Package {ExtremeCI}

Type: Package
Title: Realistic Confidence Intervals for Non-Stationary Extreme Value Statistics
Version: 0.2.1
Description: This framework provides versatile algorithms to efficiently infer confidence intervals for extreme value statistics, such as extreme quantiles and return levels, that are representative of the asymmetric uncertainty spread, using extreme value theory extrapolation and the profile likelihood (see e.g., Coles (2001) <doi:10.1007/978-1-4471-3675-0>). Unlike existing algorithms, the CI endpoints are found without the need for a strict prespecified range, can be covariate-dependent, and can be based on weighted samples. This package is motivated by Zeder et al. (2023) <doi:10.1029/2023GL104090> and by Pasche et al. (2026) <doi:10.1007/s10687-026-00536-9>.
License: GPL (≥ 3)
Encoding: UTF-8
Imports: doFuture, dplyr, evd, foreach, future, ggplot2, magrittr, rlang, stats, tibble, tidyr, tidyselect
URL: https://github.com/opasche/ExtremeCI, https://opasche.github.io/ExtremeCI/
BugReports: https://github.com/opasche/ExtremeCI/issues
Config/roxygen2/version: 8.0.0
NeedsCompilation: no
Packaged: 2026-05-07 15:21:49 UTC; pascheo
Author: Olivier C. Pasche ORCID iD [aut, cre, cph]
Maintainer: Olivier C. Pasche <olivier_pasche@alumni.epfl.ch>
Repository: CRAN
Date/Publication: 2026-05-12 19:10:02 UTC

ExtremeCI: Realistic Confidence Intervals for Non-Stationary Extreme Value Statistics

Description

This framework provides versatile algorithms to efficiently infer confidence intervals for extreme value statistics, such as extreme quantiles and return levels, that are representative of the asymmetric uncertainty spread, using extreme value theory extrapolation and the profile likelihood (see e.g., Coles (2001) doi:10.1007/978-1-4471-3675-0). Unlike existing algorithms, the CI endpoints are found without the need for a strict prespecified range, can be covariate-dependent, and can be based on weighted samples. This package is motivated by Zeder et al. (2023) doi:10.1029/2023GL104090 and by Pasche et al. (2026) doi:10.1007/s10687-026-00536-9.

Author(s)

Maintainer: Olivier C. Pasche olivier_pasche@alumni.epfl.ch (ORCID) [copyright holder]

Authors:

See Also

Useful links:

GEV parameter vector reparametrization

Description

GEV parameter vector reparametrization

Usage

GEV_change_parametrization(
  parameters,
  parametrization = c("classical", "return_level", "endpoint"),
  new_parametrization = c("classical", "return_level", "endpoint"),
  return_period = 100,
  nbloc = 1,
  nbsca = 1,
  nbsha = 1
)

Arguments

parameters

Vector of GEV parameters in the internal format.

parametrization

Current parametrization of parameters.

new_parametrization

Desired new parametrization.

return_period

Return period for the 'return_level' parameter.

nbloc

Number of location parameter coefficients (i.e. one plus the number of shape parameter covariates).

nbsca

Number of scale parameter coefficients (i.e. one plus the number of scale parameter covariates).

nbsha

Number of shape parameter coefficients (i.e. one plus the number of shape parameter covariates).

Value

The vector of GEV parameters reparametrized from parametrization to new_parametrization.

GEV covariate matrices

Description

Creates separate GEV covariate matrices from the main UI specification.

Usage

GEV_covariate_matrices(
  X = NULL,
  x_rlvl = NULL,
  loc_cols = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  parametrization = c("classical", "return_level", "endpoint"),
  out_param = parametrization
)

Arguments

X

.

x_rlvl

.

loc_cols

.

scale_cols

.

shape_cols

.

parametrization

.

out_param

.

Value

The necessary internal matrices in a list.

Space-efficient GEV covariate matrix with indices

Description

CURRENTLY UNUSED

Usage

GEV_covariate_matrix(
  X = NULL,
  x_rlvl = NULL,
  loc_cols = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  parametrization = c("classical", "return_level", "endpoint"),
  out_param = parametrization
)

Arguments

X

.

x_rlvl

.

loc_cols

.

scale_cols

.

shape_cols

.

parametrization

.

out_param

.

Value

The space-efficient GEV covariate matrix with indices in a named list.

GEV endpoint

Description

GEV endpoint

Usage

GEV_endpoint(
  loc = 0,
  scale = 1,
  shape,
  ...,
  endpoint_type = c("unrestricted", "upper", "lower")
)

Arguments

loc

Location parameter.

scale

Scale parameter.

shape

Shape parameter.

...

Other optional parameters for the internal endpoint function (Unused).

endpoint_type

Whether to compute the upper endpoint (returns Inf if shape>=0), or the unrestricted endpoint (returns the lower GEV endpoint if shape>=0).

Value

The endpoint of the specified GEV distribution.

Examples

GEV_endpoint(loc=0, scale=1, shape=-0.1, endpoint_type='upper')

GEV log likelihood

Description

GEV log likelihood

Usage

GEV_log_likelihood(Z, loc, scale, shape)

Arguments

Z

Block maxima observations.

loc

Location parameter.

scale

Scale parameter.

shape

Shape parameter.

Value

The GEV log-likelihood evaluated at the given parameters, given the data observations Y.

Internal GEV log-likelihood function format for optimisation

Description

Internal GEV log-likelihood function format for optimisation

Usage

GEV_log_likelihood_optim(
  a,
  Z,
  parametrization = c("classical", "return_level", "endpoint"),
  return_period = 100,
  locmat = as.matrix(1),
  scamat = as.matrix(1),
  shamat = as.matrix(1),
  negative = FALSE
)

Arguments

a

.

Z

.

parametrization

.

return_period

.

locmat

.

scamat

.

shamat

.

negative

.

GEV lower endpoint

Description

GEV lower endpoint

Usage

GEV_lower_endpoint(loc = 0, scale = 1, shape, ...)

Arguments

loc

Location parameter.

scale

Scale parameter.

shape

Shape parameter.

...

Unused.

Value

The lower endpoint of the specified GEV distribution (returns Inf if shape>=0).

