├── .Rbuildignore ├── .gitignore ├── .zenodo.json ├── DESCRIPTION ├── LICENSE ├── MFDFA.Rproj ├── MFDFA.pdf ├── NAMESPACE ├── R ├── MFDFA.R ├── MFXDFA.R ├── MFsim.R └── MMA.R ├── README.md ├── _config.yml └── man ├── MFDFA.Rd ├── MFXDFA.Rd ├── MFsim.Rd └── MMA.Rd /.Rbuildignore: -------------------------------------------------------------------------------- 1 | ^.*\.Rproj$ 2 | ^\.Rproj\.user$ 3 | -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | .Rproj.user 2 | .Rhistory 3 | .RData 4 | .Ruserdata 5 | -------------------------------------------------------------------------------- /.zenodo.json: -------------------------------------------------------------------------------- 1 | { 2 | "title": "MFDFA: MultiFractal Detrended Fluctuation Analysis", 3 | "publication_date": "2017-08-01", 4 | "version": "", 5 | "creators": [ 6 | { 7 | "affiliation": "Faculty of Geoscience and Environment, University of Lausanne, Lausanne, Switzerland", 8 | "name": "Laib, Mohamed", 9 | "orcid": "0000-0003-1276-1790 " 10 | }, 11 | { 12 | "affiliation": "Institute of Methodologies for Environmental Analysis, National Research Council, Italy", 13 | "name": "Telesca, Luciano", 14 | }, 15 | { 16 | "affiliation": "Faculty of Geoscience and Environment, University of Lausanne, Lausanne, Switzerland", 17 | "name": "Kanevski, Mikhail", 18 | "orcid": "0000-0001-6602-6551" 19 | } 20 | ], 21 | "keywords": [ 22 | "Time series", 23 | "Multifractal" 24 | ], 25 | "license": "GPL-3.0", 26 | "upload_type": "R_package" 27 | } 28 | -------------------------------------------------------------------------------- /DESCRIPTION: -------------------------------------------------------------------------------- 1 | Package: MFDFA 2 | Type: Package 3 | Title: MultiFractal Detrended Fluctuation Analysis 4 | Version: 1.1 5 | Authors@R: c( 6 | person("Mohamed","Laib", role=c("aut","cre"), email="laib.med@gmail.com"), 7 | person("Luciano","Telesca", role=c("aut"), email="Luciano.Telesca@imaa.cnr.it"), 8 | person("Mikhail","Kanevski", role=c("aut"), email="Mikhail.Kanevski@unil.ch")) 9 | Author: Mohamed Laib [aut, cre], 10 | Luciano Telesca [aut], 11 | Mikhail Kanevski [aut] 12 | Maintainer: Mohamed Laib 13 | Description: Contains the MultiFractal Detrended Fluctuation Analysis (MFDFA), 14 | MultiFractal Detrended Cross-Correlation Analysis (MFXDFA), and the Multiscale 15 | Multifractal Analysis (MMA). The MFDFA() function proposed in this package was 16 | used in Laib et al. ( and ). 17 | See references for more information. Interested users can find a parallel version of 18 | the MFDFA() function on GitHub. 19 | License: GPL-3 20 | Imports: numbers 21 | URL: https://mlaib.github.io 22 | Note: The originale code was in matlab, see details below. 23 | Encoding: UTF-8 24 | LazyData: true 25 | RoxygenNote: 6.1.1 26 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /MFDFA.Rproj: -------------------------------------------------------------------------------- 1 | Version: 1.0 2 | 3 | RestoreWorkspace: Default 4 | SaveWorkspace: Default 5 | AlwaysSaveHistory: Default 6 | 7 | EnableCodeIndexing: Yes 8 | UseSpacesForTab: Yes 9 | NumSpacesForTab: 2 10 | Encoding: UTF-8 11 | 12 | RnwWeave: Sweave 13 | LaTeX: pdfLaTeX 14 | 15 | AutoAppendNewline: Yes 16 | StripTrailingWhitespace: Yes 17 | 18 | BuildType: Package 19 | PackageUseDevtools: Yes 20 | PackageInstallArgs: --no-multiarch --with-keep.source 21 | PackageRoxygenize: rd,collate,namespace,vignette 22 | -------------------------------------------------------------------------------- /MFDFA.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/mlaib/MFDFA/61e42cd7aa26a1b9938b9af1d95954768a453fe1/MFDFA.pdf -------------------------------------------------------------------------------- /NAMESPACE: -------------------------------------------------------------------------------- 1 | # Generated by roxygen2: do not edit by hand 2 | 3 | export(MFDFA) 4 | export(MFXDFA) 5 | export(MFsim) 6 | export(MMA) 7 | importFrom(graphics,par) 8 | importFrom(graphics,plot) 9 | importFrom(numbers,mod) 10 | importFrom(stats,as.formula) 11 | importFrom(stats,coef) 12 | importFrom(stats,lm) 13 | importFrom(stats,predict) 14 | importFrom(stats,var) 15 | -------------------------------------------------------------------------------- /R/MFDFA.R: -------------------------------------------------------------------------------- 1 | #' MultiFractal Detrended Fluctuation Analysis 2 | #' 3 | #' Applies the MultiFractal Detrended Fluctuation Analysis (MFDFA) to time series. 4 | #' @usage MFDFA(tsx, scale, m=1, q) 5 | #' @param tsx Univariate time series (must be a vector). 6 | #' @param scale Vector of scales. There is no default value 7 | #' for this parameter, please add values. 8 | #' @param m An integer of the polynomial order for the detrending (by default m=1). 9 | #' @param q q-order of the moment. There is no default value 10 | #' for this parameter, please add values. 11 | #' 12 | #' @return A list of the following elements: 13 | #' \itemize{ 14 | #' \item \code{Hq} Hurst exponent. 15 | #' \item \code{tau_q} Mass exponent. 16 | #' \item \code{spec} Multifractal spectrum (\eqn{\alpha}{\alpha} and 17 | #' \eqn{f(\alpha)}{f(\alpha)}) 18 | #' \item \code{Fq} Fluctuation function. 19 | #' } 20 | #' 21 | #' @details The original code of this function is in Matlab, you can find it on the 22 | #' following website \href{https://ch.mathworks.com/matlabcentral/fileexchange/38262-multifractal-detrended-fluctuation-analyses?focused=5247306&tab=function}{Mathworks}. 