Multifractal Techniques Research Articles

Heart rate variability series during physical activity analyzed with multifractal techniques

Abstract Our study uses multifractal techniques to analyze time series of heart rate variability series (RR time series) during rest and physical activity. Subsets of the RR time series data were extracted for each stage. The multifractal spectra of the RR time series data were constructed using the method developed by Chhabra and Jensen. Then, parameters such as width, symmetry, and curvature of the multifractal spectra were calculated. The multifractal spectra obtained at resting stage were broad, skewed to the right, and they had small curvatures around in the maximum. On the contrary, the subsets of the RR time series obtained during physical activity were narrow, skewed to the left, and showed an increased curvature value around the maximum. In addition, these multifractal spectrum parameters shown significant differences in series of subjects who perform physical activity regularly and sedentary. Furthermore, the results obtained from this study were compared with data from previous works involving patients with congestive heart failure and healthy subjects.

Investigation on combustion instability of electronic fuel injected multi-cylinder radial piston engine through multifractal technique – A comparative analysis

The main objective of the present study was to investigate combustion instability in a multi-cylinder radial piston engine with electronic fuel injection to ensure safety and reliability, maintain fuel efficiency, reduce harmful emissions, which has a positive impact on the environment. Multi-fractal analysis showed that IMEP (mean effective pressure) fluctuations are an anti-persistent process, meaning that there is no correlation between successive events. The return plots showed significant cyclic variability and instability in neighboring cylinders. The wavelet method and multiscale entropy analysis confirmed the results of the multifractal analysis, showing changes in the characteristics of the combustion process after the introduction of perturbations. The research-based on the first return map has shown an impact on the non-uniform and complex nature of the operation of the neighboring cylinders. The statistical analysis has shown that more than 60 % is slightly negatively skewness in most of the cases, and about 67 % of the cases indicate a leptokurtic distribution of IMEP. Finally, it is concluded that, the combustion of failure of one cylinder in a nine-cylinder engine affects the operation of the other cylinders, as demonstrated through advanced multifractal analysis techniques, return plots, return maps and multiscale entropy analysis.

Anthropogenic Impacts in the Lower Stratosphere: Scale Invariant Analysis

Aircraft and rockets entered the lower stratosphere on a regular basis during World War II and have done so in increasing numbers to the present. Atmospheric testing of nuclear weapons saw radioactive isotopes in the stratosphere. Rocket launches of orbiters are projected to increase substantially in the near future. The burnup of orbiters has left signatures in the aerosol. There are proposals to attenuate incoming solar radiation by deliberate injection of artificial aerosols into the stratosphere to “geoengineer” cooling trends in surface temperature, with the aim of countering the heating effects of infrared active gases. These gases are mainly carbon dioxide from fossil burning, with additional contributions from methane, chlorofluorocarbons, nitrous oxide and the accompanying positive feedback from increasing water vapor. Residence times as a function of altitude above the tropopause are critical. The analysis of in situ data is performed using statistical multifractal techniques and combined with remotely sensed and modeled results to examine the classical radiation–photochemistry–fluid mechanics interaction that determines the composition and dynamics of the lower stratosphere. It is critical in assessing anthropogenic effects. It is argued that progress in predictive ability is driven by the continued generation of new and quantitative observations in the laboratory and the atmosphere.

Performance comparison of multifractal techniques and artificial neural networks in the construction of investment portfolios

This work aims to compare the performance of the traditional portfolios of the S&P500, Markowitz, and Sharpe with the multifractal trend fluctuation portfolios (MF-DFA) and portfolios of artificial neural networks with Student's asymmetric probability classification (ANN-t). In this study, we use daily data for S&P500 stocks between January 18, 2018, and July 12, 2022, where we backtest return and risk metrics such as annual volatility, Value at Risk, Sharpe Ratio, Sortino Ratio, Beta, and Jensen´s Alpha. For both return and risk, we obtain the results confirming that the ANN-t technique might indicate better investment entries, which contradicts the Efficient Market Hypothesis (EMH).

