gumbel: Gumbel Distribution Family Function

Description

Maximum likelihood estimation of the 2-parameter Gumbel distribution.

Usage

gumbel(llocation = "identity", lscale = "loge",
       iscale = NULL, R = NA, percentiles = c(95, 99),
       mpv = FALSE, zero = NULL)
egumbel(llocation = "identity", lscale = "loge",
        iscale = NULL, R = NA, percentiles = c(95, 99),
        mpv = FALSE, zero = NULL)

Arguments

llocation, lscale

Parameter link functions for $\mu$ and $\sigma$. See Links for more choices.

iscale

Numeric and positive. Optional initial value for $\sigma$. Recycled to the appropriate length. In general, a larger value is better than a smaller value. A NULL means an initial value is computed internally.

Numeric. Maximum number of values possible. See Details for more details.

percentiles

Numeric vector of percentiles used for the fitted values. Values should be between 0 and 100. This argument uses the argument R if assigned. If percentiles = NULL then the mean will be returned as the fitted values.

mpv

Logical. If mpv = TRUE then the median predicted value (MPV) is computed and returned as the (last) column of the fitted values. This argument is ignored if percentiles = NULL. See Details for more details.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The value (possibly values) must be from the set {1, 2} corresponding respectively to $\mu$ and $\sigma$. By default all linear/additive predicto

Value

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

Warning

When R is not given (the default) the fitted percentiles are that of the data, and not of the overall population. For example, in the example below, the 50 percentile is approximately the running median through the data, however, the data are the highest sea level measurements recorded each year (it therefore equates to the median predicted value or MPV).

Details

The Gumbel distribution is a generalized extreme value (GEV) distribution with shape parameter $\xi = 0$. Consequently it is more easily estimated than the GEV. See gev for more details.

The quantity $R$ is the maximum number of observations possible, for example, in the Venice data below, the top 10 daily values are recorded for each year, therefore $R = 365$ because there are about 365 days per year. The MPV is the value of the response such that the probability of obtaining a value greater than the MPV is 0.5 out of $R$ observations. For the Venice data, the MPV is the sea level such that there is an even chance that the highest level for a particular year exceeds the MPV. When mpv = TRUE, the column labelled "MPV" contains the MPVs when fitted() is applied to the fitted object.

The formula for the mean of a response $Y$ is $\mu+\sigma \times Euler$ where $Euler$ is a constant that has value approximately equal to 0.5772. The formula for the percentiles are (if R is not given) $\mu-\sigma \times \log[-\log(P/100)]$ where $P$ is the percentile argument value(s). If R is given then the percentiles are $\mu-\sigma \times \log[R(1-P/100)]$.

References

Yee, T. W. and Stephenson, A. G. (2007) Vector generalized linear and additive extreme value models. Extremes, 10, 1--19.

Smith, R. L. (1986) Extreme value theory based on the r largest annual events. Journal of Hydrology, 86, 27--43.

Rosen, O. and Cohen, A. (1996) Extreme percentile regression. In: Haerdle, W. and Schimek, M. G. (eds.), Statistical Theory and Computational Aspects of Smoothing: Proceedings of the COMPSTAT '94 Satellite Meeting held in Semmering, Austria, 27--28 August 1994, pp.200--214, Heidelberg: Physica-Verlag.

Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.

Examples

Run this code

# Example 1: Simulated data
gdata <- data.frame(y = rgumbel(n = 1000, loc = 100, scale = exp(1)))
fit <- vglm(y ~ 1, egumbel(perc = NULL), gdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
head(fitted(fit))
with(gdata, mean(y))

# Example 2: Venice data
(fit <- vglm(cbind(r1,r2,r3,r4,r5) ~ year, data = venice,
             gumbel(R = 365, mpv = TRUE), trace = TRUE))
head(fitted(fit))
coef(fit, matrix = TRUE)
vcov(summary(fit))  
sqrt(diag(vcov(summary(fit))))   # Standard errors


# Example 3: Try a nonparametric fit ---------------------
# Use the entire data set, including missing values
y <- as.matrix(venice[, paste("r", 1:10, sep = "")])
fit1 <- vgam(y ~ s(year, df = 3), gumbel(R = 365, mpv = TRUE),
             data = venice, trace = TRUE, na.action = na.pass)
depvar(fit1)[4:5, ] # NAs used to pad the matrix

# Plot the component functions
par(mfrow = c(2, 1), mar = c(5, 4, 0.2, 1) + 0.1, xpd = TRUE)
plot(fit1, se = TRUE, lcol = "blue", scol = "green", lty = 1,
     lwd = 2, slwd = 2, slty = "dashed")

# Quantile plot --- plots all the fitted values
par(mfrow = c(1, 1), bty = "l", mar = c(4, 4, 0.2, 3) + 0.1, xpd = TRUE, las = 1)
qtplot(fit1, mpv = TRUE, lcol = c(1, 2,5), tcol = c(1, 2,5), lwd = 2,
       pcol = "blue", tadj = 0.1, ylab = "Sea level (cm)")

# Plot the 99 percentile only
par(mfrow = c(1, 1), mar = c(3, 4, 0.2, 1) + 0.1, xpd = TRUE)
year = venice[["year"]]
matplot(year, y, ylab = "Sea level (cm)", type = "n")
matpoints(year, y, pch = "*", col = "blue")
lines(year, fitted(fit1)[,"99%"], lwd = 2, col = "red")

# Check the 99 percentiles with a smoothing spline.
# Nb. (1-0.99) * 365 = 3.65 is approx. 4, meaning the 4th order 
# statistic is approximately the 99 percentile.
par(mfrow = c(1, 1), mar = c(3, 4, 2, 1) + 0.1, xpd = TRUE, lwd = 2)
plot(year, y[, 4], ylab = "Sea level (cm)", type = "n",
     main = "Red is 99 percentile, Green is a smoothing spline")
points(year, y[, 4], pch = "4", col = "blue")
lines(year, fitted(fit1)[,"99%"], lty = 1, col = "red")
lines(smooth.spline(year, y[, 4], df = 4), col = "darkgreen", lty = 2)

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