Four Methods to Estimate the Fractal Dimension from Self-Affine Signals

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.

Quantifying the strain history of bone: spatial uniformity and self-similarity of low-magnitude strains

We hypothesize that when a broad spectrum of bone strain is considered, strain history is similar for different bones in different species. Using a data collection protocol with a fine resolution, mid-diaphyseal strains were measured in vivo for both weightbearing and non-weightbearing bones in three species: dog, sheep, and turkey, with strain information collected continuously while the animals performed their natural daily activities. The daily strain history was quantified by both counting cyclic strain events (to quantify the distribution of strains of different magnitudes) and by estimating the average spectral characteristics of the strain (to quantify the frequency content of the strain signals). Counting of the daily (12-24 h) strain events show that large strains (> 1000 microstrain) occur relatively few times a day, while very small strains (< 10 microstrain) occur thousands of times a day. The lower magnitude strains (< approximately 200 microstrain) are found to be more uniform around the bone cross-section than the higher magnitude, peak strains. Strain dynamics are found to be well described by a power-law relationship and exhibit self-similar characteristics. These data lead to the suggestion that the organization of bone tissue is driven by the continual barrage of activity spanning a wide but consistent range of frequency and amplitude, and until the mechanism of bone's mechanosensory system is fully understood, all portions of bone's strain history should be considered to possibly play a role in bone adaptation.

Generation of fractal dimension images and its application to automatic edge detection in brain MRI

We have developed four methods to generate a fractal dimension image and have applied them to the brain MRI. We have adopted four types of scanning methods, "CONVENTIONAL", "OVERLAPPING", "SYMMETRIC" and "FOLDED" to estimate the fractal dimension. The first three methods show almost the same fractal dimension images and their values were between two and three. In the "FOLDED" method, we were able to obtain the images in which the edge of a narrow region including dura and scalp surrounding the brain was selectively enhanced in the T1-weighted MRI. This is found to be a new edge-enhancing filter. We could remove the surrounding structure of the brain by using these filtered images and detect the edge of the brain surface automatically. The brain surface data can be used for various applications such as three-dimensional surface display and registration of inter-modal brain images.

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).

Non-linear behaviour of human EEG: fractal exponent versus correlation dimension in awake and sleep stages

The question of whether the finite values of the correlation dimension (D2), used as an index of EEG complexity are due to its chaotic nature or they reflect its behaviour as linearly-correlated noise, remains open. This report aims at clarifying this by measuring D2 and analysing the non-linear nature of EEG through the method of surrogate data as well as by calculating the fractal exponent (beta) via coarse graining spectral analysis (CGSA) in nine adult subjects during waking and sleep states. The results show that even if it is possible to get an estimation of D2 in all states, non-linear structure appears to be present only during slow wave sleep (SWS). EEG exhibits random fractal structure with 1/f(-beta) spectrum (1 < beta < 3) and a negative linear correlation between D2 and beta in all states except during SWS. In consequence, in those states, finite D2 values could be attributed to the fractal nature of EEG and not to the presence of low-dimensional chaos, and therefore, it the use of beta would be more appropriate to describe the complexity of EEG, due to its lower computational cost.

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.

Fractal character of the neural spike train in the visual system of the cat

We used a variety of statistical measures to identify the point process that describes the maintained discharge of retinal ganglion cells (RGC's) and neurons in the lateral geniculate nucleus (LGN) of the cat. These measures are based on both interevent intervals and event counts and include the interevent-interval histogram, rescaled range analysis, the event-number histogram, the Fano factor, Allan factor, and the periodogram. In addition, we applied these measures to surrogate versions of the data, generated by random shuffling of the order of interevent intervals. The continuing statistics reveal 1/f-type fluctuations in the data (long-duration power-law correlation), which are not present in the shuffled data. Estimates of the fractal exponents measured for RGC- and their target LGN-spike trains are similar in value, indicating that the fractal behavior either is transmitted form one cell to the other or has a common origin. The gamma-r renewal process model, often used in the analysis of visual-neuron interevent intervals, describes certain short-term features of the RGC and LGN data reasonably well but fails to account for the long-duration correlation. We present a new model for visual-system nerve-spike firings: a gamma-r renewal process whose mean is modulated by fractal binomial noise. This fractal, doubly stochastic point process characterizes the statistical behavior of both RGC and LGN data sets remarkably well.