Maximum-likelihood GEV estimate

Description

Maximum-likelihood GEV estimate

Usage

GEV_maxlik(
  Z,
  parametrization = c("classical", "return_level", "endpoint"),
  return_period = 100,
  orthogonal = FALSE,
  X = NULL,
  x_rlvl = NULL,
  loc_cols = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  out_param = parametrization,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  ...
)

Arguments

Z

Block maxima observations.

parametrization

Likelihood parametrization. Alternatives to classical substitute for the location parameter.

return_period

Return period for the 'return_level' parametrization.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits).

x_rlvl

Covariate vector at which to reparametrize for the 'return_level' or 'endpoint' parametrizations (for conditional/non-stationary fits).

loc_cols

Column indices of X to use as covariate for the (conditional) location parameter (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

out_param

Additional output parametrization (same as parametrization, by default). If out_param != parametrization, the parameters are reparametrized from parametrization to out_param after estimation, in a separate output.

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used. See stats::optim() for more details.

...

Other arguments passed to the control argument of stats::optim().

Value

The fitted maximum-likelihood GEV as a GEV_ML object, containing:

mle

The estimated maximum likelihood GEV parameters, as a named vector (expressed in parametrization).

loglik

The log-likelihood of the estimated parameters, given the data

conv

Whether the optimisation procedure converged. See the convergence output of stats::optim() for more details.

hessian

The loglikelihood hessian evaluated at the estimated parameters, given the data.

parametrization

Likelihood parametrization.

out_mle

The estimated maximum likelihood GEV parameters, reparametrized in out_param.

out_parametrization

Additional output parametrization.

GEV_paraboot_CIs_multiple (IN DEVELOPPEMENT)

Description

GEV_paraboot_CIs_multiple (IN DEVELOPPEMENT)

Usage

GEV_paraboot_CIs_multiple(
  Z,
  alpha = 0.05,
  return_period = 100,
  R = 1000,
  endpoint_type = c("upper", "lower", "unrestricted"),
  orthogonal = FALSE,
  X = NULL,
  X_rlvl = NULL,
  loc_cols = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  bootstrap_X = FALSE,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  parallel_strat = c("none", "multisession", "sequential", "multicore"),
  n_workers = NULL,
  ...
)

Arguments

Z

Block maxima observations.

alpha

.

return_period

.

R

.

endpoint_type

.

orthogonal

.

X

.

X_rlvl

.

loc_cols

.

scale_cols

.

shape_cols

.

bootstrap_X

.

hessian

.

maxit

.

method

.

parallel_strat

.

n_workers

.

...

.

Value

Parametric bootstrap CIs

Initial GEV parameter vector defaults for profile optimization

Description

Initial GEV parameter vector defaults for profile optimization

Usage

GEV_param_init(
  Z,
  nbloc,
  nbsca,
  nbsha,
  parametrization = c("classical", "return_level", "endpoint"),
  return_period
)

Arguments

Z

.

nbloc

Number of location parameter coefficients (i.e. one plus the number of shape parameter covariates).

nbsca

Number of scale parameter coefficients (i.e. one plus the number of scale parameter covariates).

nbsha

Number of shape parameter coefficients (i.e. one plus the number of shape parameter covariates).

parametrization

.

return_period

.

Value

The initial parameter values as a vector, in the correct internal format.

GEV parameter vector names

Description

GEV parameter vector names

Usage

GEV_param_names(
  nbloc = 1,
  nbsca = 1,
  nbsha = 1,
  parametrization = c("classical", "return_level", "endpoint")
)

Arguments

nbloc

.

nbsca

.

nbsha

.

parametrization

.

Value

The names of the GEV parameters in the specified internal vector format, as a vector.

GEV profile CI using binary search

Description

GEV profile CI using binary search

Usage

GEV_profile_CI(
  Z,
  parameter = c("shape", "location", "scale", "return_level", "endpoint"),
  subparam_id = 0,
  alpha = 0.05,
  return_period = 100,
  orthogonal = FALSE,
  X = NULL,
  x_rlvl = NULL,
  loc_cols = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  warmstart_table = NULL,
  init_step_pos = 100,
  init_step_neg = 10,
  tol = 0.01,
  steps_beyond_conf = 5,
  initial_MLE_para = c("classical", "same"),
  max_steps = 1000,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  verbose = 1,
  ...
)

Arguments

Z

Block maxima observations.

parameter

Parameter for which to compute the profile log-likelihood.

subparam_id

Index of the parameter coefficient for which to compute the profile log-likelihood (for conditional/non-stationary fits).

alpha

Confidence alpha for the profile likelihood confidence interval (i.e. for the confidence line on the profile plot).

return_period

Return period for the 'return_level' parameter.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits). Columns should be variables, and rows should be observations matching Y.

x_rlvl

Covariate vector at which to reparametrize for the 'return_level' or 'endpoint' parametrizations (for conditional/non-stationary fits).

loc_cols

Column indices of X to use as covariate for the (conditional) location parameter (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

warmstart_table

Evaluation table from a previous run.

init_step_pos

Initial numerical size of each evaluation step to the right, in the profile parameter's scale.

init_step_neg

Initial numerical size of each evaluation step to the left, in the profile parameter's scale.

tol

Numerical tolerance for convergence, in the profile parameter's scale.

steps_beyond_conf

Number of additional steps to take (in each direction) after the profile log-likelihood values reach below the confidence line.

initial_MLE_para

Parametrization used for the initial maximum likelihood estimate (defaults to classical, for better stability).

max_steps

Maximum number of steps taken (in each direction). If the confidence line was not reached, the corresponding confidence interval endpoint will be infinite.

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used. See stats::optim() for more details.

verbose

Verbose level, as integer.

...

Other arguments passed to the control argument of stats::optim().

Value

The GEV profile log-likelihood confidence interval for the desired parameter, with confidence line and resulting (1-alpha) confidence interval, as a GEV_profileLogLik object containing:

mle

The estimated maximum likelihood GEV parameters, as a named vector (expressed in the profile parametrization).

ci

Length-two vector containing the lower and upper endpoints of the desired profile likelihood confidence interval.

profile_loglik

Named matrix containing the profile loglikelihood value (Column 2) for each considered profile parameter value (Column 1).

conf_line

Confidence line for the desired profile likelihood confidence interval. See e.g. Coles (2001) for more details.

eval_table

Tibble (tibble::tibble()) containing the history of profile log-likelihood evaluation values, and related metadata.

param_name

Name of the profiled parameter (infered, for debugging purposes).

parameter

Name of the profiled parameter (given).

parametrization

Parametrization used for the profile likelihood.

subparam_id

Index of the parameter coefficient for which the profile log-likelihood was computed.

id_param

Index of the profile parameter in the GEV parameter vector.

