R: Generalized Extreme Value location-scale model family

gevlss {mgcv} | R Documentation

## Generalized Extreme Value location-scale model family

### Description

The `gevlss` family implements Generalized Extreme Value location scale additive models in which the location, scale and shape parameters depend on additive smooth predictors. Usable only with `gam`, the linear predictors are specified via a list of formulae.

### Usage

    gevlss(link=list("identity","identity","logit"))

### Arguments

`link` | three item list specifying the link for the location scale and shape parameters. See details.

### Details

Used with `gam` to fit Generalized Extreme Value location scale and shape models. `gam` is called with a list containing 3 formulae: the first specifies the response on the left hand side and the structure of the linear predictor for the location parameter on the right hand side. The second is one sided, specifying the linear predictor for the log scale parameter on the right hand side. The third is one sided specifying the linear predictor for the shape parameter.

Link functions `"identity"` and `"log"` are available for the location (mu) parameter. There is no choice of link for the log scale parameter (rho = log sigma). The shape parameter (xi) defaults to a modified logit link restricting its range to (-1,.5), the upper limit is required to ensure finite variance, while the lower limit ensures consistency of the MLE (Smith, 1985).

The fitted values for this family will be a three column matrix. The first column is the location parameter, the second column is the log scale parameter, the third column is the shape parameter.

This family does not produce a null deviance. Note that the distribution for xi=0 is approximated by setting xi to a small number.

The derivative system code for this family is mostly auto-generated, and the family is still somewhat experimental.

### Value

An object inheriting from class `general.family`.

### References

Smith, R.L. (1985) Maximum likelihood estimation in a class of nonregular cases. Biometrika 72(1):67-90

Wood, S.N., N. Pya and B. Saefken (2016), Smoothing parameter and model selection for general smooth models. Journal of the American Statistical Association. http://arxiv.org/abs/1511.03864

### Examples

    library(mgcv)
    Fi.gev <- function(z,mu,sigma,xi) {
    ## GEV inverse cdf.
      xi[abs(xi)<1e-8] <- 1e-8 ## approximate xi=0, by small xi
      x <- mu + ((-log(z))^-xi-1)*sigma/xi
    }

    ## simulate test data...
    f0 <- function(x) 2 * sin(pi * x)
    f1 <- function(x) exp(2 * x)
    f2 <- function(x) 0.2 * x^11 * (10 * (1 - x))^6 + 10 *
                (10 * x)^3 * (1 - x)^10
    set.seed(1)
    n <- 500
    x0 <- runif(n);x1 <- runif(n);x2 <- runif(n)
    mu <- f2(x2)
    rho <- f0(x0)
    xi <- (f1(x1)-4)/9
    y <- Fi.gev(runif(n),mu,exp(rho),xi)
    dat <- data.frame(y,x0,x1,x2);pairs(dat)

    ## fit model....
    b <- gam(list(y~s(x2),~s(x0),~s(x1)),family=gevlss,data=dat)

    ## plot and look at residuals...
    plot(b,pages=1,scale=0)
    summary(b)

    par(mfrow=c(2,2))
    mu <- fitted(b)[,1];rho <- fitted(b)[,2]
    xi <- fitted(b)[,3]
    ## Get the predicted expected response...
    fv <- mu + exp(rho)*(gamma(1-xi)-1)/xi
    rsd <- residuals(b)
    plot(fv,rsd);qqnorm(rsd)
    plot(fv,residuals(b,"pearson"))
    plot(fv,residuals(b,"response"))

[Package mgcv version 1.8-17 Index]