Monofractal Analysis Research Articles

The Morphology of Brain Neurons: The Box-Counting Method in the Quantitative Analysis of 2D Images.

This chapter begins by showing the difference between fractal geometry and fractal analysis. The text shows the difference between mathematical and natural fractals and how they are best defined by explaining the concept of fractal analysis. Furthermore, the text presents the most famous technique of fractal analysis: the box-counting method. Defining this method and showing the methodology that leads to the precise value of the fractal (i.e., the box) dimension is done by demonstrating the images of human dentate neurons. A more detailed explanation of the methodology was presented in the previous version of this chapter.This version promotes the notion of monofractal analysis and shows how three types of the same neuronal images can quantify four image properties. The results showed that monofractal parameters successfully quantified four image properties in three nuclei of the cerebellum. Finally, the author discusses the results of this chapter and previously published conclusions. The results show how the monofractal parameters discriminate images of neurons from the three nuclei of the human cerebrum. These outcomes are discussed along with the results of previous studies.

A guide to Whittle maximum likelihood estimator in MATLAB.

The assessment of physiological complexity via the estimation of monofractal exponents or multifractal spectra of biological signals is a recent field of research that allows detection of relevant and original information for health, learning, or autonomy preservation. This tutorial aims at introducing Whittle's maximum likelihood estimator (MLE) that estimates the monofractal exponent of time series. After introducing Whittle's maximum likelihood estimator and presenting each of the steps leading to the construction of the algorithm, this tutorial discusses the performance of this estimator by comparing it to the widely used detrended fluctuation analysis (DFA). The objective of this tutorial is to propose to the reader an alternative monofractal estimation method, which has the advantage of being simple to implement, and whose high accuracy allows the analysis of shorter time series than those classically used with other monofractal analysis methods.

SEM-imaging-based mapping of monofractal and multifractal patterns of the Piper krukoffii Yunck leaf surface architecture

We employed Scanning Electron Microscopy (SEM) to investigate the monofractal and multifractal characteristics of the leaf architecture of Amazon Piper krukoffii Yunck plant. By examining morphological features of the leaves using SEM imaging, we observed pentagonal to polygonal epidermal walls with fine tuberculate relief and characteristic cells over both leaf surfaces. Our Minkowski functionals analysis showed that the adaxial surface of the leaf had greater percolative behavior compared to the abaxial surface. Using monofractal analysis, we found that the adaxial side exhibited higher spatial complexity (FD = 1.6), more percolative features (FS = 0.228), and higher spatial homogeneity (β = 0.297). Although both sides of the leaf exhibited highly multifractal behavior, the abaxial surface showed a decrease in the trend of interval between heights (Δf = 0.583). These results demonstrate that the adaxial and abaxial surfaces of Piper krukoffii have a similar multifractality, but the abaxial side has less variability in vertical complexity, consistent with our monofractal analysis. Our findings prove that the spatial complexity of leaves is strongly related to their growth dynamics and can play a crucial role in the overall health, pest control, and evolutionary achievements.

Fractal analysis for low temperature physics

A short review of existing methods of monofractal analysis proposed for low temperature physics investigations is presented. Based on the nature of the algorithms used, a new classification of modern monofractal analysis methods is proposed. The properties, the advantages and the disadvantages of these methods are considered. An important new concept of second-order fractals is introduced. Using the set of simple deterministic theoretical models of both the second-order fractal signals created especially for this and the traditional fractal signals, the results of their comparison based on the wavelet analysis are shown. The peculiarities of the second-order fractals are discussed.

Simulation of Nonlinear Viscous Fingering in a Reactive Flow Displacement: A Multifractal Approach

fractal analysis of viscous fingering of a reactive miscible flow displacement in homogeneous porous media is investigated and multifractal spectrum, and fractal dimension are introduced as two essential features to characterize the irregularity of finger patterns. The Reaction of the two reactant fluids generates a miscible chemical product C in the contact zone. Considering the similarity between chemical products and coastline, monofractal and multifractal analyzes are performed. In monofractal analysis, the box-counting method is implemented on binary images and in multifractal analysis, due to the image processing; the fractal characteristics of viscous fingering instability are analyzed by means of fractal quantities such as Holder exponent, multifractal spectrum, f (α)-image and fractal dimension dynamics. Fractal analysis shows that the fractal dimension increases with time. Also, by considering five different nonlinear simulations, the results show that in the case both sides of the chemical product C are unstable, the multifractal spectrum curve has the highest peak, which means the more complex finger patterns lead to more values of fractal dimension. In addition, a comparison between different values of Ar is conducted and the results show similar behavior. However, small value of aspect ratio leads to a broader width of the multifractal spectrum curve. Furthermore, f (α)-images of concentration contour were investigated for different precisions and some undetectable finger patterns were observed in these images. It can be concluded that the use of f (α)-image represents more detailed image than concentration contours.