References

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

Multi-value conditional GEV profile likelihood confidence intervals

Description

For non-stationary models, the return level reparametrization depends on covariate values. This function repeats the profile likelihood procedure for several covariate values. It enables obtaining a return-level (or endpoint) curve, with profile-likelihood confidence bands, as a function of the covariate values.

Usage

GEV_profile_CIs_multiple(
  Z,
  parameter = c("return_level", "endpoint"),
  alpha = 0.05,
  return_period = 100,
  orthogonal = FALSE,
  X = NULL,
  X_rlvl = NULL,
  loc_cols = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  init_step_pos = 100,
  init_step_neg = 10,
  tol = 0.01,
  steps_beyond_conf = 5,
  initial_MLE_para = c("classical", "same"),
  max_steps = 10000,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  parallel_strat = c("none", "multisession", "sequential", "multicore"),
  n_workers = NULL,
  ...
)

Arguments

Z

Block maxima observations.

parameter

Parameter for which to compute the profile likelihood confidence intervals.

alpha

Confidence alpha for the profile likelihood confidence intervals.

return_period

Return period for the 'return_level' parameter.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits). Columns should be variables, and rows should be observations matching Y.

X_rlvl

Covariate matrix at which to reparametrize for the 'return_level' or 'endpoint' parametrizations (for conditional/non-stationary fits). Columns should be variables, and each row should give one covariate realization at which to reparametrize and obtain a CI.

loc_cols

Column indices of X to use as covariate for the (conditional) location parameter (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

init_step_pos

Initial numerical size of each evaluation step to the right, in the profile parameter's scale.

init_step_neg

Initial numerical size of each evaluation step to the left, in the profile parameter's scale.

tol

Numerical tolerance for convergence, in the profile parameter's scale.

steps_beyond_conf

Number of additional steps to take (in each direction) after the profile log-likelihood values reach below the confidence line.

initial_MLE_para

Parametrization used for the initial maximum likelihood estimate (defaults to classical, for better stability).

max_steps

Maximum number of steps taken (in each direction). If the confidence line was not reached, the corresponding confidence interval endpoint will be infinite.

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used. See stats::optim() for more details.

parallel_strat

Parallel strategy. One of "sequential" (default), "multisession", "multicore", or "mixed".

n_workers

A positive numeric scalar or a function specifying the maximum number of parallel futures that can be active at the same time before blocking. If a function, it is called without arguments when the future is created and its value is used to configure the workers. The function should return a numeric scalar. Defaults to future::availableCores()-1 if NULL (default), with "multicore" constraint in the relevant case. Ignored if strategy=="sequential".

...

Other arguments passed to the control argument of stats::optim().

Value

The GEV profile log-likelihood (1-alpha) confidence intervals for the desired parameter, for each desired covariate values, as a tibble::tibble(), with columns:

obs

Index of the observation (i.e. row) of X_rlvl for which the parameter estimate and CI was computed.

\verb{<parameter name>}

Conditional estimate of the parameter.

ci_down

Lower endpoint of the conditional (1-alpha) profile-likelihood confidence interval for parameter.

ci_up

Upper endpoint of the conditional (1-alpha) profile-likelihood confidence interval for parameter.

parameter

Name of the parameter for which the estimates and CIs were computed.

alpha

Confidence alpha for the profile likelihood confidence intervals.

return_period

Return period for the 'return_level' parameter (only if parameter==return_level).

GEV profile log-likelihood

Description

GEV profile log-likelihood

Usage

GEV_profile_loglik(
  val,
  Z,
  parameter = c("shape", "location", "scale", "return_level", "endpoint"),
  subparam_id = 0,
  return_period = 100,
  orthogonal = FALSE,
  X = NULL,
  x_rlvl = NULL,
  loc_cols = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  init = NULL,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  ...
)

Arguments

val

Parameter value at which to evaluate the GEV profile log-likelihood.

Z

Block maxima observations.

parameter

Parameter for which to compute the profile log-likelihood.

subparam_id

Index of the parameter coefficient for which to compute the profile log-likelihood (for conditional/non-stationary fits).

return_period

Return period for the 'return_level' parameter.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits). Columns should be variables, and rows should be observations matching Y.

x_rlvl

Covariate vector at which to reparametrize for the 'return_level' or 'endpoint' parametrizations (for conditional/non-stationary fits).

loc_cols

Column indices of X to use as covariate for the (conditional) location parameter (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

init

Optional initial values for the remaining parameter's optimisation process, in the correct internal format.

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used. See stats::optim() for more details.

...

Other arguments passed to the control argument of stats::optim().

Value

The GEV profile log-likelihood of parameter evaluated at val, given the data, as a GEV_profML object containing:

param_val

(Named) parameter value at which the GEV profile log-likelihood was evaluated.

param_name

Name of the evaluated profile likelihood parameter.

mle_other

Maximum-likelihood estimate of the other GEV parameters.

loglik

Profile GEV log-likelihood value of parameter evaluated at val, given the data.

conv

Whether the optimisation procedure converged. See the convergence output of stats::optim() for more details.

hessian

The loglikelihood hessian evaluated at the estimated parameters, given the data.

parameter

Name of the evaluated profile likelihood parameter given as argument (redundent).

parametrization

Likelihood parametrization.

subparam_id

Index of the parameter coefficient for which the profile log-likelihood was computed.

id_param

Index of the likelihood profile parameter, in the internal parameter vector format.

