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+
+
+ +
+

Maximum-likelihood estimation for the generalized Pareto model, including generalized linear modelling of each parameter. This function was adapted by Paul Northrop to include the gradient in the gpd.fit routine from ismev.

+
+ +
+

Usage

+
.gpd_2D_fit(
+  xdat,
+  threshold,
+  npy = 365,
+  ydat = NULL,
+  sigl = NULL,
+  shl = NULL,
+  siglink = identity,
+  shlink = identity,
+  siginit = NULL,
+  shinit = NULL,
+  show = TRUE,
+  method = "Nelder-Mead",
+  maxit = 10000,
+  ...
+)
+
+ +
+

Arguments

+
xdat
+

numeric vector of data to be fitted.

+ + +
threshold
+

a scalar or a numeric vector of the same length as xdat.

+ + +
npy
+

number of observations per year/block.

+ + +
ydat
+

matrix of covariates for generalized linear modelling of the parameters (or NULL (the default) for stationary fitting). The number of rows should be the same as the length of xdat.

+ + +
sigl
+

numeric vector of integers, giving the columns of ydat that contain covariates for generalized linear modelling of the scale parameter (or NULL (the default) if the corresponding parameter is stationary).

+ + +
shl
+

numeric vector of integers, giving the columns of ydat that contain covariates for generalized linear modelling of the shape parameter (or NULL (the default) if the corresponding parameter is stationary).

+ + +
siglink
+

inverse link functions for generalized linear modelling of the scale parameter

+ + +
shlink
+

inverse link functions for generalized linear modelling of the shape parameter

+ + +
siginit
+

numeric giving initial value(s) for parameter estimates. If NULL the default is sqrt(6 * var(xdat))/pi

+ + +
shinit
+

numeric giving initial value(s) for the shape parameter estimate; if NULL, this is 0.1. If using parameter covariates, then these values are used for the constant term, and zeros for all other terms.

+ + +
show
+

logical; if TRUE (default), print details of the fit.

+ + +
method
+

optimization method (see optim for details).

+ + +
maxit
+

maximum number of iterations.

+ + +
...other
+

control parameters for the optimization. These are passed to components of the control argument of optim.

+ +
+
+

Value

+ + +

a list with components

nexc
+

scalar giving the number of threshold exceedances.

+ +
nllh
+

scalar giving the negative log-likelihood value.

+ +
mle
+

numeric vector giving the MLE's for the scale and shape parameters, resp.

+ +
rate
+

scalar giving the estimated probability of exceeding the threshold.

+ +
se
+

numeric vector giving the standard error estimates for the scale and shape parameter estimates, resp.

+ +
trans
+

logical indicator for a non-stationary fit.

+ +
model
+

list with components sigl and shl.

+ +
link
+

character vector giving inverse link functions.

+ +
threshold
+

threshold, or vector of thresholds.

+ +
nexc
+

number of data points above the threshold.

+ +
data
+

data that lie above the threshold. For non-stationary models, the data are standardized.

+ +
conv
+

convergence code, taken from the list returned by optim. A zero indicates successful convergence.

+ +
nllh
+

negative log likelihood evaluated at the maximum likelihood estimates.

+ +
vals
+

matrix with three columns containing the maximum likelihood estimates of the scale and shape parameters, and the threshold, at each data point.

+ +
mle
+

vector containing the maximum likelihood estimates.

+ +
rate
+

proportion of data points that lie above the threshold.

+ +
cov
+

covariance matrix.

+ +
se
+

numeric vector containing the standard errors.

+ +
n
+

number of data points (i.e., the length of xdat).

+ +
npy
+

number of observations per year/block.

+ +
xdata
+

data that has been fitted.

+ + +
+
+

Details

+

For non-stationary fitting it is recommended that the covariates within the generalized linear models are (at least approximately) centered and scaled (i.e. the columns of ydat should be approximately centered and scaled).

+

The form of the GP model used follows Coles (2001) Eq (4.7). In particular, the shape parameter is defined so that positive values imply a heavy tail and negative values imply a bounded upper value.

