V. Detrending methods and filters
A. The impact of detrending and filters
From the above sections it is clear that spectral analysis is dependent upon the statistical characteristics of the data being analyzed. If the data are aperiodic and non-stationary major problems are encountered. In this section, we will assess the effectiveness of a variety of detrending and filtering methods to remove low frequency and aperiodic non-stationary components from the physiological time series.
Throughout this chapter we have assumed that the time series being analyzed accurately represents the highest frequency components imbedded in the signal. Of course, if the data were sampled too slowly, then high frequency components will be aliased on lower frequencies. Similarly, analog low-pass filters are frequently used to remove the impact of unwanted sources of variance (e.g., 60 Hz associated with line voltage). The analog low-pass filter will not have serious quantitative consequences, if the variances of the frequency components being filtered are small relative to the signal being studied. However, in some situations analog low-pass filters will distort high frequency processes and transform them into slower processes. The concept of a polygraph integrator functionally employs this procedure in dealing with very high frequency components like EMG.
Filters transform one time series into another. The frequency domain modification of the time series by the filter is called a transfer function. The transfer function provides information regarding the filter's ability to block, attenuate, and amplify specific frequencies within the spectrum.
An ideal filter would remove the sources of variance that confound the assessment of the periodicities of interest to the researcher. Our discussions of filters will be limited to detrending methods which remove non-stationary influences and perform other high-pass functions. A detrending filter needs to have a number of characteristics to successfully perform. First, the filter must be able to remove non- stationary trend from the time series. Second, the filter should have a sharp cut-off so the non-stationary and low frequency variations are completely removed without distorting the frequency and amplitude components of the periodicity being studied. Third, since periodic physiological activity seldom can be described by a single sine wave, the filter should have ability to remove the influences of slow processes which although periodic may have component variances in the frequency band of the periodicity of interest. Fourth, the transfer function of the filter should be predictable for all applications.
Filtering can occur in the time domain or in the frequency domain. In this section we will deal only with time domain filters. Frequency domain filters do not appear to be useful for many psychophysiological applications. Frequency domain filters perform their filtering after the data have been transformed into the frequency domain. In the frequency domain it is impossible to determine which proportion of a spectral density estimate represents the real signal and which part is a function of a non-stationary series or a slower periodic process which is not a perfect sine wave. Thus, frequency band rejection does not insure that all the variances associated with non-stationary and slow periodic (but not a perfect sine wave) will be removed.
One practical way of assessing the transfer function of a filter is observe the impact of the filter on a white noise time series. Since white noise is defined as a time series with the expected value of the spectral densities being the same for all frequencies, the spectrum of white noise should theoretically be a flat. With the knowledge of the spectral characteristics of a white noise time series, we can evaluate the transfer function of the filter on the spectrum by comparing the filtered spectral densities to the white noise spectrum. If we divide the original white noise spectrum into the filtered spectrum, we would be able to estimate the transfer function. If the numbers at specific frequencies approximate 1.0, the filter provides an accurate description of periodicity. If the numbers are less than 1.0, the filter attenuates the spectral densities. If the numbers approach 0.0, the filter rejects specific frequencies. If the numbers are greater than 1.0, the filter amplifies specific frequencies.
In Figure 11 spectra are illustrated for three detrending methods and the original white noise series. Figure 11a illustrates the spectrum of the white noise series. Note that across frequencies, spectral densities are relatively uniform and approximate the theoretical flat spectrum. The values are not identical, because the white noise series is a sample of finite duration. If we increased the number of data points or averaged over a number of spectra, the spectrum would be flatter.
B. Linear detrending
Since heart rate patterns often exhibit trends, a common method of detrending is to remove the linear influence. This is done fitting a linear regression to the data. This method removes linear influences and transforms the data set into a series with mean of approximately zero. Linear detrending has little impact on the spectrum of white noise. However, if we had superimposed the white noise series on a large linear trend, the spectral decomposition of this series would distribute most of the variance at 0.0 Hz in the spectrum. A spectrum with a relative large proportion of the variance at 0.0 Hz is characteristic of a time series with non-stationary components. If the non-stationarity is caused by a linear trend, linear detrending will remove the non-stationary influence and provide an interpretable spectrum. Unfortunately, the non-stationary influences in physiological times series are seldom fit with a linear regression. Routine application of linear and other low order polynomial fits (e.g., quadratic, cubic, etc) to the entire data set, seldom achieves the anticipated goal of removing non-stationary influences.
