MFDFA: MultiFractal Detrended Fluctuation Analysis

Description

Applies the MultiFractal Detrended Fluctuation Analysis (MFDFA) to time series.

Usage

MFDFA(tsx, scale, m=1, q)

Arguments

tsx

Univariate time series (must be a vector).

scale

Vector of scales. There is no default value for this parameter, please add values.

An integer of the polynomial order for the detrending (by default m=1).

q-order of the moment. There is no default value for this parameter, please add values.

Value

A list of the following elements:

Hq Hurst exponent.
tau_q Mass exponent.
spec Multifractal spectrum (\(\alpha\) and \(f(\alpha)\))
Fq Fluctuation function.

Details

The original code of this function is in Matlab, you can find it on the following website Mathworks.

References

J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988.

Espen A. F. Ihlen, Introduction to multifractal detrended fluctuation analysis in matlab, Frontiers in Physiology: Fractal Physiology, 3 (141),(2012) 1-18.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, H. Stanley, Multifractal detrended fluctuation analysis of nonstationary time series, Physica A: Statistical Mechanics and its Applications, 316 (1) (2002) 87 <U+2013> 114.

Kantelhardt J.W. (2012) Fractal and Multifractal Time Series. In: Meyers R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY.

M. Laib, L. Telesca and M. Kanevski, Long-range fluctuations and multifractality in connectivity density time series of a wind speed monitoring network, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018) p. 033108, Paper.

M. Laib, J. Golay, L. Telesca, M. Kanevski, Multifractal analysis of the time series of daily means of wind speed in complex regions, Chaos, Solitons & Fractals, 109 (2018) pp. 118-127, Paper.

Examples

Run this code

# NOT RUN {
# }
# NOT RUN {
## MFDFA package installation: from github ####
install.packages("devtools")
devtools::install_github("mlaib/MFDFA")

## Get the Parellel version:
devtools::source_gist("bb0c09df9593dad16ae270334ec3e7d7", filename = "MFDFA2.r")
# }
# NOT RUN {
library(MFDFA)
a<-0.9
N<-1024
tsx<-MFsim(N,a)
scale=10:100
q<--10:10
m<-1
b<-MFDFA(tsx, scale, m, q)

# }
# NOT RUN {
## Results plot ####
dev.new()
par(mai=rep(1, 4))
plot(q, b$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8,
     cex.axis=1.8, main="Hurst exponent",
     ylim=c(min(b$Hq),max(b$Hq)))
grid(col="midnightblue")
axis(1)
axis(2)

##################################
## Suggestion of output plot: ####
## Supplementary functions: #####
reset <- function(){
par(mfrow=c(1, 1), oma=rep(0, 4), mar=rep(0, 4), new=TRUE)
plot(0:1, 0:1, type="n", xlab="", ylab="", axes=FALSE)}

poly_fit<-function(x,y,n){
  formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='',collapse='+'))))
  res1<-coef(formule)
  poly.res<-res1[length(res1):1]
  allres<-list(polyfit=poly.res, model1=formule)
  return(allres)}

##################################
## Output plots: #################
dev.new()
layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE),heights=c(4, 4))
## b : mfdfa output
par(mai=rep(0.8, 4))

## 1st plot: Scaling function order Fq (q-order RMS)
p1<-c(1,which(q==0),which(q==q[length(q)]))
plot(log2(scale),log2(b$Fqi[,1]),  pch=16, col=1, axes = F, xlab = "s (days)",
     ylab=expression('log'[2]*'(F'[q]*')'), cex=1, cex.lab=1.6, cex.axis=1.6,
     main= "Fluctuation function Fq",
     ylim=c(min(log2(b$Fqi[,c(p1)])),max(log2(b$Fqi[,c(p1)]))))
lines(log2(scale),b$line[,1], type="l", col=1, lwd=2)
grid(col="midnightblue")
axis(2)
lbl<-scale[c(1,floor(length(scale)/8),floor(length(scale)/4),
             floor(length(scale)/2),length(scale))]
att<-log2(lbl)
axis(1, at=att, labels=lbl)
for (i in 2:3){
  k<-p1[i]
  points(log2(scale), log2(b$Fqi[,k]),  col=i,pch=16)
  lines(log2(scale),b$line[,k], type="l", col=i, lwd=2)
}
legend("bottomright", c(paste('q','=',q[p1] , sep=' ' )),cex=2,lwd=c(2,2,2),
 bty="n", col=1:3)

## 2nd plot: q-order Hurst exponent
plot(q, b$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8,
    cex.axis=1.8, main="Hurst exponent", ylim=c(min(b$Hq),max(b$Hq)))
grid(col="midnightblue")
axis(1, cex=4)
axis(2, cex=4)

## 3rd plot: q-order Mass exponent
plot(q, b$tau_q, col=1, axes=F, cex.lab=1.8, cex.axis=1.8,
     main="Mass exponent",
     pch=16,ylab=expression(tau[q]))
grid(col="midnightblue")
axis(1, cex=4)
axis(2, cex=4)

## 4th plot: Multifractal spectrum
plot(b$spec$hq, b$spec$Dq, col=1, axes=F, pch=16, #main="Multifractal spectrum",
     ylab=bquote("f ("~alpha~")"),cex.lab=1.8, cex.axis=1.8,
     xlab=bquote(~alpha))
grid(col="midnightblue")
axis(1, cex=4)
axis(2, cex=4)

x1=b$spec$hq
y1=b$spec$Dq
rr<-poly_fit(x1,y1,4)
mm1<-rr$model1
mm<-rr$polyfit
x2<-seq(0,max(x1)+1,0.01)
curv<-mm[1]*x2^4+mm[2]*x2^3+mm[3]*x2^2+mm[4]*x2+mm[5]
lines(x2,curv, col="red", lwd=2)
reset()
legend("top", legend="MFDFA Plots", bty="n", cex=2)
# }

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