23 | #' 24 | #' 25 | #' @examples 26 | #' 27 | #' \dontrun{ 28 | #' ## MFDFA package installation: from github #### 29 | #' install.packages("devtools") 30 | #' devtools::install_github("mlaib/MFDFA") 31 | #' 32 | #' ## Get the Parellel version: 33 | #' devtools::source_gist("bb0c09df9593dad16ae270334ec3e7d7", filename = "MFDFA2.r") 34 | #' } 35 | #' 36 | #' library(MFDFA) 37 | #' a<-0.9 38 | #' N<-1024 39 | #' tsx<-MFsim(N,a) 40 | #' scale=10:100 41 | #' q<--10:10 42 | #' m<-1 43 | #' b<-MFDFA(tsx, scale, m, q) 44 | #' 45 | #'\dontrun{ 46 | #' ## Results plot #### 47 | #' dev.new() 48 | #' par(mai=rep(1, 4)) 49 | #' plot(q, b$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 50 | #' cex.axis=1.8, main="Hurst exponent", 51 | #' ylim=c(min(b$Hq),max(b$Hq))) 52 | #' grid(col="midnightblue") 53 | #' axis(1) 54 | #' axis(2) 55 | #' 56 | #' ################################## 57 | #' ## Suggestion of output plot: #### 58 | #' ## Supplementary functions: ##### 59 | #' reset <- function(){ 60 | #' par(mfrow=c(1, 1), oma=rep(0, 4), mar=rep(0, 4), new=TRUE) 61 | #' plot(0:1, 0:1, type="n", xlab="", ylab="", axes=FALSE)} 62 | #' 63 | #' poly_fit<-function(x,y,n){ 64 | #' formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='',collapse='+')))) 65 | #' res1<-coef(formule) 66 | #' poly.res<-res1[length(res1):1] 67 | #' allres<-list(polyfit=poly.res, model1=formule) 68 | #' return(allres)} 69 | #' 70 | #' ################################## 71 | #' ## Output plots: ################# 72 | #' dev.new() 73 | #' layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE),heights=c(4, 4)) 74 | #' ## b : mfdfa output 75 | #' par(mai=rep(0.8, 4)) 76 | #' 77 | #' ## 1st plot: Scaling function order Fq (q-order RMS) 78 | #' p1<-c(1,which(q==0),which(q==q[length(q)])) 79 | #' plot(log2(scale),log2(b$Fqi[,1]), pch=16, col=1, axes = F, xlab = "s (days)", 80 | #' ylab=expression('log'[2]*'(F'[q]*')'), cex=1, cex.lab=1.6, cex.axis=1.6, 81 | #' main= "Fluctuation function Fq", 82 | #' ylim=c(min(log2(b$Fqi[,c(p1)])),max(log2(b$Fqi[,c(p1)])))) 83 | #' lines(log2(scale),b$line[,1], type="l", col=1, lwd=2) 84 | #' grid(col="midnightblue") 85 | #' axis(2) 86 | #' lbl<-scale[c(1,floor(length(scale)/8),floor(length(scale)/4), 87 | #' floor(length(scale)/2),length(scale))] 88 | #' att<-log2(lbl) 89 | #' axis(1, at=att, labels=lbl) 90 | #' for (i in 2:3){ 91 | #' k<-p1[i] 92 | #' points(log2(scale), log2(b$Fqi[,k]), col=i,pch=16) 93 | #' lines(log2(scale),b$line[,k], type="l", col=i, lwd=2) 94 | #' } 95 | #' legend("bottomright", c(paste('q','=',q[p1] , sep=' ' )),cex=2,lwd=c(2,2,2), 96 | #' bty="n", col=1:3) 97 | #' 98 | #' ## 2nd plot: q-order Hurst exponent 99 | #' plot(q, b$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 100 | #' cex.axis=1.8, main="Hurst exponent", ylim=c(min(b$Hq),max(b$Hq))) 101 | #' grid(col="midnightblue") 102 | #' axis(1, cex=4) 103 | #' axis(2, cex=4) 104 | #' 105 | #' ## 3rd plot: q-order Mass exponent 106 | #' plot(q, b$tau_q, col=1, axes=F, cex.lab=1.8, cex.axis=1.8, 107 | #' main="Mass exponent", 108 | #' pch=16,ylab=expression(tau[q])) 109 | #' grid(col="midnightblue") 110 | #' axis(1, cex=4) 111 | #' axis(2, cex=4) 112 | #' 113 | #' ## 4th plot: Multifractal spectrum 114 | #' plot(b$spec$hq, b$spec$Dq, col=1, axes=F, pch=16, #main="Multifractal spectrum", 115 | #' ylab=bquote("f ("~alpha~")"),cex.lab=1.8, cex.axis=1.8, 116 | #' xlab=bquote(~alpha)) 117 | #' grid(col="midnightblue") 118 | #' axis(1, cex=4) 119 | #' axis(2, cex=4) 120 | #' 121 | #' x1=b$spec$hq 122 | #' y1=b$spec$Dq 123 | #' rr<-poly_fit(x1,y1,4) 124 | #' mm1<-rr$model1 125 | #' mm<-rr$polyfit 126 | #' x2<-seq(0,max(x1)+1,0.01) 127 | #' curv<-mm[1]*x2^4+mm[2]*x2^3+mm[3]*x2^2+mm[4]*x2+mm[5] 128 | #' lines(x2,curv, col="red", lwd=2) 129 | #' reset() 130 | #' legend("top", legend="MFDFA Plots", bty="n", cex=2) 131 | #' } 132 | #' @references 133 | #' J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988. 134 | #' 135 | #' Espen A. F. Ihlen, Introduction to multifractal detrended fluctuation analysis 136 | #' in matlab, Frontiers in Physiology: Fractal Physiology, 3 (141),(2012) 1-18. 137 | #' 138 | #' J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, 139 | #' A. Bunde, H. Stanley, Multifractal detrended fluctuation analysis of 140 | #' nonstationary time series, Physica A: Statistical Mechanics and its 141 | #' Applications, 316 (1) (2002) 87 – 114. 142 | #' 143 | #' Kantelhardt J.W. (2012) Fractal and Multifractal Time Series. In: Meyers R. (eds) 144 | #' Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. 145 | #' 146 | #' M. Laib, L. Telesca and M. Kanevski, Long-range fluctuations and 147 | #' multifractality in connectivity density time series of a wind speed 148 | #' monitoring network, Chaos: An Interdisciplinary Journal of Nonlinear 149 | #' Science, 28 (2018) p. 033108, \href{https://aip.scitation.org/doi/10.1063/1.5022737}{Paper}. 150 | #' 151 | #' M. Laib, J. Golay, L. Telesca, M. Kanevski, Multifractal 152 | #' analysis of the time series of daily means of wind speed 153 | #' in complex regions, Chaos, Solitons & Fractals, 109 (2018) 154 | #' pp. 118-127, \href{https://www.sciencedirect.com/science/article/pii/S0960077918300699}{Paper}. 