GEV profile log-likelihood curve

Description

GEV profile log-likelihood curve

Usage

GEV_profile_loglik_curve(
  Z,
  parameter = c("shape", "location", "scale", "return_level", "endpoint"),
  subparam_id = 0,
  alpha = 0.05,
  return_period = 100,
  orthogonal = FALSE,
  X = NULL,
  x_rlvl = NULL,
  loc_cols = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  warmstart_table = NULL,
  stepsize = 0.1,
  steps_beyond_conf = 5,
  initial_MLE_para = c("classical", "same"),
  max_steps = 10000,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  ...
)

Arguments

Z

Block maxima observations.

parameter

Parameter for which to compute the profile log-likelihood.

subparam_id

Index of the parameter coefficient for which to compute the profile log-likelihood (for conditional/non-stationary fits).

alpha

Confidence alpha for the profile likelihood confidence interval (i.e. for the confidence line on the profile plot).

return_period

Return period for the 'return_level' parameter.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits). Columns should be variables, and rows should be observations matching Y.

x_rlvl

Covariate vector at which to reparametrize for the 'return_level' or 'endpoint' parametrizations (for conditional/non-stationary fits).

loc_cols

Column indices of X to use as covariate for the (conditional) location parameter (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

warmstart_table

Evaluation table from a previous run.

stepsize

Numerical size of each evaluation step, in the profile parameter's scale.

steps_beyond_conf

Number of additional steps to take (in each direction) after the profile log-likelihood values reach below the confidence line.

initial_MLE_para

Parametrization used for the initial maximum likelihood estimate (defaults to classical, for better stability).

max_steps

Maximum number of steps taken (in each direction). If the confidence line was not reached, the corresponding confidence interval endpoint will be infinite.

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used. See stats::optim() for more details.

...

Other arguments passed to the control argument of stats::optim().

Value

The GEV profile log-likelihood curve for the desired parameter, with confidence line and resulting (1-alpha) confidence interval, as a GEV_profileLogLik object containing:

mle

The estimated maximum likelihood GEV parameters, as a named vector (expressed in the profile parametrization).

ci

Length-two vector containing the lower and upper endpoints of the desired profile likelihood confidence interval.

profile_loglik

Named matrix containing the profile loglikelihood value (Column 2) for each considered profile parameter value (Column 1).

conf_line

Confidence line for the desired profile likelihood confidence interval. See e.g. Coles (2001) for more details.

eval_table

Tibble (tibble::tibble()) containing the history of profile log-likelihood evaluation values, and related metadata.

param_name

Name of the profiled parameter (infered, for debugging purposes).

parameter

Name of the profiled parameter (given).

parametrization

Parametrization used for the profile likelihood.

subparam_id

Index of the parameter coefficient for which the profile log-likelihood was computed.

id_param

Index of the profile parameter in the GEV parameter vector.

References

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

Internal function for the GEV profile likelihood

Description

Internal function for the GEV profile likelihood

Usage

GEV_profile_loglik_internal(
  val,
  Z,
  parameter = c("shape", "location", "scale", "return_level", "endpoint"),
  return_period = 100,
  locmat = as.matrix(1),
  scamat = as.matrix(1),
  shamat = as.matrix(1),
  subparam_id = 0,
  orthogonal = FALSE,
  init = NULL,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  ...
)

Arguments

val

.

Z

.

parameter

.

return_period

.

locmat

.

scamat

.

shamat

.

subparam_id

.

orthogonal

.

init

.

hessian

.

maxit

.

method

.

...

.

GEV profile parameter vector id

Description

GEV profile parameter vector id

Usage

GEV_profpar_id(
  parameter = c("shape", "location", "scale", "return_level", "endpoint"),
  nbloc = 1,
  nbsca = 1,
  nbsha = 1,
  subparam_id = 0
)

Arguments

parameter

.

nbloc

Number of location parameter coefficients (i.e. one plus the number of shape parameter covariates).

nbsca

Number of scale parameter coefficients (i.e. one plus the number of scale parameter covariates).

nbsha

Number of shape parameter coefficients (i.e. one plus the number of shape parameter covariates).

subparam_id

.

Value

The id of the desired GEV parameter in the specified internal vector format.

Compute return level from GEV parameters

Description

Compute return level from GEV parameters

Usage

GEV_return_level(loc = 0, scale = 1, shape, return_period)

Arguments

loc

Location parameter.

scale

Scale parameter.

shape

Shape parameter.

return_period

Return period for the desired return level.

Value

The return level of the specified GEV distribution with a return period of return_period. In other terms, the quantile of the GEV distribution at probability level 1 - 1/return_period.

Examples

GEV_return_level(loc=0, scale=1, shape=0.1, return_period=100)

Unrestricted GEV endpoint

Description

Unrestricted GEV endpoint

Usage

GEV_unrestricted_endpoint(loc = 0, scale = 1, shape, ...)

Arguments

loc

Location parameter.

scale

Scale parameter.

shape

Shape parameter.

...

Unused.

Value

The unrestricted endpoint of the specified GEV distribution (returns the upper GEV endpoint if shape<0, and the lower endpoint otherwise).

GEV upper endpoint

Description

GEV upper endpoint

Usage

GEV_upper_endpoint(loc = 0, scale = 1, shape, ...)

Arguments

loc

Location parameter.

scale

Scale parameter.

shape

Shape parameter.

...

Unused.

Value

The upper endpoint of the specified GEV distribution (returns Inf if shape>=0).

GPD parameter vector reparametrization

Description

GPD parameter vector reparametrization

Usage

GPD_change_parametrization(
  parameters,
  threshold = 0,
  threshold_lvl = 0,
  parametrization = c("classical", "orthogonal", "quantile", "endpoint"),
  new_parametrization = c("classical", "orthogonal", "quantile", "endpoint"),
  quantile_lvl = 1 - (1/100),
  nbsca = 1,
  nbsha = 1
)

Arguments

parameters

Vector of GPD parameters in the internal format.

threshold

GPD threshold value.

threshold_lvl

Probability level of the threshold threshold.

parametrization

Current parametrization of parameters.

new_parametrization

Desired new parametrization.

quantile_lvl

Quantile probability level for the 'quantile' parameter.

nbsca

Number of scale parameter coefficients (i.e. one plus the number of scale parameter covariates).

nbsha

Number of shape parameter coefficients (i.e. one plus the number of shape parameter covariates).

Value

The vector of GPD parameters reparametrized from parametrization to new_parametrization.

GPD covariate matrices

Description

Creates separate GPD covariate matrices from the main UI specification.

Usage

GPD_covariate_matrices(
  X = NULL,
  x_rlvl = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  parametrization = c("classical", "orthogonal", "quantile", "endpoint"),
  out_param = parametrization
)

Arguments

X

.

x_rlvl

.

scale_cols

.

shape_cols

.

parametrization

.

out_param

.

Value

The necessary internal matrices in a list.

Space-efficient GPD covariate matrix with indices

Description

CURRENTLY UNUSED

Usage

GPD_covariate_matrix(
  X = NULL,
  x_rlvl = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  parametrization = c("classical", "orthogonal", "quantile", "endpoint"),
  out_param = parametrization
)

Arguments

X

.

x_rlvl

.

scale_cols

.

shape_cols

.

parametrization

.

out_param

.

