We use some fractal analysis methods to study river flow fluctuations. The result of the Multifractal Detrended Fluctuation Analysis (MF-DFA) shows that there are two crossover timescales at s 1 × ∼ 12 and s 2 × ∼ 130 months in the fluctuation function. We discuss how the existence of the crossover timescales are related to a sinusoidal trend. The first crossover is due to the seasonal trend and the value of second one is approximately equal to the well-known cycle of sun activity. Using Fourier Detrended Fluctuation Analysis, the sinusoidal trend is eliminated. The values of Hurst exponents of the runoff water of rivers without the sinusoidal trend show a long-range correlation behavior. For the Daugava river, the value of Hurst exponent is 0.52 ± 0.01 and also we find that these fluctuations have multifractal nature. Comparing the MF-DFA results for the remaining data set of Daugava river to those for shuffled and surrogate series, we conclude that its multifractal nature is almost entirely due to the broadness of probability density function.

Fractal Analysis of River Flow Fluctuations

Mutually interacting components form complex systems and the outputs of these components are usually long-range cross-correlated. Using wavelet leaders, we propose a method of characterizing the joint multifractal nature of these long-range cross correlations, a method we call joint multifractal analysis based on wavelet leaders (MF-X-WL). We test the validity of the MF-X-WL method by performing extensive numerical experiments on the dual binomial measures with multifractal cross correlations and the bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. Both experiments indicate that MF-X-WL is capable to detect the cross correlations in synthetic data with acceptable estimating errors. We also apply the MF-X-WL method to the pairs of series from financial markets (returns and volatilities) and online worlds (online numbers of different genders and different societies) and find an intriguing joint multifractal behavior.

Joint multifractal analysis based on wavelet leaders

Multifractal detrended cross-correlation methodology is described and applied to Foreign exchange (Forex) market time series. Fluctuations of high-frequency exchange rates of eight major world currencies over 2010–2018 period are used to study cross-correlations. The study is motivated by fundamental questions in complex systems’ response to significant environmental changes and by potential applications in investment strategies, including detecting triangular arbitrage opportunities. Dominant multiscale cross-correlations between the exchange rates are found to typically occur at smaller fluctuation levels. However, hierarchical organization of ties expressed in terms of dendrograms, with a novel application of the multiscale cross-correlation coefficient, is more pronounced at large fluctuations. The cross-correlations are quantified to be stronger on average between those exchange rate pairs that are bound within triangular relations. Some pairs from outside triangular relations are, however, identified to be exceptionally strongly correlated as compared to the average strength of triangular correlations. This in particular applies to those exchange rates that involve Australian and New Zealand dollars and reflects their economic relations. Significant events with impact on the Forex are shown to induce triangular arbitrage opportunities which at the same time reduce cross-correlations on the smallest timescales and act destructively on the multiscale organization of correlations. In 2010–2018, such instances took place in connection with the Swiss National Bank intervention and the weakening of British pound sterling accompanying the initiation of Brexit procedure. The methodology could be applicable to temporal and multiscale pattern detection in any time series.

/pdf/detecting-correlations-and-triangular-arbitrage-38s6s027wk.pdf

Detecting correlations and triangular arbitrage opportunities in the Forex by means of multifractal detrended cross-correlations analysis

Multifractal detrended cross-correlation methodology is described and applied to Foreign exchange (Forex) market time series. Fluctuations of high frequency exchange rates of eight major world currencies over 2010-2018 period are used to study cross-correlations. The study is motivated by fundamental questions in complex systems' response to significant environmental changes and by potential applications in investment strategies, including detecting triangular arbitrage opportunities. Dominant multiscale cross-correlations between the exchange rates are found to typically occur at smaller fluctuation levels. However hierarchical organization of ties expressed in terms of dendrograms, with a novel application of the multiscale cross-correlation coefficient, are more pronounced at large fluctuations. The cross-correlations are quantified to be stronger on average between those exchange rate pairs that are bound within triangular relations. Some pairs from outside triangular relations are however identified to be exceptionally strongly correlated as compared to the average strength of triangular correlations.This in particular applies to those exchange rates that involve Australian and New Zealand dollars and reflects their economic relations. Significant events with impact on the Forex are shown to induce triangular arbitrage opportunities which at the same time reduce cross--correlations on the smallest time scales and act destructively on the multiscale organization of correlations. In 2010--2018 such instances took place in connection with the Swiss National Bank intervention and the weakening of British pound sterling accompanying the initiation of Brexit procedure. The methodology could be applicable to temporal and multiscale pattern detection in any time series.