155 | #' 156 | #' @importFrom stats var 157 | #' 158 | #' @export 159 | 160 | MFDFA<-function(tsx, scale, m=1, q){ 161 | if (!is.numeric(tsx) | var(tsx)==0){ 162 | stop("Check your time series") 163 | } 164 | 165 | if (m%%1 != 0){ 166 | stop("m must be an integer") 167 | } 168 | 169 | if (!is.numeric(scale) | length(scale) <= 1 | any(scale%%1 != 0)){ 170 | stop("scale must be a vector containing integers") 171 | } 172 | 173 | if (!is.numeric(q) | any(q%%1 != 0)){ 174 | stop("q must contain integers") 175 | } 176 | 177 | 178 | X<-cumsum(tsx-mean(tsx)) 179 | seg<-list() 180 | qRMS<-list() 181 | Fq<-c() 182 | Fqi<-list() 183 | Hq<-c() 184 | qRegLine<-list() 185 | RMSvi<-list() 186 | for (i in 1:length(scale)){ 187 | seg[[i]]<-floor(length(X)/scale[i]) 188 | rmvi<-c() 189 | for (vi in 1:seg[[i]]){ 190 | Index=((((vi-1)*scale[i])+1):(vi*scale[i])) 191 | polyft<-poly_fit.val( Index,X[Index], m) 192 | C<-polyft$polyfit 193 | fit<-polyft$polyval 194 | rmvi[vi]<-sqrt(mean((fit-X[Index])^2)) 195 | RMSvi[[i]]<-rmvi 196 | } 197 | 198 | 199 | for (nq in 1:length(q)){ 200 | rod<-RMSvi[[i]]^q[nq] 201 | qRMS[[nq]]<-rod 202 | Fq[nq]<-mean(qRMS[[nq]])^(1/q[nq]) 203 | 204 | } 205 | if (any(q==0)){Fq[which(q==0)]<-exp(0.5*mean(log(RMSvi[[i]]^2)))} 206 | 207 | Fqi[[i]]<-Fq 208 | 209 | } 210 | Fqi<-Reduce("rbind", Fqi) 211 | for (nq in 1:length(q)){ 212 | polyft<-poly_fit.val( log2(scale),log2(Fqi[,nq]), 1) 213 | C<-polyft$polyfit 214 | Hq[nq]<-C[1] 215 | qRegLine[[nq]]<-polyft$polyval 216 | } 217 | qRegLine<-as.data.frame(Reduce("cbind", qRegLine)) 218 | tq<-Hq*q-1 219 | hq<-diff(tq)/(q[2]-q[1]) 220 | Dq<-(q[1:(length(q)-1)]*hq)-tq[1:(length(tq)-1)] 221 | return(list(Hq=Hq, tau_q=tq, spec=data.frame(hq=hq, Dq=Dq), 222 | Fqi=Fqi, line=qRegLine))} 223 | 224 | 225 | ## intern functions: polyfit #### 226 | poly_fit.val<-function(x,y,n){ 227 | formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='',collapse='+')))) 228 | res1<-coef(formule) 229 | poly.res<-res1[length(res1):1] 230 | suppressWarnings(res2<-predict(formule, interval = "prediction")) 231 | poly.eva<-res2[,1] 232 | allres<-list(polyfit=round(poly.res,4), polyval=poly.eva) 233 | return(allres)} 234 | 235 | reset <- function(){ 236 | par(mfrow=c(1, 1), oma=rep(0, 4), mar=rep(0, 4), new=TRUE) 237 | plot(0:1, 0:1, type="n", xlab="", ylab="", axes=FALSE) 238 | } 239 | 240 | poly_fit<-function(x,y,n){ 241 | formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='',collapse='+')))) 242 | res1<-coef(formule) 243 | poly.res<-res1[length(res1):1] 244 | allres<-list(polyfit=poly.res, model1=formule) 245 | return(allres)} 246 | 247 | -------------------------------------------------------------------------------- /R/MFXDFA.R: -------------------------------------------------------------------------------- 1 | #' Multifractal detrended cross-correlation analysis 2 | #' 3 | #' Applies the MultiFractal Detrended Fluctuation cross-correlation Analysis (MFXDFA) on two time series. 4 | #' @usage MFXDFA(tsx1, tsx2, scale, m=1, q) 5 | #' @param tsx1 Univariate time series (must be a vector or a ts object). 6 | #' @param tsx2 Univariate time series (must be a vector or a ts object). 7 | #' @param scale Vector of scales. 8 | #' @param m Polynomial order for the detrending (by default m=1). 9 | #' @param q q-order of the moment. There is no default value 10 | #' for this parameter, please add values. 11 | #' @return A list of the following elements: 12 | #' \itemize{ 13 | #' \item \code{Hq} Hurst exponent. 14 | #' \item \code{h} Holder exponent. 15 | #' \item \code{Dh} Multifractal spectrum. 16 | #' \item \code{Fq} Fluctuation function in log. 17 | #' } 18 | #' 19 | #' @note The original code of this function is in Matlab, you can find it on the 20 | #' following website \href{https://ch.mathworks.com/matlabcentral/fileexchange/38262-multifractal-detrended-fluctuation-analyses?focused=5247306&tab=function}{Mathworks}. 21 | #' 22 | #' 23 | #' 24 | #' @examples 25 | #' 26 | #' library(MFDFA) 27 | #' a<-0.6 28 | #' N<-1024 29 | #' 30 | #' tsx1<-MFsim(N,a) 31 | #' b<-0.8 32 | #' N<-1024 33 | #' 34 | #' tsx2<-MFsim(N,b) 35 | #' scale=10:100 36 | #' q<--10:10 37 | #' m<-1 38 | #' 39 | #' \dontrun{ 40 | #' b<-MFXDFA(tsx1, tsx2, scale, m=1, q) 41 | #' 42 | #' ## Supplementary functions: ##### 43 | #' reset <- function(){ 44 | #' par(mfrow=c(1, 1), oma=rep(0, 4), mar=rep(0, 4), new=TRUE) 45 | #' plot(0:1, 0:1, type="n", xlab="", ylab="", axes=FALSE)} 46 | #' 47 | #' poly_fit<-function(x,y,n){ 48 | #' formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='',collapse='+')))) 49 | #' res1<-coef(formule) 50 | #' poly.res<-res1[length(res1):1] 51 | #' allres<-list(polyfit=poly.res, model1=formule) 52 | #' return(allres)} 53 | #' 54 | #' ## Plot results: ##### 55 | #' dev.new() 56 | #' layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE),heights=c(4, 4)) 57 | #' ## b : mfdfa output 58 | #' par(mai=rep(0.8, 4)) 59 | #' 60 | #' ## 1st plot: Fluctuations function 61 | #' p1<-which(q==2) 62 | #' plot(log(scale),b$Fq[,p1], pch=16, col=1, axes = FALSE, xlab = "s", 63 | #' ylab=expression('log'*'(F'[2]*')'), cex=1, cex.lab=1.6, cex.axis=1.6, 64 | #' main= "Fluctuation function F for q=2", 65 | #' ylim=c(min(b$Fq[,c(p1)]),max(b$Fq[,c(p1)]))) 66 | #' lines(log(scale),b$line[,p1], type="l", col=1, lwd=2) 67 | #' grid(col="midnightblue") 68 | #' axis(2) 69 | #' lbl<-scale[c(1,floor(length(scale)/8),floor(length(scale)/4), 70 | #' floor(length(scale)/2),length(scale))] 71 | #' att<-log(lbl) 72 | #' axis(1, at=att, labels=lbl) 73 | #' 74 | #' ## 2nd plot: q-order Hurst exponent 75 | #' plot(q, b$Hq, col=1, axes= FALSE, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 76 | #' cex.axis=1.8, main="Hurst exponent", ylim=c(min(b$Hq),max(b$Hq))) 77 | #' grid(col="midnightblue") 78 | #' axis(1, cex=4) 79 | #' axis(2, cex=4) 80 | #' 81 | #' ## 3rd plot: Spectrum 82 | #' plot(b$h, b$Dh, col=1, axes=FALSE, pch=16, main="Multifractal spectrum", 83 | #' ylab=bquote("f ("~alpha~")"),cex.lab=1.8, cex.axis=1.8, 84 | #' xlab=bquote(~alpha)) 85 | #' grid(col="midnightblue") 86 | #' axis(1, cex=4) 87 | #' axis(2, cex=4) 88 | #' 89 | #' x1=b$h 90 | #' y1=b$Dh 91 | #' rr<-poly_fit(x1,y1,4) 92 | #' mm1<-rr$model1 93 | #' mm<-rr$polyfit 94 | #' x2<-seq(min(x1),max(x1)+1,0.01) 95 | #' curv<-mm[1]*x2^4+mm[2]*x2^3+mm[3]*x2^2+mm[4]*x2+mm[5] 96 | #' lines(x2,curv, col="red", lwd=2) 97 | #' } 98 | #' 99 | #' @references 100 | #' J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988. 