Value

The space-efficient GPD covariate matrix with indices in a named list.

GPD endpoint

Description

GPD endpoint

Usage

GPD_endpoint(
  threshold = 0,
  scale = 1,
  shape,
  ...,
  endpoint_type = c("unrestricted", "upper")
)

Arguments

threshold

GPD threshold value.

scale

Scale parameter.

shape

Shape parameter.

...

Other optional parameters for the internal endpoint function (Unused).

endpoint_type

Whether to compute the upper endpoint (returns Inf if shape>=0), or the unrestricted endpoint (returns the lower GPD endpoint if shape>=0).

Value

The endpoint of the specified GPD distribution.

Examples

GPD_endpoint(threshold=0, scale=1, shape=-0.1, endpoint_type='upper')

GPD log likelihood

Description

GPD log likelihood

Usage

GPD_log_likelihood(
  Y,
  threshold,
  scale,
  shape,
  obs_weights = NULL,
  ill_defined_value = -10^6
)

Arguments

Y

Data observations.

threshold

GPD threshold value.

scale

Scale parameter.

shape

Shape parameter.

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

Value

The GPD log-likelihood evaluated at the given parameters, given the data observations Y.

Internal GPD log-likelihood function format for optimisation

Description

Internal GPD log-likelihood function format for optimisation

Usage

GPD_log_likelihood_optim(
  a,
  Y,
  threshold = 0,
  threshold_lvl = 0,
  parametrization = c("classical", "orthogonal", "quantile", "endpoint"),
  quantile_lvl = 1 - (1/100),
  scamat = as.matrix(1),
  shamat = as.matrix(1),
  negative = FALSE,
  obs_weights = NULL,
  ill_defined_value = -10^6
)

Arguments

a

.

Y

.

threshold

.

threshold_lvl

.

parametrization

.

quantile_lvl

.

scamat

.

shamat

.

negative

.

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

Maximum-likelihood GPD estimate

Description

Maximum-likelihood GPD estimate

Usage

GPD_maxlik(
  Y,
  threshold = 0,
  threshold_lvl = 0,
  parametrization = c("classical", "orthogonal", "quantile", "endpoint"),
  quantile_lvl = 1 - (1/100),
  orthogonal = FALSE,
  X = NULL,
  x_rlvl = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  out_param = parametrization,
  obs_weights = NULL,
  ill_defined_value = -10^6,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  verbose = 1,
  ...
)

Arguments

Y

Data observations.

threshold

GPD threshold value.

threshold_lvl

Probability level of the threshold threshold.

parametrization

Likelihood parametrization. Alternatives to classical substitute for the scale parameter.

quantile_lvl

Quantile probability level for the 'quantile' parametrization.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits).

x_rlvl

Covariate vector at which to reparametrize for the 'quantile' or 'endpoint' parametrizations (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

out_param

Additional output parametrization (same as parametrization, by default). If out_param != parametrization, the parameters are reparametrized from parametrization to out_param after estimation, in a separate output.

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used. See stats::optim() for more details.

verbose

Verbose level, as integer.

...

Other arguments passed to the control argument of stats::optim().

Value

The fitted maximum-likelihood GPD as a GPD_ML object, containing:

mle

The estimated maximum likelihood GPD parameters, as a named vector (expressed in parametrization).

loglik

The log-likelihood of the estimated parameters, given the data

conv

Whether the optimisation procedure converged. See the convergence output of stats::optim() for more details.

hessian

The loglikelihood hessian evaluated at the estimated parameters, given the data.

parametrization

Likelihood parametrization.

out_mle

The estimated maximum likelihood GPD parameters, reparametrized in out_param.

out_parametrization

Additional output parametrization.

Initial GPD parameter vector defaults for profile optimization

Description

Initial GPD parameter vector defaults for profile optimization

Usage

GPD_param_init(
  Y,
  threshold = 0,
  threshold_lvl = 0,
  nbsca,
  nbsha,
  parametrization = c("classical", "orthogonal", "quantile", "endpoint"),
  quantile_lvl,
  obs_weights = NULL
)

Arguments

Y

.

threshold

.

threshold_lvl

.

nbsca

.

nbsha

.

parametrization

.

quantile_lvl

.

Value

The initial parameter values as a vector, in the correct internal format.

GPD parameter vector names

Description

GPD parameter vector names

Usage

GPD_param_names(
  nbsca = 1,
  nbsha = 1,
  parametrization = c("classical", "orthogonal", "quantile", "endpoint")
)

Arguments

nbsca

.

nbsha

.

parametrization

.

Value

The names of the GPD parameters in the specified internal vector format, as a vector.

GPD profile CI using binary search

Description

GPD profile CI using binary search

Usage

GPD_profile_CI(
  Y,
  threshold = 0,
  threshold_lvl = 0,
  parameter = c("shape", "scale", "quantile", "endpoint"),
  subparam_id = 0,
  alpha = 0.05,
  quantile_lvl = 1 - (1/100),
  orthogonal = FALSE,
  X = NULL,
  x_rlvl = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  warmstart_table = NULL,
  init_step_pos = 100,
  init_step_neg = 10,
  tol = 0.01,
  steps_beyond_conf = 5,
  initial_MLE_para = c("classical", "same"),
  max_steps = 1000,
  obs_weights = NULL,
  ill_defined_value = -10^6,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  method_prof = c("default", "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  verbose = 1,
  ...
)

Arguments

Y

Data observations.

threshold

GPD threshold value.

threshold_lvl

Probability level of the threshold threshold.

parameter

Parameter for which to compute the profile log-likelihood.

subparam_id

Index of the parameter coefficient for which to compute the profile log-likelihood (for conditional/non-stationary fits).

alpha

Confidence alpha for the profile log-likelihood confidence interval (i.e. for the confidence line on the profile plot).

quantile_lvl

Quantile probability level for the 'quantile' parameter.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits). Columns should be variables, and rows should be observations matching Y.

x_rlvl

Covariate vector at which to reparametrize for the 'quantile' or 'endpoint' parametrizations (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

warmstart_table

Evaluation table from a previous run.

init_step_pos

Initial numerical size of each evaluation step to the right, in the profile parameter's scale.

init_step_neg

Initial numerical size of each evaluation step to the left, in the profile parameter's scale.

tol

Numerical tolerance for convergence, in the profile parameter's scale.

steps_beyond_conf

Number of additional steps to take (in each direction) after the profile log-likelihood values reach below the confidence line.

initial_MLE_para

Parametrization used for the initial maximum likelihood estimate (defaults to classical, for better stability).

max_steps

Maximum number of steps taken (in each direction). If the confidence line was not reached, the corresponding confidence interval endpoint will be infinite.

