Detrending moving average algorithm for multifractals

Multifractal Analysis in Neuroimaging

The characteristics of biomedical signals are not captured by conventional measures like the average amplitude of the signal. The methodologies derived from fractal geometry have been a very useful approach to study the degree of irregularity of a signal. The monofractal analysis of a signal is defined by a single power-law exponent in assuming a scale invariance in time and space. However, temporal and spatial variation in the scale-invariant structure of the biomedical signal often appears. In this case, multifractal analysis is well-suited because it is defined by a multifractal spectrum of power-law exponents. There are several approaches to the implementation of this analysis, and there are numerous ways to present these.In this chapter, we review the use of multifractal analysis for the purpose of characterizing signals in neuroimaging. After describing the tenets of multifractal analysis, we present several approaches to estimating the multifractal spectrum. Finally, we describe the applications of this spectrum on biomedical signals in the characterization of several diseases in neurosciences.

Role of vaccine in fighting the variants of COVID-19

In this paper, we investigate the effectiveness of COVID-19 vaccination in controlling the infectivity and mortality of the SARS-CoV-2. Two major variants Delta and Omicron are investigated respectively. The main method used in the research is the multifractal detrended fluctuation analysis (MF-DFA). We use Δ α as the evaluation of control effectiveness. In the transmission stages of Delta and Omicron, we observe whether Δ α shows a downward trend by gradually expanding the length of time series. Vaccine effectiveness is evaluated using a time series of newly diagnosed patients and newly reported deaths. Data samples are taken from 9 different countries. According to the obtained results, the vaccine controls infectivity and mortality of the virus in the Delta transmission stage, but infectivity control is less effective than mortality. In the Omicron transmission stage, the immune effect of the vaccine is not obvious, which may be related to the high infectivity of Omicron. However, the vaccine is still effective in controlling mortality. We also find that the immune effect of vaccine on Omicron was lower than that of Delta. Finally, we observe that the immune effect of the vaccine in 'Poland' was abnormal. By analyzing the vaccination curve, we conclude that in 'Poland', when the growth rate of vaccination rate slowed down, the immune effect of the vaccine was very poor in terms of pathogenicity and lethality. Therefore, we suggest that all countries should continue to strengthen the vaccination rate. A higher or faster growth rate of vaccination rate will help control the infectivity and mortality rate, especially in the effectiveness of controlling mortality. Our research can be used to evaluate the effectiveness of vaccines for epidemic prevention and control, the formulation of epidemic prevention measures and vaccination policies for different countries with respect to their current pandemic situation accordingly.

Determination of local geometric geoid model for Kuwait

Determining a precise local geoid is particularly important for converting the Global Navigation Satellite System (GNSS) heights to orthometric heights. The geometric method for computing the geoid has been extensively used for a comparatively small region, which, in some points, interpolates geoid heights based on GNSS-derived heights and levelling heights. Several considerations should be considered when using the geometric method to increase the accuracy of a local geoid. Kuwait is used as a test area in this paper to investigate several features of the geometric method. The achievable precision is one of these aspects, the role of the interpolation method, global geopotential models, and the influence of the topographic effect. The accuracy of the local geoid can be substantially enhanced by integrating a geopotential model with a digital terrain model of the research region. It is possible to get a precision of 2–3 cm.

Detrending Moving Average, Power Spectral Density, and Coherence: Three EEG-Based Methods to Assess Emotion Irradiation during Facial Perception

Understanding brain reactions to facial expressions can help in explaining emotion-processing and memory mechanisms. The purpose of this research is to examine the dynamics of electrical brain activity caused by visual emotional stimuli. The focus is on detecting changes in cognitive mechanisms produced by negative, positive, and neutral expressions on human faces. Three methods were used to study brain reactions: power spectral density, detrending moving average (DMA), and coherence analysis. Using electroencephalogram (EEG) recordings from 48 subjects while presenting facial image stimuli from the International Affective Picture System, the topographic representation of the evoked responses was acquired and evaluated to disclose the specific EEG-based activity patterns in the cortex. The theta and beta systems are two key cognitive systems of the brain that are activated differently on the basis of gender. The obtained results also demonstrate that the DMA method can provide information about the cortical networks’ functioning stability, so it can be coupled with more prevalent methods of EEG analysis.

Fractional Stochastic Differential Equation Approach for Spreading of Diseases

The nonlinear fractional stochastic differential equation approach with Hurst parameter H within interval H∈(0,1) to study the time evolution of the number of those infected by the coronavirus in countries where the number of cases is large as Brazil is studied. The rises and falls of novel cases daily or the fluctuations in the official data are treated as a random term in the stochastic differential equation for the fractional Brownian motion. The projection of novel cases in the future is treated as quadratic mean deviation in the official data of novel cases daily since the beginning of the pandemic up to the present. Moreover, the rescaled range analysis (RS) is employed to determine the Hurst index for the time series of novel cases and some statistical tests are performed with the aim to determine the shape of the probability density of novel cases in the future.