101 | #' 102 | #' Espen A. F. Ihlen, Introduction to multifractal detrended fluctuation analysis 103 | #' in matlab, Frontiers in Physiology: Fractal Physiology, 3 (141),(2012) 1-18. 104 | #' 105 | #' J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, 106 | #' A. Bunde, H. Stanley, Multifractal detrended fluctuation analysis of 107 | #' nonstationary time series, Physica A: Statistical Mechanics and its 108 | #' Applications, 316 (1) (2002) 87 – 114. 109 | #' 110 | #' Kantelhardt J.W. (2012) Fractal and Multifractal Time Series. In: Meyers R. (eds) Mathematics 111 | #' of Complexity and Dynamical Systems. Springer, New York, NY. 112 | #' 113 | #' @export 114 | #' 115 | 116 | MFXDFA <- function(tsx1, tsx2, scale, m=1, q){ 117 | if ((length(tsx1)==length(tsx2))==FALSE){ 118 | stop("check the length of these time series") 119 | } 120 | N <- length(tsx1) 121 | X<-cumsum(tsx1-mean(tsx1)) 122 | Y<-cumsum(tsx2-mean(tsx2)) 123 | 124 | q0 <- which(q==0) 125 | Fq <- matrix(0, nrow=length(scale), ncol=length(q)) 126 | for (ns in 1:length(scale)){ 127 | Ns <- floor(N/scale[ns]) 128 | VR <- matrix(0, nrow=Ns, ncol=length(q)) 129 | 130 | for (v in 1:Ns){ 131 | SegInd <- ((((v-1)*scale[ns])+1):(v*scale[ns])) 132 | 133 | # 1st time serie 134 | SegX <- X[SegInd] 135 | fitX <- poly_fit.val(SegInd, SegX, m)$polyval 136 | 137 | # 2nd time serie 138 | SegY <- Y[SegInd] 139 | fitY <- poly_fit.val(SegInd, SegY, m)$polyval 140 | 141 | 142 | for (nq in 1:length(q)){ 143 | VR[v,nq] <- ((sum(((SegX-fitX)^2)*((SegY-fitY)^2)))/scale[ns])^(q[nq]/4) 144 | } 145 | 146 | } 147 | for (nq in 1:length(q)){ 148 | if ((q[nq]==0)==FALSE){ 149 | Fq[ns,nq] <- ((sum(VR[,nq])+sum(VR[,nq]))/(2*Ns))^(1/q[nq]) 150 | } 151 | } 152 | Fq[ns,q0] <- (Fq[ns,(q0+1)]+Fq[ns,(q0-1)])/2 153 | 154 | 155 | } 156 | 157 | LogF <- log2(Fq) 158 | 159 | qRegLine <- list() 160 | Hq <- c() 161 | for (nq in 1:length(q)){ 162 | polyft<-poly_fit.val(log2(scale),LogF[,nq], 1) 163 | 164 | Hq[nq] <- polyft$polyfit[1] 165 | qRegLine[[nq]]<-polyft$polyval 166 | } 167 | qRegLine <- as.data.frame(Reduce("cbind", qRegLine)) 168 | tau <- (q*Hq)-1 169 | 170 | H <- diff(tau)/diff(q) 171 | Dh <- (q[-length(q)]*H)-tau[-length(tau)] 172 | h <- H-1 173 | 174 | return(list(Hq=Hq, h=h, Dh=Dh, Fq=LogF, line=qRegLine)) 175 | } 176 | 177 | 178 | 179 | 180 | 181 | 182 | poly_fit.val<-function(x,y,n){ 183 | formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='', 184 | collapse='+')))) 185 | res1<-coef(formule) 186 | poly.res<-res1[length(res1):1] 187 | suppressWarnings(res2<-predict(formule, interval = "prediction")) 188 | poly.eva<-res2[,1] 189 | allres<-list(polyfit=round(poly.res,4), polyval=poly.eva) 190 | return(allres)} 191 | -------------------------------------------------------------------------------- /R/MFsim.R: -------------------------------------------------------------------------------- 1 | #' Simulated multifractal series. 2 | #' 3 | #' Generates series using the binomial multifractal model (see references). 4 | #' @usage MFsim(N,a) 5 | #' @param N The length of the generated multifractal series. 6 | #' @param a Exponent that takes values in [0.6, 1]. 7 | #' 8 | #' @return A vector containing the multifractal series. 9 | #' 10 | #' 11 | #' @examples 12 | #' 13 | #' a<-0.9 14 | #' N<-1024 15 | #' tsx<-MFsim(N,a) 16 | #' scale=10:100 17 | #' q<--10:10 18 | #' m<-1 19 | #' b<-MFDFA(tsx, scale, m, q) 20 | #' 21 | #' dev.new() 22 | #' par(mai=rep(1, 4)) 23 | #' plot(q, b$Hq, col=1, axes= FALSE, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 24 | #' cex.axis=1.8, main="q-order Hurst exponent", ylim=c(min(b$Hq),max(b$Hq))) 25 | #' grid(col="midnightblue") 26 | #' axis(1) 27 | #' axis(2) 28 | #' 29 | #' \dontrun{ 30 | #' ## Example with Levy distribution #### 31 | #' require(rmutil) 32 | #' tsx <- rlevy(1000, 0, 1) 33 | #' scale=10:100 34 | #' q<--10:10 35 | #' m<-1 36 | #' b<-MFDFA(tsx, scale, m, q) 37 | #' 38 | #' dev.new() 39 | #' plot(q, b$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 40 | #' cex.axis=1.8, main="Hurst exponent", ylim=c(min(b$Hq),max(b$Hq))) 41 | #' grid(col="midnightblue") 42 | #' axis(1, cex=4) 43 | #' axis(2, cex=4) 44 | #' } 45 | #' 46 | #' @references 47 | #' 48 | #' J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988. 49 | #' 50 | #' E.L. Flores-Márquez, A. Ramírez-Rojas, L. Telesca, Multifractal detrended 51 | #' fluctuation analysis of earthquake magnitude series of Mexican South Pacific 52 | #' Region, Applied Mathematics and Computation, Volume 265, 2015, 53 | #' Pages 1106-1114, ISSN 0096-3003. 54 | #' 55 | #' 56 | #' @importFrom graphics par plot 57 | #' @importFrom stats as.formula coef lm predict 58 | #' @export 59 | 60 | MFsim<-function(N,a){ 61 | if (a==0.5){ 62 | warning("Generated signal is constant when a = 0.5") 63 | } 64 | m<-1:N 65 | b1<-a^(nbit(m-1)) 66 | b2<-(1-a)^(16-nbit(m-1)) 67 | XM<-b1*b2 68 | return(XM) 69 | } 70 | 71 | # intern function: nbit 72 | nbit<-function(num){ 73 | num<-as.matrix(num) 74 | s<-list() 75 | s<-apply(num, 1, intToBits) 76 | rs<-apply(s, 2, FUN = function(xa) (length(which(xa==1)))) 77 | return(rs) 78 | } 79 | -------------------------------------------------------------------------------- /R/MMA.R: -------------------------------------------------------------------------------- 1 | #' Multiscale Multifractal Analysis 2 | #' 3 | #' Applies the Multiscale Multifractal Analysis (MMA) on time series. 4 | #' @usage MMA(tsx, scale, qminmax, ovlap=0, m=2) 5 | #' @param tsx Univariate time series (must be a vector or a ts object). 6 | #' @param scale Vector of scales. 7 | #' @param qminmax Vector of two values min and max of q-order of the moment. 8 | #' @param ovlap Overlapping parameter (By default ovlap=0: no overlapping). 9 | #' @param m Polynomial order for the detrending (by defaults m=2). 