Fractal Characterization of Brass Corrosion in Cavitation Field in Seawater

Cavitation is a physical process that produces complex effects on the machines and components working in conditions where it acts. One effect is the materials-mass loss by corrosion–erosion when components are introduced into fluids under cavitation. The analysis of the damages produced by cavitation is generally performed by using different destructive and non-destructive experimental techniques. Most studies on materials’ behavior in cavitation refer to the erosion–corrosion mechanism, and very few investigate the fissure propagation by fractal methods. None have investigated the fractal characteristics of the sample surface after erosion–corrosion or the multifractal characteristics of materials’ mass variation in time in a cavitation field. Therefore, this research proposes a computational approach to determine the pattern of materials’ damages produced by ultrasound cavitation. The studied material is a brass, introduced in seawater. Fractal and multifractal techniques are applied to the series of the absolute mass loss per surface and the sample’s micrography after corrosion. Such an approach has not been utilized for such a material in similar experimental conditions. This study emphasizes that the box dimension of the series of the absolute mass loss per surface is close to one, and its behaviour is close to a non-/monofractal. It is demonstrated that the material’s surface corrosion is not uniform, and its multifractal character is highlighted by the f(α)− spectrum and the multifractal dimensions, which have the following values: the capacity dimension = 1.5969, the information dimension = 1.49836, and the correlation dimension = 1.4670.

Multifractal Characteristics on Temporal Maximum of Air Pollution Series

Presenting and describing a temporal series of air pollution data with longer time lengths provides more concise information and is, in fact, one of the simplest techniques of data reduction in a time series. However, this process can result in the loss of important information related to data features. Thus, the purpose of this study is to determine the type of data characteristics that might be lost when describing data with different time lengths corresponding to a process of data reduction. In parallel, this study proposes the application of a multifractal technique to investigate the properties on an air pollution series with different time lengths. A case study has been carried out using an air pollution index data in Klang, Malaysia. Results show that hourly air pollution series contain the most informative knowledge regarding the behaviors and characteristics of air pollution, particularly in terms of the strength of multifractality, long-term persistent correlations, and heterogeneity of variations. On the other hand, the statistical findings found that data reduction corresponding to a longer time length will change the multifractal properties of the original data.

Sedimentary and environmental conditions of Al-Rassafeh Badyieh (Area-2), Syria through aerial gamma ray spectrometry and multifractal techniques

The interpretation of aerial gamma ray spectrometry in term of geological and environmental analysis is undertaken herein through applying a specific methodology including different radioactive and statistical techniques for characterizing the sedimentary and environmental conditions of the Al-Rassafeh Badyieh (Area-2), Syria. The radioactive technique uses the uranium favorability index (UI) and alteration(F) parameters, that are evaluated through analyzing the relationships between eU, eTh, K and their ratios (eTh/eU, eU/eK, and eTh/eK) for the nine scored lithological units already determined in the study area. The statistical technique applies the non linear multifractal approach with the concentration-number model (C–N) and log-log graphs to differentiate between different radioactive ranges of UI and F parameters.The radioactive element re-distribution, the favorability as regards uranium potentiality, and the degree of uranium remobilization are separately estimated and documented for the nine scored lithological units. Those units show a limited uranium remobilization and redistribution. The sedimentary and environmental conditions of the Area-2 are clarified through analyzing the eTh/eU ratio, where marine and continental environments are indicated in the study region.

Study of Fractal and Multifractal Features of Pore Structure in Tight Sandstone Reservoirs of the Permian Lucaogou Formation, Jimsar Sag, Junggar Basin, Northwest China.