Detrending-moving-average-based multivariate regression model for nonstationary series

Dependency between a response variable and the explanatory variables is a relationship of universal concern in various real-world problems. Multivariate linear regression (MLR) is a well-known method to focus on this issue. However, it is limited to dealing with stationary variables. In this work, we develop a MLR framework based on detrending moving average (DMA) analysis to reveal the actual dependency among variables with nonstationary measures. The DMA-based MLR can generate multiscale regression coefficients, which characterize different dependent behavior at different timescales. Artificial tests show that the DMA-MLR model can successfully resist the impact of trends on the studied series and produce more accurate regression coefficients with multiscale. Furthermore, some scale-dependent statistics are developed to deduce some important relationships in three typical DMA-based MLR models, which help us to deeply understand the DMA-MLR models in theory. The application of the proposed DMA-MLR framework to Beijing's air quality index system demonstrates that fine particulate matter with diameter ≤2.5μm (PM_{2.5}) is the dominant pollutant affecting the air quality of Beijing in recent years.

A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect

Stock markets can become inefficient due to calendar anomalies known as the day-of-the-week effect. Calendar anomalies are well known in the financial literature, but the phenomena remain to be explored in econophysics. This paper uses multifractal analysis to evaluate if the temporal dynamics of market returns also exhibit calendar anomalies such as day-of-the-week effects. We apply multifractal detrended fluctuation analysis (MF-DFA) to the daily returns of market indices worldwide for each day of the week. Our results indicate that distinct multifractal properties characterize individual days of the week. Monday returns tend to exhibit more persistent behavior and richer multifractal structures than other day-resolved returns. Shuffling the series reveals that multifractality arises from a broad probability density function and long-term correlations. The time-dependent multifractal analysis shows that the Monday returns’ multifractal spectra are much wider than those of other days. This behavior is especially persistent during financial crises. The presence of day-of-the-week effects in multifractal dynamics of market returns motivates further research on calendar anomalies for distinct market regimes.

Is gold a hedge or safe haven against oil and currency market movements? A revisit using multifractal approach

We investigate gold's role as a hedge or safe haven against oil price and currency movements across calm and extreme market conditions. For the empirical analysis, we extend the intraday multifractal correlation measure developed by Madani et al. (Bankers, Markets & Investors, 163:2-13, 2020) to consider the dependence for calm and extreme movement periods across different time scales. Interestingly, we employ the rolling window method to examine the time-varying dependence between gold-oil and gold-currency in terms of calm and turmoil market conditions. Based on high frequency (5-min intervals) across the period 2017-2019, our analysis shows three interesting findings. First, gold acts as a weak (strong) hedge for oil (currency) market movements, across all agent types. Second, gold has strong safe-haven capability against extreme currency movements, and against only short time scales of oil price movements. Third, hedging strategies confirm the scale-dependent gold's role in reducing portfolio risk as a hedge or safe haven. Implications for investors, financial institutions, and policymakers are discussed.

Multifractal and cross-correlation analysis on mitochondrial genome sequences using chaos game representation

We characterized the multifractality and power-law cross-correlation of mitochondrial genomes of various species through the recently developed method which combines the chaos game representation theory and 2D-multifractal detrended cross-correlation analysis. In the present paper, we analyzed 32 mitochondrial genomes of different species and the obtained results show that all the analyzed data exhibit multifractal nature and power-law cross-correlation behaviour. Further, we performed a cluster analysis from the calculated scaling exponents to identify the class affiliation and its outcome is represented as a dendrogram. We suggest that this integrative approach may help the researchers to understand the phylogeny of any kingdom with their varying genome lengths and also this approach may find applications in characterizing the protein sequences, mRNA sequences, next-generation sequencing, and drug development, etc.

Self-organizing three-dimensional Ising model of financial markets

The three-dimensional Ising model in an external field is used as a mathematical model for price dynamics of financials market. The model allows us to test within the same framework the comparative explanatory power of rational agents versus irrational agents with respect to the stylized facts of the financial markets. We obtain the price dynamics in terms of the strength of the field that reinforces the sensitivity of the agent's sentiment to external news. The exponent of long-tail cumulative probability density is determined and satisfies the inverse of the cubic law. Furthermore, the long range memory of the model is studied using different methods to determine the Hurst index of the model. The results obtained display that the model does serve as a mathematical model for financial markets.

Analysis of fractality and complexity of the planetary K-index

The objective of this research is to explore the inherent complexities and multifractal properties of the underlying distributions in the daily Planetary K-index time series collected from NOAA Space Weather Prediction Center. In this article, non-stationary and nonlinear characteristics of the signal have been explored using Smoothed Pseudo Wigner–Ville Distribution and Delay Vector Variance algorithms, respectively, while Recurrence Plot, 0–1 test, Recurrence Quantification Analysis and correlation dimension analysis have been applied to confirm and measure the chaos in the signal under consideration. Multifractal detrending moving average has been used to evaluate the multifractality and also recognise the singularities of the signal. The result of these analyses validates the nonstationary and nonlinear characteristics of the Planetary K-index signal, while a significant presence of deterministic chaos in it has also been noticed. It has also been confirmed that the Planetary K-index exhibits multifractal nature with positive persistence. The long-range temporal association and also the large pdf are discovered to be the primary factors that contribute to the multifractal behaviour of the Kp-index.

Multivariate Multifractal Detrending Moving Average Analysis of Air Pollutants

One of the most challenging endeavors of contemporary research is to describe and analyze the dynamic behavior of time series arising from real-world systems. To address the need for analyzing long-range correlations and multifractal properties of multivariate time series, we generalize the multifractal detrended moving average algorithm (MFDMA) to the multivariate case and propose a multivariate MFDMA algorithm (MV-MFDMA). The validity and performance of the proposed algorithm are tested by conducting numerical simulations on synthetic multivariate monofractal and multifractal time series. The MV-MFDMA algorithm is then utilized to analyze raw, seasonally adjusted, and remainder components of five air pollutant time series. Results from all three cases reveal multifractal properties with persistent long-range correlations.