ФРАКТАЛЬНА РАДІОФІЗИКА. Частина 2. ФРАКТАЛЬНИЙ І МУЛЬТИФРАКТАЛЬНИЙ АНАЛІЗ СИГНАЛІВ ТА ПРОЦЕСІВ

Subject and Purpose. The subject of this paper is to review the principal methods of fractal and multifractal analysis of signals and processes, in combination with a detailed consideration of the algorithms that can provide for a successful practical implementation of the methods described. Methods and Methodology. The results presented concern modeling of both deterministic and stochastic fractal and multifractal signals and processes. The corresponding practical methods of analysis are considered, with discussion of their essential features, advantages and disadvantages, as well as of the problems of application that may exist. Results. Several approaches have been discussed as to categorizing the signals and processes within the notion of fractality. A few tens of models of deterministic and stochastic fractal or multifractal signals and processes have been analyzed in detail. Over twenty methods of monofractal analysis have been analyzed, with identifi cation of their features, advantages or disadvantages, and limits of applicability. The expediency of resorting to complex methods of monofractal analysis has also been discussed. Those methods are not based upon application of fractal analysis techniques alone but rather combine them with linear and nonlinear integral time-frequency transforms. The effectiveness of the ten most popular multifractal analysis techniques has been confirmed, with consideration of their special features, advantages and drawbacks. Conclusion. The mathematical foundations have been presented which underlie modern methods of analysis and modeling of fractal and multifractal signals and processes. The methods discussed may allow revealing a great amount of unique hidden information on the world around us.

Corrective Function Method for the Fractal Analysis

One of the main numerical characteristics used in numerous methods of fractal analysis is the corresponding fractal dimensions. The accuracy of estimating these dimensions in the vast majority of cases is quite small, which cannot satisfy, first of all, researchers-practitioners. The method of the corrective function is put forward, which makes it possible to compensate for the ever-existing nonlinearity of the dependence between the true value of the fractal dimension and its estimation, performed using the selected method of monofractal analysis of signals and processes for a known number of samples of the discrete data vector of the investigated signal. The main idea of the method is to build and apply a special correction function using a set of model fractal signals with previously known values of the fractal dimension. The mathematical bases of the new method are outlined. Features of the practical application of the corrective function method are considered on the example of the evaluation of regularization, boxing, variation and Hurst fractal dimensions. For them, the minimum values of the number of samples of the discrete data vector of the investigated signal, at which these dimensions can still be estimated, are defined. Using a set of model monofractal and multifractal signals on the example of the dynamical fractal analysis method, the effectiveness of the created method of the corrective function is shown. It is proven that due to the application of the correction function method, the maximum deviation of the estimated fractal dimension from the true known value for the specified dimensions is reduced from 25 – 55% to 5 – 7%.

Monofractal analysis of functional magnetic resonance imaging: An introductory review

The following review will aid readers in providing an overview of scale‐free dynamics and monofractal analysis, as well as its applications and potential in functional magnetic resonance imaging (fMRI) neuroscience and clinical research. Like natural phenomena such as the growth of a tree or crashing ocean waves, the brain expresses scale‐invariant, or fractal, patterns in neural signals that can be measured. While neural phenomena may represent both monofractal and multifractal processes and can be quantified with many different interrelated parameters, this review will focus on monofractal analysis using the Hurst exponent (H). Monofractal analysis of fMRI data is an advanced analysis technique that measures the complexity of brain signaling by quantifying its degree of scale‐invariance. As such, the H value of the blood oxygenation level‐dependent (BOLD) signal specifies how the degree of correlation in the signal may mediate brain functions. This review presents a brief overview of the theory of fMRI monofractal analysis followed by notable findings in the field. Through highlighting the advantages and challenges of the technique, the article provides insight into how to best conduct fMRI fractal analysis and properly interpret the findings with physiological relevance. Furthermore, we identify the future directions necessary for its progression towards impactful functional neuroscience discoveries and widespread clinical use. Ultimately, this presenting review aims to build a foundation of knowledge among readers to facilitate greater understanding, discussion, and use of this unique yet powerful imaging analysis technique.