C. Successive-difference filters
In the psychophysiological literature, it has been proposed that successive difference statistics such as the successive difference mean square (see Heslegrave, Ogilvie, and Furedy, 1979) are useful in removing linear trends and other slow sources of variation. All measures of successive differences, like other filters, have a transfer function. When the data are sampled at equal time intervals (i.e., second-by-second and not beat-by-beat), the transfer function will be consistent across all subjects and conditions. However, if this family of statistics is calculated on the sequential heart beat measures of period or rate, the transfer function is unreliable due to the variance of the sampling rate between and within subjects. The successive differencing of slow heart rate passes lower frequencies than the successive differencing of fast heart rate. Thus, the component variances of the complex heart rate pattern which are partially determined by the temporal characteristics of neural feedback are not treated consistently across subjects and conditions.
With data sampled at equal time intervals, it is interesting to observe the dramatic impact of successive differencing on the spectrum of white noise. In our example, we have sampled from the white noise process at rate of 2 Hz. All spectra decompose the time series in frequencies from 0.0 to Pi or in Figure 11 from 0.0 to 1.0 Hz. Inspection of Figure 11c illustrates an important phenomenon. The low frequencies in the spectrum are greatly attenuated, demonstrating the success that the successive- difference filter has with low frequencies. However, the successive- difference filter amplifies higher frequency components. The 1.0 Hz component of the white noise series has been amplified by a factor of 5! Knowledge of this transfer function is extremely important in interpreting data that have been successively differenced. At frequencies approximately 1/6 the sampling rate (Pi/3), the spectral densities are relatively accurate. At frequencies slower than 1/6 the sampling rate (Pi/3), the spectral densities are greatly attenuated. At frequencies faster than 1/6 the sampling rate (Pi/3), the spectral densities are amplified.
D. Summary of problems with traditional methods
As we have repeatedly discussed, detrending and filtering procedures need to be capable of removing non-stationary components and rejecting low periodic non-sinusoidal activity which may have higher frequency harmonics. The commonly employed methods of linear and successive difference filtering do not succeed. Statisticians do not have standard tools to deal with these problems. Most available time series statistical packages merely ritualize and exacerbate the existing problems by lulling the researcher into believing that he has adequately manipulated his data. Given these problems, what can the researcher do to insure that the data are appropriately analyzed?
Being most familiar with heart rate patterns, we will describe the problems and solutions associated with the extraction of an accurate measure of respiratory sinus arrhythmia. If the baseline trend is a complex function that cannot be mathematically described by a linear or low-order polynomial or by a sum of sine waves slower than the frequencies characteristic of respiratory sinus arrhythmia, the spectral composition of the baseline trend will include faster frequency components. The higher frequencies associated with the trend will leak through the detrending techniques and be superimposed on the spectral representation of the amplitude of respiratory sinus arrhythmia. Since the amplitude of the faster frequency components of the baseline trend are not known a priori and cannot be estimated a posteriori from the spectrum because they change over time when the baseline is not constant (i.e., nonstationary), high-pass filters do not eliminate all of the variance of baseline trend; a traditional high-pass filter cannot discriminate between the component variances associated with trend and the amplitude of respiratory sinus arrhythmia, if both co-exist in the same frequency band.
E. A solution to the problem: The Moving Polynomial Filter
These points focus on the problems of applying spectral technology to accurately describe rhythmic physiological processes. In some situations it may be possible to minimize the impact of a complex baseline trend on the periodic activity by analyzing relatively short epochs and fitting the remaining trend with a linear fit. This approach is based on the assumption that a complex trend may be approximated by a series of adjacent linear trends. This method will be effective, if and only if, the trend component in each epoch is primarily linear. It is, however, possible to model the complex aperiodic baseline with a series of localized polynomials (see Porges, 1985). These short-duration polynomials may be stepped through the data set. The moving polynomial filter smooths the data set by conforming to the shifting levels of the baseline. When the smoothed baseline is subtracted from the original data set, the residual time series is free from the influence of the baseline and slow periodic activity.
Figure 12 illustrates how the moving polynomial procedure functions. The top panel illustrates 60 seconds of heart period data sampled every 500 msec. from a healthy adult. Note the rhythmic oscillations occurring approximately 20 times within the 60 second data set. This oscillation is respiratory sinus arrhythmia. A graph of simultaneously recorded respiration (e.g., Figure 8a) would reflect the same periodicity.
