| Type: | Package |
| Title: | Monotonize Point and Interval Functional Estimates by Rearrangement |
| Version: | 2.1 |
| Author: | Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon |
| Maintainer: | Ivan Fernandez-Val <ivanf@bu.edu> |
| Description: | The rearrangement operator (Hardy, Littlewood, and Polya 1952) for univariate, bivariate, and trivariate point estimates of monotonic functions. The package additionally provides a function that creates simultaneous confidence intervals for univariate functions and applies the rearrangement operator to these confidence intervals. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| LazyLoad: | yes |
| Depends: | quantreg, splines |
| NeedsCompilation: | no |
| Packaged: | 2016-03-01 18:33:53 UTC; Mingli |
| Repository: | CRAN |
| Date/Publication: | 2016-03-02 01:48:30 |
Point and Interval Rearrangement
Description
This package implements the rearrangement operator (Hardy, Littlewood, and Polya 1952) for univariate, bivariate, and trivariate point estimates of monotonic functions. It additionally provides a function that creates simultaneous confidence intervals for univariate functions and applies the rearrangement operator to these confidence intervals.
Details
| Package: | Rearrangement |
| Type: | Package |
| Version: | 1.0 |
| Date: | 2011-09-11 |
| License: | GPL(>=2) |
| LazyLoad: | yes |
This package is used for rearranging both point and interval estimates of a target function. Given an original point estimate of a target function, one may use rearrangement to monotonize this estimate. One may also create simultaneous confidence interval estimates using simconboot and monotonize these estimates using rconint.
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
Maintainer: Ivan Fernandez-Val <ivanf@bu.edu>
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
Examples
##rearrangement 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)
##rconint example:
## Not run:
data(GrowthChart)
attach(GrowthChart)
nage <- 2 * pi * (age - min(age)) / (max(age) - min(age))
formula <- 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))
j <- simconboot(nage,height,lm,formula)
k <- rconint(j)
plot(k, border=NA, col='darkgray')
polygon.conint(j, border=NA, col='lightgray')
polygon.conint(k, border=NA, col='darkgray', density=50)
points(nage,height)
detach(GrowthChart)
## End(Not run)
Age and Height of White Males
Description
This data set contains age and height of US-born white males age two through twenty. Note that age is measured in months and expressed in years, and height is measured in centimeters.
Usage
data(GrowthChart)Format
A data frame with 533 observations on the following 3 variables.
sexa numeric vector. Male = 1
heighta numeric vector. Height in cm
agea numeric vector. Age in years
Source
The data consist of repeated cross sectional measurements of height and age from the 2003-2004 National Health and Nutrition Survey collected by the US National Center for Health Statistics.
Examples
data(GrowthChart)
attach(GrowthChart)
plot(age,height,pch=21,bg='gray',cex=.5,
xlab="Age (years)",ylab="Height (cms)",col='gray')
detach(GrowthChart)
Local Constant Estimator for Conditional Mean Functions
Description
Implements the local nonparametric method kernel estimator–with box kernel (default), for conditional mean functions.
Usage
lclm(x, y, h, xx)
Arguments
x |
The conditioning covariate |
y |
The response variable |
h |
The bandwidth parameter |
xx |
The points at which the function is to be estimated |
Details
The function uses a box kernel.
Value
xx |
The design points at which the evaluation occurs |
fitted.values |
The estimated function values at these design points |
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
Examples
data(GrowthChart)
attach(GrowthChart)
ages <- unique(sort(age))
lclm.fit1 <- lclm(age,height,h=1,xx=ages)
detach(GrowthChart)
Local Constant Estimator for Conditional Quantile Functions
Description
Implements the local nonparametric method kernel estimator–with box kernel (default), for conditional quantile functions.
This is a modification of Koenker's lprq (from package quantreg).
Usage
lcrq2(x, y, h, xx, tau)
Arguments
x |
The conditioning covariate |
y |
The response variable |
h |
The bandwidth parameter |
xx |
The points at which the function is to be estimated |
tau |
The quantile(s) to be estimated. This should be a list of quantiles if the function estimates the quantile process |
Details
The function uses a box kernel.
Value
xx |
The design points at which the evaluation occurs |
fitted.values |
The estimated function values at these design points |
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
See Also
Examples
require(quantreg)
data(GrowthChart)
attach(GrowthChart)
ages <- unique(sort(age))
lcq.fit1 <- lcrq2(age,height,h=1,xx=ages,tau=0.01)
detach(GrowthChart)
Lines Method for Simultaneous Confidence Intervals
Description
A method for the lines generic. It graphs both the upper and lower end-point functions of a confidence interval as lines on a plot.
