Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The time series multifractal ( upper panel ), monofractal ( middle panel ), and whitenoise ( lower panel ) with zero average ( red dashed lines ) and ±1 RMS ( red solid lines ) . All time series have equal RMS = 1, but quite different structure. RMS is only sensitive to differences in the amplitude of variation and not differences in the structure of variation. Notice the different scaling for the vertical axis of the multifractal time series.

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The computation of local fluctuations, RMS{1} , around linear (A), quadratic (B), and cubic trends (C) by Matlab code ( m = 1 , m = 2 , and m = 3 , respectively) . The red dashed line is the fitted trend, fit{v} , within eight segments of sample size 1000. The distance between the red dashed trend and the solid red lines represents ±1 RMS{1} . The local fluctuation, RMS{1} , around trends is the basic “building block” of the detrended fluctuation analysis .

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques: Blocking Assay

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The plot of overall RMS (i.e., F in Matlab code ) versus the segment sample size (i.e., scale in Matlab code ) where both F and scale are represented in log-coordinates . The scale invariant relation is indicated by the slope, H , of the regression lines, RegLine , computed by Matlab code . The slope, H , is a power law exponent called the Hurst exponent because F and scale are represented in log-coordinates. Notice that both the monofractal and multifractal time series have more apparent slow fluctuations compared to whitenoise indicated by larger amplitudes of the overall RMS on the larger scales.

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: q -order RMS Fq(nq,:) and corresponding regression line qRegLine{nq} computed by MFDFA (i.e., Matlab code and ) for time series multifractal (A), monofractal (B), and whitenoise (C) . (A) The scaling functions Fq ( blue dots ) and corresponding regression slopes Hq ( blue dashed lines ) are q -dependent. (B,C) The scaling functions Fq ( red and turqouish dots ) and regression slope Hq ( red and turqouish dashed lines ) are q -independent. (D) The q -order Hurst exponent Hq for the time series multifractal ( blue trace ), monofractal ( red trace ) and whitenoise ( turqouish trace ) where the colored dots represents the slopes Hq for q = −3, −1, 1, and 3 illustrated in (A–C) . Notice that the intercept of Hq for multifractal and monofractal time series [intercept between blue and red trace in (D) ] are close to q = 2. This intercept reflects the similarity between their overall RMS, F , in Figure .

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: Multiple representations of multifractal spectrum for multifractal ( blue traces ), monofractal ( red traces ), and whitenoise ( turquoise trace ) time series . (A) q -order Hurst exponent Hq computed in Matlab code . (B) Mass exponent tq computed in Matlab code . (C) Multifractal spectrum of Dq and hq ( upper right panels ) computed in Matlab code and plotted against each other. The arrow indicates the difference between the maximum and minimum hq that are called the multifractal spectrum width. Notice that the constant Hq for monofractal and whitenoise times series leads to a linear tq that further leads to a constant hq and Dq that, finally, are joined to become only tiny arcs in (C) .

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: Illustration of a continuum of multifractal time series with the same q -order Hurst exponent for q = 2 but with different multifractal spectrum width [compare vertical axis of the (A) and the arrow in the (B)] . Notice the growth of structural differences between the periods with small and large fluctuations as the multifractal spectrum width become larger.

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: Illustration of multifractal spectra with a right truncation ( upper right panel ) and a left truncation ( upper left panel ) . These truncations originate from the leveling of the q -order Hurst exponents for negative q ’s ( upper right panel ) and positive q ’s ( upper left panel ), respectively.

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: (A) A summary of how the local Hurst exponent Ht is estimated in Matlab code . The regression line Regfit (red center line) is the center of the spread of local RMS in log-coordinates and is equal to the regression line qRegLine{q }= = 0 in Matlab code and [see (B) ]. The minimum and maximum local Hurst exponent Ht(5,:) is the slope of the upper and lower red lines, respectively, that converge from the maximum and minimum of RMS{5} onto the regression line Regfit at the maximum scale maxL . Consequently, the local Hurst exponent Ht(ns,:) estimated by dividing the residual resRMS{ns}(v) for each time instant v by logscale(ns) (i.e., the difference between the maximal scale maxL and scale(ns) in log-coordinates) and adding the slope Hq q=0 of the regression line Regfit . (B) The scaling function Fq (blue dots) and the regression lines qRegLine{nq} (blue lines) computed by Matlab code and . All Fq lies within the envelope between the red lines for the maximum and minimum Ht(5,:) , but does not cover the entire range in the same way as the local RMS{5} in (A) . (C) The smallest scales used to compute the local Hurst exponents and the multifractal spectrum illustrated in Figure . The red dots represent the maximum RMS{ns}(1080) and minimum RMS{ns}(1199) for multiple segment sample sizes [i.e., scale(ns) ] at time instant v = 1080 and v=1199 , respectively [see Figure A] whereas blue dots represent the local fluctuations for 30 other time instants. Notice that both the horizontal and vertical axes in all panels are in log-coordinates.