The heart period data were collected while the subject was seated quietly and not involved in an experimental task. However, the pattern of the heart period data, even during these conditions, exhibited a complex trend which included contributions of slower periodic activity (e.g., Traube-Hering-Mayer wave) and aperiodic activity (e.g., acceleratory trend). A simple way of estimating the relative contributions of the various components to the total variance of the heart period pattern is to approximate the range of each component. In the example plotted in Figure 12, respiratory sinus arrhythmia has a range of about 50 msec, the Traube- Hering-Mayer wave has a range of about 50 msec, and the trend has a range of about 100 msec. Thus, even in an example in which respiratory sinus arrhythmia is visually apparent, respiratory sinus arrhythmia contributes far less than 50% of the total variance of the time series. There are situations, such as with the fetus, when heart period oscillations in the respiratory frequencies account for less than 1: of the total heart period variance (Donchin, Caton, and Porges, 1984).
The complex trend cannot be fit accurately with a linear regression over the entire data set. Moreover, the Traube-Hering-Mayer wave cannot be fit with a perfect sine wave. If spectral analysis is used to describe respiratory sinus arrhythmia, the amplitude of respiratory sinus arrhythmia may be inflated because linear detrending and frequency domain filtering would pass variance unrelated to respiratory sinus arrhythmia into the respiratory frequency band. Thus, traditional filtering strategies will inflate respiratory sinus arrhythmia amplitude. This presents the possibility that estimates of respiratory sinus arrhythmia amplitude derived from spectral density analyses may be modulated, not by changes in the cardio-vagal tone mediated by respiration, but by changes in trend and Traube-Hering-Mayer wave profiles.
It is possible that harmonics from the Traube-Hering-Mayer wave distort efforts to quantify respiratory sinus arrhythmia. The Traube-Hering-Mayer wave, associated with blood pressure feedback, has been reported in the heart period spectrum at frequencies between 0.07 and 0.10 Hz. This periodic process is far from sinusoidal and, theoretically should have higher frequency harmonics at integer frequencies in the band from 0.14 to 0.20 Hz. Coincidentally, the 0.14 to 0.20 Hz band represents normal respiration for many healthy resting adults. Thus, the quantification of respiratory sinus arrhythmia is potentially inflated by the harmonics of the slower periodic heart period activity. Since the appropriateness of the sinusoidal fit to the Traube-Hering-Mayer wave is not known prior to data collection and the shape of the wave is far from reliable within a subject, it is impossible to determine how much of the variance attributed to the respiratory frequency band is a function of respiratory sinus arrhythmia and how much is a function of the Traube-Hering-Mayer wave.
The Traube-Hering-Mayer example understates the problem of harmonics in the heart period spectrum. The heart period pattern is very complicated. The spectral densities in the frequency bands associated with both the Traube-Hering-Mayer wave and respiratory sinus arrhythmia are confounded by even slower periodic non-sinusoidal processes which reflect thermoregulatory, ultradian, and circadian influences.
The moving polynomial procedure functions as a high pass filter and partitions the time series into two uncorrelated components: a smoothed template series (see middle panel) containing the slow periodic and aperiodic activity (i.e., trends) that occur at frequencies slower than breathing; and a residual series (see bottom panel) generated by subtracting the template series from the unfiltered data that now conforms to the requirements of weak stationarity. Note how effectively the template series fits the complex aperiodic baseline and the quasi-periodic Traube-Hering-Mayer wave. The residual series clearly and accurately passes the frequency components associated with respiratory sinus arrhythmia.
The moving polynomial filter has unique performance characteristics as a filter, since it does not make specific assumptions regarding the spectral characteristics of the trend prior to detrending. The number of coefficients and order of the polynomial are critical. In the example illustrated in Figure 10, a filter was designed to pass variance associated with frequencies above 0.12 Hz. With knowledge of the transfer function of the moving polynomial procedure and the sampling rate of the time series, it was possible to design a sensitive filter which dynamically fit the changing baselevel (see Porges, 1985). The residual series may then be analyzed with spectral analysis to extract the variance associated with the respiratory frequency band.
