Fractal dimension and histogram method: Algorithm and some preliminary results of noise-like time series analysis

In the present work a methodological background for the histogram method of time series analysis is developed. Connection between shapes of smoothed histograms constructed on the basis of short segments of time series of fluctuations and the fractal dimension of the segments is studied. It is shown that the fractal dimension possesses all main properties of the histogram method. Based on it a further development of fractal dimension determination algorithm is proposed. This algorithm allows more precision determination of the fractal dimension by using the “all possible combination” method. The application of the method to noise-like time series analysis leads to results, which could be obtained earlier only by means of the histogram method based on human expert comparisons of histograms shapes.     

Fractal analysis of the uterine contractions.

The fractal dimension D may be calculated in many ways, since its strict definition, the Hausdorff definition is too complicated for practical estimation. In this paper we perform a comparative study often methods of fractal analysis of time series. In Benoit, a commercial program for fractal analysis, five methods of computing fractal dimension of time series (rescaled range analysis, power spectral analysis, roughness-length, variogram methods and wavelet method) are available. We have implemented some other algorithms for calculating D: Higuchi's fractal dimension, relative dispersion analysis, running fractal dimension, method based on mathematical morphology and method based on intensity differences. For biomedical signals results obtained by means of different algorithms are different, but consistent.     

Fractal geometry of root systems: Field observations of contrasting genotypes of common bean (Phaseolus vulgaris L.) grown under different phosphorus regimes

Root growth and architecture are important for phosphorus acquisition due to the relative immobility of P in the soil. Fractal geometry is a potential new approach to the analysis of root architecture. Substantial genetic variation in root growth and architecture has been observed in common bean. Common bean (Phaseolus vulgaris L.) genotypes with contrasting root architecture were grown under moderate and low P conditions in a field experiment. Linear and planar fractal dimension were measured by tracing root intercepts with vertical planes. Linear fractal dimension increased over time in efficient genotypes, but remained fairly constant over time in inefficient genotypes. Planar fractal dimension increased over time for all genotypes, but was higher in efficient than inefficient genotypes at the end of the experiment. Planar fractal dimension of medium P plants was found to correlate with shoot P content indicating fractal dimension to be a possible indicator for root P uptake. The increasing fractal dimension over time indicates that fractal analysis is a sensitive measure of root branching intensity. A less destructive method for acquisition of data that allows for continuous analysis of fractal geometry and thereby screening for more P efficient genotypes in the field is suggested. This method will allow the researcher to conduct fractal analysis and still complete field trials with final yield evaluation.     

Correcting for finite spatial scales of self-similarity when calculating fractal dimensions of real-world structures

Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales.Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real-world branching structures.     

Higuchi dimension of digital images

There exist several methods for calculating the fractal dimension of objects represented as 2D digital images. For example, Box counting, Minkowski dilation or Fourier analysis can be employed. However, there appear to be some limitations. It is not possible to calculate only the fractal dimension of an irregular region of interest in an image or to perform the calculations in a particular direction along a line on an arbitrary angle through the image. The calculations must be made for the whole image. In this paper, a new method to overcome these limitations is proposed. 2D images are appropriately prepared in order to apply 1D signal analyses, originally developed to investigate nonlinear time series. The Higuchi dimension of these 1D signals is calculated using Higuchi's algorithm, and it is shown that both regions of interests and directional dependencies can be evaluated independently of the whole picture. A thorough validation of the proposed technique and a comparison of the new method to the Fourier dimension, a common two dimensional method for digital images, are given. The main result is that Higuchi's algorithm allows a direction dependent as well as direction independent analysis. Actual values for the fractal dimensions are reliable and an effective treatment of regions of interests is possible. Moreover, the proposed method is not restricted to Higuchi's algorithm, as any 1D method of analysis, can be applied.     