/pdf/detecting-correlations-and-triangular-arbitrage-2wb9qlfmmy.pdf

Different variants of MFDFA technique are applied in order to investigate various (artificial and real-world) time series. Our analysis shows that the calculated singularity spectra are very sensitive to the order of the detrending polynomial used within the MFDFA method. The relation between the width of the multifractal spectrum (as well as the Hurst exponent) and the order of the polynomial used in calculation is evident. Furthermore, type of this relation itself depends on the kind of analyzed signal. Therefore, such an analysis can give us some extra information about the correlative structure of the time series being studied.

Effect of detrending on multifractal characteristics

Recently posture-induced cardiovascular changes have become a subject of study for its obvious implications. This chapter presents analysis of non-linear time series of ECG and arterial blood pressure (ABP) data from a new perspective using state-of-the-art non-linear technique for assessment of cardiac disorder induced by different human posture. Precisely two methods multifractal detrended fluctuation analysis and multifractal detrended cross-correlation analysis have been applied to ECG and ABP data to quantify degree of impact of posture on cardiovascular functions. Moreover the knowledge of cross-correlation parameters is significant as it provides information for understanding the dynamics of orthostatic stress.

Multifractal Correlation Study Between Posture and Autonomic Deregulation Using ECG and Blood Pressure Data

In this chapter we have presented how multifractal detrended cross-correlation analysis technique can be used to study Parkinson’s disease from human gait pattern of those patients when compared to those of normal people. The chapter further emphasizes that this study is important as a new novel technique whereby data from the correlation between the two feet provides status of the degree of neurodegenerative disorder. The chapter also presents how multifractal methodologies can also be applied in Huntington’s disease.

Multifractal Study of Parkinson’s and Huntington’s Diseases with Human Gait Data

Since the autonomic function affecting the sympathetic, parasympathetic nervous system changes due to epileptic seizure, the corresponding changes in cardiac signals may be effectively used as potential biomarker providing an extracerebral indicator of ictal onset in some cases. Patients suffering from epilepsy may experience serious cardiac malfunctions leading to sudden unexpected death (SUDEP). Thus the heart rate fluctuations during seizure are non-linear and extremely complex. This chapter presents a report on the analysis of ECG signals of postictal patients using a modern and rigorous chaos-based non-linear technique.

Multifractal Approach for Quantification of Autonomic Deregulation Due to Epileptic Seizure with ECG Data

Myopathies (MYO) are a group of disorders where malfunction of muscle fibers occurs for a number of reasons which results in a muscular dysfunction manifesting weakness of muscles. Neuropathies are also disorders of the peripheral nervous system for which information transmission from brain and spinal cord to every other part of the body is disturbed. For diagnosis and characterization of motor neuron disease (MND), myopathy, and neuropathy, the electromyography (EMG) is extensively used since EMG signal can be analyzed to obtain information in regard to degree of disorder. The contents of the chapter deal with the details of a rigorous and robust non-linear technique, namely, multifractal detrended fluctuation analysis, to assess the multifractal property of EMG signals of patients with neuromuscular disorders and also use of two quantitative parameters, the multifractal width, and the auto-correlation exponent as biomarker for diagnosis and prognosis of both MYO and NEURO and even for early detection of MND.

Multifractal Analysis of Electromyography Data

Epilepsy has been identified as a common disorder of central nervous system affecting a huge size of population. This chapter presents a new approach for studying EEG patterns of the human brain in different physiological and pathological states in epileptic patients and normal people with the help of multifractal detrended fluctuation analysis. The chapter also includes a brief discussion about Alzheimer’s diseases and its diagnosis techniques. Further multifractal cross-correlation study was also applied on EEG data taken from patients in both stages – during seizure and in seizure-free interval. The chapter ends with a discussion of how this method can be used as a possible biomarker of epilepsy.

Sayantan Chakraborty

Papers

Multifractal Correlation Study Between Posture and Autonomic Deregulation Using ECG and Blood Pressure Data

Multifractal Study of Parkinson’s and Huntington’s Diseases with Human Gait Data

Multifractal Approach for Quantification of Autonomic Deregulation Due to Epileptic Seizure with ECG Data

Multifractal Analysis of Electromyography Data

Multifractal Study of EEG Signal of Subjects with Epilepsy and Alzheimer’s