Evaluation of the dispersional analysis method for fractal time series

Fractal dimensions of laser doppler flowmetry time series

Laser Doppler flowmetry (LDF) provides a non-invasive method of assessing cutaneous perfusion. As the microvasculature under the probe is not defined the measured flux cannot be given absolute units, but the technique has nevertheless proved valuable for assessing relative changes in perfusion in response to physiological stress. LDF signals normally show pronounced temporal variability, both as a consequence of the pulsatile nature of blood flow and local changes in dynamic vasomotor activity. The aim of the present study was to investigate the use of methods of nonlinear analysis in characterizing temporal fluctuations in LDF signals. Data were collected under standardised conditions from the forearm of 16 normal subjects at rest, during exercise and on recovery. Surrogate data was then generated from the original time series by phase randomization. Dispersional analysis demonstrated that the LDF data was fractal with two distinct scaling regions, thus allowing the calculation of a fractal dimension which decreased significantly from 1.23 +/- 0.09 to 1.04 +/- 0.02 during exercise. By contrast, dispersional analysis of the surrogate data showed no scaling region.

Variability in isochronous tapping: higher order dependencies as a function of intertap interval

Isochronous serial interval production (ISIP) data, as from unpaced finger tapping, exhibit higher order dependencies (drift). This fact has largely been ignored by the timing literature, one reason probably being that influential timing models assume random variability. Men and women, 22-36 years old, performed a synchronization-continuation task with intertap intervals (ITI) from 0.4 s to 2.2 s. ISIP variability was partitioned into components attributable to drift and 1st-order serial correlation, and the results indicate that (a) drift contributes substantially to the dispersion for longer ITIs, (b) drift and 1st-order correlation are different functions of the ITI, and (c) drift exhibits break close to 1.0 s and 1.4 s ITI. These breaks correspond to qualitative changes in performance for other temporal tasks, which suggests common timing processes across modalities and tasks.

Surface fractal computation and its application to immunofluorescent histochemical studies of calpain and calpastatin in PC12 cells

The purpose of this report is to present a method which can be used to parameterize patterns of immunofluorescent staining in cultured neural cells. The algorithm is based on the observation that the variance in pixel intensity of the image is a power function of the magnitude of the area in immunofluorescently stained PC12 cells. This property is used to derive the fractal dimension (D) of the region of interest (ROI), and corresponds to the complexity of the pixel intensity associated with the ROI, which is analogous to a fractal surface. We show that the measure is useful in characterizing immunofluorescent staining patterns, and apply this measure to study the effects of ethanol exposure on mu-calpain and calpastatin-associated immunoreactivity. Exposure of PC12 cells to ethanol (80 mM)x48 h resulted in alterations in immunofluorescent signal (Control vs ethanol) associated with actin, calpastatin and mu-calpain: 2289+/-166 vs 1709+/-69, P<0.01; 1681+/-38 vs 2224+/-95, P<0.001; 1823+/-39 vs 2841+/-68, P<0.0001 respectively, magnitudes being pixel intensity units on a scale of 0-4095. D-values for the three proteins in the same order were: 2.32+/-0.01 vs 2.31+/-0.03, NS; 2.31+/-0.01 vs 2.32+/-0.01, NS; 2.16+/-0.03 vs 2. 24+/-0.02, P<0.01, with a possible D-value range of 2-3.

Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes

Stochastic fractal signals can be characterized by the Hurst coefficient H, which is related to the exponents of various power-law statistics characteristic of these processes. Two techniques widely used to estimate H are spectral analysis and detrended fluctuation analysis (DFA). This paper examines the analytical link between these two measures and shows that they are related through an integral transform. Numerical simulations confirm this relationship for ideal synthesized fractal signals. Their performance as estimators of H is compared based on a mean square error criterion and found to be similar. DFA measures are derived for physiological signals of heartbeat R-R intervals through the integral transform of a spectral density estimate. These agree with directly calculated DFA estimates, indicating that the relationship holds for signals with nonideal fractal properties. It is concluded that DFA and spectral measures provide equivalent characterizations of stochastic signals with long-term correlation.

