in 1992. Since then he has been with the Institute of Medical Statistics, Computer Sciences and Documentation at Jena. His research interests include stochastic processes and biomedical signal processing.

Wolfgang H. R. Miltner received the diploma in psychology in 1975, the doctor (Ph.D.) degree in social science in 1980 and the habilitation degree in medical psychology in 1989, all at the University of Tuebingen, Tuebingen, Germany. He is currently Director of the Institute of Psychology, Chair of the Department of Biological and Clinical Psychology, and Full Professor of Psychology at the Friedrich Schiller University of Jena, Jena, Germany. His research is focused on the investigation of cortical reorganization in chronic pain and stroke patients, cortical processes of learning and memory formation and error processing in healthy subjects and neurological patients, and on cortical and subcortical modulations of information processing in subjects suffering from mood disorders and anxiety disorders.

Herbert Witte (M’96) received the diploma degree in cybernetics at the Technical University of Magdeburg, Magdeberg, Germany in 1974. Afterwards he worked in neurophysiological research and received the Ph.D. degree in neurobiology from the Friedrich Schiller University Jena, Jena, Germany. In 1986 he received the Engineer of Medicine degree in biomedical engineering and the Doctor of Science (habilitation) degree for his work in neonatal monitoring. Since 1992 he has been Professor of Medical Informatics and Director of the Institute of Medical Statistics, Computer Sciences and Documentation at the Friedrich Schiller University Jena. Dr. Witte has been Chairman of the IEEE joint chapter BME (German section) since 1996.

Reinhard Bauer, for a photograph and biography, see this issue, p. 552.

Christoph Braun studied physics and biology at the University of Tuebingen, Tuebingen, Germany. After one year of work at the Department of Animal Physiology he joined the Institute of Medical Psychology at Tuebingen in 1985. There he received the Ph.D. degree in 1991. Since 1997, he has worked at the MEG-Center of the University of Tuebingen. His main interests are the study of changes in the functional organization of the human cortex by means of multichannel evoked potentials and magnetic fields.