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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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1

Holdsworth, Amber M., Nicholas K. R. Kevlahan, and David J. D. Earn. "Multifractal signatures of infectious diseases." Journal of The Royal Society Interface 9, no. 74 (March 22, 2012): 2167–80. http://dx.doi.org/10.1098/rsif.2011.0886.

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Incidence of infection time-series data for the childhood diseases measles, chicken pox, rubella and whooping cough are described in the language of multifractals. We explore the potential of using the wavelet transform maximum modulus (WTMM) method to characterize the multiscale structure of the observed time series and of simulated data generated by the stochastic susceptible-exposed-infectious-recovered (SEIR) epidemic model. The singularity spectra of the observed time series suggest that each disease is characterized by a unique multifractal signature, which distinguishes that particular disease from the others. The wavelet scaling functions confirm that the time series of measles, rubella and whooping cough are clearly multifractal, while chicken pox has a more monofractal structure in time. The stochastic SEIR epidemic model is unable to reproduce the qualitative singularity structure of the reported incidence data: it is too smooth and does not appear to have a multifractal singularity structure. The precise reasons for the failure of the SEIR epidemic model to reproduce the correct multiscale structure of the reported incidence data remain unclear.

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2

Bakucz, Peter, and Gabor Kiss. "Modeling of probable maximum values in autonomous driving." SYSTEM THEORY, CONTROL AND COMPUTING JOURNAL 1, no. 2 (December 31, 2021): 58–64. http://dx.doi.org/10.52846/stccj.2021.1.2.28.

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In this paper, we approximate the probable maximum (very rare, extremal) values of highly autonomous driving sensor signals by reviewing two methods based on dynamic time series scaling and multifractal statistics.The article is a significantly revised and modified version of the conference material ("Determination of extreme values in autonomous driving based on multifractals and dynamic scaling") presented at the conference "2021 IEEE 15th International Symposium on Applied Computational Intelligence and Informatics, SACI". The method of dynamic scaling is originally derived from statistical physics and approximates the critical interface phenomena. The time series of the vibration signal of the corner radar can be considered as a fractal surface and grow appropriately for a given scale-inverse dynamic equation. In the second method we initiate, that multifractal statistics can be useful in searching for statistical analog time series that have a similar multifractal spectrum as the original sensor time series.

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3

KLEMENT, STEPHAN, KARL W. KRATKY, and JOHANN NITTMANN. "PRACTICAL TIME-SERIES ANALYSIS WITH MULTIFRACTAL METHODS." Fractals 01, no. 03 (September 1993): 735–43. http://dx.doi.org/10.1142/s0218348x93000770.

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Time-series data of various origins are studied by analyzing their corresponding multifractal f(α)-spectral which are obtained by use of the so-called canonical method. The classes of data samples under investigation include: (a) airborne particle count data taken from an industrial cleanroom environment; (b) data generated by use of a (pseudo-)random number generator; and (c) data resulting from the iteration of the logistic map for the value r=4.0 of the control parameter, thus exhibiting chaotic behavior. From the resulting multifractal spectra, typical features of the f(α)-curve can be identified in relation to the corresponding class of original data. These findings can be of interest for various purposes. One application under consideration is the processing of microcontamination particle data recorded in high-quality cleanrooms. These are of great importance to the increasing miniaturization of semiconductor devices. In processing microcontamination particle data, the multifractal analysis can help to extract significant information from an enormous number of data to compress these data into a reasonable quantity. Another interesting aspect can be found in using the multifractal spectrum as a possible instrument for estimating the quality and performance of a random number generator.

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4

Figueirêdo, P. H., E. Nogueira, M. A. Moret, and Sérgio Coutinho. "Multifractal analysis of polyalanines time series." Physica A: Statistical Mechanics and its Applications 389, no. 10 (May 2010): 2090–95. http://dx.doi.org/10.1016/j.physa.2009.11.045.

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5

Li, Yun Fa. "Application of Multifractal Statistics Method on Time Series." Applied Mechanics and Materials 556-562 (May 2014): 4559–62. http://dx.doi.org/10.4028/www.scientific.net/amm.556-562.4559.

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The fluctuations observed in tokamaks, stellarators and linear machines were similar with turbulent plasma in fusion devices, which were stochastic system, and the application of statistics method on them is studied in depth. First, the relating theories were summarized; Second, the mathematical model of the multifractal process is analyzed; Finally, the simulation on multifractal analysis of plasma turbulence and financial time series is carried out, results show that this method can be applied in time series effectively.

