## Global Scaling Phenomena

“If you have not found the 1/f spectrum, it is because you have not waited long enough. You have not looked at low enough frequencies.”

- Machlup (1981)

The family of fluctuation analyses all start with fd_ and most of them are based on quantifying a dependency of the magnitude of fluctuations observed at different time scales.

The slope of time scale with fluctuation in log-log coordinates represents the scaling exponent, which can be transformed into an estimate of the Fractal Dimension. In casnet this conversion is performed by applying the formula’s provided in .

Let’s create some noise series:

library(casnet)
y0 <- noise_powerlaw(alpha = -2, N = 512)
y1 <- noise_powerlaw(alpha = -1, N = 512)
y2 <- noise_powerlaw(alpha =  0, N = 512)
y3 <- noise_powerlaw(alpha =  1, N = 512)
y4 <- noise_powerlaw(alpha =  2, N = 512)

plotTS_multi(data.frame(y0,y1,y2,y3,y4))

ts_list <- list(y0,y1,y2,y3,y4)
noiseNames <- c("y0: Brownian (red) noise", "y1: Pink noise", "y2: White noise", "y3: Blue noise" ,"y4: Violet noise")

### Standardised Dispersion Analysis (SDA)

In Standardised Dispersion Analysis, the time series is converted to z-scores (standardised) and the way the average standard deviation (SD) calculated in bins of a particular size scales with the bin size should be an indication of the presence of power-laws. That is, if the bins get larger and the variability decreases, there probably is no scaling relation. If the SD systematically increases either with larger bin sizes, or, in reverse, this means the fluctuations depend on the size of the bins, the size of the measurement stick.

library(cowplot)

sdaPlots <- plyr::llply(seq_along(ts_list), function(t){
fd_sda(ts_list[[t]],silent = TRUE, returnPlot = TRUE, noTitle = TRUE,
tsName = noiseNames[t])$plot }) cowplot::plot_grid(plotlist = sdaPlots, ncol = 1) To increase the resolution of the bins adjust the argument scaleResolution and/or the values of scaleMin and scaleMax, OR the value dataMin. At a resolution of 32 there appear to be 2 scaling regions t <- 2 # Pink noise fd_sda(ts_list[[t]], silent = TRUE, scaleResolution = 32, noTitle = TRUE, tsName = noiseNames[t], doPlot = TRUE) Below the argument dataMin, which defaults to 4, is used to adjust the fit range. fd_sda(ts_list[[t]], silent = TRUE, scaleResolution = 32, dataMin = 10, noTitle = TRUE, tsName = noiseNames[t], doPlot = TRUE) This is probably not a good idea… fd_sda(ts_list[[t]], silent = TRUE, scaleResolution = 2, noTitle = TRUE, tsName = noiseNames[t], doPlot = TRUE) ### Detrended Fluctuation Analysis (DFA) The procedure for Detrended Fluctuation Analysis is similar to SDA, except that within each bin, the signal is first detrended, what remains is then considered the residual variance (see e.g., Kantelhardt et al. 2002). The logic is the same, the way the average residual variance scales with the bin size should be an indication of the presence of power-laws. There are many different versions of DFA, one can choose to detrend polynomials of a higher order, or even detrend using the best fitting model, which is decided for each bin individually. See the manual pages of fd_dfa() for details. dfaPlots <- plyr::llply(seq_along(ts_list), function(t){ fd_dfa(ts_list[[t]],silent = TRUE, returnPlot = TRUE, noTitle = TRUE, tsName = noiseNames[t])$plot
})

cowplot::plot_grid(plotlist = dfaPlots, ncol = 1)

For more customization options, use function plotFD_loglog(), make sure to return the Power Law in the output by setting returnPLAW = TRUE. You could use plotFD_loglog() or create your own figure based on the data in the PLAW field of the output.

dfa0a <- fd_dfa(y0, silent = TRUE, returnPLAW = TRUE)
plotFD_loglog(dfa0a, title = "Custom title",  subtitle = "Custom subtitle", xlabel = "X", ylabel = "Y")

### Power Spectral Density Slope (PSD slope)

psd0 <- fd_psd(y0, silent = TRUE, returnPlot = TRUE, noTitle = TRUE, tsName = "y0: Brownian (red) noise")
psd1 <- fd_psd(y1, silent = TRUE, returnPlot = TRUE, noTitle = TRUE, tsName = "y1: Pink noise")
psd2 <- fd_psd(y2, silent = TRUE, returnPlot = TRUE, noTitle = TRUE, tsName = "y2: White noise")
psd3 <- fd_psd(y3, silent = TRUE, returnPlot = TRUE, noTitle = TRUE, tsName = "y3: Blue noise")
psd4 <- fd_psd(y4, silent = TRUE, returnPlot = TRUE, noTitle = TRUE, tsName = "y4: Violet noise")