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: (A) The multifractal , monofractal, and whitenoise time series (upper panel) and their local Hurst exponents Ht(:,5) computed by Matlab code (lower panel). The multifractal time series have a larger variation in the local Hurst exponents Ht(5,:) compared with the monofractal and whitenoise time series. The period with the local fluctuation of the smallest magnitude in multifractal time series contains the maximum Ht(5,:) (see Ht max in period between the black dashed lines ) whereas the period with the local fluctuation of the largest magnitudes contains the smallest Ht(5,:) (see Ht min in the period between red dashed lines ). (B) The probability distribution Ph of the local Hurst exponents Ht estimated as histograms by Matlab code for the multifractal , monofractal, and whitenoise time series. (C) The multifractal spectrum Dh estimated from distribution Ph by Matlab code for the same time series. The distribution Ph and spectrum Dh have a larger width for the multifractal time series compared to the monofractal and whitenoise time series.

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques:

Journal: Frontiers in Physiology

Article Title: Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

doi: 10.3389/fphys.2012.00141

Figure Lengend Snippet: The time series multifractal (upper panel), monofractal (middle panel), and whitenoise (lower panel) are shown as blue traces . They are examples of noise like time series used in the present tutorial. All time series contain 8000 data samples each where the sample numbers are indicated by the horizontal axis. Matlab code converts the noises ( blue traces ) to random walks ( red traces ) that have a picture-in-picture similarity (subplot in the upper panel). Notice that the time series multifractal has distinct periods with small and large fluctuations in contrast to time series monofractal and whitenoise . The aim of this section is to introduce MFDFA that quantify the structure of fluctuations within the periods with small and large fluctuations.

Article Snippet: MFDFA is simply based on the computation of local RMS for multiple segment sizes as illustrated in Section “ Multifractal Detrended Fluctuation Analysis in Matlab .” However, special issues in Section “ The Best Practice of Multifractal Detrended Fluctuation Analysis ” for the best practice of MFDFA are of paramount importance when MFDFA are employed to biomedical time series.

Techniques: Introduce

Journal: GeroScience

Article Title: Lifelong longitudinal assessment of the contribution of multi-fractal fluctuations to heart rate and heart rate variability in aging mice: role of the sinoatrial node and autonomic nervous system

doi: 10.1007/s11357-024-01267-0

Figure Lengend Snippet: Multi-scale multi-fractal detrended-fluctuation analysis of heart rate variability in aging mice. A Representative plots of overall RMS (Fq(s)) as a function of scale (in seconds) for different orders (q = − 5, − 3, − 1, 0, 1, 3, 5) at 3 month intervals in an aging mouse. B-I Phase shift surrogate analysis for a representative mouse at 6 months and 27 months of age in baseline conditions and after ANS blockade. Surface plots for MSMFDFA scale exponents of the original RR interval time series at 6 months of age in baseline conditions (B), at 6 months of age after ANS blockade (C), at 27 months of age in baseline conditions (D), and at 27 months of age after ANS blockade (E). Surface plots for MSMFDFA scale exponents after performing a 100 Fourier phase shuffled time series of the original RR interval time series (as shown in B-E) at 6 months of age in baseline conditions (F), at 6 months of age after ANS blockade (G), at 27 months of age in baseline conditions (H), and at 27 months of age after ANS blockade (I)

Article Snippet: We have recently developed and validated the use of multi-scale multi-fractal detrended fluctuation analysis (MSMFDFA) as a complexity based method to jointly assess multi-scale and multi-fractal nonlinear dynamics of HR in normal, healthy wildtype mice [ 24 ].

Techniques:

Journal: Theoretical Ecology

Article Title: Theory of early warning signals of disease emergenceand leading indicators of elimination

doi: 10.1007/s12080-013-0185-5

Figure Lengend Snippet: Performance of the statistics over a moving window for the SIS system approaching elimination, assuming that immigration occurs ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta >0$\end{document} ). Panels a , c , and e show the median statistics ( thick lines ) and 95 % prediction intervals ( shaded regions ). The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from van Kampen and Gaussian detrending. The coefficient of variation (CV) is marked in dashed green line to indicate that it was calculated from the raw time series, not the deviations from the mean. The dashed vertical line marks the time of the transcritical bifurcation in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon \rightarrow 0$\end{document} limiting case with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta =0$\end{document} . Panels b , d , and f show the corresponding ROC curves. The AUC value indicates the area under the corresponding ROC curve. All curves are above the black line , showing that the indicators behave better than chance in distinguishing between realizations that have been generated by the null and test models. However, the Gaussian filtering may not be an appropriate method to obtain the fluctuations, because it does not remove the slowly varying trend close to the transition, as exhibited by the rapid rise in the indicators. All calculations used the parameter values in Table . A bandwidth of 20 years was chosen for the Gaussian filtering

Article Snippet: The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from van Kampen and Gaussian detrending.

