Wavelet versus detrended fluctuation analysis of multifractal structures

Exploring Word-Adjacency Networks with Multifractal Time Series Analysis Techniques

A novel method of exploring linguistic networks is introduced by mapping word-adjacency networks to time series and applying multifractal analysis techniques. This approach captures the complex structural patterns of language by encoding network properties-such as clustering coefficients and node degrees-into temporal sequences. Using Alice's Adventures in Wonderland by Lewis Carroll as a case study, both traditional word-adjacency networks and extended versions that incorporate punctuation are examined. The results indicate that the time series derived from clustering coefficients, when following the natural reading order, exhibits multifractal characteristics, revealing inherent complexity in textual organization. Statistical validation confirms that observed multifractal properties arise from genuine correlations rather than from spurious effects. Extending this analysis by taking into account punctuation equally with words, however, changes the nature of the global scaling to a more convolved form that is not describable by a uniform multifractal. An analogous analysis based on the node degrees does not show such rich behaviors, however. These findings reveal a new perspective for quantitative linguistics and network science, providing a deeper understanding of the interplay between text structure and complex systems.

Multifractal dynamic changes of spontaneous brain activity in psychiatric disorders: Adult attention deficit-hyperactivity disorder, bipolar disorder, and schizophrenia

It is one of the strategies to study the complexity of spontaneous fluctuation of brain neurons based on resting-state functional magnetic resonance imaging (rs-fMRI), but the multifractal characteristics of spontaneous fluctuation of brain neurons in psychiatric diseases need to be studied. Therefore, this paper will study the multifractal spontaneous brain activity changes in psychiatric disorders using the multifractal detrended fluctuation analysis algorithm based on the UCLA datasets. Specifically: (1) multifractal characteristics in adult attention deficit-hyperactivity disorder (ADHD), bipolar disorder (BP), and schizophrenia (SCHZ); (2) the source of those multifractal characteristics. Results showed that for adult ADHD, BP, and SCHZ, all 6 functional brain regions exhibit multifractal characteristics, and the multifractal spectrum shows a reduction in bell-shaped asymmetry, unlike the intensity of healthy control (HC) asymmetry. Besides, compared with HC, the multifractal sources of all functional brain regions were fat-tail probability distribution and the long-range dependence correlation, but the intensity of fat-tail probability distribution was decreased and the long-range dependence correlation was increased. The results provide a reference for further understanding the complexity of spontaneous fluctuation of neurons in psychiatric disorders.

Punctuation Patterns in Finnegans Wake by James Joyce Are Largely Translation-Invariant

The complexity characteristics of texts written in natural languages are significantly related to the rules of punctuation. In particular, the distances between punctuation marks measured by the number of words quite universally follow the family of Weibull distributions known from survival analyses. However, the values of two parameters marking specific forms of these distributions distinguish specific languages. This is such a strong constraint that the punctuation distributions of texts translated from the original language into another adopt quantitative characteristics of the target language. All these changes take place within Weibull distributions such that the corresponding hazard functions are always increasing. Recent previous research shows that James Joyce's famous novel Finnegans Wake is subject to such an extreme distribution from the Weibull family that the corresponding hazard function is clearly decreasing. At the same time, the distances of sentence-ending punctuation marks, determining the sentence length variability, have an almost perfect multifractal organization to an extent found nowhere else in the literature thus far. In the present contribution, based on several available translations (Dutch, French, German, Polish, and Russian) of Finnegans Wake, it is shown that the punctuation characteristics of this work remain largely translation-invariant, contrary to the common cases. These observations may constitute further evidence that Finnegans Wake is a translinguistic work in this respect as well, in line with Joyce's original intention.

Quantifying patterns of punctuation in modern Chinese prose

Recent research shows that punctuation patterns in texts exhibit universal features across languages. Analysis of Western classical literature reveals that the distribution of spaces between punctuation marks aligns with a discrete Weibull distribution, typically used in survival analysis. By extending this analysis to Chinese literature represented here by three notable contemporary works, it is shown that Zipf's law applies to Chinese texts similarly to Western texts, where punctuation patterns also improve adherence to the law. Additionally, the distance distribution between punctuation marks in Chinese texts follows the Weibull model, though larger spacing is less frequent than in English translations. Sentence-ending punctuation, representing sentence length, diverges more from this pattern, reflecting greater flexibility in sentence length. This variability supports the formation of complex, multifractal sentence structures, particularly evident in Gao Xingjian's Soul Mountain. These findings demonstrate that both Chinese and Western texts share universal punctuation and word distribution patterns, underscoring their broad applicability across languages.

Quantifying multifractal anisotropy in two dimensional objects

An efficient method of exploring the effects of anisotropy in the fractal properties of 2D surfaces and images is proposed. It can be viewed as a direction-sensitive generalization of the multifractal detrended fluctuation analysis into 2D. It is tested on synthetic structures to ensure its effectiveness, with results indicating consistency. The interdisciplinary potential of this method in describing real surfaces and images is demonstrated, revealing previously unknown directional multifractality in data sets from the Martian surface and the Crab Nebula. The multifractal characteristics of Jackson Pollock's paintings are also analyzed. The results point to their evolution over the time of creation of these works.