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6

Tzanis, Chris G., Ioannis Koutsogiannis, Kostas Philippopoulos, and Nikolaos Kalamaras. "Multifractal Detrended Cross-Correlation Analysis of Global Methane and Temperature." Remote Sensing 12, no. 3 (February 7, 2020): 557. http://dx.doi.org/10.3390/rs12030557.

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Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) was applied to time series of global methane concentrations and remotely-sensed temperature anomalies of the global lower and mid-troposphere, with the purpose of investigating the multifractal characteristics of their cross-correlated time series and examining their interaction in terms of nonlinear analysis. The findings revealed the multifractal nature of the cross-correlated time series and the existence of positive persistence. It was also found that the cross-correlation in the lower troposphere displayed more abundant multifractal characteristics when compared to the mid-troposphere. The source of multifractality in both cases was found to be mainly the dependence of long-range correlations on different fluctuation magnitudes. Multifractal Detrended Fluctuation Analysis (MF-DFA) was also applied to the time series of global methane and global lower and mid-tropospheric temperature anomalies to separately study their multifractal properties. From the results, it was found that the cross-correlated time series exhibit similar multifractal characteristics to the component time series. This could be another sign of the dynamic interaction between the two climate variables.

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7

Kalisky, Tomer, Yosef Ashkenazy, and Shlomo Havlin. "Volatility of fractal and multifractal time series." Israel Journal of Earth Sciences 56, no. 1 (December 1, 2007): 47–56. http://dx.doi.org/10.1560/ijes.56.1.47.

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8

Wang, Jeen-Hwa, and Chung-Wein Lee. "Multifractal Measures of Time Series of Earthquakes." Journal of Physics of the Earth 45, no. 5 (1997): 331–45. http://dx.doi.org/10.4294/jpe1952.45.331.

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9

Turiel, Antonio, and Conrad J. Pérez-Vicente. "Multifractal geometry in stock market time series." Physica A: Statistical Mechanics and its Applications 322 (May 2003): 629–49. http://dx.doi.org/10.1016/s0378-4371(02)01830-7.

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10

Xiong, Hui, and Pengjian Shang. "Weighted multifractal analysis of financial time series." Nonlinear Dynamics 87, no. 4 (November 10, 2016): 2251–66. http://dx.doi.org/10.1007/s11071-016-3187-1.

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11

Yu, Z. G., V. Anh, R. Eastes, and D. L. Wang. "Multifractal analysis of solar flare indices and their horizontal visibility graphs." Nonlinear Processes in Geophysics 19, no. 6 (November 29, 2012): 657–65. http://dx.doi.org/10.5194/npg-19-657-2012.

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Abstract. The multifractal properties of the daily solar X-ray brightness, Xl and Xs, during the period from 1 January 1986 to 31 December 2007 which includes two solar cycles are examined using the universal multifractal approach and multifractal detrended fluctuation analysis. Then we convert these time series into networks using the horizontal visibility graph technique. Multifractal analysis of the resulting networks is performed using an algorithm proposed by us. The results from the multifractal analysis show that multifractality exists in both raw daily time series of X-ray brightness and their horizontal visibility graphs. It is also found that the empirical K(q) curves of raw time series can be fitted by the universal multifractal model. The numerical results on the raw data show that the Solar Cycle 23 is weaker than the Solar Cycle 22 in multifractality. The values of h(2) from multifractal detrended fluctuation analysis for these time series indicate that they are stationary and persistent, and the correlations in the time series of Solar Cycle 23 are stronger than those for Solar Cycle 22. Furthermore, the multifractal scaling for the networks of the time series can reflect some properties which cannot be picked up by using the same analysis on the original time series. This suggests a potentially useful method to explore geophysical data.

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12

MALI, P. "FLUCTUATION OF GOLD PRICE IN INDIA VERSUS GLOBAL CONSUMER PRICE INDEX." Fractals 22, no. 01n02 (March 2014): 1450004. http://dx.doi.org/10.1142/s0218348x14500042.

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The time series of gold price in the Indian market and the global consumer price index for the period of January 1985 to June 2013 are analyzed in terms of the multifractal detrended fluctuation analysis (MF-DFA). Multifractal variables, such as the generalized Hurst exponent, the multifractal mass exponent, the singularity spectrum, are extracted for both the series. Special emphasis is given on the possible source(s) of correlations in these series. The multifractal results are fitted to the generalized binomial multifractal model consists of only two parameters. Our analysis show that the multifractal nature of the Indian gold market time series and the global consumer price index series is due to both the long-range temporal correlation and the fat-tailed probability density function of the values. Surprisingly, the series are well described by the two-parameter binomial multifractal model used.

