QTps.Rd
This function uses the standard thin plate spline function Tps
and a algorithm based on
psuedo data to compute robust smoothers based on the Huber weight function. By modifying the
symmetry of the Huber function and changing the scale one can also approximate a quantile
smoother. This function is experimental in that is not clear how efficient the psuedo-data
algorithm is acheiving convergence to a solution.
QTps(x, Y, ..., f.start = NULL, psi.scale = NULL, C = 1, alpha = 0.5, Niterations = 100,
tolerance = 0.001, verbose = FALSE)
QSreg(x, Y, lambda = NA, f.start = NULL, psi.scale = NULL,
C = 1, alpha = 0.5, Niterations = 100, tolerance = 0.001,
verbose = FALSE)
Locations of observations.
Observations
Value of the smoothing parameter. If NA found by an approximate corss-validation criterion.
Any other arguments to pass to the Tps function, which are then passed to the Krig function. give.warnings =FALSE
can be used to turn off pesky warnings when not important (see example below).
Scaling for huber robust weighting function. (See below.) Usually it is better to leave this at 1 and
just modify the scale psi.scale
according to the size of the residuals.
The initial value for the estimated function. If NULL then the constant function at the
median of Y
will be used. NOTE: This may not be a very good starting vector and a more robust
method would be to use a local robust smoother.
The scale value for the Huber function. When C=1, this is the point where the Huber weight function will
change from quadratic to linear. Default is to use the scale .05*mad(Y)
and C=1
. Very small scales relative to the
size of the residuals will cause the estimate to approximate a quantile spline. Very large scales will yield the
ordinary least squares spline.
The quantile that is estimated by the spline. Default is .5 giving a median. Equivalently this parameter controls the slope of the linear wings in the Huber function 2*alpha
for the positive wing and 2*(1-alpha)
for the negative wing.
Maximum number of interations of the psuedo data algorithm
Convergence criterion based on the relative change in the predicted values of the function estimate. Specifically if the criterion mean(abs(f.hat.new - f.hat))/mean(abs(f.hat))
is less than tolerance
the iterations re stopped.
If TRUE intermediate results are printed out.
These are experimental functions that use the psuedo-value algorithm to compute a class of robust and quantile problems. QTps
use the Tps
function as its least squares base smoother while QSreg
uses the efficient sreg
for 1-D cubic smoothing spline models. Currently for the 1-d spline problem we recommend using the (or at least comparing to ) the old qsreg
function. QSreg
was created to produce a more readable version of the 1-d method that follows the thin plate spline format.
The Thin Plate Spline/ Kriging model through fields is: Y.k= f(x.k) = P(x.k) + Z(x.k) + e.k
with the goal of estimating the smooth function: f(x)= P(x) + Z(x)
The extension in this function is that e.k can be heavy tailed or have outliers and one would still like a robust estimate of f(x). In the quantile approximation (very small scale parameter) f(x) is an estimate of the alpha quantile of the conditional distribution of Y given x.
The algorithm is iterative and involves at each step tapering the residuals in a nonlinear way. Let psi.wght be this tapering function then given an initial estimate of f, f.hat the new data for smoothing is
Y.psuedo<- f.hat + psi.scale* psi.wght( Y - f.hat, psi.scale=psi.scale, alpha=alpha)
A thin plate spline is now estimated for these data and a new prediction for f is found. This new vector is
used to define new psuedo values. Convergence is achieved when the the subsequent estimates of f.hat do not
change between interations. The advantage of this algorithm is at every step a standard "least squares" thin
plate spline is fit to the psuedo data. Because only the observation vector is changing at each iteration
Some matrix decompositions need only be found once and the computations at each subsequent iteration are efficient.
At convergence there is some asymptotic theory to suggest that the psuedo data can be fit using the least
squares spline and the standard smoothing techinques are valid. For example one can consider looking at the
cross-validation function for the psuedo-data as a robust version to select a smoothing parameter. This approach
is different from the weighted least squared algorithm used in the qsreg
function. Also qsreg
is only
designed to work with 1-d cubic smoothing splines.
The "sigma" function indicating the departure from a pure quadratic loss function has the definition
qsreg.sigma<-function(r, alpha = 0.5, C = 1)
temp<- ifelse( r< 0, ((1 - alpha) * r^2)/C , (alpha * r^2)/C)
temp<- ifelse( r >C, 2 * alpha * r - alpha * C, temp)
temp<- ifelse( r < -C, -2 * (1 - alpha) * r - (1 - alpha) * C, temp)
temp
The derivative of this function "psi" is
qsreg.psi<- function(r, alpha = 0.5, C = 1)
temp <- ifelse( r < 0, 2*(1-alpha)* r/C, 2*alpha * r/C )
temp <- ifelse( temp > 2*alpha, 2*alpha, temp)
temp <- ifelse( temp < -2*(1-alpha), -2*(1-alpha), temp)
temp
Note that if C is very small and if alpha = .5 then psi will essentially be 1 for r > 0 and -1 for r < 0. The key feature here is that outside a ceratin range the residual is truncated to a constant value. This is similar to the Windsorizing operation in classical robust statistics.
