Gumbel Regression Family Function
Maximum likelihood estimation of the 2-parameter Gumbel distribution.
gumbel(llocation = "identitylink", lscale = "loglink",
iscale = NULL, R = NA, percentiles = c(95, 99),
mpv = FALSE, zero = NULL)
gumbelff(llocation = "identitylink", lscale = "loglink",
iscale = NULL, R = NA, percentiles = c(95, 99),
zero = "scale", mpv = FALSE)llocation, lscale |
Parameter link functions for mu and sigma.
See |
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 |
R |
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 |
mpv |
Logical. If |
zero |
A vector specifying which linear/additive predictors
are modelled as intercepts only. The value (possibly values) can
be from the set {1, 2} corresponding respectively to mu
and sigma. By default all linear/additive predictors
are modelled as a linear combination of the explanatory variables.
See |
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.
If mpv = TRUE then 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
μ+σ \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)
location-
scale*log[-log(P/100)]
where P is the percentile argument value(s).
If R is given then the percentiles are
location-
scale*log[-log(R*(1-P/100))].
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
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).
Like many other usual VGAM family functions,
gumbelff() handles (independent) multiple responses.
gumbel() can handle
more of a
multivariate response, i.e., a
matrix with more than one column. Each row of the matrix is
sorted into descending order.
Missing values in the response are allowed but require
na.action = na.pass. The response matrix needs to be
padded with any missing values. With a multivariate response
one has a matrix y, say, where
y[, 2] contains the second order statistics, etc.
T. W. Yee
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.
# Example 1: Simulated data
gdata <- data.frame(y1 = rgumbel(n = 1000, loc = 100, scale = exp(1)))
fit1 <- vglm(y1 ~ 1, gumbelff(perc = NULL), data = gdata, trace = TRUE)
coef(fit1, matrix = TRUE)
Coef(fit1)
head(fitted(fit1))
with(gdata, mean(y1))
# Example 2: Venice data
(fit2 <- vglm(cbind(r1, r2, r3, r4, r5) ~ year, data = venice,
gumbel(R = 365, mpv = TRUE), trace = TRUE))
head(fitted(fit2))
coef(fit2, matrix = TRUE)
sqrt(diag(vcov(summary(fit2)))) # Standard errors
# Example 3: Try a nonparametric fit ---------------------
# Use the entire data set, including missing values
# Same as as.matrix(venice[, paste0("r", 1:10)]):
Y <- Select(venice, "r", sort = FALSE)
fit3 <- vgam(Y ~ s(year, df = 3), gumbel(R = 365, mpv = TRUE),
data = venice, trace = TRUE, na.action = na.pass)
depvar(fit3)[4:5, ] # NAs used to pad the matrix
## Not run: # Plot the component functions
par(mfrow = c(2, 3), mar = c(6, 4, 1, 2) + 0.3, xpd = TRUE)
plot(fit3, se = TRUE, lcol = "blue", scol = "limegreen", lty = 1,
lwd = 2, slwd = 2, slty = "dashed")
# Quantile plot --- plots all the fitted values
qtplot(fit3, 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
year <- venice[["year"]]
matplot(year, Y, ylab = "Sea level (cm)", type = "n")
matpoints(year, Y, pch = "*", col = "blue")
lines(year, fitted(fit3)[, "99%"], lwd = 2, col = "orange")
# 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.
plot(year, Y[, 4], ylab = "Sea level (cm)", type = "n",
main = "Orange is 99 percentile, Green is a smoothing spline")
points(year, Y[, 4], pch = "4", col = "blue")
lines(year, fitted(fit3)[, "99%"], lty = 1, col = "orange")
lines(smooth.spline(year, Y[, 4], df = 4), col = "limegreen", lty = 2)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.