alaplace  R Documentation 
Maximum likelihood estimation of the 1, 2 and 3parameter asymmetric Laplace distributions (ALDs). The 2parameter ALD may, with trepidation and lots of skill, sometimes be used as an approximation of quantile regression.
alaplace1(tau = NULL, llocation = "identitylink",
ilocation = NULL, kappa = sqrt(tau/(1  tau)), Scale.arg = 1,
ishrinkage = 0.95, parallel.locat = TRUE ~ 0, digt = 4,
idf.mu = 3, zero = NULL, imethod = 1)
alaplace2(tau = NULL, llocation = "identitylink", lscale = "loglink",
ilocation = NULL, iscale = NULL, kappa = sqrt(tau/(1  tau)),
ishrinkage = 0.95,
parallel.locat = TRUE ~ 0,
parallel.scale = FALSE ~ 0,
digt = 4, idf.mu = 3, imethod = 1, zero = "scale")
alaplace3(llocation = "identitylink", lscale = "loglink",
lkappa = "loglink", ilocation = NULL, iscale = NULL,
ikappa = 1, imethod = 1, zero = c("scale", "kappa"))
tau, kappa 
Numeric vectors with

llocation, lscale, lkappa 
Character.
Parameter link functions for
location parameter 
ilocation, iscale, ikappa 
Optional initial values. If given, it must be numeric and values are recycled to the appropriate length. The default is to choose the value internally. 
parallel.locat, parallel.scale 
See the 
imethod 
Initialization method. Either the value 1, 2, 3 or 4. 
idf.mu 
Degrees of freedom for the cubic smoothing spline fit
applied to get an initial estimate of the location parameter.
See 
ishrinkage 
How much shrinkage is used when initializing 
Scale.arg 
The value of the scale parameter 
digt 
Passed into 
zero 
See 
These VGAM family functions implement one variant of asymmetric Laplace distributions (ALDs) suitable for quantile regression. Kotz et al. (2001) call it the ALD. Its density function is
f(y;\xi,\sigma,\kappa) = \frac{\sqrt{2}}{\sigma} \,
\frac{\kappa}{1 + \kappa^2} \,
\exp \left(  \frac{\sqrt{2}}{\sigma \, \kappa} y  \xi 
\right)
for y \leq \xi
, and
f(y;\xi,\sigma,\kappa) = \frac{\sqrt{2}}{\sigma} \,
\frac{\kappa}{1 + \kappa^2} \,
\exp \left(  \frac{\sqrt{2} \, \kappa}{\sigma} y  \xi 
\right)
for y > \xi
.
Here, the ranges are for all real y
and
\xi
, positive \sigma
and positive \kappa
.
The special case \kappa = 1
corresponds to the
(symmetric) Laplace distribution of Kotz et al. (2001).
The mean is \xi + \sigma (1/\kappa  \kappa) / \sqrt{2}
and the variance is
\sigma^2 (1 + \kappa^4) / (2 \kappa^2)
.
The enumeration of the linear/additive predictors used for
alaplace2()
is
the first location parameter followed by the first scale parameter,
then the second location parameter followed by the
second scale parameter, etc.
For alaplace3()
, only a vector response is handled
and the last (third) linear/additive predictor is for
the asymmetry parameter.
It is known that the maximum likelihood estimate of the
location parameter \xi
corresponds to the regression
quantile estimate of the classical quantile regression approach
of Koenker and Bassett (1978). An important property of the
ALD is that
P(Y \leq \xi) = \tau
where
\tau = \kappa^2 / (1 + \kappa^2)
so that
\kappa = \sqrt{\tau / (1\tau)}
.
Thus alaplace2()
might be used as an alternative to rq
in the quantreg package, although scoring is really
an unsuitable algorithm for estimation here.
Both alaplace1()
and alaplace2()
can handle
multiple responses, and the number of linear/additive
predictors is dictated by the length of tau
or
kappa
. The functions alaplace1()
and alaplace2()
can also
handle multiple responses (i.e., a matrix response)
but only with a singlevalued tau
or kappa
.
An object of class "vglmff"
(see
vglmffclass
). The object is used by modelling
functions such as vglm
and vgam
.
In the extra
slot of the fitted object are some list
components which are useful, e.g., the sample proportion of
values which are less than the fitted quantile curves.
These functions are experimental and especially subject to
change or withdrawal.
The usual MLE regularity conditions do not hold for
this distribution so that misleading inferences may result,
e.g., in the summary
and vcov
of the object. The
1parameter ALD can be approximated by extlogF1
which has continuous derivatives and is recommended over
alaplace1
.
Care is needed with tau
values which are too small,
e.g., for count data with llocation = "loglink"
and if
the sample proportion of zeros is greater than tau
.
These VGAM family functions use Fisher scoring.
Convergence may be slow and halfstepping is usual
(although one can use trace = TRUE
to see which
is the best model and then use maxit
to choose
that model)
due to the regularity conditions not holding.
Often the iterations slowly crawl towards the solution so
monitoring the convergence (set trace = TRUE
) is highly
recommended.