Review about the Application of Fractal Theory in the Research of Packaging Materials

The work is intended to summarize the recent progress in the work of fractal theory in packaging material to provide important insights into applied research on fractal in packaging materials. The fractal analysis methods employed for inorganic materials such as metal alloys and ceramics, polymers, and their composites are reviewed from the aspects of fractal feature extraction and fractal dimension calculation methods. Through the fractal dimension of packaging materials and the fractal in their preparation process, the relationship between the fractal characteristic parameters and the properties of packaging materials is discussed. The fractal analysis method can qualitatively and quantitatively characterize the fractal characteristics, microstructure, and properties of a large number of various types of packaging materials. The method of using fractal theory to probe the preparation and properties of packaging materials is universal; the relationship between the properties of packaging materials and fractal dimension will be a critical trend of fractal theory in the research on properties of packaging materials.

Multiscale adaptive multifractal analysis and its applications

To precisely analyze the fractal nature of a short-term time series under the multiscale framework, this study introduces multiscale adaptive multifractal analysis (MAMFA) combining the adaptive fractal analysis method with the multiscale multifractal analysis (MMA). MAMFA and MMA are both applied to the two kinds of simulation sequences, and the results show that the MAMFA method achieves better performances than MMA. MAMFA is also applied to the Chinese and American stock indexes and the R-R interval of heart rate data. It is found that the multifractal characteristics of stock sequences are related to the selection of the scale range s. There is a big difference in the Hurst surface's shape of Chinese and American stock indexes and Chinese stock indexes have more obvious multifractal characteristics. For the R-R interval sequence, we find that the subjects with abnormal heart rate have significant shape changes in three areas of Hurst surface compared with healthy subjects, thereby patients can be effectively distinguished from healthy subjects.

Detrending-moving-average-based bivariate regression estimator

In this work, a detrending-moving-average- (DMA) based bivariate linear regression analysis method is proposed. The method is combination of detrended moving average analysis and standard regression methodology, which allows us to estimate the scale-dependent regression coefficients for nonstationary and power-law correlated time series. By using synthetic simulations with error of estimation for different position parameter θ of detrending windows, we test our DMA-based bivariate linear regression algorithm and find that the centered detrending technique (θ=0.5) is of best performance, which provides the most accurate estimates. In addition, the estimated regression coefficients are in good agreement with the theoretical values. The center DMA-based bivariate linear regression estimator is applied to analyze the return series of Shanghai stock exchange composite index, the Hong Kong Hangseng index and the NIKKEI 225 index. The dependence among the Asian stock market across timescales is confirmed. Furthermore, two statistics based on the scale-dependent t statistic and the partial detrending-moving-average cross-correlation coefficient are used to demonstrate the significance of the dependence. The scale-dependent evaluation parameters also show that the DMA-based bivariate regression model can provide rich information than standard regression analysis.

Randomized multifractal detrended fluctuation analysis of long time series

A novel general randomized method is proposed to investigate multifractal properties of long time series. Based on multifractal temporally weighted detrended fluctuation analysis (MFTWDFA), we obtain randomized multifractal temporally weighted detrended fluctuation analysis (RMFTWDFA). The innovation of this algorithm is applying a random idea in the process of dividing multiple intervals to find the local trend. To test the performance of the RMFTWDFA algorithm, we apply it, together with the MFTWDFA, to the artificially generated time series and real genomic sequences. For three types of artificially generated time series, consistency tests are performed on the estimated h(q), and all results indicate that there is no significant difference in the estimated h(q) of the two methods. Meanwhile, for different sequence lengths, the running time of RMFTWDFA is reduced by over ten times. We use prokaryote genomic sequences with large scales as real examples, the results obtained by RMFTWDFA demonstrate that these genomic sequences show fractal characteristics, and we leverage estimated exponents to study phylogenetic relationships between species. The final clustering results are consistent with real relationships. All the results reflect that RMFTWDFA is significantly effective and timesaving for long time series, while obtaining an accuracy statistically comparable to other methods.

Probabilistic properties of detrended fluctuation analysis for Gaussian processes

Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter.

Classification of low-density EEG for epileptic seizures by energy and fractal features based on EMD

We are here to present a new method for the classification of epileptic seizures from electroencephalogram (EEG) signals. It consists of applying empirical mode decomposition (EMD) to extract the most relevant intrinsic mode functions (IMFs) and subsequent computation of the Teager and instantaneous energy, Higuchi and Petrosian fractal dimension, and detrended fluctuation analysis (DFA) for each IMF. We validated the method using a public dataset of 24 subjects with EEG signals from 22 channels and showed that it is possible to classify the epileptic seizures, even with segments of six seconds and a smaller number of channels (e.g., an accuracy of 0.93 using five channels). We were able to create a general machine-learning-based model to detect epileptic seizures of new subjects using epileptic-seizure data from various subjects, after reducing the number of instances, based on the k-means algorithm.