+
+
+

References

+

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

+
+ +
+ + +
+ + + +
+ + + + + + + diff --git a/docs/reference/dot-is.CNSD.html b/docs/reference/dot-is.CNSD.html index 3d819b6..753f71a 100644 --- a/docs/reference/dot-is.CNSD.html +++ b/docs/reference/dot-is.CNSD.html @@ -1,7 +1,7 @@ Is the matrix conditionally negative semi-definite? -Function adapted from 'is.CNSD' in the CEGO package, v 2.1.0 — .is.CNSD • mevInternal function — .mvasym.check • mevInternal function — .mvasym.check • mevMultivariate Normal distribution sampler (Rcpp version), derived using the eigendecomposition -of the covariance matrix Sigma. The function utilizes the arma random normal generator — .mvrnorm_arma • mevGenerate from Brown-Resnick process \(Y \sim {P_x}\), where -\(P_{x}\) is probability of extremal function — .rPBrownResnick • mevGenerate from extremal Husler-Reiss distribution \(Y \sim {P_x}\), where -\(P_{x}\) is probability of extremal function — .rPHuslerReiss • mevGenerate from Smith model (moving maxima) \(Y \sim {P_x}\), where -\(P_{x}\) is probability of extremal function — .rPSmith • mevGenerate from bilogistic \(Y \sim {P_x}\), where -\(P_{x}\) is the probability of extremal functions — .rPbilog • mevGenerate from extremal Dirichlet \(Y \sim {P_x}\), where \(P_{x}\) is the probability of extremal functions from the Dirichlet model of -Coles and Tawn. — .rPdir • mevGenerate from extremal Dirichlet \(Y \sim {P_x}\), where -\(P_{x}\) is the probability of extremal functions from a Dirichlet mixture — .rPdirmix • mevGenerate from extremal Student-t \(Y \sim {P_x}\), where -\(P_{x}\) is probability of extremal function — .rPexstud • mevGenerate from logistic \(Y \sim {P_x}\), where -\(P_{x}\) is probability of extremal function scaled by a Frechet variate — .rPlog • mevGenerate from negative logistic \(Y \sim {P_x}\), where -\(P_{x}\) is probability of extremal function scaled by a Frechet variate — .rPneglog • mevSamples from exceedances at site (scaled extremal function definition) — .rPsite • mevSamples from exceedances at site (scaled extremal function definition) — .rPsite • mev @@ -10,7 +10,7 @@ mev - 1.14 + 1.15 + + + + + +
+
+
+ +
+

Density function, distribution function, quantile function and +random number generation for the generalized extreme value +distribution.

+
+ +
+

Usage

+
qgev(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
+
+rgev(n, loc = 0, scale = 1, shape = 0)
+
+dgev(x, loc = 0, scale = 1, shape = 0, log = FALSE)
+
+pgev(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
+
+ +
+

Arguments

+
p
+

vector of probabilities

+ + +
loc
+

scalar or vector of location parameters whose length matches that of the input

+ + +
scale
+

scalar or vector of positive scale parameters whose length matches that of the input

+ + +
shape
+

scalar shape parameter

+ + +
lower.tail
+

logical; if TRUE (default), returns the distribution function, otherwise the survival function

+ + +
n
+

scalar number of observations

+ + +
x, q
+

vector of quantiles

+ + +
log, log.p
+

logical; if TRUE, probabilities \(p\) are given as +\(\log(p)\).

+ +
+
+

Details

+

The distribution function of a GEV distribution with parameters + loc = \(\mu\), scale = \(\sigma\) and + shape = \(\xi\) is + $$F(x) = \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}$$ + for \(1 + \xi (x - \mu) / \sigma > 0\). If \(\xi = 0\) the + distribution function is defined as the limit as \(\xi\) tends to zero.

+

The quantile function, when evaluated at zero or one, + returns the lower and upper endpoint, whether the latter is finite or not.