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13

Cabrera-Brito, Laura, German Rodriguez, Luis García-Weil, Mercedes Pacheco, Esther Perez, and Joanna J. Waniek. "Fractal Analysis of Deep Ocean Current Speed Time Series." Journal of Atmospheric and Oceanic Technology 34, no. 4 (April 2017): 817–27. http://dx.doi.org/10.1175/jtech-d-16-0098.1.

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AbstractFractal properties of deep ocean current speed time series, measured at a single-point mooring on the Madeira Abyssal Plain at 1000- and 3000-m depth, are explored over the range between one week and 5 years, by using the detrended fluctuation analysis and multifractal detrended fluctuation analysis methodologies. The detrended fluctuation analysis reveals the existence of two subranges with different scaling behaviors. Long-range temporal correlations following a power law are found in the time-scale range between approximately 50 days and 5 years, while a Brownian motion–type behavior is observed for shorter time scales. The multifractal analysis approach underlines a multifractal structure whose intensity decreases with depth. The analysis of the shuffled and surrogate versions of the original time series shows that multifractality is mainly due to long-range correlations, although there is a weak nonlinear contribution at 1000-m depth, which is confirmed by the detrended fluctuation analysis of volatility time series.

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14

DOĞANGÜN, ITIR, and GAZANFER ÜNAL. "MULTIFRACTAL BEHAVIOR IN PRECIOUS METALS: WAVELET COHERENCY AND FORECASTING BY VARIMA AND V-FARIMA MODELS." Annals of Financial Economics 14, no. 02 (April 21, 2019): 1950006. http://dx.doi.org/10.1142/s2010495219500064.

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We introduce a new approach to improve the forecasting performance by investigating the multifractal features and the dynamic correlations of return on spot prices of precious metals, namely, gold and platinum. The Hölder exponent of multifractal time series is employed to detect the critical fluctuations during the financial crises through measuring the multifractal behavior. We also consider co-movement of Hölder exponents and forecast the Hölder exponents of multifractal precious metal time series on coherent time periods. The results indicate that forecasting of multiple wavelet coherence of Hölder exponents of multifractal precious metal time series is efficiently improved by using Vector FARIMA and VARIMA models.

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15

Jiang, Chuxuan, Priya Dev, and Ross A. Maller. "A Hypothesis Test Method for Detecting Multifractal Scaling, Applied to Bitcoin Prices." Journal of Risk and Financial Management 13, no. 5 (May 20, 2020): 104. http://dx.doi.org/10.3390/jrfm13050104.

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Multifractal processes reproduce some of the stylised features observed in financial time series, namely heavy tails found in asset returns distributions, and long-memory found in volatility. Multifractal scaling cannot be assumed, it should be established; however, this is not a straightforward task, particularly in the presence of heavy tails. We develop an empirical hypothesis test to identify whether a time series is likely to exhibit multifractal scaling in the presence of heavy tails. The test is constructed by comparing estimated scaling functions of financial time series to simulated scaling functions of both an iid Student t-distributed process and a Brownian Motion in Multifractal Time (BMMT), a multifractal processes constructed in Mandelbrot et al. (1997). Concavity measures of the respective scaling functions are estimated, and it is observed that the concavity measures form different distributions which allow us to construct a hypothesis test. We apply this method to test for multifractal scaling across several financial time series including Bitcoin. We observe that multifractal scaling cannot be ruled out for Bitcoin or the Nasdaq Composite Index, both technology driven assets.

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16

WANG, JING, PENGJIAN SHANG, and WEIJIE GE. "MULTIFRACTAL CROSS-CORRELATION ANALYSIS BASED ON STATISTICAL MOMENTS." Fractals 20, no. 03n04 (September 2012): 271–79. http://dx.doi.org/10.1142/s0218348x12500259.

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We introduce a new method, multifractal cross-correlation analysis based on statistical moments (MFSMXA), to investigate the long-term cross-correlations and cross-multifractality between time series generated from complex system. Efficiency of this method is shown on multifractal series, comparing with the well-known multifractal detrended cross-correlation analysis (MFXDFA) and multifractal detrending moving average cross-correlation analysis (MFXDMA). We further apply this method on volatility time series of DJIA and NASDAQ indices, and find some interesting results. The MFSMXA has comparative performance with MFXDMA and sometimes perform slightly better than MFXDFA. Multifractal nature exists in volatility series. In addition, we find that the cross-multifractality of volatility series is mainly due to their cross-correlations, via comparing the MFSMXA results for original series with those for shuffled series.

