alaplace: Asymmetric Laplace Distribution Family Functions

Description

Maximum likelihood estimation of the 1, 2 and 3-parameter asymmetric Laplace distributions (ALDs). The 1-parameter ALD may be used for quantile regression.

Usage

alaplace1(tau = NULL, llocation = "identity", elocation = list(),
          ilocation = NULL, kappa = sqrt(tau/(1 - tau)), Scale.arg = 1,
          shrinkage.init = 0.95, parallelLocation = FALSE, digt = 4,
          dfmu.init = 3, method.init = 1, zero = NULL)
alaplace2(tau = NULL,  llocation = "identity", lscale = "loge",
          elocation = list(), escale = list(),
          ilocation = NULL, iscale = NULL, kappa = sqrt(tau/(1 - tau)),
          shrinkage.init = 0.95,
          parallelLocation = FALSE, digt = 4, sameScale = TRUE,
          dfmu.init = 3, method.init = 1, zero = "(1 + M/2):M")
alaplace3(llocation = "identity", lscale = "loge", lkappa = "loge",
          elocation = list(), escale = list(), ekappa = list(),
          ilocation = NULL, iscale = NULL, ikappa = 1,
          method.init = 1, zero = 2:3)

Arguments

tau, kappa

Numeric vectors with $0 < \tau < 1$ and $\kappa >0$. Most users will only specify tau since the estimated location parameter corresponds to the $\tau$th regression quantile, which is easier to understand. See below for det

llocation, lscale, lkappa

Character. Parameter link functions for location parameter $\xi$, scale parameter $\sigma$, asymmetry parameter $\kappa$. See Links for more choices. For example, the argument llocation

elocation, escale, ekappa

List. Extra argument for each of the links. See earg in Links for general information.

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.

parallelLocation

Logical. Should the quantiles be parallel on the transformed scale (argument llocation)? Assigning this argument to TRUE circumvents the seriously embarrassing quantile crossing problem.

sameScale

Logical. Should the scale parameters be equal? It is advised to keep sameScale=TRUE unchanged because it does not make sense to have different values for each tau value.

method.init

Initialization method. Either the value 1, 2, 3 or 4.

dfmu.init

Degrees of freedom for the cubic smoothing spline fit applied to get an initial estimate of the location parameter. See vsmooth.spline. Used only when method.init=3.

shrinkage.init

How much shrinkage is used when initializing $\xi$. The value must be between 0 and 1 inclusive, and a value of 0 means the individual response values are used, and a value of 1 means the median or mean is used. This argument is used only when

Scale.arg

The value of the scale parameter $\sigma$. This argument may be used to compute quantiles at different $\tau$ values from an existing fitted alaplace2() model (practical only if it has a single value). If the model has parallelL

digt

Passed into Round as the digits argument for the tau values; used cosmetically for labelling.

zero

See CommonVGAMffArguments for more information. Where possible, the default is to model all the $\sigma$ and $\kappa$ as an intercept-only term.

Value

An object of class "vglmff" (see vglmff-class). 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.

Warning

The 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.

Care is needed with tau values which are too small, e.g., for count data with llocation="loge" and if the sample proportion of zeros is greater than tau.

Details

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; ξ, σ, κ) = \frac{\sqrt{2}}{σ} \frac{κ}{1 + κ^{2}} \exp (- \frac{\sqrt{2}}{σ κ} | y - ξ |)

for $y \leq \xi$, and

f (y; ξ, σ, κ) = \frac{\sqrt{2}}{σ} \frac{κ}{1 + κ^{2}} \exp (- \frac{\sqrt{2} κ}{σ} | y - ξ |)

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 here is to first have all the location parameters, followed by all the scale parameters. Finally, for alaplace3(), the last one is 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 alaplace1() may be used as an alternative to rq in the quantreg package.

In general the response must be a vector or a 1-column matrix. For alaplace1() and alaplace2() the number of linear/additive predictors is dictated by the length of tau or kappa.

References

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.

Yee, T. W. (2009) Quantile regression for counts and proportions. In preparation.

Examples

Run this code

# Example 1: quantile regression with smoothing splines
adata = data.frame(x = sort(runif(n <- 500)))
mymu = function(x) exp(-2 + 6*sin(2*x-0.2) / (x+0.5)^2)
adata = transform(adata, y = rpois(n, lambda=mymu(x)))
mytau = c(0.25, 0.75); mydof = 4

fit = vgam(y ~ s(x, df=mydof), alaplace1(tau=mytau, llocation="loge",
           parallelLoc=FALSE), adata, trace=TRUE)
fitp = vgam(y ~ s(x, df=mydof), alaplace1(tau=mytau, llocation="loge",
            parallelLoc=TRUE), adata, trace=TRUE)
 
par(las=1)
mylwd = 1.5
with(adata, plot(x, jitter(y, factor=0.5), col="red",
                  main="Example 1; green: parallelLoc=TRUE",
                  ylab="y", pch="o", cex=0.75))
with(adata, matlines(x, fitted(fit), col="blue", lty="solid", lwd=mylwd))
with(adata, matlines(x, 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 ~ bs(x, df=mydof),
             family = alaplace1(tau=mytau, llocation="loge",
                                parallelLoc=TRUE),
             adata, trace=TRUE)

newtau = 0.5  # Want to refit the model with this tau value
fitp3 = vglm(y ~ 1 + offset(predict(fitp2)[,1]),
            family = alaplace1(tau=newtau, llocation="loge"),
             adata)
with(adata, plot(x, jitter(y, factor=0.5), col="red", ylab="y",
                  pch="o", cex=0.75,
                  main="Example 2; parallelLoc=TRUE"))
with(adata, matlines(x, fitted(fitp2), col="blue", lty="solid", lwd=mylwd))
with(adata, matlines(x, fitted(fitp3), col="black", lty="solid", 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=y-offsety)
    iloc = ifelse(ii==1, with(adata, median(y)), 1.0) # Well-chosen!
    mydf = ifelse(ii==1, 5, 3)  # Maybe less smoothing will help
    lloc = ifelse(ii==1, "identity", "loge")  # 2nd value must be "loge"
    fit3 = vglm(usey ~ ns(x, df=mydf), adata, trace=TRUE,
                fam=alaplace1(tau=usetau[ii], lloc=lloc, iloc=iloc))
    answer[,ii] = (if(ii==1) 0 else answer[,ii-1]) + fitted(fit3)
    adata = transform(adata, offsety=answer[,ii])
}

# Plot the results.
with(adata, plot(x, y, col="blue",
     main=paste("Noncrossing and nonparallel; tau =",
                paste(mytau, collapse=", "))))
with(adata, matlines(x, answer, col="red", lty=1))

# Zoom in near the origin.
with(adata, plot(x, y, col="blue", xlim=c(0, 0.2), ylim=0:1,
     main=paste("Noncrossing and nonparallel; tau =",
                paste(mytau, collapse=", "))))
with(adata, matlines(x, answer, col="red", lty=1))

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