+
+
+

References

+

Jenkinson, A. F. (1955) The frequency distribution of the + annual maximum (or minimum) of meteorological elements. + Quart. J. R. Met. Soc., 81, 158-171. + Chapter 3: doi:10.1002/qj.49708134804

+

Coles, S. G. (2001) An Introduction to Statistical + Modeling of Extreme Values, Springer-Verlag, London. + doi:10.1007/978-1-4471-3675-0_3

+
+
+

Author

+

Leo Belzile, with code adapted from Paul Northrop

+
+ +
+ + +
+ + + +
+ + + + + + + diff --git a/docs/reference/gevr.html b/docs/reference/gevr.html index c13993e..e1d0c87 100644 --- a/docs/reference/gevr.html +++ b/docs/reference/gevr.html @@ -1,7 +1,7 @@ Generalized extreme value distribution (return level parametrization) — gevr • mevGeneralized extreme value distribution (return level parametrization) — gevr • mevObserved information matrix for GEV distribution (return levels) — gevr.infomat • mevObserved information matrix for GEV distribution (return levels) — gevr.infomat • mev @@ -10,7 +10,7 @@ mev - 1.14 + 1.15 + + + + + +
+
+
+ +
+

Density function, distribution function, quantile function and +random number generation for the generalized Pareto +distribution.

+
+ +
+

Usage

+
pgp(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
+
+dgp(x, loc = 0, scale = 1, shape = 0, log = FALSE)
+
+qgp(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE)
+
+rgp(n, loc = 0, scale = 1, shape = 0)
+
+ +
+

Arguments

+
loc
+

location parameter.

+ + +
scale
+

scale parameter, strictly positive.

+ + +
shape
+

shape parameter.

+ + +
lower.tail
+

logical; if TRUE (default), the lower tail probability \(\Pr(X \leq x)\) is returned.

+ + +
log.p, log
+

logical; if FALSE (default), values are returned on the probability scale.

+ + +
x, q
+

vector of quantiles

+ + +
p
+

vector of probabilities

+ + +
n
+

scalar number of observations

+ +
+
+

References

+

Coles, S. G. (2001) An Introduction to Statistical + Modeling of Extreme Values, Springer-Verlag, London. + doi:10.1007/978-1-4471-3675-0_3

+
+ +
+ + +
+ + + +
+ + + + + + + diff --git a/docs/reference/gpdr.html b/docs/reference/gpdr.html index 1df4769..a50ee8d 100644 --- a/docs/reference/gpdr.html +++ b/docs/reference/gpdr.html @@ -1,7 +1,7 @@ Generalized Pareto distribution (return level parametrization) — gpdr • mevGeneralized Pareto distribution (return level parametrization) — gpdr • mevObserved information matrix for GP distribution (return levels) — gpdr.infomat • mevObserved information matrix for GP distribution (return levels) — gpdr.infomat • mev @@ -10,7 +10,7 @@ mev - 1.14 + 1.15
+ + + + + +
+
+
+ +
+

The Ramos and Ledford (2005) score test of independence is a modification of tests by Tawn (1988) and Ledford and Tawn (1996) for a logistic model parameter \(\alpha=1\); the latter two have scores with zero expectation, but the variance of the score are infinite, which produces non-regularity and yield test, once suitably normalized, that converge slowly to their asymptotic null distribution. The test, designed for bivariate samples, transforms observations to have unit Frechet margins and considers a bivariate censored likelihood approach for the logistic distribution.

+
+ +
+

Usage

+
scoreindep(xdat, p, test = c("ledford", "tawn"))
+
+ +
+

Arguments

+
xdat
+

a n by 2 matrix of observations

+ + +
p
+

probability level for the marginal threshold

+ + +
test
+

string; if tawn, only censor observations in the upper quadrant when both variables are large as in Tawn (1988), otherwise censor marginally for ledford as in Ledford and Tawn (1996).

+ +
+
+

Value

+ + +

a list with elements

stat
+

value of the score test statistic

+ +
pval
+

asymptotic p-value

+ +
test
+

test argument

+ + +
+ +
+

Examples

+
samp <- rmev(n = 1000, d = 2,
+    param = 0.99, model = "log")
+scoreindep(samp, p = 0.9)
+#> $stat
+#> [1] 0.8223361
+#> 
+#> $pval
+#> [1] 0.4108856
+#> 
+#> $test
+#> [1] 0.8223361
+#> 
+
+
+
+ + +
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