Density, distribution function, quantile function and
random generation for the Generalized Extreme Value (GEV)
distribution with parameters loc, scale and
shape.
The distribution function \(F(x) = \textrm{Pr}[X \leq x]\)
is given by
$$F(x) = \exp\left\{-[1 + \xi z]^{-1/\xi}\right\}$$
when \(\xi \neq 0\) and \(1 + \xi z > 0\), and by
$$F(x) = \exp\left\{-e^{-z}\right\}$$
for \(\xi =0\) where \(z := (x - \mu) / \sigma\) in both cases.
dGEV(
x,
loc = 0,
scale = 1,
shape = 0,
log = FALSE,
deriv = FALSE,
hessian = FALSE
)pGEV(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, deriv = FALSE)
qGEV(
p,
loc = 0,
scale = 1,
shape = 0,
lower.tail = TRUE,
deriv = FALSE,
hessian = FALSE
)
rGEV(n, loc = 0, scale = 1, shape = 0, array)
A numeric vector with length n as described in the
Details section. When deriv is TRUE, the
returned value has an attribute named "gradient" which
is a matrix with \(n\) lines and \(3\) columns containing
the derivatives. A row contains the partial derivatives of the
corresponding element w.r.t. the three parameters loc
scale and shape in that order.
Vector of quantiles.
Location parameter. Numeric vector with suitable length, see Details.
Scale parameter. Numeric vector with suitable length, see Details.
Shape parameter. Numeric vector with suitable length, see Details.
Logical; if TRUE, densities p are
returned as log(p).
Logical. If TRUE, the gradient of each
computed value w.r.t. the parameter vector is computed, and
returned as a "gradient" attribute of the result. This
is a numeric array with dimension c(n, 3) where
n is the length of the first argument, i.e. x,
p or q depending on the function.
Logical. If TRUE, the Hessian of each
computed value w.r.t. the parameter vector is computed, and
returned as a "hessian" attribute of the result. This
is a numeric array with dimension c(n, 3, 3) where
n is the length of the first argument, i.e. x,
p or depending on the function.
Logical; if TRUE (default), probabilities
are \(\textrm{Pr}[X \leq x]\), otherwise,
\(\textrm{Pr}[X>x]\).
Vector of probabilities.
Sample size.
Logical. If TRUE, the simulated values form a
numeric matrix with n columns and np rows where
np is the number of GEV parameter values. This number
is obtained by recycling the three GEV parameters vectors to a
common length, so np is the maximum of the lengths of
the parameter vectors loc, scale,
shape. This option is useful to cope with so-called
non-stationary models with GEV margins. See
Examples. The default value is TRUE if any of
the vectors loc, scale and shape has
length > 1 and FALSE otherwise.
Each of the probability function normally requires two formulas: one for the non-zero shape case \(\xi \neq 0\) and one for the zero-shape case \(\xi = 0\). However the non-zero shape formulas lead to numerical instabilities near \(\xi = 0\), especially for the derivatives w.r.t. \(\xi\). This can create problem in optimization tasks. To avoid this, a Taylor expansion w.r.t. \(\xi\) is used for \(|\xi| < \epsilon\) for a small positive \(\epsilon\). The expansion has order \(2\) for the functions (log-density, distribution and quantile), order \(1\) for their first-order derivatives and order \(0\) for the second-order derivatives.
For the d, p and q functions, the GEV
parameter arguments loc, scale and shape
are recycled in the same fashion as the classical R
distribution functions in the stats package, see e.g.,
Normal, GammaDist, ...
Let n be the maximum length of the four arguments:
x q or p and the GEV parameter arguments,
then the four provided vectors are recycled in order to have
length n. The returned vector has length n and
the attributes "gradient" and "hessian", when
computed, are arrays wich dimension: c(1, 3) and
c(1, 3, 3).
ti <- 1:10; names(ti) <- 2000 + ti
mu <- 1.0 + 0.1 * ti
## simulate 40 paths
y <- rGEV(n = 40, loc = mu, scale = 1, shape = 0.05)
matplot(ti, y, type = "l", col = "gray")
lines(ti, apply(y, 1, mean))
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