Multifractal Hopscotch in Hopscotch by Julio Cortázar

Punctuation is the main factor introducing correlations in natural language written texts and it crucially impacts their overall effectiveness, expressiveness, and readability. Punctuation marks at the end of sentences are of particular importance as their distribution can determine various complexity features of written natural language. Here, the sentence length variability (SLV) time series representing Hopscotch by Julio Cortázar are subjected to quantitative analysis with an attempt to identify their distribution type, long-memory effects, and potential multiscale patterns. The analyzed novel is an important and innovative piece of literature whose essential property is freedom of movement between its building blocks given to a reader by the author. The statistical consequences of this freedom are closely investigated in both the original, Spanish version of the novel, and its translations into English and Polish. Clear evidence of rich multifractality in the SLV dynamics, with a left-sided asymmetry, however, is observed in all three language versions as well as in the versions with differently ordered chapters.

On Multiscaling of Parkinsonian Rest Tremor Signals and Their Classification

Self-similar stochastic processes and broad probability distributions are ubiquitous in nature and in many man-made systems. The brain is a particularly interesting example of (natural) complex system where those features play a pivotal role. In fact, the controversial yet experimentally validated "criticality hypothesis" explaining the functioning of the brain implies the presence of scaling laws for correlations. Recently, we have analyzed a collection of rest tremor velocity signals recorded from patients affected by Parkinson's disease, with the aim of determining and hence exploiting the presence of scaling laws. Our results show that multiple scaling laws are required in order to describe the dynamics of such signals, stressing the complexity of the underlying generating mechanism. We successively extracted numeric features by using the multifractal detrended fluctuation analysis procedure. We found that such features can be effective for discriminating classes of signals recorded in different experimental conditions. Notably, we show that the use of medication (L-DOPA) can be recognized with high accuracy.

Characteristics of price related fluctuations in non-fungible token (NFT) market

A non-fungible token (NFT) market is a new trading invention based on the blockchain technology, which parallels the cryptocurrency market. In the present work, we study capitalization, floor price, the number of transactions, the inter-transaction times, and the transaction volume value of a few selected popular token collections. The results show that the fluctuations of all these quantities are characterized by heavy-tailed probability distribution functions, in most cases well described by the stretched exponentials, with a trace of power-law scaling at times, long-range memory, persistence, and in several cases even the fractal organization of fluctuations, mostly restricted to the larger fluctuations, however. We conclude that the NFT market-even though young and governed by somewhat different mechanisms of trading-shares several statistical properties with the regular financial markets. However, some differences are visible in the specific quantitative indicators.

Multifractal long-range dependence pattern of functional magnetic resonance imaging in the human brain at rest

Long-range dependence is a prevalent phenomenon in various biological systems that characterizes the long-memory effect of temporal fluctuations. While recent research suggests that functional magnetic resonance imaging signal has fractal property, it remains unknown about the multifractal long-range dependence pattern of resting-state functional magnetic resonance imaging signals. The current study adopted the multifractal detrended fluctuation analysis on highly sampled resting-state functional magnetic resonance imaging scans to investigate long-range dependence profile associated with the whole-brain voxels as specific functional networks. Our findings revealed the long-range dependence's multifractal properties. Moreover, long-term persistent fluctuations are found for all stations with stronger persistency in whole-brain regions. Subsets with large fluctuations contribute more to the multifractal spectrum in the whole brain. Additionally, we found that the preprocessing with band-pass filtering provided significantly higher reliability for estimating long-range dependence. Our validation analysis confirmed that the optimal pipeline of long-range dependence analysis should include band-pass filtering and removal of daily temporal dependence. Furthermore, multifractal long-range dependence characteristics in healthy control and schizophrenia are different significantly. This work has provided an analytical pipeline for the multifractal long-range dependence in the resting-state functional magnetic resonance imaging signal. The findings suggest differential long-memory effects in the intrinsic functional networks, which may offer a neural marker finding for understanding brain function and pathology.

Multifractal Analysis of the Influence of Indole-3-Acetic Acid on Fast-Activating Vacuolar (FV) Channels of Beta vulgaris L. Taproot Cells

In this paper, the multifractal properties of the ion current time series in the fast-activating vacuolar (FV) channels of Beta vulgaris L. taproot cells were investigated. These channels are permeable for only monovalent cations and mediate K+ at very low concentrations of cytosolic Ca2+ and large voltages of either polarity. Using the patch clamp technique, the currents of the FV channels in red beet taproot vacuoles were recorded and analysed by using the multifractal detrended fluctuation analysis (MFDFA) method. The activity of the FV channels depended on the external potential and was sensitive to the auxin. It was also shown that the singularity spectrum of the ion current in the FV channels is non-singular, and the multifractal parameters, i.e., the generalised Hurst exponent and the singularity spectrum, were modified in the presence of IAA. Taking into account the obtained results, it can be suggested that the multifractal properties of fast-activating vacuolar (FV) K+ channels, indicating the existence of long-term memory, should be taken into account in the molecular mechanism of the auxin-induced growth of plant cells.

