# Multifractal detrended cross-correlation analysis Applies the MultiFractal Detrended Fluctuation cross-correlation Analysis (MFXDFA) on two time series. ```r 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. ## Returns A list of the following elements: * `Hq` Hurst exponent. * `h` Holder exponent. * `Dh` Multifractal spectrum. * `Fq` Fluctuation function in log. ## Note The original code of this function is in Matlab, you can find it on the following website [Mathworks](https://ch.mathworks.com/matlabcentral/fileexchange/38262-multifractal-detrended-fluctuation-analyses?focused=5247306&tab=function). ## Examples ```r 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) ## End(Not run) ``` ## 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 – 114. Kantelhardt J.W. (2012) Fractal and Multifractal Time Series. In: Meyers R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY.

MFXDFA function