Multifractal analysis of financial markets: a review

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

Long-range temporal correlation in Auditory Brainstem Responses to Spoken Syllable/da/

The speech auditory brainstem response (sABR) is an objective clinical tool to diagnose particular impairments along the auditory brainstem pathways. We explore the scaling behavior of the brainstem in response to synthetic /da/ stimuli using a proposed pipeline including Multifractal Detrended Moving Average Analysis (MFDMA) modified by Singular Value Decomposition. The scaling exponent confirms that all normal sABR are classified into the non-stationary process. The average Hurst exponent is H = 0:77 ± 0:12 at 68% confidence interval indicating long-range correlation which shows the first universality behavior of sABR. Our findings exhibit that fluctuations in the sABR series are dictated by a mechanism associated with long-term memory of the dynamic of the auditory system in the brainstem level. The q-dependency of h(q) demonstrates that underlying data sets have multifractal nature revealing the second universality behavior of the normal sABR samples. Comparing Hurst exponent of original sABR with the results of the corresponding shuffled and surrogate series, we conclude that its multifractality is almost due to the long-range temporal correlations which are devoted to the third universality. Finally, the presence of long-range correlation which is related to the slow timescales in the subcortical level and integration of information in the brainstem network is confirmed.

Multifractal analysis of uterine electromyography signals to differentiate term and preterm conditions

In this study, an attempt has been made to identify the origin of multifractality in uterine electromyography signals and to differentiate term (gestational age > 37 weeks) and preterm (gestational age ≤ 37 weeks) conditions by multifractal detrended moving average technique. The signals obtained from a publicly available database, recorded from the abdominal surface during the second trimester, are used in this study. The signals are preprocessed and converted to shuffle and surrogate series to examine the source of multifractality. Multifractal detrended moving average algorithm is applied on all the signals. The presence of multifractality is verified using scaling exponents, and multifractal spectral features are extracted from the spectrum. The variation of multifractal features in term and preterm conditions is analyzed statistically using Student's t-test. The results of scaling exponents show that the uterine electromyography or electrohysterography signals reveal multifractal characteristics in term and preterm conditions. Further investigation indicates the existence of long-range correlation as the primary source of multifractality. Among all extracted features, strength of multifractality, exponent index, and maximum and peak singularity exponents are statistically significant ( p < 0.05) in differentiating term and preterm conditions. The coefficient of variation is found to be lower for strength of multifractality and peak singularity exponent, which reveal that these features exhibit less inter-subject variance. Hence, it appears that multifractal analysis can aid in the diagnosis of preterm or term delivery of pregnant women.

Fractal and Multifractal Properties of Electrographic Recordings of Human Brain Activity: Toward Its Use as a Signal Feature for Machine Learning in Clinical Applications

The quantification of brain dynamics is essential to its understanding. However, the brain is a system operating on multiple time scales, and characterization of dynamics across time scales remains a challenge. One framework to study such dynamics is that of fractal geometry; and currently there exist several methods for the study of brain dynamics using fractal geometry. We aim to highlight some of the practical challenges of applying fractal geometry to brain dynamics—and as a putative feature for machine learning applications, and propose solutions to enable its wider use in neuroscience. Using intracranially recorded electroencephalogram (EEG) and simulated data, we compared monofractal and multifractal methods with regards to their sensitivity to signal variance. We found that both monofractal and multifractal properties correlate closely with signal variance, thus not being a useful feature of the signal. However, after applying an epoch-wise standardization procedure to the signal, we found that multifractal measures could offer non-redundant information compared to signal variance, power (in different frequency bands) and other established EEG signal measures. We also compared different multifractal estimation methods to each other in terms of reliability, and we found that the Chhabra-Jensen algorithm performed best. Finally, we investigated the impact of sampling frequency and epoch length on the estimation of multifractal properties. Using epileptic seizures as an example event in the EEG, we show that there may be an optimal time scale (i.e., combination of sampling frequency and epoch length) for detecting temporal changes in multifractal properties around seizures. The practical issues we highlighted and our suggested solutions should help in developing robust methods for the application of fractal geometry in EEG signals. Our analyses and observations also aid the theoretical understanding of the multifractal properties of the brain and might provide grounds for new discoveries in the study of brain signals. These could be crucial for the understanding of neurological function and for the developments of new treatments.

Multifractal Analysis of Uterine Electromyography Signals for the Assessment of Progression of Pregnancy in Term Conditions

OBJECTIVES

The objectives of this paper are to examine the source of multifractality in uterine electromyography (EMG) signals and to study the progression of pregnancy in the term (gestation period > 37 weeks) conditions using multifractal detrending moving average (MFDMA) algorithm.

METHODS

The signals for the study, considered from an online database, are obtained from the surface of abdomen during the second (T1) and third trimester (T2). The existence of multifractality is tested using Hurst and scaling exponents. With the intention of identifying the origin of multifractality, the preprocessed signals are converted to shuffle and surrogate data. The original and the transformed signals are subjected to MFDMA to extract multifractal spectrum features, namely strength of multifractality, maximum, minimum, and peak singularity exponents.

RESULTS

The Hurst and scaling exponents extracted from the signals indicate that uterine EMG signals are multifractal in nature. Further analysis shows that the source of multifractality is mainly owing to the presence of long-range correlation, which is computed as 79.98% in T1 and 82.43% in T2 groups. Among the extracted features, the peak singularity exponent and strength of multifractality show statistical significance in identifying the progression of pregnancy. The corresponding coefficients of variation are found to be low, which show that these features have low intersubject variability.

