alaplace {VGAM} | R Documentation

## 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 tauth regression quantile, which is easier to understand. See below for details.
`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` can help handle count data by restricting the quantiles to be positive (use `llocation="loge"`). However, `llocation` is best left alone since the theory only works properly with the identity link.
`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 `method.init=4`.
`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 `parallelLocation = TRUE` then only the intercept need be estimated; use an offset. See below for an example.
`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.

### 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;xi,sigma,kappa) = (sqrt(2)/sigma) * (kappa/(1+ kappa^2)) * exp( -(sqrt(2) / (sigma * kappa)) * |y-xi| )

for y <= xi, and

f(y;xi,sigma,kappa) = (sqrt(2)/sigma) * (kappa/(1+ kappa^2)) * exp( - (sqrt(2) * kappa / sigma) * |y-xi| )

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 <= 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`.

### 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`.

### Note

These VGAM family functions use Fisher scoring. Convergence may be slow and half-stepping is usual (although one can use `trace=TRUE` to see which is the best model and then use `maxit` to fit that model).

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 `parallelLoc=FALSE` one can repeatedly fit a model with `alaplace1()` with one tau at a time; and for `parallelLoc=TRUE` one can refit a model with `alaplace1()` with one tau at a time but using offsets and an intercept-only 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 (2009). It does not make the strong parallelism assumption.

The functions `alaplace2()` and `laplace` differ slightly in terms of the parameterizations.

### Author(s)

Thomas W. Yee

### 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.

### See Also

`ralap`, `laplace`, `lms.bcn`, `amlnormal`.

### Examples

    # Example 1: quantile regression with smoothing splines
    alldat = data.frame(x = sort(runif(n <- 500)))
    mymu = function(x) exp(-2 + 6*sin(2*x-0.2) / (x+0.5)^2)
    alldat = transform(alldat, 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), data=alldat, trace=TRUE)
    fitp = vgam(y ~ s(x, df=mydof), alaplace1(tau=mytau, llocation="loge",
                parallelLoc=TRUE), data=alldat, trace=TRUE)

    ## Not run:
    par(las=1)
    mylwd = 1.5
    with(alldat, plot(x, jitter(y, factor=0.5), col="red",
                      main="Example 1; green: parallelLoc=TRUE",
                      ylab="y", pch="o", cex=0.75))
    with(alldat, matlines(x, fitted(fit), col="blue", lty="solid", lwd=mylwd))
    with(alldat, 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)
    ## End(Not run)
    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),
                 data=alldat, 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"),
                 data=alldat)
    ## Not run:
    with(alldat, plot(x, jitter(y, factor=0.5), col="red", ylab="y",
                      pch="o", cex=0.75,
                      main="Example 2; parallelLoc=TRUE"))
    with(alldat, matlines(x, fitted(fitp2), col="blue", lty="solid", lwd=mylwd))
    with(alldat, matlines(x, fitted(fitp3), col="black", lty="solid", lwd=mylwd))
    ## End(Not run)

    # 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(alldat), length(mytau)) # Stores the quantiles
    alldat = transform(alldat, offsety=y*0)
    usetau = mytau
    for(ii in 1:length(mytau)) {
    #   cat("\n\nii =", ii, "\n")
        alldat = transform(alldat, usey=y-offsety)
        iloc = ifelse(ii==1, with(alldat, 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), data=alldat, trace=TRUE,
                    fam=alaplace1(tau=usetau[ii], lloc=lloc, iloc=iloc))
        answer[,ii] = (if(ii==1) 0 else answer[,ii-1]) + fitted(fit3)
        alldat = transform(alldat, offsety=answer[,ii])
    }

    # Plot the results.
    ## Not run:
    with(alldat, plot(x, y, col="blue",
         main=paste("Noncrossing and nonparallel; tau =",
                    paste(mytau, collapse=", "))))
    with(alldat, matlines(x, answer, col="red", lty=1))

    # Zoom in near the origin.
    with(alldat, plot(x, y, col="blue", xlim=c(0, 0.2), ylim=0:1,
         main=paste("Noncrossing and nonparallel; tau =",
                    paste(mytau, collapse=", "))))
    with(alldat, matlines(x, answer, col="red", lty=1))
    ## End(Not run)

[Package VGAM version 0.7-9 Index]