Prediction of the oculocardiac reflex from pre-operative linear and nonlinear heart rate dynamics in children

This study was aimed to determine whether pre-operatively measured linear and nonlinear analysis of heart rate variability might predict the occurrence of the oculocardiac reflex (OCR) or other arrhythmia during strabismus surgery in children (n = 185, mean (SD) age = 8.09 (3.31) years). We compared time- and frequency-domain, and nonlinear dynamic indices of pre-operatively measured RR interval data between the OCR-positive group (maximum heart rate decrement = 20 beat.min-1 during the traction of extraocular muscle, n = 54), OCR-negative group (< 20 beat x min(-1), n = 78) and arrhythmia-positive group (all other arrhythmias; n = 53). pNN50, rMSSD, high-frequency power and nonlinear prediction error were significantly lower in the OCR-positive and arrhythmia-positive groups than in the OCR-negative group. Discriminant analysis using these indices could correctly identify 39/54 (72.2%) OCR-positive patients. Some pre-operatively measured indices of linear and nonlinear heart rate variability, especially when used in combination, are valuable for predicting significant bradycardia during strabismus surgery in children.

Correlation structure of end-expiratory lung volume in anesthetized rats with intact upper airway

The correlation structure of breath-to-breath fluctuations of end-expiratory lung volume (EEV) was studied in anesthetized rats with intact airways subjected to positive and negative transrespiratory pressure (i.e., PTRP and NTRP, correspondingly). The Hurst exponent, H, was estimated from EEV fluctuations using modified dispersional analysis. We found that H for EEV was 0.5362 +/- 0.0763 and 0.6403 +/- 0.0561 with PTRP and NTRP, respectively (mean +/- SD). Both H were significantly different from those obtained after random shuffling of the original time series. Also, H with NTRP was significantly greater than that with PTRP (P = 0.029). We conclude that in rats breathing through the upper airway, a positive long-term correlation is present in EEV that is different between PTRP and NTRP.

Evaluating maximum likelihood estimation methods to determine the Hurst coeficient

A maximum likelihood estimation method implemented in S-PLUS (S-MLE) to estimate the Hurst coefficient (H) is evaluated. The Hurst coefficient, with 0.5 < H <1, characterizes long memory time series by quantifying the rate of decay of the autocorrelation function. S-MLE was developed to estimate H for fractionally differenced (fd) processes. However, in practice it is difficult to distinguish between fd processes and fractional Gaussian noise (fGn) processes. Thus, the method is evaluated for estimating H for both fd and fGn processes. S-MLE gave biased results of H for fGn processes of any length and for fd processes of lengths less than 2(10). A modified method is proposed to correct for this bias. It gives reliable estimates of H for both fd and fGn processes of length greater than or equal to 2(11).

Determining the Hurst exponent of fractal time series and its application to electrocardiographic analysis

An alternative regression-based method for estimating the Hurst coefficient of a fractal time series is proposed. A formal mathematical description of the methodology is presented. The geometric relationship of the algorithm to the family of self-similar fractal curves is outlined. The computational structure of the algorithm is optimal for generation of real-time estimates of H. We show that the method can be applied to biologically-derived time series such as the cardiac interbeat interval and we obtain estimates of H from several diverse electrocardiographic data sets.

A study of speech fractal dimensions

The fractal dimension (D) of a signal gives an estimate of its degree of freedom, which allows estimation of its fluctuations. Using 16 kHz time sampling and the box counting method we studied the Ds of some of the main stationary parts of French speech, the phonemes [a], [e], [i], [o], [y], pronounced 4 times by 10 males and 10 females. Our study demonstrated that the stationary signal of vowels is not fractal, but may, at the smallest scale, provide a kind of signature for each vowel, though the present categorization is not totally significant. Since the box counting method objectifies and quantifies the roughness of the signal, this procedure may be useful for clinical applications. In case of dysphonia, moreover, these signatures could be perhaps be included in the speech signal processing of cochlear implants.

Non-linear cardiac dynamics and morning dip: an unsound circadian rhythm

The frequency of sudden cardiac death increases in the morning. The relationship between decreased complexity of heart rate dynamics and sudden cardiac death has been documented. An understanding of circadian variation in the complexity of cardiac dynamics may be important to predict and prevent sudden cardiac death. Dynamic 24-h electrocardiographic recordings were obtained from 30 healthy ambulant subjects aged 41-50 years, and the digitized data were partitioned into sections of 30 min duration. For each section, four indexes obtained from separate algorithms of non-linear dynamics of the RR interval--modified correlation dimension, Lyapunov exponent, approximate entropy, and fractal dimension--were calculated. Normalized low-(0.04-0.15 hertz) and high-frequency (> 0.15 hertz) components were also calculated. All four indexes of non-linear dynamics showed a remarkably similar circadian rhythm: a prominent morning dip preceded by a steep decline during the late night, a recovery during the evening and a peak around midnight. In the morning, the low-frequency component rose rapidly with concomitant reduction in the high-frequency component. The complexity of cardiac dynamics decreases significantly in the morning, and this may contribute to the ominously increased rate of cardiac death in the morning hours.