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used for the initial maximum likelihood optimisation. See stats::optim() for more details.

method_prof

The optimisation method to be used for the profile likelihood optimisation. See stats::optim() for more details.

verbose

Verbose level, as integer.

...

Other arguments passed to the control argument of stats::optim().

Value

The GPD profile log-likelihood confidence interval for the desired parameter, with confidence line and resulting (1-alpha) confidence interval, as a GPD_profileLogLik object containing:

mle

The estimated maximum likelihood GPD parameters, as a named vector (expressed in the profile parametrization).

ci

Length-two vector containing the lower and upper endpoints of the desired profile likelihood confidence interval.

profile_loglik

Named matrix containing the profile loglikelihood value (Column 2) for each considered profile parameter value (Column 1).

conf_line

Confidence line for the desired profile likelihood confidence interval. See e.g. Coles (2001) for more details.

eval_table

Tibble (tibble::tibble()) containing the history of profile log-likelihood evaluation values, and related metadata.

param_name

Name of the profiled parameter (infered, for debugging purposes).

parameter

Name of the profiled parameter (given).

parametrization

Parametrization used for the profile likelihood.

subparam_id

Index of the parameter coefficient for which the profile log-likelihood was computed.

id_param

Index of the profile parameter in the GPD parameter vector.

References

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

Multi-value conditional GPD profile likelihood confidence intervals

Description

For non-stationary models, the quantile reparametrization depends on covariate values. This function repeats the profile likelihood procedure for several covariate values. It enables obtaining a quantile (or endpoint) curve, with profile-likelihood confidence bands, as a function of the covariate values.

Usage

GPD_profile_CIs_multiple(
  Y,
  threshold = 0,
  threshold_lvl = 0,
  parameter = c("quantile", "endpoint"),
  alpha = 0.05,
  quantile_lvl = 1 - (1/100),
  orthogonal = FALSE,
  X = NULL,
  X_rlvl = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  init_step_pos = 100,
  init_step_neg = 10,
  tol = 0.01,
  steps_beyond_conf = 5,
  initial_MLE_para = c("classical", "same"),
  max_steps = 10000,
  obs_weights = NULL,
  ill_defined_value = -10^6,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  method_prof = c("default", "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  parallel_strat = c("none", "multisession", "sequential", "multicore"),
  n_workers = NULL,
  ...
)

Arguments

Y

Data observations.

threshold

GPD threshold value.

threshold_lvl

Probability level of the threshold threshold.

parameter

Parameter for which to compute the profile likelihood confidence intervals.

alpha

Confidence alpha for the profile likelihood confidence intervals.

quantile_lvl

Quantile probability level for the 'quantile' parameter.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits). Columns should be variables, and rows should be observations matching Y.

X_rlvl

Covariate matrix at which to reparametrize for the 'quantile' or 'endpoint' parametrizations (for conditional/non-stationary fits). Columns should be variables, and each row should give one covariate realization at which to reparametrize and obtain a CI.

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

init_step_pos

Initial numerical size of each evaluation step to the right, in the profile parameter's scale.

init_step_neg

Initial numerical size of each evaluation step to the left, in the profile parameter's scale.

tol

Numerical tolerance for convergence, in the profile parameter's scale.

steps_beyond_conf

Number of additional steps to take (in each direction) after the profile log-likelihood values reach below the confidence line.

initial_MLE_para

Parametrization used for the initial maximum likelihood estimate (defaults to classical, for better stability).

max_steps

Maximum number of steps taken (in each direction). If the confidence line was not reached, the corresponding confidence interval endpoint will be infinite.

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used for the initial maximum likelihood optimisation. See stats::optim() for more details.

method_prof

The optimisation method to be used for the profile likelihood optimisation. See stats::optim() for more details.

parallel_strat

Parallel strategy. One of "sequential" (default), "multisession", "multicore", or "mixed".

n_workers

A positive numeric scalar or a function specifying the maximum number of parallel futures that can be active at the same time before blocking. If a function, it is called without arguments when the future is created and its value is used to configure the workers. The function should return a numeric scalar. Defaults to future::availableCores()-1 if NULL (default), with "multicore" constraint in the relevant case. Ignored if strategy=="sequential".

...

Other arguments passed to the control argument of stats::optim().

Value

The GPD profile log-likelihood (1-alpha) confidence intervals for the desired parameter, for each desired covariate values, as a tibble::tibble(), with columns:

obs

Index of the observation (i.e. row) of X_rlvl for which the parameter estimate and CI was computed.

\verb{<parameter name>}

Conditional estimate of the parameter.

ci_down

Lower endpoint of the conditional (1-alpha) profile-likelihood confidence interval for parameter.

ci_up

Upper endpoint of the conditional (1-alpha) profile-likelihood confidence interval for parameter.

parameter

Name of the parameter for which the estimates and CIs were computed.

alpha

Confidence alpha for the profile likelihood confidence intervals.

quantile_lvl

Quantile probability level for 'quantile' parameter (only if parameter==quantile).

GPD profile log-likelihood

Description

GPD profile log-likelihood

Usage

GPD_profile_loglik(
  val,
  Y,
  threshold = 0,
  threshold_lvl = 0,
  parameter = c("shape", "scale", "quantile", "endpoint"),
  subparam_id = 0,
  quantile_lvl = 1 - (1/100),
  orthogonal = FALSE,
  X = NULL,
  x_rlvl = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  obs_weights = NULL,
  ill_defined_value = -10^6,
  init = NULL,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  ...
)

Arguments

val

Parameter value at which to evaluate the GPD profile log-likelihood.

Y

Data observations.

threshold

GPD threshold value.

threshold_lvl

Probability level of the threshold threshold.

parameter

Parameter for which to compute the profile log-likelihood.

subparam_id

Index of the parameter coefficient for which to compute the profile log-likelihood (for conditional/non-stationary fits).

quantile_lvl

Quantile probability level for the 'quantile' parameter.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits). Columns should be variables, and rows should be observations matching Y.

x_rlvl

Covariate vector at which to reparametrize for the 'quantile' or 'endpoint' parametrizations (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

init

Optional initial values for the remaining parameter's optimisation process, in the correct internal format.