CONCLUSION

It appears that the multifractal analysis can help in investigating the progressive changes in uterine muscle contractions during pregnancy.

Equation-based model for the stock market

We propose a stock market model which is investigated in the forms of difference and differential equations whose variables correspond to the demand or supply of each agent and to the price. In the model, agents are driven by the behavior of their trust contact network as well by fundamental analysis. By means of the deterministic version of the model, the connection between such drive mechanisms and the price is analyzed: imitation behavior promotes market instability, finitude of resources is associated to stock index stability, and high sensitivity to the fair price provokes price oscillations. Long-range correlations in the price temporal series and heavy-tailed distribution of returns are observed for the version of the model which considers different proposals for stochasticity of microeconomic and macroeconomic origins.

Direct determination approach for the multifractal detrending moving average analysis

In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent τ(q) is related to the partition function and the multifractal spectrum f(α) can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional p-model, the two-dimensional p-model, and the fractional Brownian motions. We find that both approaches have comparable performances to unveil the fractal and multifractal nature. In other words, without loss of accuracy, the multifractal spectrum f(α) can be directly determined using the new approach with less computation cost. We also apply the new MF-DMA approach to the volatility time series of stock prices and confirm the presence of multifractality.

Decoding the Morphological Differences between Himalayan Glacial and Fluvial Landscapes Using Multifractal Analysis

Himalayas is the home to nearly 10,000 glaciers which are mostly located at high and inaccessible region. Digital Elevation Model (DEM) can be effective in the study of these glaciers. This paper aims at providing an automated distinction of glacial and fluvial morphologies using multifractal technique. We have studied the variation of elevation profile of Glacial and Fluvial landscapes using Multifractal Detrended Fluctuation Analysis (MFDFA). Glacial landscapes reveal more complex structure compared to the fluvial landscapes as indicated by fractal parameters degree of multifractality, asymmetry index.

Temporal and spatial correlation patterns of air pollutants in Chinese cities

As a huge threat to the public health, China’s air pollution has attracted extensive attention and continues to grow in tandem with the economy. Although the real-time air quality report can be utilized to update our knowledge on air quality, questions about how pollutants evolve across time and how pollutants are spatially correlated still remain a puzzle. In view of this point, we adopt the PMFG network method to analyze the six pollutants’ hourly data in 350 Chinese cities in an attempt to find out how these pollutants are correlated temporally and spatially. In terms of time dimension, the results indicate that, except for O₃, the pollutants have a common feature of the strong intraday patterns of which the daily variations are composed of two contraction periods and two expansion periods. Besides, all the time series of the six pollutants possess strong long-term correlations, and this temporal memory effect helps to explain why smoggy days are always followed by one after another. In terms of space dimension, the correlation structure shows that O₃ is characterized by the highest spatial connections. The PMFGs reveal the relationship between this spatial correlation and provincial administrative divisions by filtering the hierarchical structure in the correlation matrix and refining the cliques as the tinny spatial clusters. Finally, we check the stability of the correlation structure and conclude that, except for PM₁₀ and O₃, the other pollutants have an overall stable correlation, and all pollutants have a slight trend to become more divergent in space. These results not only enhance our understanding of the air pollutants’ evolutionary process, but also shed lights on the application of complex network methods into geographic issues.

Decomposing Multifractal Crossovers

Physiological processes-such as, the brain's resting-state electrical activity or hemodynamic fluctuations-exhibit scale-free temporal structuring. However, impacts common in biological systems such as, noise, multiple signal generators, or filtering by transport function, result in multimodal scaling that cannot be reliably assessed by standard analytical tools that assume unimodal scaling. Here, we present two methods to identify breakpoints or crossovers in multimodal multifractal scaling functions. These methods incorporate the robust iterative fitting approach of the focus-based multifractal formalism (FMF). The first approach (moment-wise scaling range adaptivity) allows for a breakpoint-based adaptive treatment that analyzes segregated scale-invariant ranges. The second method (scaling function decomposition method, SFD) is a crossover-based design aimed at decomposing signal constituents from multimodal scaling functions resulting from signal addition or co-sampling, such as, contamination by uncorrelated fractals. We demonstrated that these methods could handle multimodal, mono- or multifractal, and exact or empirical signals alike. Their precision was numerically characterized on ideal signals, and a robust performance was demonstrated on exemplary empirical signals capturing resting-state brain dynamics by near infrared spectroscopy (NIRS), electroencephalography (EEG), and blood oxygen level-dependent functional magnetic resonance imaging (fMRI-BOLD). The NIRS and fMRI-BOLD low-frequency fluctuations were dominated by a multifractal component over an underlying biologically relevant random noise, thus forming a bimodal signal. The crossover between the EEG signal components was found at the boundary between the δ and θ bands, suggesting an independent generator for the multifractal δ rhythm. The robust implementation of the SFD method should be regarded as essential in the seamless processing of large volumes of bimodal fMRI-BOLD imaging data for the topology of multifractal metrics free of the masking effect of the underlying random noise.