Nonlinear fractal dynamics of human colonic pressure activity based upon the box-counting method

The computational fractal dimension of human colonic pressure activity acquired by a telemetric capsule robot under normal physiological conditions was studied using the box-counting method. The fractal dimension is a numeric value that quantifies to measure how rough the signal is from nonlinear dynamics, rather than its amplitude or other linear statistical features. The colonic pressure activities from the healthy subject during three typical periods were analysed. The results showed that the activity might be fractal with a non-integer fractal dimension after it being integrated over time using the cumsum method, which was never revealed before. Moreover, the activity (after it being integrated) acquired soon after wakening up was the roughest (also the most complex one) with the largest fractal dimension, closely followed by that acquired during sleep with that acquired long time after awakening up (in the daytime) ranking third with the smallest fractal dimension. Fractal estimation might provide a new method to learn the nonlinear dynamics of human gastrointestinal pressure recordings.     

铅染毒大鼠脑电信号的分形特性研究

利用混沌时间序列的分析方法,对铅中毒情况下的大鼠的脑电波进行了测试和分析,求出了其分形维数。从研究结果中得出了铅中毒情况与脑电信号的分形维数之间的关系,结果表明,铅中毒会引起脑电信号的分形维数的显著变化。通过这一工作,能够为临床上重金属中毒的早期诊断提供新的方便、灵敏的指标。     

Analysis of fractal dimension of O2A glial cells differentiating in vitro

Fractal dimension is a quantitative measure of morphological complexity. Glial cells of the oligodendrocyte-type 2 astrocyte (O2A) lineage exhibit increasing morphological complexity as they differentiate in vitro. Enriched populations of O2A progenitor cells isolated from neonatal rat cerebral hemispheres or optic nerves were allowed to differentiate in vitro, and their fractal dimensions were measured over time. The fractal dimensions of the maturing cells correlated with perceived complexity; cells with elaborate process branching had larger fractal dimensions than cells with a simpler morphology. An analysis of changes in fractal dimension revealed distinct rates of growth for both oligodendrocytes and type 2 astrocytes. The fractal dimension remained constant over a 10-fold range in optical magnification, demonstrating that cultured O2A glial cells exhibit self-similarity, a defining characteristic of fractal objects. These results illustrate that fractal dimension analysis of maturing cell populations is a useful method for quantitatively describing the process of cell differentiation.     

Fractal dimension analysis of factor X activation in the presence of tissue factor-factor viia complex in a continuous flow reactor

Summary The characteristics of a phospholipid surface are of major importance in the activation of factor X in the presence of tissue factor-factor VIIa (TF-VIIa) complex. A possible tool which provides a measure of the surface corrugation and roughness is the fractal dimension analysis. This paper uses the fractal characterization of a phospholipid surface to develop a model for analyzing surface based enzymatic reaction data. The modeling indicates that the fractal dimension (D) of a phospholipid surface is a function of the wall shear rate. The results also indicate that the fractal dimension of the phospholipid surface decreases from approximately 2.9 to 1.4 as the wall shear rate increases from 50 to 1600 sec^–1. At the same time the factor Xa production increases from 1.9 to 5.8 pmoles/ (min. cm^–2).The results of the fractal dimension analysis clearly indicates that the surface roughness of a phospholipid surface may have a significant effect on factor X activation.     

Fractal methods to analyze ion channel kinetics

We describe the traditional nonfractal and the new fractal methods used to analyze the currents through ion channels in the cell membrane. We discuss the hidden assumptions used in these methods and how those assumptions lead to different interpretations of the same experimental data. The nonfractal methods assumed that channel proteins have a small number of discrete states separated by fixed energy barriers. The goal was to determine the parameters of the kinetic diagram, which are the number of states, the pathways between them, and the kinetic rate constants of those pathways. The discovery that these data have fractal characteristics suggested that fractal approaches might provide more appropriate tools to analyze and interpret these data. The fractal methods determine the characteristics of the data over a broad range of time scales and how those characteristics depend on the time scale at which they are measured. This is done by using a multiscale method to accurately determine the probability density function over many time scales and by determining how the effective kinetic rate constant, the probability of switching states, depends on the effective time scale at which it is measured. These fractal methods have led to new information about the physical properties of channel proteins in terms of the number of conformational substates, the distribution of energy barriers between those states, and how those energy barriers change with time. The new methods developed from the fractal paradigm shifted the analysis of channel data from determining the parameters of a kinetic diagram to determining the physical properties of channel proteins in terms of the distribution of energy barriers and/or their time dependence.     