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used. See stats::optim() for more details.

...

Other arguments passed to the control argument of stats::optim().

Value

The GPD profile log-likelihood of parameter evaluated at val, given the data, as a GPD_profML object containing:

param_val

(Named) parameter value at which the GPD profile log-likelihood was evaluated.

param_name

Name of the evaluated profile likelihood parameter.

mle_other

Maximum-likelihood estimate of the other GPD parameters.

loglik

Profile GPD log-likelihood value of parameter evaluated at val, given the data.

conv

Whether the optimisation procedure converged. See the convergence output of stats::optim() for more details.

hessian

The loglikelihood hessian evaluated at the estimated parameters, given the data.

parameter

Name of the evaluated profile likelihood parameter given as argument (redundent).

parametrization

Likelihood parametrization.

subparam_id

Index of the parameter coefficient for which the profile log-likelihood was computed.

id_param

Index of the likelihood profile parameter, in the internal parameter vector format.

GPD profile log-likelihood curve

Description

GPD profile log-likelihood curve

Usage

GPD_profile_loglik_curve(
  Y,
  threshold = 0,
  threshold_lvl = 0,
  parameter = c("shape", "scale", "quantile", "endpoint"),
  subparam_id = 0,
  alpha = 0.05,
  quantile_lvl = 1 - (1/100),
  orthogonal = FALSE,
  X = NULL,
  x_rlvl = NULL,
  scale_cols = NULL,
  shape_cols = NULL,
  warmstart_table = NULL,
  stepsize = 0.1,
  steps_beyond_conf = 5,
  initial_MLE_para = c("classical", "same"),
  max_steps = 10000,
  obs_weights = NULL,
  ill_defined_value = -10^6,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  method_prof = c("default", "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  ...
)

Arguments

Y

Data observations.

threshold

GPD threshold value.

threshold_lvl

Probability level of the threshold threshold.

parameter

Parameter for which to compute the profile log-likelihood.

subparam_id

Index of the parameter coefficient for which to compute the profile log-likelihood (for conditional/non-stationary fits).

alpha

Confidence alpha for the profile log-likelihood confidence interval (i.e. for the confidence line on the profile plot).

quantile_lvl

Quantile probability level for the 'quantile' parameter.

orthogonal

DEPRECATED.

X

Covariate matrix (for conditional/non-stationary fits). Columns should be variables, and rows should be observations matching Y.

x_rlvl

Covariate vector at which to reparametrize for the 'quantile' or 'endpoint' parametrizations (for conditional/non-stationary fits).

scale_cols

Column indices of X to use as covariate for the (conditional) scale parameter (for conditional/non-stationary fits).

shape_cols

Column indices of X to use as covariate for the (conditional) shape parameter (for conditional/non-stationary fits).

warmstart_table

Evaluation table from a previous run.

stepsize

Numerical size of each evaluation step, in the profile parameter's scale.

steps_beyond_conf

Number of additional steps to take (in each direction) after the profile log-likelihood values reach below the confidence line.

initial_MLE_para

Parametrization used for the initial maximum likelihood estimate (defaults to classical, for better stability).

max_steps

Maximum number of steps taken (in each direction). If the confidence line was not reached, the corresponding confidence interval endpoint will be infinite.

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

hessian

Logical. Should a numerically differentiated Hessian matrix be returned? See stats::optim() for more details.

maxit

The maximum number of iterations. See stats::optim() for more details.

method

The optimisation method to be used for the initial maximum likelihood optimisation. See stats::optim() for more details.

method_prof

The optimisation method to be used for the profile likelihood optimisation. See stats::optim() for more details.

...

Other arguments passed to the control argument of stats::optim().

Value

The GPD profile log-likelihood curve for the desired parameter, with confidence line and resulting (1-alpha) confidence interval, as a GPD_profileLogLik object containing:

mle

The estimated maximum likelihood GPD parameters, as a named vector (expressed in the profile parametrization).

ci

Length-two vector containing the lower and upper endpoints of the desired profile likelihood confidence interval.

profile_loglik

Named matrix containing the profile loglikelihood value (Column 2) for each considered profile parameter value (Column 1).

conf_line

Confidence line for the desired profile likelihood confidence interval. See e.g. Coles (2001) for more details.

eval_table

Tibble (tibble::tibble()) containing the history of profile log-likelihood evaluation values, and related metadata.

param_name

Name of the profiled parameter (infered, for debugging purposes).

parameter

Name of the profiled parameter (given).

parametrization

Parametrization used for the profile likelihood.

subparam_id

Index of the parameter coefficient for which the profile log-likelihood was computed.

id_param

Index of the profile parameter in the GPD parameter vector.

References

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

Internal function for the GPD profile likelihood

Description

Internal function for the GPD profile likelihood

Usage

GPD_profile_loglik_internal(
  val,
  Y,
  threshold = 0,
  threshold_lvl = 0,
  parameter = c("shape", "scale", "quantile", "endpoint"),
  quantile_lvl = 1 - (1/100),
  scamat = as.matrix(1),
  shamat = as.matrix(1),
  subparam_id = 0,
  orthogonal = FALSE,
  obs_weights = NULL,
  ill_defined_value = -10^6,
  init = NULL,
  hessian = TRUE,
  maxit = 1e+06,
  method = c("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Brent"),
  verbose = 1,
  ...
)

Arguments

val

.

Y

.

threshold

.

threshold_lvl

.

parameter

.

quantile_lvl

.

scamat

.

shamat

.

subparam_id

.

orthogonal

.

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

init

.

hessian

.

maxit

.

method

.

verbose

Verbose level, as integer.

...

.

GPD profile parameter vector id

Description

GPD profile parameter vector id

Usage

GPD_profpar_id(
  parameter = c("shape", "scale", "quantile", "endpoint"),
  nbsca = 1,
  nbsha = 1,
  subparam_id = 0
)

Arguments

parameter

.

nbsca

.

nbsha

.

subparam_id

.

Value

The id of the desired GPD parameter in the specified internal vector format.

Compute extreme quantile from GPD parameters

Description

Compute extreme quantile from GPD parameters

Usage

GPD_quantiles(quantile_lvl, threshold_lvl, threshold, scale, shape)

Arguments

quantile_lvl

Probability level of the desired extreme quantile.

threshold_lvl

Probability level of the GPD threshold.

threshold

GPD threshold value.

scale

Value(s) for the GPD scale parameter.

shape

Value(s) for the GPD shape parameter.