Usage
## S3 method for class 'conint'
lines(x, ...)
Arguments
x |
object of class |
... |
further arguments to |
Details
This is intended for plotting confidence intervals produced by the output of simconboot or rconint.
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
See Also
lines,plot.conint,points.conint
Examples
data(GrowthChart)
attach(GrowthChart)
nage <- 2*pi*(age-min(age))/(max(age)-min(age))
formula<-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))
j<-simconboot(nage,height,lm,formula)
plot(nage,height,pch=21,bg='gray',cex=.5,xlab="Age (years)",ylab="Height (cms)",col='gray',xaxt='n')
axis(1, at = seq(-2*pi*min(age)/(max(age)-min(age)),
2*pi+1, by=5*2*pi/(max(age)-min(age))), label = seq(0, max(age)+1, by=5))
lines(j)
detach(GrowthChart)
Local Linear Regression Methods for Conditional Mean Functions
Description
Implements the local nonparametric method, local linear regression estimator with box kernel (default), for conditional mean functions.
Usage
lplm(x, y, h, xx)
Arguments
x |
The conditioning covariate |
y |
The response variable |
h |
The bandwidth parameter |
xx |
The points at which the function is to be estimated |
Details
The function uses a box kernel.
Value
xx |
The design points at which the evaluation occurs |
fitted.values |
The estimated function values at these design points |
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
Examples
data(GrowthChart)
attach(GrowthChart)
ages <- unique(sort(age))
lplm.fit1 <- lplm(age,height,h=1,xx=ages)
detach(GrowthChart)
Local Linear Regression Methods for Conditional Quantile Functions
Description
Implements the local nonparametric method, local linear regression estimator with box kernel (default), for conditional quantile functions.
This is a modification of Koenker's lprq ( from package quantreg).
Usage
lprq2(x, y, h, xx, tau)
Arguments
x |
The conditioning covariate |
y |
The response variable |
h |
The bandwidth parameter |
xx |
The points at which the function is to be estimated |
tau |
The quantile(s) to be estimated. This should be a list of quantiles if the function estimates the quantile process |
Details
The function uses a box kernel.
Value
xx |
The design points at which the evaluation occurs |
fitted.values |
The estimated function values at these design points |
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
Examples
require(quantreg)
data(GrowthChart)
attach(GrowthChart)
ages <- unique(sort(age))
llq.fit1 <- lprq2(age,height,h=1,xx=ages,tau=0.2)
detach(GrowthChart)
Plot Method for Simultaneous Confidence Intervals
Description
A method for the plot generic. It graphs both the upper and lower end-point functions of a confidence interval as an unfilled polygon.
Usage
## S3 method for class 'conint'
plot(x, border, col, ...)
Arguments
x |
object of class |
border, col |
same usage as in |
... |
further arguments to |
Details
This is intended for plotting confidence intervals produced by the output of simconboot or rconint.
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
See Also
plot, lines.conint, points.conint
Examples
data(GrowthChart)
attach(GrowthChart)
nage <- 2 * pi * (age - min(age)) / (max(age) - min(age))
formula <- 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))
j<-simconboot(nage,height,lm,formula)
plot(j)
points(nage,height)
detach(GrowthChart)
Points Method for Simultaneous Confidence Intervals
Description
A method for the points generic. It graphs both the upper and lower end-point functions of a confidence interval as points on a plot.
Usage
## S3 method for class 'conint'
points(x, ...)
Arguments
x |
object of class |
... |
further arguments to |
Details
This is intended for plotting confidence intervals produced by the output of simconboot or rconint.
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
See Also
points, plot.conint, lines.conint
Examples
data(GrowthChart)
attach(GrowthChart)
nage <- 2 * pi * (age - min(age)) / (max(age) - min(age))
formula<-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))
j <- simconboot(nage,height,lm,formula)
plot(nage,height,pch=21,bg='gray',cex=.5,xlab="Age (years)",ylab="Height (cms)",col="gray")
points(j)
detach(GrowthChart)
polygon Method for Simultaneous Confidence Intervals
Description
polygon.conint graphs both the upper and lower end-point functions of a confidence interval as a standard polygon on a plot.
Usage
polygon.conint(x, ...)
Arguments
x |
object of class |
... |
further arguments to |
Details
This is intended for plotting confidence intervals produced by the output of simconboot or rconint.