10 | #' @return A matrix with three columns (q-order, scale (s), and the scale exponent). 11 | #' 12 | #' @note The original code of this function is in Matlab, you can find it on the 13 | #' following website \href{https://physionet.org/physiotools/mma/}{Physionet}. See 14 | #' references below. 15 | #' 16 | #' 17 | #' 18 | #' @examples 19 | #' 20 | #' \dontrun{ 21 | #' library(MFDFA) 22 | #' library(plotly) 23 | #' library(plot3D) 24 | #' 25 | #' a<-0.6 26 | #' N<-800 27 | #' tsx<-MFsim(N,a) 28 | #' scale=10:100 29 | #' res<-MMA(tsx, scale, qminmax=c(-10,10), ovlap=0, m=2) 30 | #' 31 | #' ## Visualisation 1: 32 | #' S_exponent <- matrix(res[,3], nrow=length(unique(res[,1])), ncol=length(min(scale):(max(scale)/5))) 33 | #' m_scale <- unique(res[,2]) 34 | #' q <- unique(res[,1]) 35 | #' plot_ly() %>% add_surface(x = ~m_scale, y = ~q, 36 | #' z = ~S_exponent) 37 | #' 38 | #' ## Visualisation 2: 39 | #' image2D(S_exponent, xlab="q", ylab="scale", axes=F) 40 | #' axis(1, seq(0,1,0.1), round(quantile(q, seq(0, 1, 0.1)), 2)) 41 | #' axis(2, seq(0,1,0.1), round(quantile(m_scale, seq(0, 1, 0.1)), 2)) 42 | #' } 43 | #' 44 | #' @references 45 | #' J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988. 46 | #' 47 | #' J. Gieraltowski, J. J. Zebrowski, and R. Baranowski, 48 | #' Multiscale multifractal analysis of heart rate variability recordings 49 | #' http://dx.doi.org/10.1103/PhysRevE.85.021915 50 | #' 51 | #' Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG, 52 | #' Mietus JE, Moody GB, Peng C-K, Stanley HE. PhysioBank, PhysioToolkit, 53 | #' and PhysioNet: Components of a New Research Resource for Complex 54 | #' Physiologic Signals. Circulation 101(23):e215-e220. 55 | #' 56 | #' J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, 57 | #' A. Bunde, H. Stanley, Multifractal detrended fluctuation analysis of 58 | #' nonstationary time series, Physica A: Statistical Mechanics and its 59 | #' Applications, 316 (1) (2002) 87 – 114. 60 | #' 61 | #' J. Gierałtowski, J. J. Żebrowski, and R. Baranowski, "Multiscale 62 | #' multifractal analysis of heart rate variability recordings with a 63 | #' large number of occurrences of arrhythmia," Phys. Rev. E 85, 021915 (2012) 64 | #' 65 | #' 66 | #' @importFrom numbers mod 67 | #' @export 68 | #' 69 | #' 70 | 71 | MMA <- function(tsx, scale, qminmax, ovlap=0, m=2){ 72 | qrange <- seq(qminmax[1], qminmax[2], 0.1) 73 | qrange[which(qrange==0)]<-0.0001 74 | prof = cumsum(tsx) 75 | slength = length(prof) 76 | fqs<-c() 77 | for (i in scale){ 78 | if (ovlap==1){ 79 | vec <- 0:(i-1) 80 | ind <- 1:(slength-i+1) 81 | A<-matrix(rep(vec, each = length(ind)), length(ind), length(vec)) 82 | coorxy <- apply(A, 2, FUN=function(x) (x+ind)) 83 | } else { 84 | nd<-(slength-mod(slength,i)) 85 | nc<-(slength-mod(slength,i))/i 86 | coorxy<-matrix(1:nd, nc, i, byrow = TRUE) 87 | } 88 | segments <- apply(coorxy, 2, FUN=function(x)(prof[x])) 89 | xbs <- 1:i 90 | f2nis <- c() 91 | 92 | for (j in 1:nrow(segments)){ 93 | seg <- segments[j,] 94 | ft <- poly_fit.val(xbs, seg, m) 95 | fit<-ft$polyval 96 | f2nis <-c(f2nis, mean((seg- fit)^2)) 97 | 98 | } 99 | 100 | for (qq in qrange){ 101 | fqs<- c(fqs, qq, i, mean(f2nis^(qq/2))^(1/qq)) 102 | } 103 | 104 | 105 | } 106 | 107 | fqs1 <- matrix(fqs, ncol=3, byrow = T) 108 | fqs1 <- matrix(append(fqs1, log(fqs1[,2])), ncol=4) 109 | fqs1 <- matrix(append(fqs1, log(fqs1[,3])), ncol=5) 110 | 111 | 112 | hqs <- c() 113 | 114 | for (i in min(scale):(max(scale)/5)){ 115 | for (qq in qrange){ 116 | tempfqs <- fqs1[which(fqs1[,1]==qq & fqs1[,2] >= i & fqs1[,2] <= 5*i),] 117 | hft <- poly_fit.val(tempfqs[,4], tempfqs[,5], 1) 118 | hqs <- c(hqs, qq, i, as.numeric(hft$polyfit[1])) 119 | } 120 | } 121 | 122 | hqs1 <- matrix(hqs, ncol=3, byrow = T) 123 | colnames(hqs1)<- c("q", "s", "S_Exp") 124 | return(hqs1) 125 | } 126 | 127 | poly_fit.val<-function(x,y,n){ 128 | formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='',collapse='+')))) 129 | res1<-coef(formule) 130 | poly.res<-res1[length(res1):1] 131 | suppressWarnings(res2<-predict(formule, interval = "prediction")) 132 | poly.eva<-res2[,1] 133 | allres<-list(polyfit=round(poly.res,4), polyval=poly.eva) 134 | return(allres)} 135 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | ## MFDFA: MultiFractal Detrended Fluctuation Analysis for Time Series 2 | Applies the MultiFractal Detrended Fluctuation Analysis (MFDFA) to time series. The package contains some suggestion plot of the MFDFA results. 3 | 4 | The MFDFA R library is now available on CRAN. Further update will be added soon. 5 | 6 | A new file is available [Here](https://gist.github.com/mlaib/bb0c09df9593dad16ae270334ec3e7d7). It proposes the MFDFA with a parallel version (MFDFA2.R). Useful for long time series. It can be used as the first one with same parameters. It uses (N-1) of CPU cores of your computer. 7 | 8 | Use the following to get it: 9 | ```{r} 10 | devtools::source_gist("bb0c09df9593dad16ae270334ec3e7d7", filename = "MFDFA2.r") 11 | ``` 12 | 13 | ENJOY ... 