Supraspinal effects on the fractal correlation in human H-reflex

In our previous study, 1/f beta-type power spectrum with the spectral exponent beta significantly greater than zero was found in the variability of soleus H-reflex amplitudes. This result indicated that the H-reflex variability was time-correlated owing to fractal characteristics. Furthermore, it was also suggested that the fractal characteristics were generated at the spinal level. The purpose of the present study was to investigate whether the fractal nature of the H-reflex variability was influenced by the loss of supraspinal input. Six healthy normal subjects and seven patients with spinal cord injury participated in this study. Soleus H-reflexes were evoked every 1 s from both legs simultaneously (stimulation intensity: motor threshold) and 1050 successive amplitudes of the H-reflex were recorded. The H-reflex sequence evoked from each leg was analyzed by "coarse graining spectral analysis" to calculate the spectral exponent beta. The value of beta was used to evaluate the level of time-correlation (fractal correlation). Cross-spectral analysis was used to evaluate the degree of synchronization between the H-reflex sequences evoked from both legs. The beta values for normal subjects (0.84 +/- 0.33, left leg; 0.88 +/- 0.34. right leg) were significantly greater (P < 0.001) than those for patients (0.31 +/- 0.18, left leg; 0.32 +/- 0.14, right leg), suggesting that the H-reflex sequences for normal subjects were more time-correlated than for patients. In the frequency range less than 0.2 Hz, the coherence of both legs was high (0.41 +/- 0.14) for normal subjects as compared to 0.20 +/- 0.12 for patients (P < 0.005). In this frequency range, the phase was almost 0 rad for normal subjects, indicating that the H-reflex variabilities of both legs were synchronized. These results suggested that (1) the strong fractal correlation observed in the H-reflex sequences for normal subjects was associated with supraspinal input, and (2) such supraspinal input had equal influence on the reflex arcs of the soleus of both legs.

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.

Fractal character of the electrocardiogram: distinguishing heart-failure and normal patients

Statistical analysis of the sequence of heartbeats can provide information about the state of health of the heart. We used a variety of statistical measures to identify the form of the point process that describes the human heartbeat. These measures are based on both interevent intervals and counts, and include the interevent-interval histogram, interval-based periodogram, rescaled range analysis, the event-number histogram, Fano-factor, Allan Factor, and generalized-rate-based periodogram. All of these measures have been applied to data from both normal and heart-failure patients, and various surrogate versions thereof. The results show that almost all of the interevent-interval and the long-term counting statistics differ in statistically significant ways for the two classes of data. Several measures reveal 1/f-type fluctuations (long-duration power-law correlation). The analysis that we have conducted suggests the use of a conveniently calculated, quantitative index, based on the Allan factor, that indicates whether a particular patient does or does not suffer from heart failure. The Allan factor turns out to be particularly useful because it is easily calculated and is jointly responsive to both short-term and long-term characteristics of the heartbeat time series. A phase-space reconstruction based on the generalized heart rate is used to obtain a putative attractor's capacity dimension. Though the dependence of this dimension on the embedding dimension is consistent with that of a low-dimensional dynamical system (with a larger apparent dimension for normal subjects), surrogate-data analysis shows that identical behavior emerges from temporal correlation in a stochastic process. We present simulated results for a purely stochastic integrate-and-fire model, comprising a fractal-Gaussian-noise kernel, in which the sequence of heartbeats is determined by level crossings of fractional Brownian motion. This model characterizes the statistical behavior of the human electrocardiogram remarkably well, properly accounting for the behavior of all of the measures studied, over all time scales.