Multifractal-spectrum shape parameters for characterizing distribution and evolution of multiple cracks in concrete structures

The evolution of cracks in reinforced concrete (RC) structures may trigger critical failure modes of the entire structure. In order to adequately assess the damage in RC structures, it is essential to characterize the distribution and evolution of cracks. Compared with existing quantitative methods of describing concrete cracks, multifractal analysis (MUTFA) is an emerging and sophisticated tool for characterizing the complexity and irregularity of crack distributions in RC structures. Despite the fact that MUTFA embodies a greater capability than monofractal analysis (MONFA), the uncertainty of the mechanism of the multifractal spectrum for representing concrete cracks is a major limitation in using MUTFA to depict cracks. To address this drawback, this study illustrates the difference between the features of MUTFA and MONFA as well as proposes a multifractal-spectrum shape parameter to characterize the complexity and irregularity of fractal-like cracks. The advantages of MUTFA in portraying concrete cracks are demonstrated in two typical cracking scenarios of concrete structures, namely, crack distributions in shear-controlled concrete beams and crack patterns in thin and lightly-reinforced concrete walls. The results of the study show that the proposed multifractal-spectrum shape parameter offsets the deficiency of traditional multifractal parameters in revealing the heterogeneity of crack distributions. The results demonstrate that MUTFA is competent in distinguishing the subtle differences between two similar distributions of concrete cracks, and it provides a path to assess damage in concrete structures.

Fractal-Based Analysis of fMRI BOLD Signal During Naturalistic Viewing Conditions

Background: Temporal fractals are characterized by prominent scale-invariance and self-similarity across time scales. Monofractal analysis quantifies this scaling behavior in a single parameter, the Hurst exponent (H). Higher H reflects greater correlation in the signal structure, which is taken as being more fractal. Previous fMRI studies have observed lower H during conventional tasks relative to resting state conditions, and shown that H is negatively correlated with task difficulty and novelty. To date, no study has investigated the fractal dynamics of BOLD signal during naturalistic conditions.Methods: We performed fractal analysis on Human Connectome Project 7T fMRI data (n = 72, 41 females, mean age 29.46 ± 3.76 years) to compare H across movie-watching and rest.Results: In contrast to previous work using conventional tasks, we found higher H values for movie relative to rest (mean difference = 0.014; p = 5.279 × 10−7; 95% CI [0.009, 0.019]). H was significantly higher in movie than rest in the visual, somatomotor and dorsal attention networks, but was significantly lower during movie in the frontoparietal and default networks. We found no cross-condition differences in test-retest reliability of H. Finally, we found that H of movie-derived stimulus properties (e.g., luminance changes) were fractal whereas H of head motion estimates were non-fractal.Conclusions: Overall, our findings suggest that movie-watching induces fractal signal dynamics. In line with recent work characterizing connectivity-based brain state dynamics during movie-watching, we speculate that these fractal dynamics reflect the configuring and reconfiguring of brain states that occurs during naturalistic processing, and are markedly different than dynamics observed during conventional tasks.

Microstructural and Energy-Dispersive X-ray Analyses on Argon Ion Implantations in Tantalum Thin Films for Microelectronic Substrates

In the present study, the microstructural and statistical properties of unimplanted in comparison to argon ion-implanted tantalum-based thin film surface structures are investigated for potential application in microelectronic thin film substrates. In the study, the argon ions were implanted at the energy of 30 keV and the doses of 1 × 1017, 3 × 1017, and 7 × 1017 (ion/cm2) at an ambient temperature. Two primary goals have been pursued in this study. First, by using atomic force microscopy (AFM) analysis, the roughness of samples, before and after implantation, has been studied. The corrosion apparatus wear has been used to compare resistance against tantalum corrosion for all samples. The results show an increase in resistance against tantalum corrosion after the argon ion implantation process. After the corrosion test, scanning electron microscopy (SEM) analysis was applied to study the sample morphology. The elemental composition of the samples was characterized by using energy-dispersive X-ray (EDX) analysis. Second, the statisticalcharacteristics of both unimplanted and implanted samples, using the monofractal analysis with correlation function and correlation length of samples, were studied. The results show, however, that all samples are correlated and that the variation of ion doses has a negligible impact on the values of correlation lengths. Moreover, the study of height distribution and higher-order moments show the deviation from Gaussian distribution. The calculations of the roughness exponent and fractal dimension indicates that the implanted samples are the self-affine fractal surfaces.

