MFDFA (version 1.1)

MFXDFA: Multifractal detrended cross-correlation analysis

Description

Applies the MultiFractal Detrended Fluctuation cross-correlation Analysis (MFXDFA) on two time series.

Usage

MFXDFA(tsx1, tsx2, scale, m=1, q)

Arguments

tsx1

Univariate time series (must be a vector or a ts object).

tsx2

Univariate time series (must be a vector or a ts object).

scale

Vector of scales.

m

Polynomial order for the detrending (by default m=1).

q

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.

• h Holder exponent.

• Dh Multifractal spectrum.

• Fq Fluctuation function in log.

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.

Examples

# NOT RUN {
library(MFDFA)
a<-0.6
N<-1024

tsx1<-MFsim(N,a)
b<-0.8
N<-1024

tsx2<-MFsim(N,b)
scale=10:100
q<--10:10
m<-1

# }
# NOT RUN {
b<-MFXDFA(tsx1, tsx2, scale, m=1, q)

## 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)}

## Plot results: #####
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: Fluctuations function
p1<-which(q==2)
plot(log(scale),b$Fq[,p1],  pch=16, col=1, axes = FALSE, xlab = "s",
     ylab=expression('log'*'(F'[2]*')'), cex=1, cex.lab=1.6, cex.axis=1.6,
     main= "Fluctuation function F for q=2",
     ylim=c(min(b$Fq[,c(p1)]),max(b$Fq[,c(p1)])))
lines(log(scale),b$line[,p1], 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<-log(lbl)
axis(1, at=att, labels=lbl)

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

## 3rd plot: Spectrum
plot(b$h, b$Dh, col=1, axes=FALSE, 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$h
y1=b$Dh
rr<-poly_fit(x1,y1,4)
mm1<-rr$model1
mm<-rr$polyfit
x2<-seq(min(x1),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)
# }
# NOT RUN {
# }