The primary factor impacting the tight sandstone reservoirs and fluid flow capacity represents the pore-throat structure. On the basis of petrophysical characteristics test, scanning electron microscopy (SEM), and casting thin-section examination of tight sandstone reservoir specimens from the Permian Lucaogou Formation in Jimsar Sag, Junggar Basin., the full-size pore-throat parameters and distribution characteristics were determined by constant-rate mercury injection (CRMI) analysis. Using fractal theory and multifractal theory, the pore architectures of sandstone pores are analyzed. Mercury intrusion capillary pressure (MICP) is used to compute the dimensions of fractals using various fractal models and multifractal characteristics. Analysis is done on the relationships between tight sandstone pore architectures and fractal and multifractal characteristics. According to the research, a network of tightly packed sandstone pores may be assessed using the dimensions of fractals computed from a 3D capillary model. When displacement pressure is increased, the dimensions of fractals rise; when permeability, pore-throat diameter, and variable coefficient are increased, it falls. Tight sandstone pores exhibit multifractal features, according to the multifractal analysis, and multifractal parameters may depict the size, concentration, and asymmetry of the pore size distribution (PSD). Sandstone’s PSD is comparable when its multifractal parameters (Δα, Δf, α0, α1, α2) are identical. Pore diameters of tight sandstone are positively connected with information dimensions D1 and correlation dimensions D2, and information dimensions D1 have a greater impact on PSD than correlation dimensions D2. Additionally, the 3D capillary model’s dimensions of fractals and D1 exhibit a substantial negative association. These findings play a significant guiding role in the quantitative characterization of unconventional reservoir pore structures. The multifractal technique is effective to define the heterogeneity of the sandstone pore system and to differentiate between distinct PSD in heterogeneity.

Complexity analysis of three-dimensional stochastic discrete fracture networks with fractal and multifractal techniques

Systematic analysis of the complexity of fracture systems, especially for three-dimensional (3D) fracture networks, is largely insufficient. In this work, we generate different fracture networks with various geometries with a stochastic discrete fracture network method. The fractal dimension (D) and the singularity variation in a multifractal spectrum (Δα) are utilized to quantify the complexity of fracture networks in different aspects (spatial filling and heterogeneity). Influential factors of complexity, including geometrical fracture properties and system size, are then systematically studied. We generalize the analysis by considering two critical (percolative and over-percolative) stages of fracture networks. At the first stage, κ (fracture orientation) is the most significant parameter for D, following a (fracture length) and L (system size). FD (fracture positions) has a weak correlation with D but a strong correlation with Δα. At the second stage, the sensitivity results of each geometrical parameter and the system size are the same as in stage one for D. For Δα, κ and FD become more significant. For both stages, there is a weak finite-size effect for D and no finite-size effect for Δα. Therefore, a large fracture system is more suitable for a stable fractal dimension estimation, but no requirement for the estimation of Δα. D and Δα are almost independent. Therefore, they can separately quantify different aspects of complexity.

Fusion of Lineament Factor (LF) Map Analysis and Multifractal Technique for Massive Sulfide Copper Exploration: The Sahlabad Area, East Iran

Fault systems are characteristically one of the main factors controlling massive sulfide mineralization. The main objective of this study was to investigate the relationship between fault systems and host lithology with massive sulfide copper mineralization in the Sahlabad area, South Khorasan province, east of Iran. Subsequently, the rose diagram analysis, Fry analysis, lineament factor (LF) map analysis and multifractal technique were implemented for geological and geophysical data. Airborne geophysical analysis (aeromagnetometric data) was executed to determine the presence of intrusive and extrusive masses associated with structural systems. Accordingly, the relationship between the formation boundaries and the fault system was understood. Results indicate that the NW-SE fault systems are controlling the lithology of the host rock for copper mineralization in the Sahlabad area. Hence, the NW-SE fault systems are consistent with the main trend of lithological units related to massive sulfide copper mineralization in the area. Additionally, the distance of copper deposits, mines and indices in the Sahlabad area with fault systems was calculated and interpreted. Fieldwork results confirm that the NW-SE fault systems are entirely matched with several massive sulfide copper mineralizations in the area. This study demonstrates that the fusion of lineament factor (LF) map analysis and multifractal technique is a valuable and inexpensive approach for exploring massive sulfide mineralization in metallogenic provinces.