Quantitative method to study the network formation of endothelial cells in response to tumor angiogenic factors

To study the network formation of endothelial cells (ECs) in an extracellular matrix (ECM) environment, we have devised an EC aggregation-type model based on a diffusion limited cluster aggregation model (DLCA), where clusters of particles diffuse and stick together upon contact. We use this model to quantify EC differentiation into cord-like structures by comparing experimental and simulation data. Approximations made with the DLCA model, when combined with experimental kinetics and cell concentration results, not only allow us to quantify cell differentiation by a pseudo diffusion coefficient, but also measure the effects of tumor angiogenic factors (TAFs) on the formation of cord-like structures by ECs. We have tested our model by using an in vitro assay, where we record EC aggregation by analysing time-lapse images that provide us with the evolution of the fractal dimension measure through time. We performed these experiments for various cell concentrations and TAFs (e.g. EVG, FGF-b, and VEGF). During the first six hours of an experiment, ECs aggregate quickly. The value of the measured fractal dimension decreases with time until reaching an asymptotic value that depends solely on the EC concentration. In contrast, the kinetics depend on the nature of TAFs. The experimental and simulation results correlate with each other in regards to the fractal dimension and kinetics, allowing us to quantify the influence of each TAF by a pseudo diffusion coefficient. We have shown that the shape, kinetic aggregation, and fractal dimension of the EC aggregates fit into an in vitro model capable of reproducing the first stage of angiogenesis. We conclude that the DLCA model, combined with experimental results, is a highly effective assay for the quantification of the kinetics and network characteristics of ECs embedded in ECM proteins. Finally, we present a new method that can be used for studying the effect of angiogenic drugs in in vitro assays.     

Evaluation of self-similar features in time series of serum growth hormone and prolactin levels by fractal analysis: Effect of delayed sleep and complexity of diurnal variation

We assayed the diurnal concentrations of growth hormone (GH) and prolactin (PRL) in 6 healthy male volunteers to evaluate the self-similar features in the time series of each hormone on the basis of fractal theory and to determine the fractal dimension as an index of the complexity of the diurnal variation. In addition, we assessed the effects of a 6-hour delay in the sleep period on the complexity of the diurnal variaton of these hormones. There was a statistically significant fractal feature in the serum levels of GH both under the nocturnal-sleep and delayed-sleep conditions in all subjects. The time series of the serum PRL concentrations also showed a statistically significant fractal feature under the nocturnal-sleep and delayed-sleep conditions in all subjects. The fractal dimensions of the patterns of the GH or PRL levels were 1.879 and 1.929 or 1.754 and 1.785 under the nocturnal-sleep and delayed-sleep conditions, respectively. Two-way ANOVA revealed no significant difference in the fractal dimension between the two sleep conditions but did reveal a significant difference between the fractal dimensions of the GH and PRL levels. These results showed (1) that delayed sleep had no significant effect on the complexity of the diurnal pattern of these hormones, and (2) that the diurnal pattern of the GH levels was more complex than that of the PRL levels.     

Use of the fractal dimension for the analysis of electroencephalographic time series

 Electroencephalogram (EEG) traces corresponding to different physiopathological conditions can be characterized by their fractal dimension, which is a measure of the signal complexity. Generally this dimension is evaluated in the phase space by means of the attractor dimension or other correlated parameters. Nevertheless, to obtain reliable values, long duration intervals are needed and consequently only long-term events can be analysed; also much calculation time is required. To analyse events of brief duration in real-time mode and to apply the results obtained directly in the time domain, thus providing an easier interpretation of fractal dimension behaviour, in this work we optimize and propose a new method for evaluating the fractal dimension. Moreover, we study the robustness of this evaluation in the presence of white or line noises and compare the results with those obtained with conventional spectral methods. The non-linear analysis carried out allows us to investigate relevant EEG events shorter than those detectable by means of other linear and non-linear techniques, thus achieving a better temporal resolution. An interesting link between the spectral distribution and the fractal dimension value is also pointed out. Received: 21 November 1996 / Accepted in revised form: 1 July 1997     