Effects of maternal alcohol intake on fractal properties in human fetal breathing dynamics

Fractal methods have been found to be useful in characterizing biomedical signals. The use of fractal estimation requires the estimation of parameter H, which is directly related to the fractal dimension D. Here, we propose a new approach which is a combination of the wavelet transform and fractal estimators to characterize the human fetal breathing signals before and after the intake of two glasses of wine by a mother. This study was performed on 26 fetuses. The variances of the wavelet coefficients were estimated at each scale. The slope of the representation on a logarithmic plot from the scales 5 to 1 was found to be increased after alcohol intake. Our results suggested that fetal breathing rates have a rough structure before the alcohol intake and a smooth structure after alcohol intake.

Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods

Precise reference signals are required to evaluate methods for characterizing a fractal time series. Here we use fGp (fractional Gaussian process) to generate exact fractional Gaussian noise (fGn) reference signals for one-dimensional time series. The average autocorrelation of multiple realizations of fGn converges to the theoretically expected autocorrelation. Two methods commonly used to generate fractal time series, an approximate spectral synthesis (SSM) method and the successive random addition (SRA) method, do not give the correct correlation structures and should be abandoned. Time series from fGp were used to test how well several versions of rescaled range analysis (R/S) and dispersional analysis (Disp) estimate the Hurst coefficient (0 < H < 1.0). Disp is unbiased for H < 0.9 and series length N ≥ 1024, but underestimates H when H > 0.9. R/S-detrended overestimates H for time series with H < 0.7 and underestimates H for H > 0.7. Estimates of H(Ĥ) from all versions of Disp usually have lower bias and variance than those from R/S. All versions of dispersional analysis, Disp, now tested on fGp, are better than we previously thought and are recommended for evaluating time series as long-memory processes.

Evaluating scaled windowed variance methods for estimating the Hurst coefficient of time series

Three-scaled windowed variance methods (standard, linear regression detrended, and brdge detrended) for estimating the Hurst coefficient (H) are evaluated. The Hurst coefficient, with 0 < H < 1, characterizes self-similar decay in the time-series autocorrelation function. The scaled windowed variance methods estimate H for fractional Brownian motion (fBm) signals which are cumulative sums of fractional Gaussian noise (fGn) signals. For all three methods both the bias and standard deviation of estimates are less than 0.05 for series having N ≥ 2(9) points. Estimates for short series (N < 2(8)) are unreliable. To have a 0.95 probability of distinguishing between two signals with true H differing by 0.1, more than 2(15) points are needed. All three methods proved more reliable (based on bias and variance of estimates) than Hurst's rescaled range analysis, periodogram analysis, and autocorrelation analysis, and as reliable as dispersional analysis. The latter methods can only be applied to fGn or differences of fBm, while the scaled windowed variance methods must be applied to fBm or cumulative sums of fGn.

A comparison of analytical methods for the study of fractional Brownian motion

Fractional Brownian motion (FBM) provides a useful model for many physical phenomena demonstrating long-term dependencies and l/f-type spectral behavior. In this model, only one parameter is necessary to describe the complexity of the data, H, the Hurst exponent. FBM is a nonstationary random function not well suited to traditional power spectral analysis however. In this paper we discuss alternative methods for the analysis of FBM, in the context of real-time biomedical signal processing. Regression-based methods utilizing the power spectral density (PSD), the discrete wavelet transform (DWT), and dispersive analysis (DA) are compared for estimation accuracy and precision on synthesized FBM datasets. The performance of a maximum likelihood estimator for H, theoretically the best possible estimator, are presented for reference. Of the regression-based methods, it is found that the estimates provided by the DWT method have better accuracy and precision for H > 0.5, but become biased for low values of H. The DA method is most accurate for H < 0.5 for a 256-point data window size. The PSD method was biased for both H < 0.5 and H > 0.5.