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17

Livi, Lorenzo, Enrico Maiorino, Antonello Rizzi, and Alireza Sadeghian. "On the Long-Term Correlations and Multifractal Properties of Electric Arc Furnace Time Series." International Journal of Bifurcation and Chaos 26, no. 01 (January 2016): 1650007. http://dx.doi.org/10.1142/s0218127416500073.

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In this paper, we study long-term correlations and multifractal properties elaborated from time series of three-phase current signals from an industrial electric arc furnace. Implicit sinusoidal trends are suitably detected by considering the scaling of the fluctuation functions. Time series are then filtered via a Fourier-based analysis to remove such strong periodicities. In the filtered time series we detected long-term, positive correlations. The presence of positive correlations is in agreement with the typical V–I characteristic (hysteresis) of the electric arc furnace, thus providing a sound physical justification for the memory effects found in the current time series. The multifractal signature is strong enough in the filtered time series to be effectively classified as multifractal.

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18

Kojić, Milena, Petar Mitić, Marko Dimovski, and Jelena Minović. "Multivariate Multifractal Detrending Moving Average Analysis of Air Pollutants." Mathematics 9, no. 7 (March 25, 2021): 711. http://dx.doi.org/10.3390/math9070711.

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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.

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19

YUAN, XIAOHUI, BIN JI, YANBIN YUAN, YUEHUA HUANG, XIANSHAN LI, and WENWU LI. "MULTIFRACTAL DETRENDED FLUCTUATION ANALYSIS OF ELECTRIC LOAD SERIES." Fractals 23, no. 02 (May 28, 2015): 1550010. http://dx.doi.org/10.1142/s0218348x15500103.

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Multifractal detrended fluctuation analysis (MF-DFA) method is applied to analyze the daily electric load time series. The results of the MF-DFA show that there are three crossover timescales at seven days, 15 days and 365 days approximately in the fluctuation function. Also we find that these fluctuations have multifractal nature with long range correlation behavior. The multifractal singularity spectrum of the daily electric load series has been fitted by the quadratic function model. Comparing the MF-DFA results of the original load series with those of shuffled and surrogate series, it concludes that the multifractal characteristics of the daily electric load time series is due to both broadness of the probability density function and long-range correlation, and the long-range correlation is dominant.

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20

YALAMOVA, ROSSITSA. "EMPIRICAL TESTING OF MULTIFRACTALITY OF FINANCIAL TIME SERIES BASED ON WTMM." Fractals 17, no. 03 (September 2009): 323–32. http://dx.doi.org/10.1142/s0218348x09004508.

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The multifractal spectrum calculated with wavelet transform modulus maxima (WTMM) provides information on the higher moments of market returns distribution and the multiplicative cascade of volatilities. This paper applies a wavelet based methodology for calculation of the multifractal spectrum of financial time series. WTMM methodology provides a better measure of risk changes compared to the structure function approach. It is well founded in applied mathematics and physics with little popularity among finance researchers.

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21

Karperien, Audrey L., Herbert F. Jelinek, and Helmut Ahammer. "Multifractal formalism in image and time series analysis." Banach Center Publications 109 (2016): 23–45. http://dx.doi.org/10.4064/bc109-0-3.

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22

Shimizu, Yu, Markus Barth, Christian Windischberger, Ewald Moser, and Stefan Thurner. "Wavelet-based multifractal analysis of fMRI time series." NeuroImage 22, no. 3 (July 2004): 1195–202. http://dx.doi.org/10.1016/j.neuroimage.2004.03.007.

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23

Kantelhardt, Jan W., Stephan A. Zschiegner, Eva Koscielny-Bunde, Shlomo Havlin, Armin Bunde, and H. Eugene Stanley. "Multifractal detrended fluctuation analysis of nonstationary time series." Physica A: Statistical Mechanics and its Applications 316, no. 1-4 (December 2002): 87–114. http://dx.doi.org/10.1016/s0378-4371(02)01383-3.

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24

Bacry, E., J. Delour, and J. F. Muzy. "Modelling financial time series using multifractal random walks." Physica A: Statistical Mechanics and its Applications 299, no. 1-2 (October 2001): 84–92. http://dx.doi.org/10.1016/s0378-4371(01)00284-9.

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25

Drożdż, S., J. Kwapień, P. Oświecimka, and R. Rak. "Quantitative features of multifractal subtleties in time series." EPL (Europhysics Letters) 88, no. 6 (December 1, 2009): 60003. http://dx.doi.org/10.1209/0295-5075/88/60003.