Structural complexity in mussel beds: the fractal geometry of surface topography

Seafloor topographic complexity is ecologically important because it provides habitat structure and alters boundary-layer flow over the bottom. Despite its importance, there is little agreement on how to define and measure surface complexity. The purpose of this investigation was to utilize fractal geometry of vertical cross-section profiles to characterize the surface topography of the soft-bottom mussel bed (Mytilus edulis L.) at Bob's Cove, ME, USA. Mussels there have been shown previously to have spatially ordered fractal characteristics in the horizontal plane. Two hypotheses were tested. The first was that the bed surface is fractal over the spatial scale of 1.44-200 mm, with fractal dimension less than or equal to 1.26, the value for the Koch curve, our model for bed profiles. The second was that bed surface topography (i.e., in vertical profile) is less complex than the mussel bed spatial pattern (i.e., aerial view in the horizontal plane). Both hypotheses were supported. Cross-sections of plaster casts of the bed produced 88 surface profiles, all of which were fractal over the entire spatial scale of more two orders of magnitude employed in the analysis. Fractal dimension values (D) for individual profiles ranged from 1.031 to 1.310. Fractal dimensions of entire casts ranged up to mean (1.242+/-0.046) and median (1.251) values similar to 1.26, the theoretical value of the Koch curve. The bed surface was less complex than the bed spatial pattern because every profile had D<1.36, the smallest value previously obtained from aerial views of the bed. The investigation demonstrated for the first time that surface topography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractal dimension that can result from other profiling methods reported in the literature. The results demonstrate that fractal dimension can be useful in the analysis of habitat space and water flow over any irregular seafloor surface because it incorporates the size, shape, and scale of roughness elements into a simple, numerical metric.     

The Quasi-Fractal Structure of Fish Brain Neurons

The box-counting method for calculating the fractal dimension (D) with the ImageJ 1.20s software is used as a tool for quantitative analysis of the neuronal morphology in the fish brain. The fractal dimension was determined for several types of neurons in the brain of two teleost species, Pholidapus dybowskii and Oncorhynchus keta. These results were compared with those obtained for some neurons of the human brain. The fractal (fractional) dimension (D), as a quantitative index of filling of two-dimensional space by the black and white image of a cell, is shown to vary from 1.22 to 1.72 depending on the type of neuron. The fractal dimension reaches its maximum in less specialized neurons that carry out a number of different functions. On the other hand, highly specialized neurons display a relatively low fractal dimension. Thus, the fractal dimension serves as a numerical measure of the spatial complexity of the neuron and correlates with the morphofunctional organization of the cell.     

A fractal model for the characterization of mycelial morphology

A new technique based on a fractal model has been developed for the quantification of the macroscopic morophology of mycelia. The morphological structuring is treated as a fractal object, and the fractal dimension, determined by an ultrasonic scattering procedure developed for the purpose, serves as a quantitative morphological index. Experimental observations reported earlier and simulations of mycelial growth, carried out using a probabilistic-geometric growth model developed for the purpose, both validate the applicability of the fractal model. In experiments with three different species, the fractal dimensions of pelletous structures were found to be in the range 1.45-2.0 and those of filamentous structures were in the range 1.9-2.7, with values around 2.0 representing mixed morphologies. Fractal dimensions calculated from simulated mycelia are in rough agreement with these ranges. The fractal dimension is also found to be relatively insensitive to the biomass concentration, as seen by dilution of the original broths. The relation between morphology and filtration properties of the broths has also been studied. The fractal dimension shows a strong correlation with the index of cake compressibility and with the Kozeny constant, two filtration parameters that are known to be morphology dependent. This technique could thus be used to develop correlations between the morphology, represented by the fractal dimension, and important morphology-dependent process variables. (c) 1993 John Wiley & Sons, Inc.     

Fractal dimension of birds population sizes time series

Information about fractal dimension is collected so that it can be applied to time series interpreting Hurst coefficient. The population size of a species is modelled as a dynamic system. The Hurst coefficient is calculated for these times series. A computer programme has been elaborated to compute the Hurst exponent of time series using the algorithms of range increment, second order moment increment and local second order moment increment. It has been applied to time series of birds' populations.