Techniques: Generated

Journal: Theoretical Ecology

Article Title: Theory of early warning signals of disease emergenceand leading indicators of elimination

doi: 10.1007/s12080-013-0185-5

Figure Lengend Snippet: Performance of the statistics over a moving window for the SIR system approaching elimination, assuming that immigration occurs ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta >0$\end{document} ). Panels a , c , and e show the median statistics ( thick lines ) and 95 % prediction intervals ( shaded regions ). The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from van Kampen and Gaussian detrending. The coefficient of variation is marked in green line to indicate that it was calculated from the raw time series, not deviations from the mean. The dashed vertical line marks the time of the transcritical bifurcation in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon \rightarrow 0$\end{document} limiting case with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta =0$\end{document} . Panels b , d , and f show the ROC curves. The AUC value indicates the area under the corresponding ROC curve. All curves are above the black line , showing that the indicators behave better than chance in distinguishing between realizations that have been generated by the null and test models. A bandwidth of 5 years was chosen for the Gaussian filtering. All calculations used the parameter values in Table

Article Snippet: The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from van Kampen and Gaussian detrending.

Techniques: Generated

Journal: Theoretical Ecology

Article Title: Theory of early warning signals of disease emergenceand leading indicators of elimination

doi: 10.1007/s12080-013-0185-5

Figure Lengend Snippet: Performance of the statistics over a moving window for the SIR system rapidly approaching elimination, assuming that immigration occurs ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta >0$\end{document} ). Panels a , c , and e show the median statistics ( thick lines ) and 95 % prediction intervals ( shaded regions ). The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from the van Kampen and Gaussian detrending. The coefficient of variation (CV) is marked in dashed green line to indicate that it was calculated from the raw time series, not deviations from the mean. The dashed vertical line marks the time of the transcritical bifurcation in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\epsilon \rightarrow 0$\end{document} limiting case with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta =0$\end{document} . Panels b , d , and f show the ROC curves. The AUC value indicates the area under the corresponding ROC curve. All curves are above the black line , showing that the indicators behave better than chance in distinguishing between realizations that have been generated by the null and test models. However, the variance performs poorly compared with the lag-1 autocorrelation and CV. A bandwidth of 5 years was chosen for the Gaussian filtering. All calculations used the parameter values in Table except for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p_{0} = 1/30~\text {year}^{-1}$\end{document}

Article Snippet: The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from van Kampen and Gaussian detrending.

Techniques: Generated

Journal: Theoretical Ecology

Article Title: Theory of early warning signals of disease emergenceand leading indicators of elimination

doi: 10.1007/s12080-013-0185-5

Figure Lengend Snippet: Performance of the statistics over a moving window for the SIR system approaching emergence, assuming that immigration occurs ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta >0$\end{document} ). Panels a , c , and e show the median statistics ( thick lines ) and 95 % prediction intervals (shaded regions). The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from van Kampen and Gaussian detrending. The coefficient of variation is marked in green ( dashed ) line to indicate that it was calculated from the raw time series, not deviations from the mean. The dashed vertical line marks the time of the transcritical bifurcation. Panels b , d , and f shows the ROC curves. The AUC value indicates the area under the corresponding ROC curve. None of median statistics exhibit a marked trend. In addition, the ROC curves indicate that the distributions of Kendall’s \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau $\end{document} overlap greatly, suggesting that it is difficult to reliably distinguish between a stable system and one undergoing a critical transition using the lag-1 autocorrelation coefficient. The coefficient of variation also performs poorly, as predicted theoretically. Due to the bifurcation delay, it is not surprising that it is difficult to distinguish between null and test replicates. A bandwidth of 20 months was selected for the Gaussian filtering. All calculations used the parameter values in Table

Article Snippet: The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from van Kampen and Gaussian detrending.

Techniques:

Journal: Theoretical Ecology

Article Title: Theory of early warning signals of disease emergenceand leading indicators of elimination

doi: 10.1007/s12080-013-0185-5

Figure Lengend Snippet: Performance of the statistics over a moving window for the SIS system approaching endemicity, assuming that immigration occurs ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\eta >0$\end{document} ). Panels a , c , and e show the median statistics ( thick lines ) and 95 % prediction intervals ( shaded regions ). The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from the van Kampen detrending and Gaussian filtering. The coefficient of variation is marked in green ( dashed ) line to indicate that it was calculated from the raw time series, not deviations from the mean. The dashed vertical line marks the time of the transcritical bifurcation. Panels b , d , and f shows the ROC curves. The AUC value indicates the area under the corresponding ROC curve. There are no marked trends in the median statistics, although the variance exhibits a slight increasing trend. The ROC curves show that the distributions of Kendall’s \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau $\end{document} overlap greatly, suggesting that it is difficult to distinguish between a quasi-stationary system and one approaching an emergence threshold. Due to the bifurcation delay, it is not surprising that it is difficult to distinguish between null and test replicates. A bandwidth of 10 years was selected for the Gaussian filtering. All calculations used the parameter values in Table

Article Snippet: The lag-1 autocorrelation and variance have been calculated from the fluctuations obtained from van Kampen and Gaussian detrending.

Techniques: