Dynamical variety of shapes in financial multifractality

The concept of multifractality offers a powerful formal tool to filter out multitude of the most relevant characteristics of complex time series. The related studies thus far presented in the scientific literature typically limit themselves to evaluation of whether or not a time series is multifractal and width of the resulting singularity spectrum is considered a measure of the degree of complexity involved. However, the character of the complexity of time series generated by the natural processes usually appears much more intricate than such a bare statement can reflect. As an example, based on the long-term records of S&P500 and NASDAQ - the two world leading stock market indices - the present study shows that they indeed develop the multifractal features, but these features evolve through a variety of shapes, most often strongly asymmetric, whose changes typically are correlated with the historically most significant events experienced by the world economy. Relating at the same time the index multifractal singularity spectra to those of the component stocks that form this index reflects the varying degree of correlations involved among the stocks.Comment: 26 pages, 10 figure

Multifractal and Network Analysis of Phase Transition

Many models and real complex systems possess critical thresholds at which the systems shift from one sate to another. The discovery of the early warnings of the systems in the vicinity of critical point are of great importance to estimate how far a system is from a critical threshold. Multifractal Detrended Fluctuation analysis (MF-DFA) and visibility graph method have been employed to investigate the fluctuation and geometrical structures of magnetization time series of two-dimensional Ising model around critical point. The Hurst exponent has been confirmed to be a good indicator of phase transition. Increase of the multifractality of the time series have been observed from generalized Hurst exponents and singularity spectrum. Both Long-term correlation and broad probability density function are identified to be the sources of multifractality of time series near critical regime. Heterogeneous nature of the networks constructed from magnetization time series have validated the fractal properties of magnetization time series from complex network perspective. Evolution of the topology quantities such as clustering coefficient, average degree, average shortest path length, density, assortativity and heterogeneity serve as early warnings of phase transition. Those methods and results can provide new insights about analysis of phase transition problems and can be used as early warnings for various complex systems.Comment: 23 pages, 11 figure

Fractal and multifractal analysis of complex networks: Estonian network of payments

Complex networks have gained much attention from different areas of knowledge in recent years. Particularly, the structures and dynamics of such systems have attracted considerable interest. Complex networks may have characteristics of multifractality. In this study, we analyze fractal and multifractal properties of a novel network: the large scale economic network of payments of Estonia, where companies are represented by nodes and the payments done between companies are represented by links. We present a fractal scaling analysis and examine the multifractal behavior of this network by using a sandbox algorithm. Our results indicate the existence of multifractality in this network and consequently, the existence of multifractality in the Estonian economy. To the best of our knowledge, this is the first study that analyzes multifractality of a complex network of payments.Comment: 13 page

Wavelet-based discrimination of isolated singularities masquerading as multifractals in detrended fluctuation analyses

The robustness of two widespread multifractal analysis methods, one based on detrended fluctuation analysis and one on wavelet leaders, is discussed in the context of time-series containing non-uniform structures with only isolated singularities. Signals generated by simulated and experimentally-realized chaos generators, together with synthetic data addressing particular aspects, are taken into consideration. The results reveal essential limitations affecting the ability of both methods to correctly infer the non-multifractal nature of signals devoid of a cascade-like hierarchy of singularities. Namely, signals harboring only isolated singularities are found to artefactually give rise to broad multifractal spectra, resembling those expected in the presence of a well-developed underlying multifractal structure. Hence, there is a real risk of incorrectly inferring multifractality due to isolated singularities. The careful consideration of local scaling properties and the distribution of Hölder exponent obtained, for example, through wavelet analysis, is indispensable for rigorously assessing the presence or absence of multifractality.Fil: Oswiecimka, Pawel. Polish Academy of Sciences; ArgentinaFil: Drozdz, Stanislaw. Cracow University of Technology. Faculty of Materials Engineering and Physics; PoloniaFil: Frasca, Mattia. University of Catania. Department of Electrical Electronic and Computer Engineering; ItaliaFil: Gebarowski, Robert. Cracow University of Technology. Faculty of Materials Engineering and Physics; PoloniaFil: Yoshimura, Natsue. Tokyo Institute of Technology. Institute of Innovative Research. FIRST; JapónFil: Zunino, Luciano José. Universidad Nacional de La Plata. Facultad de Ingeniería. Departamento de Ciencias Básicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Centro de Investigaciones Ópticas. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Centro de Investigaciones Ópticas. Universidad Nacional de La Plata. Centro de Investigaciones Ópticas; ArgentinaFil: Minati, Ludovico. Universita degli Studi di Trento; Italia. Polish Academy of Sciences; Argentin

Multi-Fractal Spectral Analysis of the 1987 Stock Market Crash

The multifractal model of asset returns captures the volatility persistence of many financial time series. Its multifractal spectrum computed from wavelet modulus maxima lines provides the spectrum of irregularities in the distribution of market returns over time and thereby of the kind of uncertainty or randomness in a particular market. Changes in this multifractal spectrum display distinctive patterns around substantial market crashes or drawdowns. In other words, the kinds of singularities and the kinds of irregularity change in a distinct fashion in the periods immediately preceding and following major market drawdowns. This paper focuses on these identifiable multifractal spectral patterns surrounding the stock market crash of 1987. Although we are not able to find a uniquely identifiable irregularity pattern within the same market preceding different crashes at different times, we do find the same uniquely identifiable pattern in various stock markets experiencing the same crash at the same time. Moreover, our results suggest that all such crashes are preceded by a gradual increase in the weighted average of the values of the Lipschitz regularity exponents, under low dispersion of the multifractal spectrum. At a crash, this weighted average irregularity value drops to a much lower value, while the dispersion of the spectrum of Lipschitz exponents jumps up to a much higher level after the crash. Our most striking result, therefore, is that the multifractal spectra of stock market returns are not stationary. Also, while the stock market returns show a global Hurst exponent of slight persistence 0.5Financial Markets, Persistence, Multi-Fractal Spectral Analysis, Wavelets

Multifractal detrended fluctuation analysis of temperature in Spain (1960–2019)

Datos de investigación disponibles en: http://www.aemet.es/es/datos_abiertos/AEMET_OpenDataIn the last decades, an ever-growing number of studies are focusing on the extreme weather conditions related to the climate change. Some of them are based on multifractal approaches, such as the Multifractal Detrended Fluctuation Analysis (MF-DFA), which has been used in this work. Daily diurnal temperature range (DTR), maximum, minimum and mean temperature from five coastal and five mainland stations in Spain have been analyzed. For comparison, two periods of 30 years have been considered: 1960–1989 and 1990–2019. By using the MF-DFA method, generalized Hurst exponents and multifractal spectra have been obtained. Outcomes corroborate that all these temperature variables have multifractal nature and show changes in multifractal properties between both periods. Also, Hurst exponents values indicate that all time series exhibit long-range correlations and a stationary behavior. Coastal locations exhibit in general wider spectra for minimum and mean temperature than for maximum and DTR, in both periods. On the contrary, the mainland ones do not show this pattern. Also, width from multifractal spectra of these two variables (minimum and mean temperature) is shortened in the last period in almost every case. To authors’ mind, changes in multifractal features might be related to the climate change experienced in the studied region. Furthermore, reduction of spectra width for minimum and mean temperature implies a decrease of the complexity of these temperature variables between both studied periods. Finally, the wider spectra found in coastal stations might be useful as a discriminator element to improve climate models