library(pracma) # Basic functions for the Tetration distributions (TD, ITD and OTD) # It is a work in progress, with no guarantee. # A package is to be made available in the forthcoming months, cowritten with my colleague Frédéric Bertrand. # Feel free to report blatant errors or suggestions at yann.dijoux@utt.fr # Yann Dijoux, M2S Team, Université de Technologie de Troyes, May 9th, 2020 PhiCont<- function(z,gam) { stopifnot(is.numeric(c(z,gam))) Q<-rbind(c(0.500000000000000,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-0.0833333333333333,0.250000000000000,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(0.0208333333333333,-0.104166666666667,0.125000000000000,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-0.00555555555555642,0.0416666666666679,-0.0902777777777777,0.0625000000000000,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(0.00127314814816160,-0.0157986111111015,0.0515046296296404,-0.0668402777777750,0.0312500000000004,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-0.000148809523778937,0.00526620370345654,-0.0258680555555202,0.0506365740740762,-0.0453125000000050,0.0156250000000000,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-4.54695771168190e-05,-0.00133308531849252,0.0113136574059354,-0.0321180555561114,0.0434317129627857,-0.0290364583333371,0.00781249999999911,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(1.99790465558181e-05,0.000158110109623522,-0.00405953759036493,0.0174816743892734,-0.0336202739281362,0.0340108989199734,-0.0178943452380764,0.00390624999999822,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(1.13215937744826e-05,1.98300695046783e-05,0.00106265651993454,-0.00798717301222496,0.0220179880561773,-0.0313159079123579,0.0249550471262410,-0.0107166108629997,0.00195312500000266,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-1.13206915557384e-05,2.81203538179398e-05,-0.000188057310879231,0.00292184669524431,-0.0121852848678827,0.0240907720872201,-0.0267906803492224,0.0174372597357433,-0.00627919353490825,0.000976562499999112,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-1.76951289176941e-06,-3.98308038711548e-05,9.28491353988648e-05,-0.000857435166835785,0.00558566674590111,-0.0156721975654364,0.0237718045245856,-0.0214808784949128,0.0117297621782200,-0.00361631686075725,0.000488281249988454,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(7.21216201782227e-06,2.38418579101563e-06,-9.96589660644531e-05,0.000321388244628906,-0.00214344263076782,0.00857207179069519,-0.0178075991570950,0.0216702790930867,-0.0163663738640025,0.00765414017223520,-0.00205391667668664,0.000244140624994671,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-7.62939453125000e-06,-1.52587890625000e-05,-2.28881835937500e-05,-0.000265121459960938,0.000832557678222656,-0.00399208068847656,0.0112962722778320,-0.0184075683355331,0.0185526404529810,-0.0119658703915775,0.00487200206407579,-0.00115322830583864,0.000122070312475131,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(4.57763671875000e-05,0.000488281250000000,-0.000976562500000000,0.000671386718750000,-0.000488281250000000,0.00173187255859375,-0.00622558593750000,0.0132861137390137,-0.0176562368869782,0.0150940008461475,-0.00845607509836555,0.00303759692906169,-0.000641314146605509,6.10351563210543e-05,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-0.00219726562500000,0.0439453125000000,0.0644531250000000,0.0498046875000000,0.00244140625000000,0.00463867187500000,0.00415039062500000,-0.00770568847656250,0.0142803192138672,-0.0159293413162231,0.0117721408605576,-0.00580776389688253,0.00186104120803066,-0.000353733671090595,3.05175781676326e-05,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(-0.125000000000000,-0.187500000000000,0.0937500000000000,