On the application of deep learning and multifractal techniques to classify emotions and instruments using Indian Classical Music

Music is often considered as the language of emotions. The way it stimulates the emotional appraisal across people from different communities, culture and demographics has long been known and hence categorizing on the basis of emotions is indeed an intriguing basic research area. Indian Classical Music (ICM) is famous for its ambiguous nature, i.e. its ability to evoke a number of mixed emotions through only a single musical narration, and hence classifying evoked emotions from ICM becomes a more challenging task. With the rapid advancements in the field of Deep Learning, this Music Emotion Recognition (MER) task is becoming more and more relevant and robust, hence can be applied to one of the most challenging test case i.e. classifying emotions elicited from ICM. In this paper we present a new dataset called JUMusEmoDB which presently has 1600 audio clips (approximately 30 s each) where 400 clips each correspond to happy, sad, calm and anxiety emotional scales. The initial annotations and emotional classification of the database was done based on an emotional rating test (5-point Likert scale) performed by 100 participants. The clips have been taken from different conventional ‘raga’ renditions played in two Indian stringed instruments – sitar and sarod by eminent maestros of ICM and digitized in 44.1 kHz frequency. The ragas, which are unique to ICM, are described as musical structures capable of inducing different moods or emotions. For supervised classification purposes, we have used Convolutional Neural Network (CNN) based architectures (resnet50, mobilenet v2.0, squeezenet v1.0 and a proposed ODE-Net) on corresponding music spectrograms of the 6400 sub-clips (where every clip was segmented into 4 sub-clips) which contain both time as well as frequency domain information. Along with emotion classification, instrument classification based response was also attempted on the same dataset using the CNN based architectures. In this context, a nonlinear technique, Multifractal Detrended Fluctuation Analysis (MFDFA) was also applied on the musical clips to classify them on the basis of complexity values extracted from the method. The initial classification accuracy obtained from the applied methods are quite inspiring and have been corroborated with ANOVA results to determine the statistical significance. This type of CNN based classification algorithm using a rich corpus of Indian Classical Music is unique even in the global perspective and can be replicated in other modalities of music also. The link to this newly developed dataset has been provided in the dataset description section of the paper. This dataset is still under development and we plan to include more data containing other emotional as well as instrumental entities into consideration.

Multifractal analysis of chest CT images of patients with the 2019 novel coronavirus disease (COVID-19)

The present study was done to evaluate chest computed tomography (CT) images of patients with 2019 novel coronavirus disease (COVID-19) by multifractal technique as a new method to find a way for comparing lung infection quantitatively and identifying progression pattern of the disease. The multifractal spectra extracted from analysis of CT images showed that these spectra were correlated with lung infection amount and disease progression so that, multifractal parameters (αmin, αmax, ∆α, f(αmin), f(αmax), ∆f(α), α(q = 0), and f(α) max) were strongly dependent on amount of lung infection. The results demonstrated that multifractality of chest CT images was increased with the increase in lung infection in patients. The interesting and promising result was that capacity dimension (D0) as a new diagnostic parameter varied linearly with progression and reduction of lung infection. A critical value was found for D0, according to which patients with D0 lower than 1.4 can be healed by treatment. Therefore, herein, a way was found for quantitative assessment of lung infection of patients with COVID-19 by analyzing chest CT images using the multifractal method. This method can be very effective for physicians in diagnosis and treatment of pneumonia caused by COVID-19 and timely identification of therapeutic effects.

Multifractal analysis of ultrasonically machined surfaces of cylindrical quartz crystals: the effect of the abrasive grits

Since synthetic quartz is essential to produce 3D resonators for numerous applications in precision electronics, in this work the surface topography of cylindrical quartz bars is investigated using the multifractal technique. The cylindrical bars were manufactured with ultrasonic machining using five SiC grits ranging from 6 to 50 μm. The machined surfaces were initially characterized by contact profilometry and scanning electron microscopy (SEM). The multifractality of the machined surfaces was scrutinized using a box-counting method applied to the images obtained with 500X magnification. The multifractal spectrum indicated that the fractal dimension f(α) and the width of the fractal spectrum Δα are dependent on the grit size, but this dependence is not monotonic. The lowest (negative) value for Δf(α) was found for 25 μm grits indicating that for these grits the lower frequency events (grooves with tens μm width occurring along the USM direction) control the surface topography much more than high-frequency events related to brittle microcracking. The abrasive wear due to the continuous slurry recycling in lateral tool-workpiece interfaces contributed to smooth the groove texture as well as the sharpness of microscopic indentations, which remained observed on the surfaces machined with 50 μm grits. The opposite paths observed for the arithmetical mean deviation of the measured profile (Ra) and Δf(α) parameters with the cutting rate measured for each grit size were valuable to differentiate flat-rough and unlevelled-rough topographies in quartz bars.