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26

García-Marín, A. P., J. Estévez, F. J. Jiménez-Hornero, and J. L. Ayuso-Muñoz. "Multifractal analysis of validated wind speed time series." Chaos: An Interdisciplinary Journal of Nonlinear Science 23, no. 1 (March 2013): 013133. http://dx.doi.org/10.1063/1.4793781.

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27

Movahed, M. Sadegh, G. R. Jafari, F. Ghasemi, Sohrab Rahvar, and M. Reza Rahimi Tabar. "Multifractal detrended fluctuation analysis of sunspot time series." Journal of Statistical Mechanics: Theory and Experiment 2006, no. 02 (February 9, 2006): P02003. http://dx.doi.org/10.1088/1742-5468/2006/02/p02003.

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28

Lee, Hojin, and Woojin Chang. "Multifractal regime detecting method for financial time series." Chaos, Solitons & Fractals 70 (January 2015): 117–29. http://dx.doi.org/10.1016/j.chaos.2014.11.006.

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29

MARTINO, WILLIAM, та MICHAEL FRAME. "MULTIFRACTAL MEASURES OF TIME SERIES: f(α) SURFACES". International Journal of Bifurcation and Chaos 20, № 08 (серпень 2010): 2453–70. http://dx.doi.org/10.1142/s0218127410027106.

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Binning the data points of a time series and associating a contraction map with each bin gives rise to a driven IFS representation of the time series. Varying the bins changes the driven IFS, sometimes in complex ways difficult to parse. From the transition matrix for any particular binning we can plot an f(α) curve. Assembling these curves as the bins change gives a surface, which we call the f(α) surface. We use properties of this surface to investigate time series from iterating logistic and tent maps, and also time series of financial data.

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30

DAI, MEIFENG, SHUXIANG SHAO, JIANYU GAO, YU SUN, and WEIYI SU. "MIXED MULTIFRACTAL ANALYSIS OF CRUDE OIL, GOLD AND EXCHANGE RATE SERIES." Fractals 24, no. 04 (December 2016): 1650046. http://dx.doi.org/10.1142/s0218348x16500468.

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The multifractal analysis of one time series, e.g. crude oil, gold and exchange rate series, is often referred. In this paper, we apply the classical multifractal and mixed multifractal spectrum to study multifractal properties of crude oil, gold and exchange rate series and their inner relationships. The obtained results show that in general, the fractal dimension of gold and crude oil is larger than that of exchange rate (RMB against the US dollar), reflecting a fact that the price series in gold and crude oil are more heterogeneous. Their mixed multifractal spectra have a drift and the plot is not symmetric, so there is a low level of mixed multifractal between each pair of crude oil, gold and exchange rate series.

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31

Zhang, Xike, Gui Zhang, Luo Qiu, Bo Zhang, Yurong Sun, Zifan Gui, and Qiuwen Zhang. "A Modified Multifractal Detrended Fluctuation Analysis (MFDFA) Approach for Multifractal Analysis of Precipitation in Dongting Lake Basin, China." Water 11, no. 5 (April 28, 2019): 891. http://dx.doi.org/10.3390/w11050891.

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Multifractal detrended fluctuation analysis (MFDFA) method can examine higher-dimensional fractal and multifractal characteristics hidden in time series. However, removal of local trends in MFDFA is based on discontinuous polynomial fitting, resulting in pseudo-fluctuation errors. In this paper, we propose a two-stage modified MFDFA for multifractal analysis. First, an overlap moving window (OMW) algorithm is introduced to divide time series of the classic MFDFA method. Second, detrending by polynomial fitting local trend in traditional MFDFA is replaced by ensemble empirical mode decomposition (EEMD)-based local trends. The modified MFDFA is named OMW-EEMD-MFDFA. Then, the performance of the OMW-EEMD-MFDFA method is assessed by extensive numeric simulation experiments based on a p-model of multiplicative cascading process. The results show that the modified OMW-EEMD-MFDFA method performs better than conventional MFDFA and OMW-MFDFA methods. Lastly, the modified OMW-EEMD-MFDFA method is applied to explore multifractal characteristics and multifractal sources of daily precipitation time series data at the Mapoling and Zhijiang stations in Dongting Lake Basin. Our results showed that the scaling properties of the daily precipitation time series at the two stations presented a long-range correlation, showing a long-term persistence of the previous state. The strong q-dependence of H ( q ) and τ ( q ) indicated strong multifractal characteristics in daily precipitation time series data at the two stations. Positive Δ f values demonstrate that precipitation may have a local increasing trend. Comparing the generalized Hurst exponent and the multifractal strength of the original precipitation time series data with its shuffled and surrogate time series data, we found that the multifractal characteristics of the daily precipitation time series data were caused by both long-range correlations between small and large fluctuations and broad probability density function, but the broad probability density function was dominant. This study may be of practical and scientific importance in regional precipitation forecasting, extreme precipitation regulation, and water resource management in Dongting Lake Basin.