Genuine multifractality in time series is due to temporal correlations

Based on the mathematical arguments formulated within the multifractal detrended fluctuation analysis (MFDFA) approach it is shown that, in the uncorrelated time series from the Gaussian basin of attraction, the effects resembling multifractality asymptotically disappear for positive moments when the length of time series increases. A hint is given that this applies to the negative moments as well and extends to the Lévy stable regime of fluctuations. The related effects are also illustrated and confirmed by numerical simulations. This documents that the genuine multifractality in time series may only result from the long-range temporal correlations, and the fatter distribution tails of fluctuations may broaden the width of the singularity spectrum only when such correlations are present. The frequently asked question of what makes multifractality in time series-temporal correlations or broad distribution tails-is thus ill posed. In the absence of correlations only the bifractal or monofractal cases are possible. The former corresponds to the Lévy stable regime of fluctuations while the latter to the ones belonging to the Gaussian basin of attraction in the sense of the central limit theorem.

Efficiency and herding analysis in gold-backed cryptocurrencies

This study analyses and compares the behavior of the gold-backed, conventional cryptocurrency, and gold markets capable of detecting the existence of herding and deducing the efficiency degree. In addition, this empirical work tried to examine the COVID-19 pandemic's influence on both cryptocurrency performances. This work developed a new method that discloses herding biases using persistence and efficiency metrics. Besides, this paper investigated the nonlinear dynamic properties of the gold-backed, conventional cryptocurrencies and Gold by estimating the Multifractal Detrended Fluctuation Analysis (MFDFA). It also assessed the inefficiency of these markets through an efficiency index (IEI) and tested the effect of COVID-19 on their dynamics. The findings of this investigation indicate that the gold-backed cryptocurrency (X8X) is the most efficient market in the long-term trading market. However, the conventional cryptocurrency market (Bitcoin) is the most efficient on the short trade horizon. Besides, gold-backed cryptocurrency markets present a smaller level of herding behavior than conventional cryptocurrencies on tall scales. Nevertheless, we noted the positive and negative effects of the pandemic on each cryptocurrency market dynamics. To the best of the authors' knowledge, this study is the first investigation that uses multifractal analysis to quantify the impact of the COVID-19 spread on gold-backed cryptocurrencies and detects the presence of herding behavior.

Analysis of inter-transaction time fluctuations in the cryptocurrency market

We analyze tick-by-tick data representing major cryptocurrencies traded on some different cryptocurrency trading platforms. We focus on such quantities like the inter-transaction times, the number of transactions in time unit, the traded volume, and volatility. We show that the inter-transaction times show long-range power-law autocorrelations. These lead to multifractality expressed by the right-side asymmetry of the singularity spectra f ( α ) indicating that the periods of increased market activity are characterized by richer multifractality compared to the periods of quiet market. We also show that neither the stretched exponential distribution nor the power-law-tail distribution is able to model universally the cumulative distribution functions of the quantities considered in this work. For each quantity, some data sets can be modeled by the former and some data sets by the latter, while both fail in other cases. An interesting, yet difficult to account for, observation is that parallel data sets from different trading platforms can show disparate statistical properties.

Multifractal Cross-Correlations of Bitcoin and Ether Trading Characteristics in the Post-COVID-19 Time

Unlike price fluctuations, the temporal structure of cryptocurrency trading has seldom been a subject of systematic study. In order to fill this gap, we analyse detrended correlations of the price returns, the average number of trades in time unit, and the traded volume based on high-frequency data representing two major cryptocurrencies: bitcoin and ether. We apply the multifractal detrended cross-correlation analysis, which is considered the most reliable method for identifying nonlinear correlations in time series. We find that all the quantities considered in our study show an unambiguous multifractal structure from both the univariate (auto-correlation) and bivariate (cross-correlation) perspectives. We looked at the bitcoin–ether cross-correlations in simultaneously recorded signals, as well as in time-lagged signals, in which a time series for one of the cryptocurrencies is shifted with respect to the other. Such a shift suppresses the cross-correlations partially for short time scales, but does not remove them completely. We did not observe any qualitative asymmetry in the results for the two choices of a leading asset. The cross-correlations for the simultaneous and lagged time series became the same in magnitude for the sufficiently long scales.

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.

Multifractal Company Market: An Application to the Stock Market Indices

Using the multiscale normalized partition function, we exploit the multifractal analysis based on directly measurable shares of companies in the market. We present evidence that markets of competing firms are multifractal/multiscale. We verified this by (i) using our model that described the critical properties of the company market and (ii) analyzing a real company market defined by the S&P500 index. As the valuable reference case, we considered a four-group market model that skillfully reconstructs this index’s empirical data. We point out that a four-group company market organization is universal because it can perfectly describe the essential features of the spectrum of dimensions, regardless of the analyzed series of shares. The apparent differences from the empirical data appear only at the level of subtle effects.

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.

