qsreg: Quantile spline regression

Description

Uses a penalized likelihood approach to estimate the conditional quantile function for regression data. This method is only implemented for univariate data. For the pairs (X,Y) the conditional quantile, f(x), is P( Y

Usage

qsreg(x, y, lam=NA, maxit=50, maxit.cv=10, tol=1e-7,  
offset=0, 
sc=sqrt(var(y)) * 1e-05, alpha=0.5, wt=rep(1, length(x)), cost=1, 
nstep.cv=100, hmin=NA, hmax=NA, trmin=2 * 1.05, trmax=0.95 * 
length(unique(x)))

Arguments

lam

Values of the smoothing parameter. If omitted is found by GCV based on the the quantile criterion

maxit

Maximum number of iterations used to estimate each quantile spline.

maxit.cv

Maximum number of iterations to find GCV minimum.

tol

Tolerance for convergence when computing quantile spline.

cost

Cost value used in the GCV criterion. Cost=1 is the usual GCV denominator.

offset

Constant added to the effective degrees of freedom in the GCV function.

Scale factor for rounding out the absolute value function at zero to a quadratic. Default is a small scale to produce something more like quantiles. Scales on the order of the residuals will result is a robust regression fit using the Huber weight func

alpha

Quantile to be estimated. Default is find the median.

Weight vector default is constant values. Passing nonconstant weights is a pretty strange thing to do.

nstep.cv

Number of points used in CV grid search

hmin

Minimum value of log( lambda) used for GCV grid search.

hmax

Maximum value of log( lambda) used for GCV grid search.

Value

trmin trmaxDefine the minimum and maximum values for the CV grid search in terms of the effective number of parameters. (see hmin, hmax) Object of class qsreg with many arguments similar to a sreg object. One difference is that cv.grid has five columns the last being the number of iterations for convergence at each value of lambda.

Details

This is an experimental function to find the smoothing parameter for a quantile spline using a more appropriate criterion than mean squared error prediction. The quantile spline is found by an iterative algorithm using weighted least squares cubic splines. At convergence the estimate will also be a weighted natural cubic spline but the weights will depend on the estimate. Of course these weights are crafted so that the resulting spline is an estimate of the alpha quantile instead of the mean. CV as function of lambda can be strange so it should be plotted.

Examples

Run this code

# fit a CV  quantile spline 
fit50<- qsreg(rat.diet$t,rat.diet$con)
# (default is .5 so this is an estimate of the conditional median) 
# control group of rats. 
plot( fit50) 
predict( fit50) 
# predicted values at data points 
xg<- seq(0,110,,50) 
plot( fit50$x, fit50$y) 
lines( xg, predict( fit50, xg))

# the data 
# The global GCV function is weird so it would be better to use the local 
# minima with largest lambda instead of this default value.  
# one should should consider redoing the three quantile fits in this 
# example after looking at the cv functions and choosing a good value for 
#lambda 
# for example 
lam<- fit50$cv.grid[,1] 
tr<- fit50$cv.grid[,2] 
# lambda close to df=6  
lambda.good<- max(lam[tr>=6]) 
fit50.subjective<-qsreg(rat.diet$t,rat.diet$con, lam= lambda.good)
fit10<-qsreg(rat.diet$t,rat.diet$con, alpha=.1, nstep.cv=200)
fit90<-qsreg(rat.diet$t,rat.diet$con, alpha=.9, nstep.cv=200)
# spline fits at 50 equally spaced points 
sm<- cbind( 

predict( fit10, xg),   
predict( fit50.subjective, xg),   
predict( fit90, xg))  
 
# and now zee data ... 
plot( fit50$x, fit50$y) 
# and now zee quantile splines at 10\% 50\% and 90\%.  
# 
matlines( xg, sm, col=c( 3,4,3), lty=1) # the spline