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32

Schmitt, F., D. Schertzer, S. Lovejoy, and Y. Brunet. "Empirical study of multifractal phase transitions in atmospheric turbulence." Nonlinear Processes in Geophysics 1, no. 2/3 (September 30, 1994): 95–104. http://dx.doi.org/10.5194/npg-1-95-1994.

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Abstract. We study atmospheric wind turbulence in the framework of universal multifractals, using several medium resolution (10 Hz) time series. We cut these original time series into 704 scale invariant realizations. We then compute the moment scaling exponent of the energy flux K(q) for 4 and 704 realizations, in order to study qualitative difference between strong and weak events associated with multifractal phase transitions. We detect a first order multifractal phase transition of the energy flux at statistical moment of order qD ≈ 2.4 ± 0.2: this means that when the number of realizations increases, moments order q ≥; qD diverge. These results are confirmed by the study of probability distributions, and wind structure functions. A consequence of these findings is that it is no use to compare different cascade models in turbulence by using the high order wind structure functions, because a linear part will always be encountered for high enough order moments. Another important implication for multifractal studies of turbulence is that the asymptotic slope of the scaling moment function is purely a function of sample size and diverges with it; it implies the same for D∞, which has often be considered as finite.

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Huang, Jingjing, and Danlei Gu. "Multiscale Multifractal Detrended Cross-Correlation Analysis of High-Frequency Financial Time Series." Fluctuation and Noise Letters 18, no. 03 (July 16, 2019): 1950014. http://dx.doi.org/10.1142/s0219477519500147.

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In order to obtain richer information on the cross-correlation properties between two time series, we introduce a method called multiscale multifractal detrended cross-correlation analysis (MM-DCCA). This method is based on the Hurst surface and can be used to study the non-linear relationship between two time series. By sweeping through all the scale ranges of the multifractal structure of the complex system, it can present more information than the multifractal detrended cross-correlation analysis (MF-DCCA). In this paper, we use the MM-DCCA method to study the cross-correlations between two sets of artificial data and two sets of 5[Formula: see text]min high-frequency stock data from home and abroad. They are SZSE and SSEC in the Chinese market, and DJI and NASDAQ in the US market. We use Hurst surface and Hurst exponential distribution histogram to analyze the research objects and find that SSEC, SZSE and DJI, NASDAQ all show multifractal properties and long-range cross-correlations. We find that the fluctuation of the Hurst surface is related to the positive and negative of [Formula: see text], the change of scale range, the difference of national system, and the length of time series. The results show that the MM-DCCA method can give more abundant information and more detailed dynamic processes.

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Olsson, J. "Limits and characteristics of the multifractal behaviour of a high-resolution rainfall time series." Nonlinear Processes in Geophysics 2, no. 1 (March 31, 1995): 23–29. http://dx.doi.org/10.5194/npg-2-23-1995.

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Abstract. The multifractal properties of a 2-year time series of 8-min rainfall intensity observations are investigated. The empirical probability distribution function suggests a hyperbolic intermittency with divergence of moment of order greater than 2. The power spectrum E(f) of the series obeys a power law form E(f)=f -0.66 in the range of scales 8 min to approximately 3 days. The variation of the average statistical moments with scale shows that the series is characterized by a multifractal behaviour between 8 min and approximately 11 days. The multifractal parameters associated with universality were estimated to be α=0.63 and C1=0,44 by using the Double Trace Moment, DTM, technique. The moment scaling functions obtained from the empirical values and the universal expression are in good agreement in the approximate range 1≤q≤3. Outside of this range, however, differences exist which may be related to either limitations of the data or an inexact estimation of the parameters by DTM. The evident multifractal nature of rainfall time series is encouraging since it may lead to new and improved ways of processing rainfall data used in hydrological calculations.

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Quang Dang Khoa, Truong, and Vo Van Toi. "Multifractals Properties on the Near Infrared Spectroscopy of Human Brain Hemodynamic." Mathematical Problems in Engineering 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/670761.