Multifractal Characteristics Analysis Based on Slope Distribution Probability in the Yellow River Basin, China

Multifractal theory provides a reliable method for the scientific quantification of the geomorphological features of basins. However, most of the existing research has investigated small and medium-sized basins rather than complex and large basins. In this study, the Yellow River Basin and its sub-basins were selected as the research areas, and the generalized fractal dimension and multifractal spectrum were computed and analyzed with a multifractal technique based on the slope distribution probability. The results showed that the Yellow River Basin and its sub-basins exhibit clear multifractal characteristics, which indicates that the multifractal theory can be applied well to the analysis of large-scale basin geomorphological features. We also concluded that the region with the most uneven terrain is the Yellow River Downstream Basin with the “overhanging river”, followed by the Weihe River Basin, the Yellow River Mainstream Basin, and the Fenhe River Basin. Multifractal analysis can reflect the geomorphological feature information of the basins comprehensively with the generalized fractal dimension and the multifractal spectrum. There is a strong correlation between some common topographic parameters and multifractal parameters, and the correlation coefficients between them are greater than 0.8. The results provide a scientific basis for analyzing the geomorphic characteristics of large-scale basins and for the further research of the morphogenesis of the forms.

Multifractal Characteristics of Geomagnetic Field Fluctuations for the Northern and Southern Hemispheres at Swarm Altitude.

This paper explores the spatial variations of the statistical scaling features of low to high latitude geomagnetic field fluctuations at Swarm altitude. The data for this study comes from the vector field magnetometer onboard Swarm A satellite, measured at low resolution (1 Hz) for one year (from 9 March 2016, to 9 March 2017). We estimated the structure-function scaling exponents using the p-leaders discrete wavelet multifractal technique, from which we obtained the singularity spectrum related to the magnetic fluctuations in the North-East-Center (NEC) coordinate system. From this estimation, we retain just the maximal fractal subset, associated with the Hurst exponent H. Here we present thresholding for two levels of the Auroral Electrojet index and almost the whole northern and southern hemispheres, the Hurst exponent, the structure-function scaling exponent of order 2, and the multifractal p-exponent width for the geomagnetic fluctuations. The latter quantifies the relevance of the multifractal property. Sometimes, we found negative values of H, suggesting a behavior similar to wave breaking or shocklet-like propagating front. Furthermore, we found some asymmetries in the magnetic field turbulence between the northern and southern hemispheres. These estimations suggest that different turbulent regimes of the geomagnetic field fluctuations exist along the Swarm path.

Radioactivity and heat production characterization by natural gamma ray well logging and multi-fractal techniques in banting district, Malaysia

Natural gamma ray well logging technique is used to characterize the radioactivity (GR) laterally and vertically in Banting district, SW of Malaysia. Seven drilled boreholes, along N–S profile with their natural gamma ray records are utilized to compute the heat production (HP) parameter, based on the Bucker and Rybach relationship.The analysis of 3467 measured points in those boreholes indicates that GR varies between 6.24 API and 358.4 API with an average of 79.95 API, while HP varies between 0.086 and 5.65 μw/m3 with an average of 1.25 μw/m3.The multi-fractal Concentration-Number (C–N) is used to characterize the radioactivity and heat production variations and to isolate different GR and HP populations in the study region. The high radioactivity and heat production ranges are mainly related to the silty clay layers, accompanied by uranium and thorium.