Information Transfer and Multifractal Analysis of EEG in Mild Blast-Induced TBI

Mild, blast-induced traumatic brain injury (mbTBI) is a common combat brain injury characterized by typically normal neuroimaging findings, with unpredictable future cognitive recovery. Traditional methods of electroencephalography (EEG) analysis (e.g., spectral analysis) have not been successful in detecting the degree of cognitive and functional impairment in mbTBI. We therefore collected resting state EEG (5 minutes, 64 leads) from twelve patients with a history of mbTBI, along with repeat neuropsychological testing (D-KEFS Tower test) to compare two new methods for analyzing EEG (multifractal detrended fluctuation analysis (MF-DFA) and information transfer modeling (ITM)) with spectral analysis. For MF-DFA, we extracted relevant parameters from the resultant multifractal spectrum from all leads and compared with traditional power by frequency band for spectral analysis. For ITM, because the number of parameters from each lead far exceeded the number of subjects, we utilized a reduced set of 10 leads which were compared with spectral analysis. We utilized separate 30 second EEG segments for training and testing statistical models based upon regression tree analysis. ITM and MF-DFA models both generally had improved accuracy at correlating with relevant measures of cognitive performance as compared to spectral analytic models ITM and MF-DFA both merit additional research as analytic tools for EEG and cognition in TBI.

Fractality and Variability in Canonical and Non-Canonical English Fiction and in Non-Fictional Texts

This study investigates global properties of three categories of English text: canonical fiction, non-canonical fiction, and non-fictional texts. The central hypothesis of the study is that there are systematic differences with respect to structural design features between canonical and non-canonical fiction, and between fictional and non-fictional texts. To investigate these differences, we compiled a corpus containing texts of the three categories of interest, the Jena Corpus of Expository and Fictional Prose (JEFP Corpus). Two aspects of global structure are investigated, variability and self-similar (fractal) patterns, which reflect long-range correlations along texts. We use four types of basic observations, (i) the frequency of POS-tags per sentence, (ii) sentence length, (iii) lexical diversity, and (iv) the distribution of topic probabilities in segments of texts. These basic observations are grouped into two more general categories, (a) the lower-level properties (i) and (ii), which are observed at the level of the sentence (reflecting linguistic decoding), and (b) the higher-level properties (iii) and (iv), which are observed at the textual level (reflecting comprehension/integration). The observations for each property are transformed into series, which are analyzed in terms of variance and subjected to Multi-Fractal Detrended Fluctuation Analysis (MFDFA), giving rise to three statistics: (i) the degree of fractality ( H ), (ii) the degree of multifractality ( D ), i.e., the width of the fractal spectrum, and (iii) the degree of asymmetry ( A ) of the fractal spectrum. The statistics thus obtained are compared individually across text categories and jointly fed into a classification model (Support Vector Machine). Our results show that there are in fact differences between the three text categories of interest. In general, lower-level text properties are better discriminators than higher-level text properties. Canonical fictional texts differ from non-canonical ones primarily in terms of variability in lower-level text properties. Fractality seems to be a universal feature of text, slightly more pronounced in non-fictional than in fictional texts. On the basis of our results obtained on the basis of corpus data we point out some avenues for future research leading toward a more comprehensive analysis of textual aesthetics, e.g., using experimental methodologies.

Network Analysis of Cross-Correlations on Forex Market during Crises. Globalisation on Forex Market

Within the paper, the problem of globalisation during financial crises is analysed. The research is based on the Forex exchange rates. In the analysis, the power law classification scheme (PLCS) is used. The study shows that during crises cross-correlations increase resulting in significant growth of cliques, and also the ranks of nodes on the converging time series network are growing. This suggests that the crises expose the globalisation processes, which can be verified by the proposed analysis.

Applications of Distributed-Order Fractional Operators: A Review

Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

Exploring the multi-fractal nature of the air flow and pollutant dispersion in a turbulent urban atmosphere and its implications for long range pollutant transport

This work investigates the multi-fractal nature of a turbulent urban atmosphere using high-resolution atmospheric data. Meteorological and concentration measurements of passive and reactive pollutants collected over a 3-year period in a sub-urban high-Reynolds number atmospheric field were analyzed. Scaling laws characterizing the self-similarity and thereby depicting the multi-fractal nature are determined by calculating the singularity spectra, where a range of Hölder exponents, h, are estimated. In doing so, the complexity of the urban atmosphere entailing different stability regimes was addressed. Using the Monin-Obukhov length (L_MO) as a marker of atmospheric stability and thereby an indication of the magnitude of anisotropy, we find where and how self-similarity is manifested relative to the different regimes and we estimate corresponding appropriate scaling laws. We find that the wind speed obeys the -5/3 law suggested by Kolmogorov only when the atmosphere lies within the stable regime as defined by Monin-Obukhov theory. Specifically, when the ratio of the atmospheric boundary layer height (H_b.l) over L_MO is greater than 15, and at the same time, the ratio of the height above ground of the wind measurements (z₀) over L_MO is higher than 3 (i.e., in stable regime), then the singularity spectra of wind speed time series indicate that the dominant Hölder exponent, h_max, coincides with Kolmogorov's second hypothesis. On the contrary under unstable regimes in the atmosphere where the anisotropy is approached, different scaling laws are estimated. In detail, when z₀/L_MO<0, the dominant Hölder exponent, h_max, of the singularity spectra of the wind speed time series is either negative or close to zero, which is an indication of an impulse-like singularity, that is associated with rapid changes. For the ambient temperature and air quality measurements such as of carbon monoxide and particulate matter concentrations, it was found that they obey different laws, which are related with the long-term correlation of their data fluctuation.