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Nonlinear physics presents us with a perplexing variety of complicated fractal objects and strange sets. Naturally one wishes to characterize the objects and describe the events occurring on them. Moreover, most time series found in “real-life” applications appear quite noisy. Therefore, at almost every point in time, they cannot be approximated either by the Taylor series or by the Fourier series of just a few terms. Many experimental time series have fractal features and display singular behavior, the so-called singularities. The multifractal spectrum quantifies the degree of fractals in the processes generating the time series. A novel definition is proposed called full-width Hölder exponents that indicate maximum expansion of multifractal spectrum. The obtained results have demonstrated the multifractal structure of near-infrared spectroscopy time series and the evidence for brain imagery activities.

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de Montera, L., L. Barthès, C. Mallet, and P. Golé. "The Effect of Rain–No Rain Intermittency on the Estimation of the Universal Multifractals Model Parameters." Journal of Hydrometeorology 10, no. 2 (April 1, 2009): 493–506. http://dx.doi.org/10.1175/2008jhm1040.1.

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Abstract The multifractal properties of rain are investigated within the framework of universal multifractals. The database used in this study includes measurements performed over several months in different locations by means of a disdrometer, the dual-beam spectropluviometer (DBS). An assessment of the effect of the rain–no rain intermittency shows that the analysis of rain-rate time series may lead to a spurious break in the scaling and to erroneous parameters. The estimation of rain multifractal parameters is, therefore, performed on an event-by-event basis, and they are found to be significantly different from those proposed in scientific literature. In particular, the parameter H, which has often been estimated to be 0, is more likely to be 0.53, thus meaning that rain is a fractionally integrated flux (FIF). Finally, a new model is proposed that simulates high-resolution rain-rate time series based on these new parameters and on a simple threshold.

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Aguilar-Molina, Ana María, Fernando Angulo-Brown, and Alejandro Muñoz-Diosdado. "Multifractal Spectrum Curvature of RR Tachograms of Healthy People and Patients with Congestive Heart Failure, a New Tool to Assess Health Conditions." Entropy 21, no. 6 (June 11, 2019): 581. http://dx.doi.org/10.3390/e21060581.

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We calculate the multifractal spectra of heartbeat RR-interval time series (tachograms) of healthy subjects and patients with congestive heart failure (CHF). From these time series, we obtained new subseries of 6 h durations when healthy persons and patients were asleep and awake respectively. For each time series and subseries, we worked out the multifractal spectra with the Chhabra and Jensen method and found that their graphs have different shapes for CHF patients and healthy persons. We suggest to measure two parameters: the curvature around the maximum and the symmetry for all these multifractal spectra graphs, because these parameters were different for healthy and CHF subjects. Multifractal spectra of healthy subjects tend to be right skewed especially when the subjects are asleep and the curvature around the maximum is small compared with the curvature around the maximum of the CHF multifractal spectra; that is, the spectra of patients tend to be more pointed around the maximum. In CHF patients, we also have encountered differences in the curvature of the multifractal spectra depending on their respective New York Heart Association (NYHA) index.

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ZHAO, XIAOJUN, PENGJIAN SHANG, and QIUYUE JIN. "MULTIFRACTAL DETRENDED CROSS-CORRELATION ANALYSIS OF CHINESE STOCK MARKETS BASED ON TIME DELAY." Fractals 19, no. 03 (September 2011): 329–38. http://dx.doi.org/10.1142/s0218348x11005415.

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Multifractal detrended cross-correlation analysis (MF-DXA) has been developed to detect the long-range power-law cross-correlation of two simultaneous series. However, the synchronization of underlying data can not be guaranteed integrated by a variety of factors. We artificially imbed a time delay in considered series and study its influence on the multifractal cross-correlation analysis. Time delay is found to affect the multifractal characterization, where a larger time delay causes a weaker multifractality. We also propose an alternative modification on MF-DXA to make the process more robust. The logarithmic return and volatility of Chinese stock indices show cross-correlation scaling behavior and strong multifractality by MF-DXA as well as singularity spectrum analysis.

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39

HUANG, ZHENG-WEN, CHUN-QIONG LIU, KAI SHI, and BIN ZHANG. "MONOFRACTAL AND MULTIFRACTAL SCALING ANALYSIS OF pH TIME SERIES FROM DONGTING LAKE INLET AND OUTLET." Fractals 18, no. 03 (September 2010): 309–17. http://dx.doi.org/10.1142/s0218348x10004981.