-0.718750000000000,-0.0468750000000000,-0.0156250000000000,0.0234375000000000,0.00170898437500000,-0.0101318359375000,0.0141830444335938,-0.0136723518371582,0.00885996222496033,-0.00389338564127684,0.00112325476948172,-0.000193744228454307,1.52587891655287e-05,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),c(4.12500000000000,15.5000000000000,57.5000000000000,47,-3.50000000000000,2.37500000000000,1.31250000000000,0.820312500000000,0.00976562500000000,-0.0168457031250000,0.0131683349609375,-0.0112361907958984,0.00646859407424927,-0.00255608744919300,0.000669227913022041,-0.000105469729533070,7.62939440335231e-06,0,0,0,0,0,0,0,0,0,0,0,0,0,0)) 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in 1:50) { dn=dn/(1+res) res<-log1p(res) } entier<-round(gam) gammaF<-gam-entier inter<-0:(n-1) VectQi<-Q%*%(gammaF^inter) Qprime<-t(t(cbind(Q, zeros(n,1))[1:n,2:(n+1)])*1:n) VectQprimei<-Qprime%*%(gammaF^inter) Phizg<-res+res^2*gammaF*horner(rev(VectQi),res)$y dPz=1+res*gammaF*horner(rev(VectQi*(inter+2)),res)$y dPg=res^2*horner(rev(VectQi),res)$y+res^2*gammaF*horner(rev(VectQprimei),res)$y dp<-0 for (i in 1:(50+entier)) { dp<-dp+Phizg #pour deriv Phizg<-expm1(Phizg) } dPhiz<-dPz*dn*exp(dp) dPhig<-dPg*exp(dp) result<-list("z"=Phizg,"dz"=dPhiz,"dg"=dPhig) return(result) } PhiContExp<- function(z,gam){ #seuil à 700 plutot que zero? Phiz<-z*0 dPhiz<-z*0 Phin<-PhiCont(exp(z[z<0]),gam) Phiz[z<0]<-Phin$z dPhiz[z<0]<-Phin$dz*exp(z[z<0]) Phip<-PhiCont(z[z>=0]+log(1+exp(-z[z>=0])),gam+1) Phiz[z>=0]<-Phip$z dPhiz[z>=0]<-Phip$dz/(1+exp(-z[z>=0])) result<-list("z"=Phiz,"dz"=dPhiz) return(result) } LogPhiCont<- function(z,gam){ if(gam<0){return(log(PhiCont(z,gam)$z))} else{ phi<- PhiCont(z,gam-1)$z return(phi+log(1-exp(-phi))) } } VerifParam<- function(UFP=NULL,UF=NULL,NUFP=0,NUF=0,NUM=1,PRO=0.5,BOO=TRUE) { if(anyNA(c(NUM,PRO))){warning("Vectors contain NA/NaN")} AllNum<-c(UFP,UF,NUM[!is.na(NUM)],PRO[!is.na(NUM)]) flag<-is.vector(AllNum,mode = "numeric")&&all(is.finite(c(0,UFP,UF)))&&(length(UFP)==NUFP)&&(length(UF)==NUF)&&all(UFP>0)&&(!any(PRO[is.finite(PRO)]>1|PRO[is.finite(PRO)]<0))&&is.logical(BOO) if(!is.logical(flag)){warning("********************Revoir code**************************")} if(!isTRUE(flag)){warning("incorrect arguments")} return(isTRUE(flag)) } pTetration<- function(x,tetration=0,shapein=1, shapeout=1, scalein=1, scaleout=1, location=0,lower.tail = TRUE){ flag<-VerifParam(c(shapein,shapeout,scalein,scaleout),c(tetration,location),4,2,x,BOO=lower.tail) stopifnot(flag) res<-x cond<-((x>=location)&is.finite(x))%in%T res[cond]<- exp(-(scalein/scaleout*PhiCont(((x[cond]-location)/scalein)^shapein,tetration)$z)^(shapeout/shapein)) res[(x==Inf)%in%T]<-0 res[(x=location)&is.finite(x))%in%T phi<-PhiCont(((x[cond]-location)/scalein)^shapein,tetration) res[cond]<- exp(-(scalein/scaleout*phi$z)^(shapeout/shapein))*shapeout/scaleout*((x[cond]-location)/scalein)^(shapein-1)*phi$dz*((scalein/scaleout*phi$z))^(shapeout/shapein-1) res[(x==Inf)%in%T]<-0 res[(x0)&(p<1))%in%T res[cond]<-location+scalein*(PhiCont(scaleout/scalein*(log1p(p[cond]/(1-p[cond])))^(shapein/shapeout),-tetration)$z)^(1/shapein) res[(p==1)%in%T]<-Inf res[(p==0)%in%T]<-location return(res) } rTetration<- function(n,tetration=0,shapein=1, shapeout=1, scalein=1, scaleout=1, location=0){ return(qTetration(runif(n),tetration,shapein,shapeout,scalein,scaleout,location)) } hTetration<- function(x,tetration=0,shapein=1, shapeout=1, scalein=1, scaleout=1, location=0){ flag<-VerifParam(c(shapein,shapeout,scalein,scaleout),c(tetration,location),4,2,x) stopifnot(flag) hres<-x Hres<-x cond<-((x>=location)&is.finite(x))%in%T phi<-PhiCont(((x[cond]-location)/scalein)^shapein,tetration) common<-((scalein/scaleout*phi$z)) Hres[cond]<- (common)^(shapeout/shapein) hres[cond]<- shapeout/scaleout*((x[cond]-location)/scalein)^(shapein-1)*phi$dz*(common)^(shapeout/shapein-1) hres[(x==Inf)%in%T]<-ifelse(tetration<0,0,ifelse((tetration>0)|(shapeout>1),Inf, ifelse(shapeout<1,0,1/scaleout))) hres[(x0)&(x[3]>0)){ Like<- log(dITetration(X,x[1],x[2],x[3])) return(-sum(Like[is.finite(Like)]))} else{return(Inf)} } stopifnot(all(is.numeric(X))) method <- match.arg(method, c("mle", "lse")) data.name <- deparse(substitute(X)) if ((bad.obs <- sum(!