Complex multifractal nature in Mycobacterium tuberculosis genome

The mutifractal and long range correlation (C(r)) properties of strings, such as nucleotide sequence can be a useful parameter for identification of underlying patterns and variations. In this study C(r) and multifractal singularity function f(α) have been used to study variations in the genomes of a pathogenic bacteria Mycobacterium tuberculosis. Genomic sequences of M. tuberculosis isolates displayed significant variations in C(r) and f(α) reflecting inherent differences in sequences among isolates. M. tuberculosis isolates can be categorised into different subgroups based on sensitivity to drugs, these are DS (drug sensitive isolates), MDR (multi-drug resistant isolates) and XDR (extremely drug resistant isolates). C(r) follows significantly different scaling rules in different subgroups of isolates, but all the isolates follow one parameter scaling law. The richness in complexity of each subgroup can be quantified by the measures of multifractal parameters displaying a pattern in which XDR isolates have highest value and lowest for drug sensitive isolates. Therefore C(r) and multifractal functions can be useful parameters for analysis of genomic sequences.

Asymptotic scaling properties and estimation of the generalized Hurst exponents in financial data

We propose a method to measure the Hurst exponents of financial time series. The scaling of the absolute moments against the aggregation horizon of real financial processes and of both uniscaling and multiscaling synthetic processes converges asymptotically towards linearity in log-log scale. In light of this we found appropriate a modification of the usual scaling equation via the introduction of a filter function. We devised a measurement procedure which takes into account the presence of the filter function without the need of directly estimating it. We verified that the method is unbiased within the errors by applying it to synthetic time series with known scaling properties. Finally we show an application to empirical financial time series where we fit the measured scaling exponents via a second or a fourth degree polynomial, which, because of theoretical constraints, have respectively only one and two degrees of freedom. We found that on our data set there is not clear preference between the second or fourth degree polynomial. Moreover the study of the filter functions of each time series shows common patterns of convergence depending on the momentum degree.

Analyzing the influence of curl speed in fatiguing biceps brachii muscles using sEMG signals and multifractal detrended moving average algorithm

In this work, an attempt has been made to analyze surface electromyography (sEMG) signals of fatiguing biceps brachii muscles at different curl speeds using multifractal detrended moving average (MFDMA) algorithm. For this purpose, signals are recorded from fifty eight healthy subjects while performing curl exercise at their comfortable speed until fatigue. The signals of first and last curls are considered as nonfatigue and fatigue conditions, respectively. Further, the number of curls performed by each subject and the endurance time is used for computing the normalized curl speed. The signals are grouped into fast, medium and slow using curl speeds. The curl segments are subjected to MFDMA to derive degree of multifractality (DOM), maximum singularity exponent (MXE) and exponent length multifractality index (EMX). The results show that multifractal features are able to differentiate sEMG signals in fatiguing conditions. The multifractality increased with faster curls as compared with slower curl speed by 12%. High statistical significance is observed using EMX and DOM values between curl speed and fatigue conditions. It appears that this method of analyzing sEMG signals with curl speed can be useful in understanding muscle dynamics in varied neuromuscular conditions and sports medicine.

Multifractality and Network Analysis of Phase Transition

Many models and real complex systems possess critical thresholds at which the systems shift dramatically from one sate to another. The discovery of early-warnings in the vicinity of critical points are of great importance to estimate how far the systems are away from the critical states. Multifractal Detrended Fluctuation analysis (MF-DFA) and visibility graph method have been employed to investigate the multifractal and geometrical properties of the magnetization time series of the two-dimensional Ising model. Multifractality of the time series near the critical point has been uncovered from the generalized Hurst exponents and singularity spectrum. Both long-term correlation and broad probability density function are identified to be the sources of multifractality. Heterogeneous nature of the networks constructed from magnetization time series have validated the fractal properties. Evolution of the topological quantities of the visibility graph, along with the variation of multifractality, serve as new early-warnings of phase transition. Those methods and results may provide new insights about the analysis of phase transition problems and can be used as early-warnings for a variety of complex systems.

Multifractals embedded in short time series: An unbiased estimation of probability moment

An exact estimation of probability moments is the base for several essential concepts, such as the multifractals, the Tsallis entropy, and the transfer entropy. By means of approximation theory we propose a new method called factorial-moment-based estimation of probability moments. Theoretical prediction and computational results show that it can provide us an unbiased estimation of the probability moments of continuous order. Calculations on probability redistribution model verify that it can extract exactly multifractal behaviors from several hundred recordings. Its powerfulness in monitoring evolution of scaling behaviors is exemplified by two empirical cases, i.e., the gait time series for fast, normal, and slow trials of a healthy volunteer, and the closing price series for Shanghai stock market. By using short time series with several hundred lengths, a comparison with the well-established tools displays significant advantages of its performance over the other methods. The factorial-moment-based estimation can evaluate correctly the scaling behaviors in a scale range about three generations wider than the multifractal detrended fluctuation analysis and the basic estimation. The estimation of partition function given by the wavelet transform modulus maxima has unacceptable fluctuations. Besides the scaling invariance focused in the present paper, the proposed factorial moment of continuous order can find its various uses, such as finding nonextensive behaviors of a complex system and reconstructing the causality relationship network between elements of a complex system.