Membrane potential fluctuations of human T-lymphocytes have fractal characteristics of fractional Brownian motion

The voltage across the cell membrane of human T-lymphocyte cell lines was recorded by the whole cell patch clamp technique. We studied how this voltage fluctuated in time and found that these fluctuations have fractal characteristics. We used the Hurst rescaled range analysis and the power spectrum of the increments of the voltage (sampled at 0.01-sec intervals) to characterize the time correlations in these voltage fluctuations. Although there was great variability in the shape of these fluctuations from different cells, they all could be represented by the same fractal form. This form displayed two different regimes. At short lags, the Hurst exponent H = 0.76 +/- 0.05 (SD) and, at long lags, H = 0.26 +/- 0.04 (SD). This finding indicated that, over short time intervals, the correlations were persistent (H > 0.5), that is, increases in the membrane voltage were more likely to be followed by additional increases. However, over long time intervals, the correlations were antipersistent (H < 0.5), that is, increases in the membrane voltage were more likely to be followed by voltage decreases. Within each time regime, the increments in the fluctuations had characteristics that were consistent with those of fractional Gaussian noise (fGn), and the membrane voltage as a function of time had characteristics that were consistent with those of fractional Brownian motion (fBm).

Correlation in stimulated respiratory neural noise

Noise in spontaneous respiratory neural activity of the neonatal rat isolated brainstem-spinal cord preparation stimulated with acetylcholine (ACh) exhibits positive correlation. Neural activity from the C4 (phrenic) ventral spinal rootlet, integrated and corrected for slowly changing trend, is interpreted as a fractal record in time by rescaled range, relative dispersional, and power spectral analyses. The Hurst exponent H measured from time series of 64 consecutive signal levels recorded at 2 s intervals during perfusion of the preparation with artificial cerebrospinal fluid containing ACh at concentrations 62.5 to 1000 &mgr;M increases to a maximum of 0.875+/-0.087 (SD) at 250 &mgr;M ACh and decreases with higher ACh concentration. Corrections for bias in measurement of H were made using two different kinds of simulated fractional Gaussian noise. Within limits of experimental procedure and short data series, we conclude that in the presence of added ACh of concentration 250 to 500 &mgr;M, noise which occurs in spontaneous respiratory-related neural activity in the isolated brainstem-spinal cord preparation observed at uniform time intervals exhibits positive correlation. (c) 1995 American Institute of Physics.

Characterization of the cellular microstructure of ocular lens using 2D power law analysis

Power law analysis provides a quantitative method for characterization of spatial fluctuations in the cellular microstructure of the ocular lens. In the power law analysis, Fourier components of the spatial fluctuations are computed, and the relationship between the amplitude, A, and spatial frequency, f, of the components is defined by a power law function: [formula, see text]. The exponent of the function, beta, defines the scaling of the amplitude of the Fourier components as a function of spatial frequency. We performed two-dimensional power law analysis on electron micrographs of lens cells ranging from transparent to opaque. We identified two values of power law exponent, beta, for the spatial fluctuations of all lens cells, one for low- and a second for high-spatial frequencies. In the low-spatial frequency region, the value of beta was in the range of 0.53 to 1.33, for transparent and opaque cells. In the high-spatial frequency region, the value of beta increased from 2.78 for transparent lens cells to 3.60 for opaque lens cells. The power law analysis provides a new method for quantitative characterization of the spatial fluctuations in the microstructure of transparent and opaque lens cells.

Evaluation of the dispersional analysis method for fractal time series

Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is < 0.5 for rough anticorrelated series, > 0.5 for positively correlated series, and = 0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 - D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (> 0.8) and low H (< 0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H. Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.

The use of fractal analysis to reveal remodelling in human alveolar bone following the placement of dental implants

In order to confirm the presence of adequate bone support for implants it is necessary to examine the surrounding bone at intervals. While this may be attempted by looking for changes in serial radiographs, such comparisons are inaccurate unless standardized procedures are followed which allow subtraction of consecutive digital images. As image textures are less susceptible to magnification distortion, it was decided to examine the fractal dimensions of successive radiographs of bone after implant placement. All available panoramic radiographs for each of 18 patients who had received fixed implant-supported prostheses were digitized. A window of bone adjacent and distal to the most posterior implant was defined as the region of interest; the fractal dimension of the image was calculated. Linear regression was used to investigate whether there were any significant shifts in fractal dimension during the recall period after implantation. A significant increase in fractal dimension was found during the period up to 2 yr after implantation (p < 0.001). The most pronounced increase was in the region of bone around the neck of the implant. An increase in orientation of the image in a direction oblique to the implant was also found during the same period. These changes are consistent with models derived from finite-element analysis that predict the relation between trabecular architecture and strain. One subject's radiographs had a significant negative regression slope, which further monitoring may reveal was an early sign of implant failure. The satisfactory remodelling of bone in response to implant placement may be monitored using a texture analysis of routine orthopantomograms.