The order and duration of the polynomial are selected to bend into slow moving trends without distorting the frequency band of interest. In our research on respiratory sinus arrhythmia we have selected a cubic order polynomial with a duration of 10.5 seconds for the research with human adults and a cubic order polynomial with a duration of 4.2 seconds for the research with human neonates. Other duration cubic polynomials have been used when we have studied other subject populations and other physiological rhythms. Figure 12a illustrates the original heart period signal of an adult subject, Figure 12b illustrates the smoothed series, and Figure 12c illustrates the residual filtered series.
Figure 11d illustrates the impact of a cubic moving polynomial filter with a duration of 10.5 seconds on the white noise time series. Note that the spectral densities at the low frequencies are totally rejected and the high frequencies are not amplified. The theoretical transfer function for this filter indicates slight attenuation at the low end of the respiratory frequency band (i.e., 0.12 Hz and 0.14 Hz) and fidelity at frequencies above .16 Hz. Although longer duration polynomials would provide greater fidelity of a sine wave at the slow respiratory frequencies, it would compromise the analysis by passing influences of slower periodic processes such as the Traube-Hering-Mayer wave. In fact, it has been demonstrated that with extremely slow breathing, respiratory sinus arrhythmia and the Traube-Hering-Mayer wave merge and become extremely difficult to interpret (Kitney, 1986). The attenuation of respiratory sinus arrhythmia at the slowest respiratory frequencies is of little concern and has only negligible impact on estimates of respiratory sinus arrhythmia, because respiratory sinus arrhythmia occurs over a band of frequencies and the spectral densities are windowed.
We can now evaluate the influence of the moving polynomial filter on our respiratory and heart period data presented in 8a. Recall that in Figure 8b the peak of heart period spectrum within the respiratory frequencies did not match the peak of the respiratory spectrum although the periodicity seem synchronous in Figure 8a. We speculated that this difference could be an example of how frequency components of aperiodic and slow periodic non-sinusoidal influences could distribute variance in higher frequencies and distort the estimate of respiratory sinus arrhythmia. In Figure 13a the detrended times series of the heart period data are presented with synchronously monitored respiration. Note that the removal of trend in the heart period also changes the values of the heart period series. Now the series has a mean of approximately zero and the values of the residual series reflect deviations in msec from the moving baseline trend that was subtracted from the data. In Figure 13b the spectrum of the filtered heart period data now exhibits a peak coincidental with the peak of the respiration data. The filter has successfully rejected the low frequencies and removed most of the aperiodic influences which had distorted, in Figure 8b, the peak of heart period spectrum in the respiratory frequency band. Linear detrending of the heart period data would also have produced a spectrum similar to 8b with a distorted peak within the respiratory frequencies.
The moving polynomial filter can also be used to enhance bivariate methods. For example, when the simultaneously recorded respiration and heart period data illustrated in Figure 8a are transformed by the moving polynomial filter, the cross-correlogram becomes more regular. As illustrated in Figure 7b, the cross-correlations now oscillate between +/- .75 and +/- .50. the cross-correlogram. Moreover, the moving polynomial filter appears not to influence the phase of highly coherent processes. As illustrated in Figure 9b, the phase between respiration and heart period at the dominant respiratory frequency is virtually identical to the phase of the unfiltered time series.
We could also assess the impact of a filter by applying it to a time series consisting of a sine wave of known amplitude superimposed on a periodic non-sinusoidal pattern. We could simulate two processes: one periodic but non-sinusoidal at a frequency similar to the Traube-Hering- Mayer wave; and the other a perfect sine wave at a frequency similar to respiration. In our example, we will use a 0.16 Hz sine wave superimposed on a 0.08 Hz periodic non-sinusoidal pattern. The 0.08 Hz pattern was generated by transforming values of a simulated sine wave to 0.0 when they were negative. This procedure will produce harmonics at integer multiples of 0.08 Hz. This periodic non-sinusoidal pattern is illustrated in Figure 14a. The sine wave is illustrated in Figure 14b. The combined process is illustrated in Figure 14c. We know the real spectral characteristics of the sine wave by calculating a spectrum separately on the sine wave.
In Figure 15a the spectrum of the 0.08 Hz periodic non-sinusoidal process is illustrated. Note that the non-sinusoidal characteristics of this periodic process result in component variances being distributed to the first harmonic, 0.16 Hz. In Figure 15b the spectrum of the 0.16 Hz sine wave is illustrated. In Figure 15c the spectrum of the combined signal is illustrated. Inspection of this figure without knowledge of the underlying components would result in overestimating of the spectral densities associated with the 0.16 Hz.