14 | 15 | ![alt text](https://github.com/mlaib/mlaib.github.io/blob/master/FunTseries.png) 16 | 17 | #### Version 18 | 1.1 19 | 20 | #### Authors 21 | Mohamed Laib, Luciano Telesca and Mikhail Kanevski 22 | 23 | #### Maintainer 24 | Mohamed Laib [mohamed.laib (at) unil.ch] or 25 | [laib.med (at) gmail.com] 26 | 27 | #### URL 28 | [https://cran.r-project.org/package=MFDFA](https://cran.r-project.org/package=MFDFA) 29 | 30 | [https://mlaib.github.io/MFDFA/](https://mlaib.github.io/MFDFA/) 31 | 32 | [https://mlaib.github.io](https://mlaib.github.io) 33 | 34 | 35 | 36 | #### License 37 | GPL-3 38 | 39 | [![Downloads from the RStudio CRAN mirror](http://cranlogs.r-pkg.org/badges/grand-total/MFDFA)](http://cran.rstudio.com/package=MFDFA) 40 | 41 | #### Note 42 | If the codes are used in scientific publications please cite the following: 43 | 44 | * M. Laib, L. Telesca, 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 (3), 033108. [Paper](https://www.researchgate.net/publication/319121707_Long-range_fluctuations_and_multifractality_in_connectivity_density_time_series_of_a_wind_speed_monitoring_network) 45 | 46 | * 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](https://www.researchgate.net/publication/320223480_Multifractal_analysis_of_the_time_series_of_daily_means_of_wind_speed_in_complex_regions) 47 | 48 | ### MFDFA package installation: from github 49 | ```{r} 50 | install.packages("devtools") 51 | devtools::install_github("mlaib/MFDFA") 52 | library(MFDFA) 53 | ``` 54 | 55 | #### Example 56 | ```{r} 57 | a<-0.9 58 | N<-1024 59 | tsx<-MFsim(N,a) 60 | scale=10:100 61 | q<--10:10 62 | m<-1 63 | mfdfa<-MFDFA(tsx, scale, m, q) 64 | ``` 65 | 66 | #### Results plot 67 | ```{r} 68 | dev.new() 69 | par(mai=rep(1, 4)) 70 | plot(q, mfdfa$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 71 | cex.axis=1.8, main="Hurst exponent", 72 | ylim=c(min(mfdfa$Hq),max(mfdfa$Hq))) 73 | grid(col="midnightblue") 74 | axis(1) 75 | axis(2) 76 | ``` 77 | 78 | #### Little comparison 79 | ```{r} 80 | library(MFDFA) 81 | a<-0.9 82 | N<-10000 83 | tsx<-MFsim(N,a) 84 | 85 | scale=10:1000 86 | q<--10:10 87 | m<-1 88 | system.time(mfdfa<-MFDFA(tsx, scale, m, q)) 89 | # ~ 47.60 s 90 | 91 | devtools::source_gist("bb0c09df9593dad16ae270334ec3e7d7", filename = "MFDFA2.r") 92 | system.time(mfdfa<-MFDFA2(tsx, scale, m, q)) 93 | # ~ 12s 94 | ``` 95 | -------------------------------------------------------------------------------- /_config.yml: -------------------------------------------------------------------------------- 1 | theme: jekyll-theme-cayman -------------------------------------------------------------------------------- /man/MFDFA.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/MFDFA.R 3 | \name{MFDFA} 4 | \alias{MFDFA} 5 | \title{MultiFractal Detrended Fluctuation Analysis} 6 | \usage{ 7 | MFDFA(tsx, scale, m=1, q) 8 | } 9 | \arguments{ 10 | \item{tsx}{Univariate time series (must be a vector).} 11 | 12 | \item{scale}{Vector of scales. There is no default value 13 | for this parameter, please add values.} 14 | 15 | \item{m}{An integer of the polynomial order for the detrending (by default m=1).} 16 | 17 | \item{q}{q-order of the moment. There is no default value 18 | for this parameter, please add values.} 19 | } 20 | \value{ 21 | A list of the following elements: 22 | \itemize{ 23 | \item \code{Hq} Hurst exponent. 24 | \item \code{tau_q} Mass exponent. 25 | \item \code{spec} Multifractal spectrum (\eqn{\alpha}{\alpha} and 26 | \eqn{f(\alpha)}{f(\alpha)}) 27 | \item \code{Fq} Fluctuation function. 28 | } 29 | } 30 | \description{ 31 | Applies the MultiFractal Detrended Fluctuation Analysis (MFDFA) to time series. 32 | } 33 | \details{ 34 | The original code of this function is in Matlab, you can find it on the 35 | following website \href{https://ch.mathworks.com/matlabcentral/fileexchange/38262-multifractal-detrended-fluctuation-analyses?focused=5247306&tab=function}{Mathworks}. 36 | } 37 | \examples{ 38 | 39 | \dontrun{ 40 | ## MFDFA package installation: from github #### 41 | install.packages("devtools") 42 | devtools::install_github("mlaib/MFDFA") 43 | 44 | ## Get the Parellel version: 45 | devtools::source_gist("bb0c09df9593dad16ae270334ec3e7d7", filename = "MFDFA2.r") 46 | } 47 | 48 | library(MFDFA) 49 | a<-0.9 50 | N<-1024 51 | tsx<-MFsim(N,a) 52 | scale=10:100 53 | q<--10:10 54 | m<-1 55 | b<-MFDFA(tsx, scale, m, q) 56 | 57 | \dontrun{ 58 | ## Results plot #### 59 | dev.new() 60 | par(mai=rep(1, 4)) 61 | plot(q, b$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 62 | cex.axis=1.8, main="Hurst exponent", 63 | ylim=c(min(b$Hq),max(b$Hq))) 64 | grid(col="midnightblue") 65 | axis(1) 66 | axis(2) 67 | 68 | ################################## 69 | ## Suggestion of output plot: #### 70 | ## Supplementary functions: ##### 71 | reset <- function(){ 72 | par(mfrow=c(1, 1), oma=rep(0, 4), mar=rep(0, 4), new=TRUE) 73 | plot(0:1, 0:1, type="n", xlab="", ylab="", axes=FALSE)} 74 | 75 | poly_fit<-function(x,y,n){ 76 | formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='',collapse='+')))) 77 | res1<-coef(formule) 78 | poly.res<-res1[length(res1):1] 79 | allres<-list(polyfit=poly.res, model1=formule) 80 | return(allres)} 81 | 82 | ################################## 83 | ## Output plots: ################# 84 | dev.new() 85 | layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE),heights=c(4, 4)) 86 | ## b : mfdfa output 87 | par(mai=rep(0.8, 4)) 88 | 89 | ## 1st plot: Scaling function order Fq (q-order RMS) 90 | p1<-c(1,which(q==0),which(q==q[length(q)])) 91 | plot(log2(scale),log2(b$Fqi[,1]), pch=16, col=1, axes = F, xlab = "s (days)", 92 | ylab=expression('log'[2]*'(F'[q]*')'), cex=1, cex.lab=1.6, cex.axis=1.6, 93 | main= "Fluctuation function Fq", 94 | ylim=c(min(log2(b$Fqi[,c(p1)])),max(log2(b$Fqi[,c(p1)])))) 95 | lines(log2(scale),b$line[,1], type="l", col=1, lwd=2) 96 | grid(col="midnightblue") 97 | axis(2) 98 | lbl<-scale[c(1,floor(length(scale)/8),floor(length(scale)/4), 99 | floor(length(scale)/2),length(scale))] 100 | att<-log2(lbl) 101 | axis(1, at=att, labels=lbl) 102 | for (i in 2:3){ 103 | k<-p1[i] 104 | points(log2(scale), log2(b$Fqi[,k]), col=i,pch=16) 105 | lines(log2(scale),b$line[,k], type="l", col=i, lwd=2) 106 | } 107 | legend("bottomright", c(paste('q','=',q[p1] , sep=' ' )),cex=2,lwd=c(2,2,2), 108 | bty="n", col=1:3) 109 | 110 | ## 2nd plot: q-order Hurst exponent 111 | plot(q, b$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 112 | cex.axis=1.8, main="Hurst exponent", ylim=c(min(b$Hq),max(b$Hq))) 113 | grid(col="midnightblue") 114 | axis(1, cex=4) 115 | axis(2, cex=4) 116 | 117 | ## 3rd plot: q-order Mass exponent 118 | plot(q, b$tau_q, col=1, axes=F, cex.lab=1.8, cex.axis=1.8, 119 | main="Mass exponent", 120 | pch=16,ylab=expression(tau[q])) 121 | grid(col="midnightblue") 122 | axis(1, cex=4) 123 | axis(2, cex=4) 124 | 125 | ## 4th plot: Multifractal spectrum 126 | plot(b$spec$hq, b$spec$Dq, col=1, axes=F, pch=16, #main="Multifractal spectrum", 127 | ylab=bquote("f ("~alpha~")"),cex.lab=1.8, cex.axis=1.8, 128 | xlab=bquote(~alpha)) 129 | grid(col="midnightblue") 130 | axis(1, cex=4) 131 | axis(2, cex=4) 132 | 133 | x1=b$spec$hq 134 | y1=b$spec$Dq 135 | rr<-poly_fit(x1,y1,4) 136 | mm1<-rr$model1 137 | mm<-rr$polyfit 138 | x2<-seq(0,max(x1)+1,0.01) 139 | curv<-mm[1]*x2^4+mm[2]*x2^3+mm[3]*x2^2+mm[4]*x2+mm[5] 140 | lines(x2,curv, col="red", lwd=2) 141 | reset() 142 | legend("top", legend="MFDFA Plots", bty="n", cex=2) 143 | } 144 | } 145 | \references{ 146 | J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988. 147 | 148 | Espen A. F. Ihlen, Introduction to multifractal detrended fluctuation analysis 149 | in matlab, Frontiers in Physiology: Fractal Physiology, 3 (141),(2012) 1-18. 150 | 151 | J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, 152 | A. Bunde, H. Stanley, Multifractal detrended fluctuation analysis of 153 | nonstationary time series, Physica A: Statistical Mechanics and its 154 | Applications, 316 (1) (2002) 87 – 114. 155 | 156 | Kantelhardt J.W. (2012) Fractal and Multifractal Time Series. In: Meyers R. (eds) 157 | Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. 158 | 159 | M. Laib, L. Telesca and M. Kanevski, Long-range fluctuations and 160 | multifractality in connectivity density time series of a wind speed 161 | monitoring network, Chaos: An Interdisciplinary Journal of Nonlinear 162 | Science, 28 (2018) p. 033108, \href{https://aip.scitation.org/doi/10.1063/1.5022737}{Paper}. 163 | 164 | M. Laib, J. Golay, L. Telesca, M. Kanevski, Multifractal 165 | analysis of the time series of daily means of wind speed 166 | in complex regions, Chaos, Solitons & Fractals, 109 (2018) 167 | pp. 118-127, \href{https://www.sciencedirect.com/science/article/pii/S0960077918300699}{Paper}. 168 | } 169 | -------------------------------------------------------------------------------- /man/MFXDFA.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/MFXDFA.R 3 | \name{MFXDFA} 4 | \alias{MFXDFA} 5 | \title{Multifractal detrended cross-correlation analysis} 6 | \usage{ 7 | MFXDFA(tsx1, tsx2, scale, m=1, q) 8 | } 9 | \arguments{ 10 | \item{tsx1}{Univariate time series (must be a vector or a ts object).} 11 | 12 | \item{tsx2}{Univariate time series (must be a vector or a ts object).} 13 | 14 | \item{scale}{Vector of scales.} 15 | 16 | \item{m}{Polynomial order for the detrending (by default m=1).} 17 | 18 | \item{q}{q-order of the moment. There is no default value 19 | for this parameter, please add values.} 20 | } 21 | \value{ 22 | A list of the following elements: 23 | \itemize{ 24 | \item \code{Hq} Hurst exponent. 25 | \item \code{h} Holder exponent. 26 | \item \code{Dh} Multifractal spectrum. 27 | \item \code{Fq} Fluctuation function in log. 28 | } 29 | } 30 | \description{ 31 | Applies the MultiFractal Detrended Fluctuation cross-correlation Analysis (MFXDFA) on two time series. 32 | } 33 | \note{ 34 | The original code of this function is in Matlab, you can find it on the 35 | following website \href{https://ch.mathworks.com/matlabcentral/fileexchange/38262-multifractal-detrended-fluctuation-analyses?focused=5247306&tab=function}{Mathworks}. 36 | } 37 | \examples{ 38 | 39 | library(MFDFA) 40 | a<-0.6 41 | N<-1024 42 | 43 | tsx1<-MFsim(N,a) 44 | b<-0.8 45 | N<-1024 46 | 47 | tsx2<-MFsim(N,b) 48 | scale=10:100 49 | q<--10:10 50 | m<-1 51 | 52 | \dontrun{ 53 | b<-MFXDFA(tsx1, tsx2, scale, m=1, q) 54 | 55 | ## Supplementary functions: ##### 56 | reset <- function(){ 57 | par(mfrow=c(1, 1), oma=rep(0, 4), mar=rep(0, 4), new=TRUE) 58 | plot(0:1, 0:1, type="n", xlab="", ylab="", axes=FALSE)} 59 | 60 | poly_fit<-function(x,y,n){ 61 | formule<-lm(as.formula(paste('y~',paste('I(x^',1:n,')', sep='',collapse='+')))) 62 | res1<-coef(formule) 63 | poly.res<-res1[length(res1):1] 64 | allres<-list(polyfit=poly.res, model1=formule) 65 | return(allres)} 66 | 67 | ## Plot results: ##### 68 | dev.new() 69 | layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE),heights=c(4, 4)) 70 | ## b : mfdfa output 71 | par(mai=rep(0.8, 4)) 72 | 73 | ## 1st plot: Fluctuations function 74 | p1<-which(q==2) 75 | plot(log(scale),b$Fq[,p1], pch=16, col=1, axes = FALSE, xlab = "s", 76 | ylab=expression('log'*'(F'[2]*')'), cex=1, cex.lab=1.6, cex.axis=1.6, 77 | main= "Fluctuation function F for q=2", 78 | ylim=c(min(b$Fq[,c(p1)]),max(b$Fq[,c(p1)]))) 79 | lines(log(scale),b$line[,p1], type="l", col=1, lwd=2) 80 | grid(col="midnightblue") 81 | axis(2) 82 | lbl<-scale[c(1,floor(length(scale)/8),floor(length(scale)/4), 83 | floor(length(scale)/2),length(scale))] 84 | att<-log(lbl) 85 | axis(1, at=att, labels=lbl) 86 | 87 | ## 2nd plot: q-order Hurst exponent 88 | plot(q, b$Hq, col=1, axes= FALSE, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 89 | cex.axis=1.8, main="Hurst exponent", ylim=c(min(b$Hq),max(b$Hq))) 90 | grid(col="midnightblue") 91 | axis(1, cex=4) 92 | axis(2, cex=4) 93 | 94 | ## 3rd plot: Spectrum 95 | plot(b$h, b$Dh, col=1, axes=FALSE, pch=16, main="Multifractal spectrum", 96 | ylab=bquote("f ("~alpha~")"),cex.lab=1.8, cex.axis=1.8, 97 | xlab=bquote(~alpha)) 98 | grid(col="midnightblue") 99 | axis(1, cex=4) 100 | axis(2, cex=4) 101 | 102 | x1=b$h 103 | y1=b$Dh 104 | rr<-poly_fit(x1,y1,4) 105 | mm1<-rr$model1 106 | mm<-rr$polyfit 107 | x2<-seq(min(x1),max(x1)+1,0.01) 108 | curv<-mm[1]*x2^4+mm[2]*x2^3+mm[3]*x2^2+mm[4]*x2+mm[5] 109 | lines(x2,curv, col="red", lwd=2) 110 | } 111 | 112 | } 113 | \references{ 114 | J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988. 