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
See Also
Examples
## Not run: data(GrowthChart)
attach(GrowthChart)
nage <- 2 * pi * (age - min(age)) / (max(age) - min(age))
formula <- 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))
j<-simconboot(nage,height,lm,formula)
plot(nage,height,pch=21,bg='gray',cex=0.5,xlab="Age (years)",
ylab="Height (cms)",col='gray',xaxt='n')
axis(1, at = seq(-2*pi*min(age)/(max(age)-min(age)),
2*pi+1,by=5*2*pi/(max(age)-min(age))), label = seq(0, max(age)+1, by=5))
polygon.conint(j, border=NA, col='darkgray')
detach(GrowthChart)
## End(Not run)
Rearrangement of Simultaneous Confidence Intervals
Description
Uses rearrangement to apply the rearrangement operator to objects of class conint.
Usage
rconint(x, n = 100, stochastic = FALSE, avg = TRUE)
Arguments
x |
object of class |
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 outputed |
Details
Implements the rearrangement operator of rearrangement on simultaneous confidence intervals. Intended for use on output of simconboot.
Value
An object of class conint with the following elements:
x |
the original x data |
y |
the original y data |
sortedx |
the original x data, sorted with repeated elements removed |
Lower |
the rearranged lower end-point function. Represented as a vector of values corresponding to sortedx |
Upper |
the rearranged upper end-point function. Represented as a vector of values corresponding to sortedx |
cef |
the corresponding estimates |
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
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.
See Also
Examples
## Not run:
data(GrowthChart)
attach(GrowthChart)
nage <- 2 * pi * (age - min(age)) / (max(age) - min(age))
formula <- 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))
j <- simconboot(nage,height,lm,formula)
k <- rconint(j)
plot(k, border=NA, col='darkgray',xlab = 'Age (years)',ylab = 'Height (cms)',xaxt = "n")
axis(1, at = seq(-2*pi*min(age)/(max(age)-min(age)),
2*pi+1,by=5*2*pi/(max(age)-min(age))), label = seq(0, max(age)+1, by=5))
polygon.conint(j, border=NA, col='lightgray')
polygon.conint(k, border=NA, col='darkgray', density=50)
points(nage,height,col='gray80')
legend(min(nage),max(height),c("95% CI Original",
"95% CI Rearranged"),lty=c(1,1),lwd=c(2,2),
col=c("lightgray","darkgray"),bty="n");
detach(GrowthChart)
## End(Not run)
Rearrangement
Description
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.
Usage
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).
Value
rearrangement returns a matrix or array of equivalent dimension and size to y that is monotonically increasing in all of its dimensions.
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
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
See Also
Examples
##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)
Simultaneous Confidence Interval Estimation using Bootstrap
Description
simconboot obtains a simultaneous confidence interval for a function. It estimates the lower and upper endpoint functions of the interval by bootstrap.
Usage
simconboot(x,y,estimator,formula,B=200,alpha=0.05,sampsize=length(x),
seed=8,colInt=c(5:39)/2,...)
Arguments
x |
a numerical vector of x values |
y |
a numerical vector of y values |
estimator |
estimator to be used in regression |
formula |
formula to be used in the estimator |
B |
an integer with the number of bootstrap repetitions |
alpha |
a real number between 0 and 1 reflecting the desired confidence level |
sampsize |
an integer with the sample size of each bootstrap repetition |
seed |
if desired, seed to be set for the random number generator |
colInt |
the points to be evaluated when ploting |
... |
further arguments to be passed to the estimator |
Details
estimator can be any of a set of standard regression models, most commonly lm or rq (from package quantreg) for global
estimators and the built-in functions lclm, lplm, lcrq2, lprq2 for local estimators.
Note: formula=0 for all the local estimators.
Value
An object of class conint with the following elements:
x |
the original x data |
y |
the original y data |
sortedx |
the original x data, sorted with repeated elements removed |
Lower |
the lower endpoint function. Represented as a vector of values corresponding to sortedx |
Upper |
the upper endpoint function. Represented as a vector of values corresponding to sortedx |
cef |
the corresponding estimates |
Author(s)
Wesley Graybill, Mingli Chen, Victor Chernozhukov, Ivan Fernandez-Val, Alfred Galichon
See Also
Examples
data(GrowthChart)
attach(GrowthChart)
nage <- 2 * pi * (age - min(age)) / (max(age) - min(age))
nages <- unique(sort(nage))
formula <- 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))
j <- simconboot(nage,height,lm,formula)
plot(j, border=NA, col='darkgray',xlab = 'Age (years)',ylab = 'Height (cms)',xaxt = "n")
axis(1, at = seq(-2*pi*min(age)/(max(age)-min(age)), 2*pi+1,
by=5*2*pi/(max(age)-min(age))), label = seq(0, max(age)+1, by=5))
points(nage,height)
lines(nages, j$cef, lty=2, col='green')
detach(GrowthChart)