Time and frequency domain characteristics of detrending-operation-based scaling analysis: Exact DFA and DMA frequency responses

We develop a general framework to study the time and frequency domain characteristics of detrending-operation-based scaling analysis methods, such as detrended fluctuation analysis (DFA) and detrending moving average (DMA) analysis. In this framework, using either the time or frequency domain approach, the frequency responses of detrending operations are calculated analytically. Although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations, the time domain approach presented here is applicable to both linear and nonlinear detrending operations. Furthermore, using the relationship between the time and frequency domain representations of the frequency responses, the frequency domain characteristics of nonlinear detrending operations can be obtained. Based on the calculated frequency responses, it is possible to establish a direct connection between the root-mean-square deviation of the detrending-operation-based scaling analysis and the power spectrum for linear stochastic processes. Here, by applying our methods to DFA and DMA, including higher-order cases, exact frequency responses are calculated. In addition, we analytically investigate the cutoff frequencies of DFA and DMA detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale.

Analysis of concentric and eccentric contractions in biceps brachii muscles using surface electromyography signals and multifractal analysis

Muscle contractions can be categorized into isometric, isotonic (concentric and eccentric) and isokinetic contractions. The eccentric contractions are very effective for promoting muscle hypertrophy and produce larger forces when compared to the concentric or isometric contractions. Surface electromyography signals are widely used for analyzing muscle activities. These signals are nonstationary, nonlinear and exhibit self-similar multifractal behavior. The research on surface electromyography signals using multifractal analysis is not well established for concentric and eccentric contractions. In this study, an attempt has been made to analyze the concentric and eccentric contractions associated with biceps brachii muscles using surface electromyography signals and multifractal detrended moving average algorithm. Surface electromyography signals were recorded from 20 healthy individuals while performing a single curl exercise. The preprocessed signals were divided into concentric and eccentric cycles and in turn divided into phases based on range of motion: lower (0°–90°) and upper (>90°). The segments of surface electromyography signal were subjected to multifractal detrended moving average algorithm, and multifractal features such as strength of multifractality, peak exponent value, maximum exponent and exponent index were extracted in addition to conventional linear features such as root mean square and median frequency. The results show that surface electromyography signals exhibit multifractal behavior in both concentric and eccentric cycles. The mean strength of multifractality increased by 15% in eccentric contraction compared to concentric contraction. The lowest and highest exponent index values are observed in the upper concentric and lower eccentric contractions, respectively. The multifractal features are observed to be helpful in differentiating surface electromyography signals along the range of motion as compared to root mean square and median frequency. It appears that these multifractal features extracted from the concentric and eccentric contractions can be useful in the assessment of surface electromyography signals in sports medicine and training and also in rehabilitation programs.

Detrending moving average algorithm: Frequency response and scaling performances

The Detrending Moving Average (DMA) algorithm has been widely used in its several variants for characterizing long-range correlations of random signals and sets (one-dimensional sequences or high-dimensional arrays) over either time or space. In this paper, mainly based on analytical arguments, the scaling performances of the centered DMA, including higher-order ones, are investigated by means of a continuous time approximation and a frequency response approach. Our results are also confirmed by numerical tests. The study is carried out for higher-order DMA operating with moving average polynomials of different degree. In particular, detrending power degree, frequency response, asymptotic scaling, upper limit of the detectable scaling exponent, and finite scale range behavior will be discussed.

Fast algorithm for scaling analysis with higher-order detrending moving average method

Among scaling analysis methods based on the root-mean-square deviation from the estimated trend, it has been demonstrated that centered detrending moving average (DMA) analysis with a simple moving average has good performance when characterizing long-range correlation or fractal scaling behavior. Furthermore, higher-order DMA has also been proposed; it is shown to have better detrending capabilities, removing higher-order polynomial trends than original DMA. However, a straightforward implementation of higher-order DMA requires a very high computational cost, which would prevent practical use of this method. To solve this issue, in this study, we introduce a fast algorithm for higher-order DMA, which consists of two techniques: (1) parallel translation of moving averaging windows by a fixed interval; (2) recurrence formulas for the calculation of summations. Our algorithm can significantly reduce computational cost. Monte Carlo experiments show that the computational time of our algorithm is approximately proportional to the data length, although that of the conventional algorithm is proportional to the square of the data length. The efficiency of our algorithm is also shown by a systematic study of the performance of higher-order DMA, such as the range of detectable scaling exponents and detrending capability for removing polynomial trends. In addition, through the analysis of heart-rate variability time series, we discuss possible applications of higher-order DMA.

Weighted multifractal cross-correlation analysis based on Shannon entropy

In this paper, we propose a modification of multifractal cross-correlation analysis based on statistical moments (MFSMXA) method, called weighted MFSMXA method based on Shannon entropy (W-MFSMXA), to investigate cross-correlations and cross-multifractality between time series. Robustness of this method is verified by numerical experiments with both artificial and stock returns series. Results show that the proposed W-MFSMXA method not only keep the multifractal structure unchanged, but contains more significant information of series compared to the previous MFSMXA method. Furthermore, analytic formulas of the binomial multifractal model are generated for W-MFSMXA. Theoretical analysis and finite-size effect test demonstrate that W-MFSMXA slightly outperforms MFSMXA for relatively shorter series. We further generate the scaling exponent ratio to describe the relation of two methods, whose profile is found approximating a centrosymmetric hyperbola. Cross-multifractality is found in returns series but then destroyed after being shuffled as a consequence of the removed long memory in separate series.