115 | 116 | Espen A. F. Ihlen, Introduction to multifractal detrended fluctuation analysis 117 | in matlab, Frontiers in Physiology: Fractal Physiology, 3 (141),(2012) 1-18. 118 | 119 | J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, 120 | A. Bunde, H. Stanley, Multifractal detrended fluctuation analysis of 121 | nonstationary time series, Physica A: Statistical Mechanics and its 122 | Applications, 316 (1) (2002) 87 – 114. 123 | 124 | Kantelhardt J.W. (2012) Fractal and Multifractal Time Series. In: Meyers R. (eds) Mathematics 125 | of Complexity and Dynamical Systems. Springer, New York, NY. 126 | } 127 | -------------------------------------------------------------------------------- /man/MFsim.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/MFsim.R 3 | \name{MFsim} 4 | \alias{MFsim} 5 | \title{Simulated multifractal series.} 6 | \usage{ 7 | MFsim(N,a) 8 | } 9 | \arguments{ 10 | \item{N}{The length of the generated multifractal series.} 11 | 12 | \item{a}{Exponent that takes values in [0.6, 1].} 13 | } 14 | \value{ 15 | A vector containing the multifractal series. 16 | } 17 | \description{ 18 | Generates series using the binomial multifractal model (see references). 19 | } 20 | \examples{ 21 | 22 | a<-0.9 23 | N<-1024 24 | tsx<-MFsim(N,a) 25 | scale=10:100 26 | q<--10:10 27 | m<-1 28 | b<-MFDFA(tsx, scale, m, q) 29 | 30 | dev.new() 31 | par(mai=rep(1, 4)) 32 | plot(q, b$Hq, col=1, axes= FALSE, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 33 | cex.axis=1.8, main="q-order Hurst exponent", ylim=c(min(b$Hq),max(b$Hq))) 34 | grid(col="midnightblue") 35 | axis(1) 36 | axis(2) 37 | 38 | \dontrun{ 39 | ## Example with Levy distribution #### 40 | require(rmutil) 41 | tsx <- rlevy(1000, 0, 1) 42 | scale=10:100 43 | q<--10:10 44 | m<-1 45 | b<-MFDFA(tsx, scale, m, q) 46 | 47 | dev.new() 48 | plot(q, b$Hq, col=1, axes= F, ylab=expression('h'[q]), pch=16, cex.lab=1.8, 49 | cex.axis=1.8, main="Hurst exponent", ylim=c(min(b$Hq),max(b$Hq))) 50 | grid(col="midnightblue") 51 | axis(1, cex=4) 52 | axis(2, cex=4) 53 | } 54 | 55 | } 56 | \references{ 57 | J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988. 58 | 59 | E.L. Flores-Márquez, A. Ramírez-Rojas, L. Telesca, Multifractal detrended 60 | fluctuation analysis of earthquake magnitude series of Mexican South Pacific 61 | Region, Applied Mathematics and Computation, Volume 265, 2015, 62 | Pages 1106-1114, ISSN 0096-3003. 63 | } 64 | -------------------------------------------------------------------------------- /man/MMA.Rd: -------------------------------------------------------------------------------- 1 | % Generated by roxygen2: do not edit by hand 2 | % Please edit documentation in R/MMA.R 3 | \name{MMA} 4 | \alias{MMA} 5 | \title{Multiscale Multifractal Analysis} 6 | \usage{ 7 | MMA(tsx, scale, qminmax, ovlap=0, m=2) 8 | } 9 | \arguments{ 10 | \item{tsx}{Univariate time series (must be a vector or a ts object).} 11 | 12 | \item{scale}{Vector of scales.} 13 | 14 | \item{qminmax}{Vector of two values min and max of q-order of the moment.} 15 | 16 | \item{ovlap}{Overlapping parameter (By default ovlap=0: no overlapping).} 17 | 18 | \item{m}{Polynomial order for the detrending (by defaults m=2).} 19 | } 20 | \value{ 21 | A matrix with three columns (q-order, scale (s), and the scale exponent). 22 | } 23 | \description{ 24 | Applies the Multiscale Multifractal Analysis (MMA) on time series. 25 | } 26 | \note{ 27 | The original code of this function is in Matlab, you can find it on the 28 | following website \href{https://physionet.org/physiotools/mma/}{Physionet}. See 29 | references below. 30 | } 31 | \examples{ 32 | 33 | \dontrun{ 34 | library(MFDFA) 35 | library(plotly) 36 | library(plot3D) 37 | 38 | a<-0.6 39 | N<-800 40 | tsx<-MFsim(N,a) 41 | scale=10:100 42 | res<-MMA(tsx, scale, qminmax=c(-10,10), ovlap=0, m=2) 43 | 44 | ## Visualisation 1: 45 | S_exponent <- matrix(res[,3], nrow=length(unique(res[,1])), ncol=length(min(scale):(max(scale)/5))) 46 | m_scale <- unique(res[,2]) 47 | q <- unique(res[,1]) 48 | plot_ly() \%>\% add_surface(x = ~m_scale, y = ~q, 49 | z = ~S_exponent) 50 | 51 | ## Visualisation 2: 52 | image2D(S_exponent, xlab="q", ylab="scale", axes=F) 53 | axis(1, seq(0,1,0.1), round(quantile(q, seq(0, 1, 0.1)), 2)) 54 | axis(2, seq(0,1,0.1), round(quantile(m_scale, seq(0, 1, 0.1)), 2)) 55 | } 56 | 57 | } 58 | \references{ 59 | J. Feder, Fractals, Plenum Press, New York, NY, USA, 1988. 60 | 61 | J. Gieraltowski, J. J. Zebrowski, and R. Baranowski, 62 | Multiscale multifractal analysis of heart rate variability recordings 63 | http://dx.doi.org/10.1103/PhysRevE.85.021915 64 | 65 | Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG, 66 | Mietus JE, Moody GB, Peng C-K, Stanley HE. PhysioBank, PhysioToolkit, 67 | and PhysioNet: Components of a New Research Resource for Complex 68 | Physiologic Signals. Circulation 101(23):e215-e220. 69 | 70 | J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, 71 | A. Bunde, H. Stanley, Multifractal detrended fluctuation analysis of 72 | nonstationary time series, Physica A: Statistical Mechanics and its 73 | Applications, 316 (1) (2002) 87 – 114. 74 | 75 | J. Gierałtowski, J. J. Żebrowski, and R. Baranowski, "Multiscale 76 | multifractal analysis of heart rate variability recordings with a 77 | large number of occurrences of arrhythmia," Phys. Rev. E 85, 021915 (2012) 78 | } 79 | --------------------------------------------------------------------------------