Value

The quantile value at probability level quantile_lvl.

Examples

GPD_quantiles(quantile_lvl=0.999, threshold_lvl=0.95, threshold=0, scale=1, shape=0.1)

Unrestricted GPD endpoint

Description

Unrestricted GPD endpoint

Usage

GPD_unrestricted_endpoint(threshold = 0, scale = 1, shape, ...)

Arguments

threshold

GPD threshold value.

scale

Scale parameter.

shape

Shape parameter.

...

Unused.

Value

The unrestricted endpoint of the specified GPD distribution (returns the upper GPD endpoint if shape<0, and the lower endpoint otherwise).

GPD upper endpoint

Description

GPD upper endpoint

Usage

GPD_upper_endpoint(threshold = 0, scale = 1, shape, ...)

Arguments

threshold

GPD threshold value.

scale

Scale parameter.

shape

Shape parameter.

...

Unused.

Value

The upper endpoint of the specified GPD distribution (returns Inf if shape>=0).

End the currently set doFuture strategy

Description

Resets the default strategy using future::plan("default").

Usage

end_doFuture_strategy()

Get doFuture operator

Description

Get doFuture operator

Usage

get_doFuture_operator(
  strategy = c("sequential", "multisession", "multicore", "mixed")
)

Arguments

strategy

One of "sequential" (default), "multisession", "multicore", or "mixed".

Value

Returns the appropriate operator to use in a foreach::foreach() loop. The %do% operator is returned if strategy=="sequential". Otherwise, the %dopar% operator is returned.

Optimisation evaluation step for the GEV profile likelihood

Description

Optimisation evaluation step for the GEV profile likelihood

Usage

optim_step_GEV_profile(
  a,
  val,
  Z,
  parametrization = c("classical", "return_level", "endpoint"),
  id_param,
  return_period = 100,
  locmat = as.matrix(1),
  scamat = as.matrix(1),
  shamat = as.matrix(1),
  negative = TRUE,
  orthogonal = FALSE
)

Arguments

a

.

val

.

Z

.

parametrization

.

id_param

.

return_period

.

locmat

.

scamat

.

shamat

.

negative

.

orthogonal

.

Optimisation evaluation step for the GPD profile likelihood

Description

Optimisation evaluation step for the GPD profile likelihood

Usage

optim_step_GPD_profile(
  a,
  val,
  Y,
  threshold = 0,
  threshold_lvl = 0,
  parametrization = c("classical", "orthogonal", "quantile", "endpoint"),
  id_param,
  quantile_lvl = 1 - (1/100),
  scamat = as.matrix(1),
  shamat = as.matrix(1),
  negative = TRUE,
  orthogonal = FALSE,
  obs_weights = NULL,
  ill_defined_value = -10^6
)

Arguments

a

.

val

.

Y

.

threshold

.

threshold_lvl

.

parametrization

.

id_param

.

quantile_lvl

.

scamat

.

shamat

.

negative

.

orthogonal

.

obs_weights

Optional observation weights for weighted likelihood.

ill_defined_value

Value to return if the arguments are out of support (e.g. negative scale, or non-positive arguments to logarithms).

Non-stationary confidence bands plot

Description

Plot the estimated conditional quantile, return level, or endpoint, with confidence bands for non-stationary models.

Usage

plot_data_quantile_ci(
  quantiles,
  q_down,
  q_up,
  time_index = seq_along(quantiles),
  x_label = "X",
  y_label = "Y",
  Y = NULL,
  event_index = NULL,
  obs_label = "Fitting obs.",
  event_label = "Event",
  legend.position = "bottom"
)

Arguments

quantiles

Vector of estimated quantiles, return levels, or endpoints.

q_down

Vector of lower confidence band values for quantiles.

q_up

Vector of upper confidence band values for quantiles.

time_index

Vector of time indices corresponding to the quantiles. Defaults to seq_along(quantiles).

x_label

Label for the x-axis. Defaults to "X".

y_label

Label for the y-axis. Defaults to "Y".

Y

(Optional) Vector of observations to add to the plot and compare to the quantiles.

event_index

(Optional) Index of an event (observation) to be highlighted.

obs_label

Label for the fitting observations. Defaults to "Fitting obs.".

event_label

(Optional) Label for the highlighted event observation. Defaults to "Event".

legend.position

Position of the legend to the side of the plot. Can be one of "bottom" (default), "right", "top", "left", or "none".

Value

A ggplot2::ggplot() object showing the estimated conditional quantile with confidence bands.

Profile likelihood curve plot

Description

Plot the profile likelihood curve for the desired parameter with the confidence line.

Usage

plot_profile_loglik_curve(
  profile_fct_object,
  prop_below = NULL,
  legend.position = "bottom"
)

Arguments

profile_fct_object

Object returned by either GPD_profile_loglik_curve() or GEV_profile_loglik_curve().

prop_below

(Optional) Proportion of the distance below the confidence line to show in the plot, to crop y-axis.

legend.position

Position of the legend to the side of the plot. Can be one of "bottom" (default), "right", "top", "left", or "none".

Value

A ggplot2::ggplot() object showing the profile likelihood curve for the desired parameter with the confidence line.

Set a doFuture execution strategy

Description

Set a doFuture execution strategy

Usage

set_doFuture_strategy(
  strategy = c("sequential", "multisession", "multicore", "mixed"),
  n_workers = NULL
)

Arguments

strategy

One of "sequential" (default), "multisession", "multicore", or "mixed".

n_workers

A positive numeric scalar or a function specifying the maximum number of parallel futures that can be active at the same time before blocking. If a function, it is called without arguments when the future is created and its value is used to configure the workers. The function should return a numeric scalar. Defaults to future::availableCores()-1 if NULL (default), with "multicore" constraint in the relevant case. Ignored if strategy=="sequential".

Value

The corresponding get_doFuture_operator() operator to use in a foreach::foreach() loop.

Insert value in vector

Description

Insert value in vector

Usage

vector_insert(vect, val, ind)

Arguments

vect

A 1-D vector.

val

A value to insert in the vector.

ind

The index at which to insert the value in the vector, must be an integer between 1 and length(vect) + 1.

Value

A 1-D vector of length length(vect) + 1, with val inserted at position ind in the original vect.