Figure 16 illustrates the relative impact on the spectrum of the three filtering procedures. Linear detrending, illustrated in Figure 16a, produces a spectrum similar to the non-detrended spectrum in Figure 15c. Since there was no linear trend, this was expected. The successive- difference filter removed most of the variance at 0.08 Hz and greatly attenuated the variance estimates at 0.16 Hz. These results would have been anticipated with knowledge of the transfer function for the successive difference filter. In contrast to the linear detrending and successive difference filtering (see Figure 16b), the moving polynomial filter illustrated in 14c removed the variances associated with the periodic activity at 0.08 Hz and also removed the harmonic variances associated with this process. The variance estimates at 0.16 Hz are very similar to the actual variances produced when the 0.16 Hz sine is analyzed by itself (see Figure 15b).
Similar to the above example, we can apply the various filtering procedures to the heart period data illustrated in Figures 3a and 8a. In Figure 17a the spectrum of the undetrended data are characterized by spectral densities at the lowest frequency and by the broad peak in the respiratory frequency band. In Figure 17b the spectrum for the linear detrended data are characterized by an attenuation of the deterministic component at the lowest frequency confirming that the data were non- stationary and had a major linear trend. The respiratory peak remains broad. In Figure 17c the spectrum for the successive differenced data demonstrate the impact of the filters transfer function. The low frequencies have been totally rejected and although the peak frequency is more similar to the peak of the respiratory spectrum illustrated in Figure 2c and 8c, the amplitude of the peak has been greatly attenuated. In Figure 15d the spectrum for the moving polynomial filtered data is illustrated. Note the rejection of low frequency components and the accurate manifestation of the amplitude and frequency of the higher frequency components.
F. Alternative methods to describe amplitude of periodicities
In many physiological and psychophysiological experiments descriptive methods have been used to quantify the amplitude of oscillations. For example, respiratory sinus arrhythmia has been quantified with a variety of heart rate variability measures including peak-to-trough calculations and differencing methods (e.g., beat-to-beat variability, short term variability) as well as the spectral techniques previously described. In virtually all of these cases the problems associated with aperiodicity, non-stationarity, and periodic non-sinusoidal patterns have been totally neglected.
One of the most common methods is the peak-to-trough measure (Katona and Jih, 1975; Hirsch and Bishop, 1981, Grossman and Wientjes, 1986). Although this method has been successful in demonstrating systematic changes in the amplitude and frequency of respiratory sinus arrhythmia due to breathing maneuvers and vagal manipulations, the method has severe quantitative vulnerabilities.
In employing the peak-to-trough measure to quantify respiratory sinus arrhythmia, the researcher measures the difference between a maximum and a minimum point associated with a specific respiratory cycle. Or, the researcher may employ a trough-to-peak measure. In this case the change is evaluated during the "up-phase" of the cycle, rather than during the "down-phase" of the cycle. The researcher assumes that the peak-to-trough measure accurately reflects the instantaneous influence of the cardiac vagus on the heart (i.e., parasympathetic tone).
If the variability of the heart period activity were solely determined by respiratory sinus arrhythmia, measurement of either the "up-phase" or "down-phase" would provide an extremely accurate measure of respiratory sinus arrhythmia. However, respiratory sinus arrhythmia is usually superimposed on a complex trend and other slower periodic processes which are not perfect sine waves (e.g., Traube-Hering-Mayer wave).
The "peak-to-trough" method is a poor measure of both average and instantaneous parasympathetic tone, because of two critical quantitative problems. First, the impact of non-stationarity associated with the complex trends will stretch and compress the peak-to-trough measure, amplifying and attenuating the measure when the amplitude of the underlying process is not changing. Second, the impact of slow periodic processes which are not perfect sine waves such as the Traube-Hering-Mayer wave will result in different durations in which the peak-to-trough measure would be stretched and compressed. Thus, the trend and slower periodic processes would significantly contribute variance to the peak-to- trough measure.
Although there are problems with many of the detrending methods used in the application of spectral analyses (see above), the literature on "peak- to-trough" measurements is devoid of attempts to remove trends. As described above, the presence of trends would stretch the peak-to-trough during periods when the trend is ascending and compress the peak-to-trough during periods when the trend is descending.
