alsqreg {VGAM}R Documentation

Asymmetric Least Squares Quantile Regression

Description

Quantile regression using asymmetric least squares error loss.

Usage

alsqreg(w.als=1, parallel=FALSE, lexpectile = "identity",
        eexpectile = list(), iexpectile = NULL,
        method.init=1, digw=4)

Arguments

w.als Numeric, a vector of positive constants controlling the percentiles. The larger the value the larger the fitted percentile value (the proportion of points below the ``w-regression plane''). The default value of unity results in the ordinary least squares (OLS) solution.
parallel If w.als has more than one value then this argument allows the quantile curves to differ by the same amount as a function of the covariates. Setting this to be TRUE should force the quantile curves to not cross (although they may not cross anyway). See CommonVGAMffArguments for more information.
lexpectile, eexpectile, iexpectile See CommonVGAMffArguments for more information.
method.init Integer, either 1 or 2 or 3. Initialization method. Choose another value if convergence fails.
digw Passed into Round as the digits argument for the w.als values; used cosmetically for labelling.

Details

This method was proposed by Efron (1991) and full details can be obtained there. Equation numbers below refer to that article. The model is essentially a linear model (see lm), however, the asymmetric squared error loss function for a residual r is r^2 if r <= 0 and w*r^2 if r > 0. The solution is the set of regression coefficients that minimize the sum of these over the data set, weighted by the weights argument (so that it can contain frequencies). Newton-Raphson estimation is used here.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

Note

On fitting, the extra slot has list components "w.als" and "percentile". The latter is the percent of observations below the ``w-regression plane'', which is the fitted values.

One difficulty is finding the w.als value giving a specified percentile. One solution is to fit the model within a root finding function such as uniroot; see the example below.

For alsqreg objects, methods functions for the generic functions qtplot and cdf have not been written yet.

See the note in amlpoisson on the jargon, including expectiles and regression quantiles.

The deviance slot computes the total asymmetric squared error loss (2.5). If w.als has more than one value then the value returned by the slot is the sum taken over all the w.als values.

This VGAM family function could well be renamed amlnormal() instead, given the other function names amlpoisson, amlbinomial, etc.

Author(s)

Thomas W. Yee

References

Efron, B. (1991) Regression percentiles using asymmetric squared error loss. Statistica Sinica, 1, 93–125.

See Also

amlpoisson, amlbinomial, amlexponential, bminz, alaplace1, lms.bcn and similar variants are alternative methods for quantile regression.

Examples

# Example 1
data(bminz)
o = with(bminz, order(age))
bminz = bminz[o,]  # Sort by age
(fit = vglm(BMI ~ bs(age), fam=alsqreg(w.als=0.1), data=bminz))
fit@extra  # Gives the w value and the percentile
coef(fit)
coef(fit, matrix=TRUE)

## Not run: 
# Quantile plot
with(bminz, plot(age, BMI, col="blue", main=
     paste(round(fit@extra$percentile, dig=1),
           "expectile-percentile curve")))
with(bminz, lines(age, c(fitted(fit)), col="black"))
## End(Not run)


# Example 2
# Find the w values that give the 25, 50 and 75 percentiles
findw = function(w, percentile=50) {
    fit2 = vglm(BMI ~ bs(age), fam=alsqreg(w=w), data=bminz)
    fit2@extra$percentile - percentile
}
## Not run: 
# Quantile plot
with(bminz, plot(age, BMI, col="blue", las=1, main=
     "25, 50 and 75 expectile-percentile curves"))
## End(Not run)
for(myp in c(25,50,75)) {
# Note: uniroot() can only find one root at a time
    bestw = uniroot(f=findw, interval=c(1/10^4, 10^4), percentile=myp)
    fit2 = vglm(BMI ~ bs(age), fam=alsqreg(w=bestw$root), data=bminz)
## Not run: 
    with(bminz, lines(age, c(fitted(fit2)), col="red"))
## End(Not run)
}


# Example 3; this is Example 1 but with smoothing splines and
# a vector w and a parallelism assumption.
data(bminz)
o = with(bminz, order(age))
bminz = bminz[o,]  # Sort by age
fit3 = vgam(BMI ~ s(age, df=4), fam=alsqreg(w=c(.1,1,10), parallel=TRUE),
            data=bminz, trac=TRUE)
fit3@extra # The w values, percentiles and weighted deviances

# The linear components of the fit; not for human consumption:
coef(fit3, matrix=TRUE)

## Not run: 
# Quantile plot
with(bminz, plot(age, BMI, col="blue", main=
     paste(paste(round(fit3@extra$percentile, dig=1), collapse=", "),
           "expectile-percentile curves")))
with(bminz, matlines(age, fitted(fit3), col=1:fit3@extra$M, lwd=2))
with(bminz, lines(age, c(fitted(fit )), col="black")) # For comparison
## End(Not run)

[Package VGAM version 0.7-7 Index]