Multifractal study of TiO2 thin films deposited by MO-CVD method: The role of precursor amount and substrate temperature

Since TiO2 thin film is an important semiconductor with many applications in various fields, in this work we studied surface morphology of TiO2 thin films deposited by metal-organic chemical vapor deposition method using multifractal technique. In order to do this, AFM images of prepared TiO2 thin films were analyzed by multifractal method and the effects of the precursor amount and substrate temperature on multifractality of The TiO2 thin films surface were investigated. Multifractal nature of surface of TiO2 thin films deposited with different amounts of precursor at two different positions of substrate with different temperatures was detected using multifractal analysis of the AFM images. The multifractal spectrum f(α) obtained from AFM images indicated that the Δf and width of spectrum are functions dependent on precursor amount and substrate temperature. The obtained data showed that local fluctuation and hight distribution increase with increasing in temperature of substrate for high amount of precursor. Also, it was observed that the substrate temperature has no effect on width of multifractal spectrum at low amount of precursor. Our results clearly showed that morphological complexity of the TiO2 surface and the effect of the precursor amount and temperature of substrate are satisfactorily described by perform multifractal analysis on AFM images. These results can be used in the design and manufacture of interface layer for application in laptops, planer device, transparent electrode, solar cell, sensors and optical devices.

Application of Finite Difference Method in Multifractal Techniques to Identify Pb–Zn Anomalies: a Case Study from Khomein, Iran

Separation of geochemical anomalies from background is an important and essential issue in the exploration of mineral deposits. Statistical methods are used to separation of anomaly from background. Fractal/multifractal and singularity methods are very suitable for the separation of weak anomalies. For interpolation, various methods could be used such as most distance-weighting methods, kriging, spline interpolation, interpolating polynomials, and finite difference methods. This paper addressed application of multifractal techniques together with finite difference procedure to detect the anomalous regions. Fractal models such as concentration–area, spectrum–area, and singularity index model were selected to identify Pb–Zn anomalies. In this research, 170 samples from 33 boreholes were taken in the Haft-Savaran area and analyzed for determination of Pb–Zn anomaly. Singularity method was a suitable method for separating Pb and Zn anomalies. Also, the results showed that the geochemical anomaly has occurred along NE–SW trend and is consistent with the sulfide mineralization process and the faults of the region.

Aspects of hydrothermal alteration assemblages around Bakwani-Bagwa, Northwestern Nigeria; Insights from multifractal analysis of measured thresholds for mineral exploration.

This study is aimed at identifying highly altered zones by applying concentration area (C-A) fractal model technique on Landsat ETM+ image of area around Bakwani-Bagwa, Northwestern Nigeria. Multifractal analysis was used to quantify degree of alteration on satellite images processed using crosta technique. A log–log plot was used to display the multifractal characteristics for clay and iron altered images and identify threshold values. Maximum and minimum threshold values obtained from fractal analysis are 1.74 and 4.01 for clay altered images and 0.607 and 3.04 for iron altered images. High clay and iron alteration zones accounts for 5.89% and 7.84%, covering an area of 22.28km2 and 29.65km2 respectively. Delineating highly altered zones from these images reveals higher alteration largely within silicified sheared rocks and areas around faults. The current study proved the multifractal processing techniques to be valuable mapping tools for geological reconnaissance which can be extended to a large area of the Nigerian basement and provide preliminary hydrothermal alteration maps that are accurate as well as cost and time effective.

A multifractal edge detector

Significant research has been conducted into image edge detection, usually focusing on gradient and higher order derivative approaches. Recent development in the field suggests the incorporation of scale factor in filter design; this is to increase robustness against noise and to facilitate the process of generating multilevel edge maps. While multifractal analysis technique uses scale factors, few researchers examine the effectiveness of this technique in image edge map generation due to its poor efficiency. Herein, we propose F-MED, a fast multifractal edge detector for grayscale images based on generating an edge map from different low frequency versions of the original image by means of frequency domain Gaussian low pass filters. The quality of results is evaluated subjectively via visual assessment and quantitatively using a set of full-reference and non-reference measures. Results show that the proposed F-MED can be leveraged to support not only high efficiency, but also quality improvement and high robustness against blurring effect and Rayleigh noise.