Complexity in Economic and Social Systems: Cryptocurrency Market at around COVID-19

Social systems are characterized by an enormous network of connections and factors that can influence the structure and dynamics of these systems. Among them the whole economical sphere of human activity seems to be the most interrelated and complex. All financial markets, including the youngest one, the cryptocurrency market, belong to this sphere. The complexity of the cryptocurrency market can be studied from different perspectives. First, the dynamics of the cryptocurrency exchange rates to other cryptocurrencies and fiat currencies can be studied and quantified by means of multifractal formalism. Second, coupling and decoupling of the cryptocurrencies and the conventional assets can be investigated with the advanced cross-correlation analyses based on fractal analysis. Third, an internal structure of the cryptocurrency market can also be a subject of analysis that exploits, for example, a network representation of the market. In this work, we approach the subject from all three perspectives based on data from a recent time interval between January 2019 and June 2020. This period includes the peculiar time of the Covid-19 pandemic; therefore, we pay particular attention to this event and investigate how strong its impact on the structure and dynamics of the market was. Besides, the studied data covers a few other significant events like double bull and bear phases in 2019. We show that, throughout the considered interval, the exchange rate returns were multifractal with intermittent signatures of bifractality that can be associated with the most volatile periods of the market dynamics like a bull market onset in April 2019 and the Covid-19 outburst in March 2020. The topology of a minimal spanning tree representation of the market also used to alter during these events from a distributed type without any dominant node to a highly centralized type with a dominating hub of USDT. However, the MST topology during the pandemic differs in some details from other volatile periods.

Multibranch multifractality and the phase transitions in time series of mean interevent times

Empirical time series of interevent or waiting times are investigated using a modified Multifractal Detrended Fluctuation Analysis operating on fluctuations of mean detrended dynamics. The core of the extended multifractal analysis is the nonmonotonic behavior of the generalized Hurst exponent h(q)-the fundamental exponent in the study of multifractals. The consequence of this behavior is the nonmonotonic behavior of the coarse Hölder exponent α(q) leading to multibranchedness of the spectrum of dimensions. The Legendre-Fenchel transform is used instead of the routinely used canonical Legendre (single-branched) contact transform. Thermodynamic consequences of the multibranched multifractality are revealed. These are directly expressed in the language of phase transitions between thermally stable, metastable, and unstable phases. These phase transitions are of the first and second orders according to Mandelbrot's modified Ehrenfest classification. The discovery of multibranchedness is tantamount in significance to extending multifractal analysis.

EEG multifractal analysis correlates with cognitive testing scores and clinical staging in mild cognitive impairment

Alzheimer's disease and mild cognitive impairment are increasingly prevalent global health concerns in aging industrialized societies. There are only limited non-invasive biomarkers for the cognitive and functional impairment associated with dementia. Multifractal analysis of EEG has recently been proposed as having the potential to be an improved method of quantitative EEG analysis compared to existing techniques (e.g., spectral analysis). We utilized an existing database of a study of healthy elderly patients (N = 20) who were assessed with cognitive testing (Folstein Mini Mental Status Exam; MMSE) and resting state EEG (4 leads). Each subject's EEG was separated into two 30 s tracings for training and testing a statistical model against the MMSE scores. We compared multifractal detrended fluctuation analysis (MF-DFA) against Fourier Transform (FT) in the ability to produce an accurate classification and regression trees estimator for the testing EEG segments. The MF-DFA-based statistical model MMSE estimation strongly correlated with the actual MMSE when applied to the test EEG parameter dataset, whereas the corresponding FT-based model did not. Using a standardized cutoff value for MMSE-based clinical staging, the MF-DFA-based statistical model was both sensitive and specific for clinical staging of both mild Alzheimer's disease and mild cognitive impairment. MF-DFA shows promise as a method of quantitative EEG analysis to accurately estimate cognition in Alzheimer's disease.

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.

Ordered Avalanches on the Bethe Lattice

We discuss deterministic sequences of avalanches on a directed Bethe lattice. The approach is motivated by the phenomenon of self-organized criticality. Grains are added only at one node of the network. When the number of grains at any node exceeds a threshold b, each of k out-neighbors gets one grain. The probability of an avalanche of size s is proportional to s−τ. When the avalanche mass is conserved (k=b), we get τ=1. For an application of the model to social phenomena, the conservation condition can be released. Then, the exponent τ is found to depend on the model parameters; τ ≈ log(b)/log(k). The distribution of the time duration of avalanches is exponential. Multifractal analysis of the avalanche sequences reveals their strongly non-uniform fractal organization. Maximal value of the singularity strength αmax in the bifractal spectrum is found to be 1/τ.