Surface characterization of Cu-doped indium sulfide thin films

In the present work, Cu-doped indium sulfide thin films (In2S3:Cu) have been produced by chemical bath deposition (CBD) method on the glass substrates. The deposited films have been post-annealed at 300 °C temperature for about one hour. The morphological properties of produced thin films have been investigated analytically before and after the Cu doping process. To this end, the atomic force microscopy analysis has been used to study the height fluctuations of these rough thin films. By the monofractal analysis, the height distribution, and the higher-order moments (skewness and kurtosis) of the surface height have been studied. The deviation from the Gaussian distribution has been shown by the results. Also, the correlation function and therefore the correlation length of the produced surfaces have been investigated. The numerical calculations revealed that the morphology and the statistical properties of thin films depend on the growth method. The power spectral density has been evaluated, and the achieved results of higher-order moments of the surface height have been confirmed. The skewness values of thin films heights are nonzero; moreover, the obtained kurtosis values illustrate the mild peak respect to Gaussian distribution. The Hurst exponent and thus the fractal dimensions of surfaces have been estimated.

2D Images of Astrocytes in the Human Principal Olivary Nucleus: Monofractal Analysis of the Morphology

Introduction: Current knowledge postulated glia as active participants in various metabolic processes within nervous tissue. The most numerous glial cells were astrocytes, and qualitative analysis divided them into two types based on their anatomical locations: fibrous and protoplasmic. The main goal of this research was to examine the morphological difference between types, analyzing four features of the image. The secondary objective of this research was to explore their morphology through maturation and aging. Materials and Methods: The material originated from bilateral sections of the human principal olivary nucleus, without disorders in the central nervous system. The brains were taken from 30 human cadavers (35 - 90 years) and cut into samples corresponding to dimensions of the principal olivary nucleus. A light microscope digitized the histological preparations. The selection of 294 images was analyzed by monofractal parameters derived from the box-counting. These parameters quantified four image properties (size, shape, complexity and homogeneity) of the glial body or whole glial cell. Results: The first results showed that images of two types of astrocytes were significantly different (p < 0.05 and higher) in all properties of whole cells. The second results examined the differences between three age groups in both types of astrocytes. The differences between groups were more evident for protoplasmic than fibrous (nine vs. three parameters). Conclusions: The main limitation of this study lies in the fact that the quantification was performed only by fractal analysis techniques. Nevertheless, a detailed monofractal analysis of astrocytes was performed for the first time. Thus, although this study can be seen as an improvement of the previous qualitative results, future research will provide the complete procedure of the image analysis.

The study of mechanical and statistical properties of nitrogen ion-implanted Tantalum bulk

In the present study, structural and mechanical properties of Tantalum samples which affected by nitrogen ion implantation have been investigated. The monofractal analysis of the samples has been carried out. These samples have been implanted with ions at different doses of 5 × 1017 and 7 × 1017 at energy 30 keV. For characterizing the samples, atomic force microscopy (AFM), EDX images analysis, and corrosion measurements, have been used. By varying the ion energy, the statistical characteristics will vary. In order to analyse the morphology of produced rough samples, the correlation function of samples have been studied and their correlation length is evaluated. Based on the statistical concept the power spectral density the height distribution, and the higher-order moments (skewness and kurtosis) of the surface height have been measured. These charactristics are the important criterions for the the monofractal evaluation. Our numerical calculations based on experimental data show the deviation from gaussian distribution. Also, the fractal dimension of the films has been calculated based on rougness exponent results. Indeed, irradiating the surfaces with energetic ions results in producing the self-affine fractal surfaces. This process via an erosion process affects the statistical characteristics which have been calculated for different samples. Because of variation in the morphology of rough samples, the corrosion of samples shows a non-uniform behavior.

Fractal characteristics of tall tower wind speeds in Missouri

The Hurst exponent H is used to determine the measure of predictability of a time series. The value between 0 and 1 with 0.5 representative of a random or uncorrelated series, H>0.5 and H<0.5 reflect a data set which is persistent and anti-persistent respectively. The fractal dimension can be given from the Hurst exponent. The fractal dimension is a factor of the complexity of which the system is being repeated at various scales. If the fractal dimension does not change with scale it is deemed monofractal if not, multifractal. The Hurst exponents were determined in this study using the Rescale Range Analysis (R/S Analysis) and Multifractal Detrended Fluctuation Analysis (MF-DFA) for monofractal and multifractal investigations respectively. These methods were applied to daily 10 min wind speed time series data for the year 2009 from three locations within Missouri: Columbia, Neosho and Blanchard for three tall tower stations. The results obtained from the monofractal analysis showed minor variations in the Hurst exponents for the three stations and heights for all the months in 2009. These values ranged from 0.7 to 0.9 and its corresponding fractal dimension was ranged between 1.3 and 1.1. The results for the MF-DFA showed that the wind speed time series were multifractal in nature as the Hurst exponents were functions of the scaling parameters. Also, the plots of the Renyi Exponent were non-linear for the stations and the various channels; this is representative of multifractal signals. The fractal dimensions of the time series using multifractal analysis were determined to be greater than these values determined using monofractal analysis. However, there were no indications of consistent increases in the complexity of the systems’ multifractality with increasing heights for the various stations’ tall towers.