Instead, extlogF1
is recommended.
For large data sets it is a very good idea to keep the length of
tau
/kappa
low to avoid large memory requirements.
Then
for parallel.locat = FALSE
one can repeatedly fit a model with
alaplace1()
with one \tau
at a time;
and
for parallel.locat = TRUE
one can refit a model with
alaplace1()
with one \tau
at a time but
using offsets and an interceptonly model.
A second method for solving the noncrossing quantile problem is illustrated below in Example 3. This is called the accumulative quantile method (AQM) and details are in Yee (2015). It does not make the strong parallelism assumption.
The functions alaplace2()
and laplace
differ slightly in terms of the parameterizations.
Thomas W. Yee
Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46, 33–50.
Kotz, S., Kozubowski, T. J. and Podgorski, K. (2001). The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance, Boston: Birkhauser.
ralap
,
laplace
,
extlogF1
,
CommonVGAMffArguments
,
lms.bcn
,
amlnormal
,
sc.studentt2
,
simulate.vlm
.
## Not run:
# Example 1: quantile regression with smoothing splines
set.seed(123); adata < data.frame(x2 = sort(runif(n < 500)))
mymu < function(x) exp(2 + 6*sin(2*x0.2) / (x+0.5)^2)
adata < transform(adata, y = rpois(n, lambda = mymu(x2)))
mytau < c(0.25, 0.75); mydof < 4
fit < vgam(y ~ s(x2, df = mydof), data=adata, trace=TRUE, maxit = 900,
alaplace2(tau = mytau, llocat = "loglink",
parallel.locat = FALSE))
fitp < vgam(y ~ s(x2, df = mydof), data = adata, trace=TRUE, maxit=900,
alaplace2(tau = mytau, llocat = "loglink", parallel.locat = TRUE))
par(las = 1); mylwd < 1.5
with(adata, plot(x2, jitter(y, factor = 0.5), col = "orange",
main = "Example 1; green: parallel.locat = TRUE",
ylab = "y", pch = "o", cex = 0.75))
with(adata, matlines(x2, fitted(fit ), col = "blue",
lty = "solid", lwd = mylwd))
with(adata, matlines(x2, fitted(fitp), col = "green",
lty = "solid", lwd = mylwd))
finexgrid < seq(0, 1, len = 1001)
for (ii in 1:length(mytau))
lines(finexgrid, qpois(p = mytau[ii], lambda = mymu(finexgrid)),
col = "blue", lwd = mylwd)
fit@extra # Contains useful information
# Example 2: regression quantile at a new tau value from an existing fit
# Nb. regression splines are used here since it is easier.
fitp2 < vglm(y ~ sm.bs(x2, df = mydof), data = adata, trace = TRUE,
alaplace1(tau = mytau, llocation = "loglink",
parallel.locat = TRUE))
newtau < 0.5 # Want to refit the model with this tau value
fitp3 < vglm(y ~ 1 + offset(predict(fitp2)[, 1]),
alaplace1(tau = newtau, llocation = "loglink"), adata)
with(adata, plot(x2, jitter(y, factor = 0.5), col = "orange",
pch = "o", cex = 0.75, ylab = "y",
main = "Example 2; parallel.locat = TRUE"))
with(adata, matlines(x2, fitted(fitp2), col = "blue",
lty = 1, lwd = mylwd))
with(adata, matlines(x2, fitted(fitp3), col = "black",
lty = 1, lwd = mylwd))
# Example 3: noncrossing regression quantiles using a trick: obtain
# successive solutions which are added to previous solutions; use a log
# link to ensure an increasing quantiles at any value of x.
mytau < seq(0.2, 0.9, by = 0.1)
answer < matrix(0, nrow(adata), length(mytau)) # Stores the quantiles
adata < transform(adata, offsety = y*0)
usetau < mytau
for (ii in 1:length(mytau)) {
# cat("\n\nii = ", ii, "\n")
adata < transform(adata, usey = yoffsety)
iloc < ifelse(ii == 1, with(adata, median(y)), 1.0) # Wellchosen!
mydf < ifelse(ii == 1, 5, 3) # Maybe less smoothing will help
fit3 < vglm(usey ~ sm.ns(x2, df = mydf), data = adata, trace = TRUE,
alaplace2(tau = usetau[ii], lloc = "loglink", iloc = iloc))
answer[, ii] < (if(ii == 1) 0 else answer[, ii1]) + fitted(fit3)
adata < transform(adata, offsety = answer[, ii])
}
# Plot the results.
with(adata, plot(x2, y, col = "blue",
main = paste("Noncrossing and nonparallel; tau = ",
paste(mytau, collapse = ", "))))
with(adata, matlines(x2, answer, col = "orange", lty = 1))
# Zoom in near the origin.
with(adata, plot(x2, y, col = "blue", xlim = c(0, 0.2), ylim = 0:1,
main = paste("Noncrossing and nonparallel; tau = ",
paste(mytau, collapse = ", "))))
with(adata, matlines(x2, answer, col = "orange", lty = 1))
## End(Not run)
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