Signatures of the Crypto-Currency Market Decoupling from the Forex

Based on the high-frequency recordings from Kraken, a cryptocurrency exchange and professional trading platform that aims to bring Bitcoin and other cryptocurrencies into the mainstream, the multiscale cross-correlations involving the Bitcoin (BTC), Ethereum (ETH), Euro (EUR) and US dollar (USD) are studied over the period between 1 July 2016 and 31 December 2018. It is shown that the multiscaling characteristics of the exchange rate fluctuations related to the cryptocurrency market approach those of the Forex. This, in particular, applies to the BTC/ETH exchange rate, whose Hurst exponent by the end of 2018 started approaching the value of 0.5, which is characteristic of the mature world markets. Furthermore, the BTC/ETH direct exchange rate has already developed multifractality, which manifests itself via broad singularity spectra. A particularly significant result is that the measures applied for detecting cross-correlations between the dynamics of the BTC/ETH and EUR/USD exchange rates do not show any noticeable relationships. This could be taken as an indication that the cryptocurrency market has begun decoupling itself from the Forex.

Multifractal detrended fluctuation analysis of continuous neural time series in primate visual cortex

BACKGROUND

Local field potential (LFP) recordings have become an important tool to study the activity of populations of neurons. The functional activity of LFPs is usually compared with the activity of neighboring single spike neurons with sampling rates much higher than those of the continuous field potential channel (5 kHz). However, comparison of these signals generated with the lower sampling rate technique is important.

NEW METHOD

In this study, we provide an analysis of extracellular field potential time series using the sophisticated nonlinear multifractal detrended fluctuation analysis (MF-DFA). Using the MF-DFA, we demonstrate that the integral of the singularity spectrum is a powerful new method to measure the response tuning of spikes in the continuous field potential channel.

RESULTS

Results show that the spikes in the continuous channel at frequency ranges above the LFP component signals were consistently tuned similar to those in the spike channel. Our results also show that using a low-pass filter (<250 Hz), which is commonly applied as a preprocessing step to insulate LFPs from spikes, significantly influences the nonlinearity of the multifractal time series.

COMPARISON WITH EXISTING METHODS AND CONCLUSIONS

Our approach for inferring the tuning curve of spiking activity from the continuous channel has some advantages compared to conventional methods such as spike trains. The MF-DFA does not require any preprocessing of the raw signal data and makes no assumptions about the time series characteristics. This method is robust and can be applied to short time series of continuous raw signals.

Bitcoin market route to maturity? Evidence from return fluctuations, temporal correlations and multiscaling effects

Based on 1-min price changes recorded since year 2012, the fluctuation properties of the rapidly emerging Bitcoin market are assessed over chosen sub-periods, in terms of return distributions, volatility autocorrelation, Hurst exponents, and multiscaling effects. The findings are compared to the stylized facts of mature world markets. While early trading was affected by system-specific irregularities, it is found that over the months preceding April 2018 all these statistical indicators approach the features hallmarking maturity. This can be taken as an indication that the Bitcoin market, and possibly other cryptocurrencies, carry concrete potential of imminently becoming a regular market, alternative to the foreign exchange. Since high-frequency price data are available since the beginning of trading, the Bitcoin offers a unique window into the statistical characteristics of a market maturation trajectory.

Nanoscale Correlated Disorder in Out-of-Equilibrium Myelin Ultrastructure

Ultrastructural fluctuations at nanoscale are fundamental to assess properties and functionalities of advanced out-of-equilibrium materials. We have taken myelin as a model of supramolecular assembly in out-of-equilibrium living matter. Myelin sheath is a simple stable multilamellar structure of high relevance and impact in biomedicine. Although it is known that myelin has a quasi-crystalline ultrastructure, there is no information on its fluctuations at nanoscale in different states due to limitations of the available standard techniques. To overcome these limitations, we have used scanning micro X-ray diffraction, which is a unique non-invasive probe of both reciprocal and real space to visualize statistical fluctuations of myelin order of the sciatic nerve of Xenopus laevis. The results show that the ultrastructure period of the myelin is stabilized by large anticorrelated fluctuations at nanoscale, between hydrophobic and hydrophilic layers. The ratio between the total thickness of hydrophilic and hydrophobic layers defines the conformational parameter, which describes the different states of myelin. Our key result is that myelin in its out-of-equilibrium functional state fluctuates point-to-point between different conformations showing a correlated disorder described by a Levy distribution. As the system approaches the thermodynamic equilibrium in an aged state, the disorder loses its correlation degree and the structural fluctuation distribution changes to Gaussian. In a denatured state at low pH, it changes to a completely disordered stage. Our results aim to clarify the degradation mechanism in biological systems by associating these states with ultrastructural dynamic fluctuations at nanoscale.

Characterizing scaling properties of complex signals with missed data segments using the multifractal analysis

The scaling properties of complex processes may be highly influenced by the presence of various artifacts in experimental recordings. Their removal produces changes in the singularity spectra and the Hölder exponents as compared with the original artifacts-free data, and these changes are significantly different for positively correlated and anti-correlated signals. While signals with power-law correlations are nearly insensitive to the loss of significant parts of data, the removal of fragments of anti-correlated signals is more crucial for further data analysis. In this work, we study the ability of characterizing scaling features of chaotic and stochastic processes with distinct correlation properties using a wavelet-based multifractal analysis, and discuss differences between the effect of missed data for synchronous and asynchronous oscillatory regimes. We show that even an extreme data loss allows characterizing physiological processes such as the cerebral blood flow dynamics.