Fractal and multifractal characteristics of swift heavy ion induced self-affine nanostructured BaF2 thin film surfaces

Fractal and multifractal characteristics of self-affine surfaces of BaF2 thin films, deposited on crystalline Si ⟨1 1 1⟩ substrate at room temperature, were studied. Self-affine surfaces were prepared by irradiation of 120 MeV Ag(9+) ions which modified the surface morphology at nanometer scale. The surface morphology of virgin thin film and those irradiated with different ion fluences are characterized by atomic force microscopy technique. The surface roughness (interface width) shows monotonic decrease with ion fluences, while the other parameters, such as lateral correlation length, roughness exponent, and fractal dimension, did not show either monotonic decrease or increase in nature. The self-affine nature of the films is further confirmed by autocorrelation function. The power spectral density of thin films surfaces exhibits inverse power law variation with spatial frequency, suggesting the existence of fractal component in surface morphology. The multifractal detrended fluctuation analysis based on the partition function approach is also performed on virgin and irradiated thin films. It is found that the partition function exhibits the power law behavior with the segment size. Moreover, it is also seen that the scaling exponents vary nonlinearly with the moment, thereby exhibiting the multifractal nature.

Detrended partial cross-correlation analysis of two nonstationary time series influenced by common external forces

When common factors strongly influence two power-law cross-correlated time series recorded in complex natural or social systems, using detrended cross-correlation analysis (DCCA) without considering these common factors will bias the results. We use detrended partial cross-correlation analysis (DPXA) to uncover the intrinsic power-law cross correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces. The DPXA method is a generalization of the detrended cross-correlation analysis that takes into account partial correlation analysis. We demonstrate the method by using bivariate fractional Brownian motions contaminated with a fractional Brownian motion. We find that the DPXA is able to recover the analytical cross Hurst indices, and thus the multiscale DPXA coefficients are a viable alternative to the conventional cross-correlation coefficient. We demonstrate the advantage of the DPXA coefficients over the DCCA coefficients by analyzing contaminated bivariate fractional Brownian motions. We calculate the DPXA coefficients and use them to extract the intrinsic cross correlation between crude oil and gold futures by taking into consideration the impact of the U.S. dollar index. We develop the multifractal DPXA (MF-DPXA) method in order to generalize the DPXA method and investigate multifractal time series. We analyze multifractal binomial measures masked with strong white noises and find that the MF-DPXA method quantifies the hidden multifractal nature while the multifractal DCCA method fails.

The effect of the underlying distribution in Hurst exponent estimation

In this paper, a heavy-tailed distribution approach is considered in order to explore the behavior of actual financial time series. We show that this kind of distribution allows to properly fit the empirical distribution of the stocks from S&P500 index. In addition to that, we explain in detail why the underlying distribution of the random process under study should be taken into account before using its self-similarity exponent as a reliable tool to state whether that financial series displays long-range dependence or not. Finally, we show that, under this model, no stocks from S&P500 index show persistent memory, whereas some of them do present anti-persistent memory and most of them present no memory at all.

Multifractal detrended moving average analysis for texture representation

Multifractal detrended moving average analysis (MF-DMA) is recently employed to detect long-range correlation and multifractal nature in stationary and non-stationary time series. In this paper, we propose a method to calculate the generalized Hurst exponent for each pixel of a surface based on MF-DMA, which we call the MF-DMA-based local generalized Hurst exponent. These exponents form a matrix, which we denote by LHq. These exponents are similar to the multifractal detrended fluctuation analysis (MF-DFA)-based local generalized Hurst exponent. The performance of the calculated LHq is tested for two synthetic multifractal surfaces and ten randomly chosen natural textures with analytical solutions under three cases, namely, backward (θ = 0), centered (θ = 0.5), and forward (θ = 1) with different q values and different sub-image sizes. Two sets of comparison segmentation experiments between the three cases of the MF-DMA-based LHq and the MF-DFA-based LHq show that the MF-DMA-based LHq is superior to the MF-DFA-based LHq. In addition, the backward MF-DMA algorithm is more efficient than the centered and forward algorithms. An interest finding is that the LHq with q < 0 outperforms the LHq with q > 0 in characterizing the image features of natural textures for both the MF-DMA and MF-DFA algorithms.

Modified multidimensional scaling approach to analyze financial markets

Detrended cross-correlation coefficient (σDCCA) and dynamic time warping (DTW) are introduced as the dissimilarity measures, respectively, while multidimensional scaling (MDS) is employed to translate the dissimilarities between daily price returns of 24 stock markets. We first propose MDS based on σDCCA dissimilarity and MDS based on DTW dissimilarity creatively, while MDS based on Euclidean dissimilarity is also employed to provide a reference for comparisons. We apply these methods in order to further visualize the clustering between stock markets. Moreover, we decide to confront MDS with an alternative visualization method, "Unweighed Average" clustering method, for comparison. The MDS analysis and "Unweighed Average" clustering method are employed based on the same dissimilarity. Through the results, we find that MDS gives us a more intuitive mapping for observing stable or emerging clusters of stock markets with similar behavior, while the MDS analysis based on σDCCA dissimilarity can provide more clear, detailed, and accurate information on the classification of the stock markets than the MDS analysis based on Euclidean dissimilarity. The MDS analysis based on DTW dissimilarity indicates more knowledge about the correlations between stock markets particularly and interestingly. Meanwhile, it reflects more abundant results on the clustering of stock markets and is much more intensive than the MDS analysis based on Euclidean dissimilarity. In addition, the graphs, originated from applying MDS methods based on σDCCA dissimilarity and DTW dissimilarity, may also guide the construction of multivariate econometric models.