(X.ok <- is.finite(X)))) > 0) { X <- X[X.ok] warning(paste(sep = "",bad.obs, " observation",ifelse(bad.obs==1,character(1),"s")," with NA/NaN/Inf in ",data.name," removed.")) } n <- length(X) if (n < 3 || length(unique(X)) < 3) stop(paste("'X' must contain at least 3 non-missing distinct values")) if (!is.vector(plot.pos.con, mode = "numeric") || length(plot.pos.con) != 1 || plot.pos.con < 0 || plot.pos.con > 1) stop("'plot.pos.con' must be a numeric scalar between 0 and 1") ABS<-log(sort(X)) n<-length(X) Femp <- ppoints(n, a = plot.pos.con) estim<-optimize(f=ResiduITetration,interval=c(-2,2)) ORD<-LogPhiCont(-log(1-Femp),-estim$minimum) mylm<-lm(ORD~ABS) reslse<-as.vector(c(estim$minimum,(mylm$coefficients[2]),(exp(-mylm$coefficients[1]/mylm$coefficients[2])))) switch(method, mle = { estim<-fminsearch(logLITetration,reslse) dist.params<-c(tetration=estim$xmin[1], shape=estim$xmin[2], scale=estim$xmin[3]) }, lse = { dist.params<-c(tetration=reslse[1], shape=reslse[2], scale=reslse[3]) }) ret.list <- list(distribution = "I Tetration", sample.size = n, parameters = dist.params, n.param.est = 3, method = method, data.name = data.name, bad.obs = bad.obs) oldClass(ret.list) <- "estimate" ret.list } eOTetration<- function(X, method="mle", plot.pos.con = 0.375){ ResiduOTetration<- function(x){ if (max(abs(x))<1.2){ ABS<-LogPhiCont(sort(Xsim),x) mylm<-lm(ORD~ABS) return(-summary(mylm)$r.squared) } else{return(Inf)} } logLOTetration<- function(x){ if((abs(x[1]<2))&(x[2]>0)&(x[3]>0)){ Like<- log(dOTetration(X,x[1],x[2],x[3])) return(-sum(Like[is.finite(Like)]))} else{return(Inf)} } stopifnot(all(is.numeric(X))) method <- match.arg(method, c("mle", "lse")) data.name <- deparse(substitute(X)) if ((bad.obs <- sum(!(X.ok <- is.finite(X)))) > 0) { X <- X[X.ok] warning(paste(sep = "",bad.obs, " observation",ifelse(bad.obs==1,character(1),"s")," with NA/NaN/Inf in ",data.name," removed.")) } n <- length(X) if (n < 3 || length(unique(X)) < 3) stop(paste("'X' must contain at least 3 non-missing distinct values")) if (!is.vector(plot.pos.con, mode = "numeric") || length(plot.pos.con) != 1 || plot.pos.con < 0 || plot.pos.con > 1) stop("'plot.pos.con' must be a numeric scalar between 0 and 1") n<-length(X) Femp <- ppoints(n, a = plot.pos.con) ORD<-log(-log(1-Femp)) estim<-optimize(f=ResiduOTetration,interval=c(-2,2)) ABS<-LogPhiCont(sort(Xsim),estim$minimum) mylm<-lm(ORD~ABS) reslse<-as.vector(c(estim$minimum,(mylm$coefficients[2]),(exp(-mylm$coefficients[1]/mylm$coefficients[2])))) switch(method, mle = { estim<-fminsearch(logLOTetration,reslse) dist.params<-c(tetration=estim$xmin[1], shape=estim$xmin[2], scale=estim$xmin[3]) }, lse = { dist.params<-c(tetration=reslse[1], shape=reslse[2], scale=reslse[3]) }) ret.list <- list(distribution = "O Tetration", sample.size = n, parameters = dist.params, n.param.est = 3, method = method, data.name = data.name, bad.obs = bad.obs) oldClass(ret.list) <- "estimate" ret.list } eTetration<- function(X){ locationstart<-min(X) paramstart<-eITetration(sort(X)[2:length(X)]-locationstart,method="mle")$parameters logLTetration<- function(x){ gam<-x[1] shapein<-x[2] shapeout<-x[3] scalein<-x[4] scaleout<-x[5] location<-x[6] if (abs(gam)<2 & shapein>0&shapeout>0&scalein>0&scaleout>0&location