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.

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.

Atypical transistor-based chaotic oscillators: Design, realization, and diversity

In this paper, we show that novel autonomous chaotic oscillators based on one or two bipolar junction transistors and a limited number of passive components can be obtained via random search with suitable heuristics. Chaos is a pervasive occurrence in these circuits, particularly after manual adjustment of a variable resistor placed in series with the supply voltage source. Following this approach, 49 unique circuits generating chaotic signals when physically realized were designed, representing the largest collection of circuits of this kind to date. These circuits are atypical as they do not trivially map onto known topologies or variations thereof. They feature diverse spectra and predominantly anti-persistent monofractal dynamics. Notably, we recurrently found a circuit comprising one resistor, one transistor, two inductors, and one capacitor, which generates a range of attractors depending on the parameter values. We also found a circuit yielding an irregular quantized spike-train resembling some aspects of neural discharge and another one generating a double-scroll attractor, which represent the smallest known transistor-based embodiments of these behaviors. Through three representative examples, we additionally show that diffusive coupling of heterogeneous oscillators of this kind may give rise to complex entrainment, such as lag synchronization with directed information transfer and generalized synchronization. The replicability and reproducibility of the experimental findings are good.

Nonlinear temperature effects on multifractal complexity of metabolic rate of mice

Complex physiological dynamics have been argued to be a signature of healthy physiological function. Here we test whether the complexity of metabolic rate fluctuations in small endotherms decreases with lower environmental temperatures. To do so, we examine the multifractal temporal scaling properties of the rate of change in oxygen consumption r(VO₂), in the laboratory mouse Mus musculus, assessing their long range correlation properties across seven different environmental temperatures, ranging from 0 °C to 30 °C. To do so, we applied multifractal detrended fluctuation analysis (MF-DFA), finding that r(VO₂) fluctuations show two scaling regimes. For small time scales below the crossover time (approximately 10² s), either monofractal or weak multifractal dynamics are observed depending on whether T_a < 15 °C or T_a > 15 °C respectively. For larger time scales, r(VO₂) fluctuations are characterized by an asymptotic scaling exponent that indicates multifractal anti-persistent or uncorrelated dynamics. For both scaling regimes, a generalization of the multiplicative cascade model provides very good fits for the Renyi exponents τ(q), showing that the infinite number of exponents h(q) can be described by only two independent parameters, a and b. We also show that the long-range correlation structure of r(VO₂) time series differs from randomly shuffled series, and may not be explained as an artifact of stochastic sampling of a linear frequency spectrum. These results show that metabolic rate dynamics in a well studied micro-endotherm are consistent with a highly non-linear feedback control system.

Multifractal cross-correlation effects in two-variable time series of complex network vertex observables

We investigate the scaling of the cross-correlations calculated for two-variable time series containing vertex properties in the context of complex networks. Time series of such observables are obtained by means of stationary, unbiased random walks. We consider three vertex properties that provide, respectively, short-, medium-, and long-range information regarding the topological role of vertices in a given network. In order to reveal the relation between these quantities, we applied the multifractal cross-correlation analysis technique, which provides information about the nonlinear effects in coupling of time series. We show that the considered network models are characterized by unique multifractal properties of the cross-correlation. In particular, it is possible to distinguish between Erdös-Rényi, Barabási-Albert, and Watts-Strogatz networks on the basis of fractal cross-correlation. Moreover, the analysis of protein contact networks reveals characteristics shared with both scale-free and small-world models.

Multiscale multifractal detrended-fluctuation analysis of two-dimensional surfaces

Two-dimensional (2D) multifractal detrended fluctuation analysis (MF-DFA) has been used to study monofractality and multifractality on 2D surfaces, but when it is used to calculate the generalized Hurst exponent in a fixed time scale, the presence of crossovers can bias the outcome. To solve this problem, multiscale multifractal analysis (MMA) was recent employed in a one-dimensional case. MMA produces a Hurst surface h(q,s) that provides a spectrum of local scaling exponents at different scale ranges such that the positions of the crossovers can be located. We apply this MMA method to a 2D surface and identify factors that influence the results. We generate several synthesized surfaces and find that crossovers are consistently present, which means that their fractal properties differ at different scales. We apply MMA to the surfaces, and the results allow us to observe these differences and accurately estimate the generalized Hurst exponents. We then study eight natural texture images and two real-world images and find (i) that the moving window length (WL) and the slide length (SL) are the key parameters in the MMA method, that the WL more strongly influences the Hurst surface than the SL, and that the combination of WL=4 and SL=4 is optimal for a 2D image; (ii) that the robustness of h(2,s) to four common noises is high at large scales but variable at small scales; and (iii) that the long-term correlations in the images weaken as the intensity of Gaussian noise and salt and pepper noise is increased. Our findings greatly improve the performance of the MMA method on 2D surfaces.