cowplot::plot_grid(plotlist = list(psd0$plot,psd1$plot,psd2$plot,psd3$plot,psd4plot), ncol = 1) ### Windowed Analysis: Brownian noise to white noise library(tidyverse) set.seed(1234) y <- rnorm(1024) y[513:1024] <- cumsum(y[513:1024]) id <- ts_windower(y = y, win = 256, step = 1, alignment = "r") DFAseries <- plyr::ldply(id, function(w){ fd <- fd_dfa(y[w], silent = TRUE) return(fdfitRange$FD) }) df_FD <- data.frame(time = 1:1024, y = y, FD = c(rep(NA,255), DFAseries$DFAout$PLAW$size.log2[fitRange])) %>%
pivot_longer(cols = 2:3, names_to = "Variable", values_to = "Value")

df_FD$Variable <- relevel(factor(df_FD$Variable), ref = "y")

ggplot(df_FD, aes(x = time, y = Value)) +
geom_line() +
geom_hline(data = df_FD %>% filter(Variable == "FD"), aes(yintercept = c(1.1)), colour = "brown") +
geom_hline(data = df_FD %>% filter(Variable == "FD"), aes(yintercept = c(1.5)), colour = "grey") +
geom_vline(xintercept = 512, linetype = 2) +
facet_grid(Variable~., scales = "free_y") +
theme_bw()

## Local Scaling Phenomena: Multi fractal DFA

Function fd_mfdfa() should reproduce results similar to the Matlab code provided by .

### Mono fractal

set.seed(33)

# White noise
fd_mfdfa(noise_powerlaw(alpha = 0, N=4096), doPlot = TRUE)
>
>
> (mf)dfa:  Sample rate was set to 1.

>
> ~~~o~~o~~casnet~~o~~o~~~
>
>  Multifractal Detrended FLuctuation Analysis
>
>   Spec_AUC Spec_Width Spec_CVplus Spec_CVmin Spec_CVtot Spec_CVasymm
> 1   0.0615     0.0649     0.00597     0.0524     0.0435       -0.795
>
>
> ~~~o~~o~~casnet~~o~~o~~~
# Pink noise
fd_mfdfa(noise_powerlaw(alpha = -1, N=4096), doPlot = TRUE)
>
>
> (mf)dfa:  Sample rate was set to 1.

>
> ~~~o~~o~~casnet~~o~~o~~~
>
>  Multifractal Detrended FLuctuation Analysis
>
>   Spec_AUC Spec_Width Spec_CVplus Spec_CVmin Spec_CVtot Spec_CVasymm
> 1    0.202       0.22      0.0868     0.0883      0.087     -0.00877
>
>
> ~~~o~~o~~casnet~~o~~o~~~

### ‘Multi’ fractal

Function fd_mfdfa()

# 'multi' fractal
N <- 2048
y <- rowSums(data.frame(elascer(noise_powerlaw(N=N, alpha = -2)), elascer(noise_powerlaw(N=N, alpha = -.5))*c(rep(.2,512),rep(.5,512),rep(.7,512),rep(1,512))))

fd_mfdfa(y=y, doPlot = TRUE)
>
>
> (mf)dfa:  Sample rate was set to 1.

>
> ~~~o~~o~~casnet~~o~~o~~~
>
>  Multifractal Detrended FLuctuation Analysis
>
>   Spec_AUC Spec_Width Spec_CVplus Spec_CVmin Spec_CVtot Spec_CVasymm
> 1    0.226      0.284       0.176      0.202      0.189      -0.0696
>
>
> ~~~o~~o~~casnet~~o~~o~~~

For more information on how to use the output from multi-fractal DFA in you studies see e.g. .

### References

Hasselman, Fred. 2013. “When the Blind Curve Is Finite: Dimension Estimation and Model Inference Based on Empirical Waveforms.” Frontiers in Physiology 4: 75.
———. 2015. “Classifying Acoustic Signals into Phoneme Categories: Average and Dyslexic Readers Make Use of Complex Dynamical Patterns and Multifractal Scaling Properties of the Speech Signal.” PeerJ 3: e837.
Ihlen, Espen A F. 2012. “Introduction to Multifractal Detrended Fluctuation Analysis in Matlab.” Frontiers in Physiology. http://www.ncbi.nlm.nih.gov/pmc/articles/pmc3366552/.
Kantelhardt, Jan W, S Zschiegner, E Koscielnybunde, S Havlin, A Bunde, and H Stanley. 2002. “Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series.” Physica A: Statistical Mechanics and Its Applications 316 (1-4): 87114. https://doi.org/10.1016/S0378-4371(02)01383-3.
Kelty-Stephen, Damian G., Kinga Palatinus, Elliot Saltzman, and James a. Dixon. 2013. “A Tutorial on Multifractality, Cascades, and Interactivity for Empirical Time Series in Ecological Science.” Ecological Psychology 25 (1): 162. http://www.tandfonline.com/doi/abs/10.1080/10407413.2013.753804.
Machlup, S. 1981. “Earthquakes, Thunderstorms and Other 1/f Noises.” In, edited by P. H. E. Meijer, R. D. Mountain, and R. J. Soulen, 614:157–60. Washington, DC: National Bureau of Standards. https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nbsspecialpublication614.pdf#page=169.