# Rearrangement Monotonize a step function by rearrangement. Returns a matrix or array of points which are monotonic, or a monotonic function performing linear (or constant) interpolation. ```r rearrangement(x,y,n=1000,stochastic=FALSE,avg=TRUE,order=1:length(x)) ``` ## Arguments - `x`: a list or data frame, the entries of which are vectors containing the x values corresponding to the fitted y values - `y`: a vector, matrix, or three-dimensional array containing the fitted values of a model, typically the result of a regression - `n`: an integer denoting the number of sample points desired - `stochastic`: logical. If TRUE, stochastic sampling will be used - `avg`: logical. If TRUE, the average rearrangement will be computed and outputted - `order`: a vector containing the desired permutation of the elements of 1:length(x). The rearrangement will be performed in the order specified if avg= FALSE, otherwise all the possible orderings are computed and the average rearrangement is reported ## Details This function applies this rearrangement operator of Hardy, Littlewood, and Polya (1952) to the estimate of a monotone function. Note: `rearrangement` currently only operates on univariate, bivariate, and trivariate regressions (that is, length(x)<=3). ## Returns `rearrangement` returns a matrix or array of equivalent dimension and size to y that is monotonically increasing in all of its dimensions. ## References Chernozhukov, V., I. Fernandez-Val, and a. Galichon. 2009. Improving point and interval estimators of monotone functions by rearrangement. Biometrika 96 (3): 559-575. Chernozhukov, V., I. Fernandez-Val, and a. Galichon. 2010. Quantile and Probability Curves Without Crossing. Econometrica 78(3): 1093-1125. Hardy, G.H., J.E. Littlewood, and G. Polya, Inequalities,2nd ed, Cambridge U. Press,1952 ## Author(s) Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon ## See Also `rconint`, `quantile` ## Examples ```r ##Univariate example: library(splines) data(GrowthChart) attach(GrowthChart) ages <- unique(sort(age)) aknots <- c(3, 5, 8, 10, 11.5, 13, 14.5, 16, 18) splines_age <- bs(age,kn=aknots) sformula <- height~splines_age sfunc <- approxfun(age,lm(sformula)$fitted.values) splreg <- sfunc(ages) rsplreg <- rearrangement(list(ages),splreg) plot(age,height,pch=21,bg='gray',cex=.5,xlab="Age (years)",ylab="Height (cms)", main="CEF (Regression Splines)",col='gray') lines(ages,splreg,col='red',lwd=3) lines(ages,rsplreg,col='blue',lwd=2) legend("topleft",c('Original','Rearranged'),lty=1,col=c('red','blue'),bty='n') detach(GrowthChart) ##Bivariate example: ## Not run: library(quantreg) data(GrowthChart) attach(GrowthChart) ages <- unique(sort(age)) taus <- c(1:999)/1000 nage <- 2 * pi * (age - min(age)) / (max(age) - min(age)) nages <- 2 * pi * (ages - min(ages)) / (max(ages) - min(ages)) fform <- height ~ I(sin(nage))+I(cos(nage))+I(sin(2*nage))+I(cos(2*nage))+ I(sin(3*nage))+I(cos(3*nage))+I(sin(4*nage))+I(cos(4*nage)) ffit <- rq(fform, tau = taus) fcoefs <- t(ffit$coef) freg <- rbind(1, sin(nages), cos(nages), sin(2*nages), cos(2*nages),sin(3*nages), cos(3*nages), sin(4*nages), cos(4*nages) ) fcqf <- crossprod(t(fcoefs),freg) rrfcqf <- rearrangement(list(taus,ages),fcqf, avg=TRUE) tdom <-c(1,10*c(1:99),999) adom <-c(1,5*c(1:floor(length(ages)/5)), length(ages)) par(mfrow=c(2,1)) persp(taus[tdom],ages[adom],rrfcqf[tdom,adom],xlab='quantile', ylab='age',zlab='height',col='lightgreen',theta=315,phi=25,shade=.5) title("CQP: Average Quantile/Age Rearrangement") contour(taus,ages,rrfcqf,xlab='quantile',ylab='age',col='green', levels=10*c(ceiling(min(fcqf)/10):floor(max(fcqf)/10))) title("CQP: Contour (RR-Quantile/Age)") detach(GrowthChart) ## End(Not run) ```

rearrangement function