To illustrate the above problem, we have selected a sine wave with an amplitude of 50 (e.g., +/- 50 msec) oscillating at a frequency of .16 Hz (see Figure 18a). This sine wave is similar in amplitude and frequency to respiratory sinus arrhythmia in the healthy human adult and is identical to the sine wave used in our earlier example demonstrating the effectiveness of the moving polynomial filter (see Figure 14a). The peak- to-trough measure for this sine wave is 100 msec. In our examples, to realistically approximate physiological characteristics, the trend has linear and higher order components. When the sine wave is superimposed either on an ascending trend (see Figure 18b) or a descending trend (see Figure 18c), the amplitude characteristics of the sine wave are changed. When the trend is relatively stable, the sine wave is accurately represented. When the trend is ascending the peak-to-trough is compressed from 100 msec to approximately 73 msec. When the trend is descending the peak-to-trough is stretched from 100 msec to approximately 134 msec. Note that, not only is the peak-to-trough stretched or compressed by the trend, but the trend contributes instability (i.e., increased variance) to the peak-to-trough measure.
If the amplitude of the sine wave were lowered and superimposed on the trend illustrated in Figure 18b, the trend would contribute even a greater percentage of the measured variance of the time series. The trend would also contribute more to the relative variability of the peak-to-trough measure. Thus, when the underlying periodic process accounts for only a small percentage of the total variance, the peak-to-trough method provides a greatly distorted estimate of the amplitude of the periodic process. In contrast, when the underlying periodic process accounts for most of variance, the peak-to-trough method will provide an accurate measure. Special populations who have low amplitude of respiratory sinus arrhythmia, such as diabetics, head injury patients, human fetuses, heart transplant patients and risk infants, are extremely vulnerable to these methods.
To illustrate the impact of detrending on the peak-to-trough measure, we have applied various methods to the ascending time series illustrated in Figure 18b. Figure 19a represents the time series with the linear trend removed. Although a major influence of trend is removed, the impact of the trend is still visible and is reflected in the variability in the peak-to-trough measures which range between 127 msec. and 88 msec. Figure 19b depicts the time series detrended by successive differences. Note the massive attenuation and variability of the peak-to-trough measure. Figure 19c illustrates the time series detrended by the moving polynomial filter. Note the stable and accurate representation of the peak-to-trough characteristics of the original sine wave (see Figure 18a).
The peak-to-trough method is not inherently a poor technique. It is accurate when applied to a time series consisting of a single oscillatory process. It is a useful method when applied to a complex time series which has been appropriately filtered (see Figure 18c). When the complex trends and slower periodic activity are effectively filtered, the method will provide an extremely accurate estimate of average or instantaneous parasympathetic tone by quantifying respiratory sinus arrhythmia. However, the method is severely compromised when time series reflect complex processes. Similar to the spectral technology described above, the problem with the peak-to-trough measure is not with the method but with the inappropriate application of the method to complex time series.
High correlations among various methods of measuring heart rate variability do not confirm that the measures behave the same. For example, in our research we investigated in healthy full-term neonates the influence of sleep state on measures of heart rate variability. The amplitude of respiratory sinus arrhythmia derived from the moving polynomial filter (i.e., vagal tone) was highly correlated with the standard deviation of the heart period during quiet sleep (r = .88) and active sleep (r = .78). However, the two measures behaved differently across the sleep states. During active sleep the overall heart period variability was higher than during quiet sleep. In contrast, the amplitude of respiratory sinus arrhythmia derived from the moving polynomial filter reflected the hypothesized increase in parasympathetic tone associated with quiet sleep relative to active sleep. The magnitude of the sleep state effect in the analysis of variance, although in different directions for the two variables, was large, accounting for 29% of the variance in the moving polynomial measure and 43% in the standard deviation measure. This analysis demonstrates that high within condition correlations do not provide information regarding the behavior of the derived variables. Also, it demonstrates that a component of the total variance (i.e., respiratory sinus arrhythmia) behaves differently than the composite standard deviation measure.
In the same study we successive differenced our data and calculated the mean successive difference. Successive differencing removes the impact of slow frequencies and amplifies the variances of fast frequencies. In this example, the mean successive difference behaved like the moving polynomial respiratory sinus arrhythmia measure. It was higher during quiet sleep than during active sleep. However, the effect size was much smaller, accounting for only 15% of the variance. In the above examples, we can see that successful filtering of our data provides the possibility of extracting physiological variables which closely parallel theoretical constructs of variables like cardiac vagal tone.
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