Statistical analysis of thin films produced by random shaped clusters

The morphology and stochastic properties of thin films, grown by deposition of random shaped clusters, are investigated. Considering the geometric constraints for deposition process, the thin films are grown by using the Monte Carlo simulation method. Based on the statistical concept, the monofractal analysis of the porous films is evaluated. The correlation function, correlation length, fractal dimension, height distribution and the higher-order moments (skewness and kurtosis) of the surface height and power spectral density are studied. The results show nonzero skewness and deviation from the Gaussian distribution. Also, the effect of variation of the substrate and cluster size on the correlation length is studied. For large substrates, the correlation length and its dependence on cluster length increase.

Cryptocurrency momentum effect: DFA and MF-DFA analysis

Cryptocurrency has experienced the skyrocketing and falling back in 2018. Beyond the hype, the specific price movements of different cryptocurrencies should be investigated in a more careful way. Since the cryptocurrency market is a non-linear complex system which are not suitable analyzed by tradition methods, this paper introduces methods from econophysics. Mono-fractal analysis (detrended fluctuation analysis, DFA) is applied to investigate the price movement. Further, multi-fractal fluctuation detrended analysis (MF-DFA) is used for robustness test. Through analyzing four representative cryptocurrencies, our paper finds a strong momentum effect in BTC and ETH market, and a reversion effect in XRP and EOS when large fluctuation occurs. These findings may provide a reference for trading strategy in alternative asset allocations.

The Pan-Cytokeratin Staining Intensity and Fractal Computational Analysis of Breast Tumor Malignant Growth Patterns Prognosticate the Occurrence of Distant Metastasis.

Improved prognosis of breast cancer outcome could prolong patient survival by reliable identification of patients at high risk of metastasis occurrence which could benefit from more aggressive treatments. Based on such clinical need, we prognostically evaluated the malignant cells in breast tumors, as the obvious potential source of unexploited prognostic information. The patient group was homogeneous, without any systemic treatments or lymph node spread, with smaller tumor size (pT1/2) and a long follow-up. Epithelial cells were labeled with AE1/AE3 pan-cytokeratin antibody cocktail and comprehensively analyzed. Monofractal and multifractal analyses were applied for quantification of distribution, shape, complexity and texture of malignant cell clusters, while mean pixel intensity and total area were measures of the pan-cytokeratin immunostaining intensity. The results surprisingly indicate that simple binary images and monofractal analysis provided better prognostic information then grayscale images and multifractal analysis. The key findings were that shapes and distribution of malignant cell clusters (by binary fractal dimension; AUC = 0.29), their contour shapes (by outline fractal dimension; AUC = 0.31) and intensity of the pan-cytokeratin immunostaining (by mean pixel intensity; AUC = 0.30) offered significant performance in metastasis risk prognostication. The results reveal an association between the lower pan-cytokeratin staining intensity and the high metastasis risk. Another interesting result was that multivariate analysis could confirm the prognostic independence only for fractal but not for immunostaining intensity features. The obtained results reveal several novel and unexpected findings highlighting the independent prognostic efficacy of malignant cell cluster distribution and contour shapes in breast tumors.

Mono-fractal analysis of CuXO films: Texture and phase distribution

Several studies on thermally-grown copper oxide films dealing with their crystalline properties and main phases as functions of the growth temperature have been published; notwithstanding, few research works have proposed a description of the surface morphology of the films using mono-fractal analysis. In this work, copper oxide films were grown by direct thermal oxidation in atmospheric air at different temperatures and times to obtain experimental proof of the phases present in the films by XRD, FTIR and Raman. The textural properties of the copper oxide films were analyzed from atomic force microscopy (AFM) images. The box counting and information fractal dimensions were used to perform the mono-fractal analysis to establish a correlation between the textural properties and the phases of the CuxO films.

Fractal Analysis of the Agricultural Products Prices Time Series

It is very necessary to research the fluctuation of agricultural products prices as well as the data character. In this paper, the wholesale price of agricultural product is considered. The monofractal analysis and multifractal analysis methods are introduced to study wholesale price. We select the appropriate algorithm to calculate the fractal characteristics of the celery wholesale price. Results show that the price is monofractal and multifractal. The methods of prediction and risk assessment based on fractal analysis are available for agricultural product price.