Another advantage of the psuedo data algotrithm is that at convergence one can just apply all the usual generic functions from Tps to the psuedo data fit. For example, predict, surface, print, etc. Some additional components are added to the Krig/Tps object, however, for information about the iterations and original data. Note that currently these are not reported in the summaries and printing of the output object.
A Krig
object with additional components:
Original Y values
A vector giving the convergence criterion at each iteration.
If TRUE then convergence criterion was less than the tolerance value.
Scaling factor used for the psi.wght function.
Value of alpha.
Oh, Hee-Seok, Thomas CM Lee, and Douglas W. Nychka. "Fast nonparametric quantile regression with arbitrary smoothing methods." Journal of Computational and Graphical Statistics 20.2 (2011): 510-526.
qsreg
data(ozone2)
x<- ozone2$lon.lat
y<- ozone2$y[16,]
# Smoothing fixed at 50 df
look1<- QTps( x,y, psi.scale= 15, df= 50)
#> Warning: 6 missing value(s) removed from data
if (FALSE) {
# Least squares spline (because scale is so large)
look2<- QTps( x,y, psi.scale= 100, df= 50)
#
y.outlier<- y
# add in a huge outlier.
y.outlier[58]<- 1e5
look.outlier1<- QTps( x,y.outlier, psi.scale= 15, df= 50,
give.warnings= FALSE)
# least squares spline.
look.outlier2<- QTps( x,y.outlier, psi.scale=100 , df= 50,
give.warnings= FALSE)
#
set.panel(2,2)
surface( look1)
title("robust spline")
surface( look2)
title("least squares spline")
surface( look.outlier1, zlim=c(0,250))
title("robust spline w/outlier")
points( rbind(x[58,]), pch="+")
surface( look.outlier2, zlim=c(0,250))
title("least squares spline w/outlier")
points( rbind(x[58,]), pch="+")
set.panel()
}
# some quantiles
look50 <- QTps( x,y, psi.scale=.5,)
#> Warning: 6 missing value(s) removed from data
look75 <- QTps( x,y,f.start= look50$fitted.values, alpha=.75)
#> Warning: 6 missing value(s) removed from data
# a simulated example that finds some different quantiles.
if (FALSE) {
set.seed(123)
N<- 400
x<- matrix(runif( N), ncol=1)
true.g<- x *(1-x)*2
true.g<- true.g/ mean( abs( true.g))
y<- true.g + .2*rnorm( N )
look0 <- QTps( x,y, psi.scale=10, df= 15)
look50 <- QTps( x,y, df=15)
look75 <- QTps( x,y,f.start= look50$fitted.values, df=15, alpha=.75)
}
if (FALSE) {
# this example tests the quantile estimate by Monte Carlo
# by creating many replicate points to increase the sample size.
# Replicate points are used because the computations for the
# spline are dominated by the number of unique locations not the
# total number of points.
set.seed(123)
N<- 80
M<- 200
x<- matrix( sort(runif( N)), ncol=1)
x<- matrix( rep( x[,1],M), ncol=1)
true.g<- x *(1-x)*2
true.g<- true.g/ mean( abs( true.g))
errors<- .2*(rexp( N*M) -1)
y<- c(matrix(true.g, ncol=M, nrow=N) + .2 * matrix( errors, ncol=M, nrow=N))
look0 <- QTps( x,y, psi.scale=10, df= 15)
look50 <- QTps( x,y, df=15)
look75 <- QTps( x,y, df=15, alpha=.75)
bplot.xy(x,y, N=25)
xg<- seq(0,1,,200)
lines( xg, predict( look0, x=xg), col="red")
lines( xg, predict( look50, x=xg), col="blue")
lines( xg, predict( look75, x=xg), col="green")
}
if (FALSE) {
# A comparison with qsreg
qsreg.fit50<- qsreg(rat.diet$t,rat.diet$con, sc=.5)
lam<- qsreg.fit50$cv.grid[,1]
df<- qsreg.fit50$cv.grid[,2]
M<- length(lam)
CV<-rep( NA, M)
M<- length( df)
fhat.old<- NULL
for ( k in M:1){
temp.obj<- QTps(rat.diet$t,rat.diet$con, f.start=fhat.old, psi.scale=.5, tolerance=1e-6,
verbose=FALSE, df= df[k],
give.warnings=FALSE)
# avoids warnings from Krig search on lambda
cat(k, " ")
CV[k] <- temp.obj$Qinfo$CV.psuedo
fhat.old<- temp.obj$fitted.values
}
plot( df, CV, type="l", lwd=2)
# psuedo data estimate
points( qsreg.fit50$cv.grid[,c(5,6)], col="blue")
# alternative CV estimate via reweighted LS
points( qsreg.fit50$cv.grid[,c(2,3)], col="red")
}