Evaluating rescaled ranged analysis for time series

Rescaled range analysis is a means of characterizing a time series or a one-dimensional (1-D) spatial signal that provides simultaneously a measure of variance and of the long-term correlation or "memory," The trend-corrected method is based on the statistical self-similarity in the signal: in the standard approach one measures the ratio R/S on the range R of the sum of the deviations from the local mean divided by the standard deviation S from the mean. For fractal signals R/S is a power law function of the length tau of each segment of the set of segments into which the data set has been divided. Over a wide range of tau's the relationship is: R/S = a tau H, where kappa is a scalar and the H is the Hurst exponent. (For a 1-D signal f(t), the exponent H = 2 - D, with D being the fractal dimension.) The method has been tested extensively on fractional Brownian signals of known H to determine its accuracy, bias, and limitations. R/S tends to give biased estimates of H, too low for H > 0.72, and too high for H < 0.72. Hurst analysis without trend correction differs by finding the range R of accumulation of differences from the global mean over the total period of data accumulation, rather than from the mean over each tau. The trend-corrected method gives better estimates of H on Brownian fractal signals of known H when H > or = 0.5, that is, for signals with positive correlations between neighboring elements. Rescaled range analysis has poor convergence properties, requiring about 2,000 points for 5% accuracy and 200 for 10% accuracy. Empirical corrections to the estimates of H can be made by graphical interpolation to remove bias in the estimates. Hurst's 1951 conclusion that many natural phenomena exhibit not random but correlated time series is strongly affirmed.

Rescaled range analysis of resting respiration

Fluctuations in resting depth of breathing (tidal volume) at constant breathing rate in the anesthetized adult rat exhibit fractal properties when analyzed by a rescaled range method characterized by a mean (+/-SD) exponent H=0.83+/-0.02 and 0.92+/-0.03 with and without sighs, respectively, for up to 400 breaths. Values of H determined from shuffled tidal volumes and simulated tidal volumes taken randomly from a Gaussian distribution of mean and variance approximating that of the actual data are consistent with the expected value of H=0.5 for an independent random process with finite variances. An empirical description is proposed to predict the change in H with length of time record.

Fractal correlation in human H-reflex

The amplitude of the H-reflex has been known to have considerable variability even if the intensity of the stimulation is held constant. However, previous studies largely ignored the time-dependent profile of this variability. Recent mathematical analyses have shown that some seemingly irregular biological signals have fractal properties. A fractal time series is characterized by the property of self-similarity (self-affinity), and has long-range time correlation. The aim of this study was to investigate the question of whether the fluctuation of H-reflex was fractal with strong time-correlation. Soleus H-reflexes were evoked in five healthy subjects at two levels of stimulation intensity [1.2 MT (motor threshold) and 0.9 MT] every 1 s and 1050 successive amplitudes of H-wave and M-wave were recorded twice. The sequences of the H-wave and the M-wave amplitudes were analyzed by "coarse graining spectral analysis" to calculate the percentage of random fractal components in the sequences (%Fractal) and the spectral exponent beta. The %Fractal values of both sequences were above 90% [H-wave: 93.3 +/- 2.3% (1.2 MT), 91.6 +/- 3.1% (0.9 MT); M-wave: 94.3 +/- 3.3%; mean +/- SD]. Nonflat power spectra of the fractal components were observed for the H-wave sequences regardless of the stimulation intensity [beta = 0.75 +/- 0.26 (1.2 MT), 0.80 +/- 0.39 (0.9 MT)], indicating that the sequences were strongly time correlated. On the other hand, the M-wave sequences had a flatter spectrum (beta = 0.26 +/- 0.14) which was close to uncorrelated white noise. These results indicated that: (1) the fractal correlation found in the H-wave sequences was caused neither by the conduction through nerve fibers nor by the transmission at the neuromuscular junction, because the M-wave sequence had a significantly weaker time correlation, and (2) antidromic impulses in a motor nerve induced by the stimulation made a minor contribution to the generation of fractal correlation in the H-wave sequences, because it was preserved when the stimulation intensity was below MT. It was suggested that the fractal correlation in human H-reflex was generated at the synaptic connections to alpha-motoneurons in the spinal cord.