Quantitative Assessment of Heart Rate Dynamics during Meditation: An ECG Based Study with Multi-Fractality and Visibility Graph

The cardiac dynamics during meditation is explored quantitatively with two chaos-based non-linear techniques viz. multi-fractal detrended fluctuation analysis and visibility network analysis techniques. The data used are the instantaneous heart rate (in beats/minute) of subjects performing Kundalini Yoga and Chi meditation from PhysioNet. The results show consistent differences between the quantitative parameters obtained by both the analysis techniques. This indicates an interesting phenomenon of change in the complexity of the cardiac dynamics during meditation supported with quantitative parameters. The results also produce a preliminary evidence that these techniques can be used as a measure of physiological impact on subjects performing meditation.

Effect of muscular fatigue on fractal upper limb coordination dynamics and muscle synergies

Rehabilitation exercises cause fatigue because tasks are repetitive. Therefore, inevitable human motion performance changes occur during the therapy. Although traditionally fatigue is considered an event that occurs in the musculoskeletal level, this paper studies whether fatigue can be regarded as context that influences lower-dimensional motor control organization and coordination at neural level. Non Negative Factorization Matrix (NNFM) and Detrended Fluctuations Analysis (DFA) are the tools used to analyze the changes in the coordination of motor function when someone is affected by fatigue. The study establishes that synergies remain fairly stable with the onset of fatigue, but the fatigue affects the dynamical coordination understood as a cognitive process. These results have been validated with 9 healthy subjects for three representative exercises for upper limb: biceps, triceps and deltoid.

Detrended fluctuation analysis made flexible to detect range of cross-correlated fluctuations

The detrended cross-correlation coefficient ρ(DCCA) has recently been proposed to quantify the strength of cross-correlations on different temporal scales in bivariate, nonstationary time series. It is based on the detrended cross-correlation and detrended fluctuation analyses (DCCA and DFA, respectively) and can be viewed as an analog of the Pearson coefficient in the case of the fluctuation analysis. The coefficient ρ(DCCA) works well in many practical situations but by construction its applicability is limited to detection of whether two signals are generally cross-correlated, without the possibility to obtain information on the amplitude of fluctuations that are responsible for those cross-correlations. In order to introduce some related flexibility, here we propose an extension of ρ(DCCA) that exploits the multifractal versions of DFA and DCCA: multifractal detrended fluctuation analysis and multifractal detrended cross-correlation analysis, respectively. The resulting new coefficient ρ(q) not only is able to quantify the strength of correlations but also allows one to identify the range of detrended fluctuation amplitudes that are correlated in two signals under study. We show how the coefficient ρ(q) works in practical situations by applying it to stochastic time series representing processes with long memory: autoregressive and multiplicative ones. Such processes are often used to model signals recorded from complex systems and complex physical phenomena like turbulence, so we are convinced that this new measure can successfully be applied in time-series analysis. In particular, we present an example of such application to highly complex empirical data from financial markets. The present formulation can straightforwardly be extended to multivariate data in terms of the q-dependent counterpart of the correlation matrices and then to the network representation.

Detecting and interpreting distortions in hierarchical organization of complex time series

Hierarchical organization is a cornerstone of complexity and multifractality constitutes its central quantifying concept. For model uniform cascades the corresponding singularity spectra are symmetric while those extracted from empirical data are often asymmetric. Using selected time series representing such diverse phenomena as price changes and intertransaction times in financial markets, sentence length variability in narrative texts, Missouri River discharge, and sunspot number variability as examples, we show that the resulting singularity spectra appear strongly asymmetric, more often left sided but in some cases also right sided. We present a unified view on the origin of such effects and indicate that they may be crucially informative for identifying the composition of the time series. One particularly intriguing case of this latter kind of asymmetry is detected in the daily reported sunspot number variability. This signals that either the commonly used famous Wolf formula distorts the real dynamics in expressing the largest sunspot numbers or, if not, that their dynamics is governed by a somewhat different mechanism.

Multifractal properties of a closed contour: a peek beyond the shape analysis

In recent decades multifractal analysis has been successfully applied to characterize the complex temporal and spatial organization of such diverse natural phenomena as heartbeat dynamics, the dendritic shape of neurons, retinal vessels, rock fractures, and intricately shaped volcanic ash particles. The characterization of multifractal properties of closed contours has remained elusive because applying traditional methods to their quasi-one-dimensional nature yields ambiguous answers. Here we show that multifractal analysis can reveal meaningful and sometimes unexpected information about natural structures with a perimeter well-defined by a closed contour. To this end, we demonstrate how to apply multifractal detrended fluctuation analysis, originally developed for the analysis of time series, to an arbitrary shape of a given study object. In particular, we show the application of the method to fish otoliths, calcareous concretions located in fish's inner ear. Frequently referred to as the fish's "black box", they contain a wealth of information about the fish's life history and thus have recently attracted increasing attention. As an illustrative example, we show that a multifractal approach can uncover unexpected relationships between otolith contours and size and age of fish at maturity.