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The water pH series from Dongting Lake Inlet and Outlet in China are analyzed by detrended fluctuation analysis (DFA), spectral analysis and multifractal methods. The results show that these pH series are characterized by long-term memory, 1/f noise and multifractal scaling, and these characteristics have obvious difference between the Lake Inlet and Outlet. The comparison results show that monofractal (DFA exponent) and multifractal (Δα, Δf, B) parameters can be quantitative dynamical indexes reflecting the capability of anti-acidification of Dongting Lake. Furthermore, we investigate the frequency-size distribution of pH series from Dongting Lake Inlet and Outlet. Our findings suggest that water pH is an example of a self-organized criticality (SOC) process. Based on concept of self-organized ctiticality, we analysis the cause that different scale-free power-law behavior between pH series from Dongting Lake Inlet and Outlet. This work can be helpful to improvement of modeling of lake water quality.

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40

Jizba, P., and J. Korbel. "Techniques for multifractal spectrum estimation in financial time series." International Journal of Design & Nature and Ecodynamics 10, no. 3 (September 30, 2015): 261–66. http://dx.doi.org/10.2495/dne-v10-n3-261-266.

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41

Wang, Qizhen. "Multifractal characterization of air polluted time series in China." Physica A: Statistical Mechanics and its Applications 514 (January 2019): 167–80. http://dx.doi.org/10.1016/j.physa.2018.09.065.

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Fan, Qingju, Shuanggui Liu, and Kehao Wang. "Multiscale multifractal detrended fluctuation analysis of multivariate time series." Physica A: Statistical Mechanics and its Applications 532 (October 2019): 121864. http://dx.doi.org/10.1016/j.physa.2019.121864.

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Huang, Jingjing, and Pengjian Shang. "Multiscale multifractal diffusion entropy analysis of financial time series." Physica A: Statistical Mechanics and its Applications 420 (February 2015): 221–28. http://dx.doi.org/10.1016/j.physa.2014.11.009.

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Ihlen, Espen A. F. "Multifractal analyses of response time series: A comparative study." Behavior Research Methods 45, no. 4 (March 23, 2013): 928–45. http://dx.doi.org/10.3758/s13428-013-0317-2.

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45

Zhou, Fang-Xin, Sheng Wang, Guo-Sheng Han, Shan Jiang, and Zu-Guo Yu. "Randomized multifractal detrended fluctuation analysis of long time series." Chaos: An Interdisciplinary Journal of Nonlinear Science 30, no. 5 (May 2020): 053113. http://dx.doi.org/10.1063/1.5139620.

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46

Wesfreid, Eva, Véronique L. Billat, and Yves Meyer. "Multifractal analysis of heartbeat time series in human races." Applied and Computational Harmonic Analysis 18, no. 3 (May 2005): 329–35. http://dx.doi.org/10.1016/j.acha.2004.12.005.

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47

Sadegh Movahed, M., G. R. Jafari, F. Ghasemi, Sohrab Rahvar, and M. Reza Rahimi Tabar. "Erratum: Multifractal detrended fluctuation analysis of sunspot time series." Journal of Statistical Mechanics: Theory and Experiment 2011, no. 09 (September 22, 2011): E09001. http://dx.doi.org/10.1088/1742-5468/2011/09/e09001.

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48

Yang, XiaoDong, AiJun He, Yong Zhou, and XinBao Ning. "Multifractal mass exponent spectrum of complex physiological time series." Chinese Science Bulletin 55, no. 19 (July 2010): 1996–2003. http://dx.doi.org/10.1007/s11434-010-3276-3.

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49

Kalamaras, N., K. Philippopoulos, D. Deligiorgi, C. G. Tzanis, and G. Karvounis. "Multifractal scaling properties of daily air temperature time series." Chaos, Solitons & Fractals 98 (May 2017): 38–43. http://dx.doi.org/10.1016/j.chaos.2017.03.003.

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50

FAN, QINGJU, SHUANGGUI LIU, and KEHAO WANG. "DETECTING THE AUTO-CORRELATION BETWEEN DAILY TEMPERATURE AND RELATIVE HUMIDITY TIME SERIES." Fractals 27, no. 02 (March 2019): 1950003. http://dx.doi.org/10.1142/s0218348x19500038.

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Multifractal detrended fluctuation analysis (MF-DFA) for bivariate series has been used to study the auto-correlation between temperature and relative humidity series in Wuhan city, China. The results show that long-range persistence auto-correlation exists between the temperature and relative humidity series and the auto-correlation has multifractal characteristics. For the two climate records, the contribution of single series to multifractality is analyzed by utilizing chi square ([Formula: see text]) test. By comparing the chi square test statistics of original series with those of shuffled and surrogate series, we conclude that the relative humidity is more responsible for the multifractality due to its long-range correlation, and the temperature and relative humidity series almost have the same degree of contributions to the multifractality due to a fatness of probability density function (PDF) correlation. On the whole, the relative humidity series has dominant effect in the auto-correlation.

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