Regional (or local) parameter and quantile estimation

Description

Calculates regional (or local) parameters of a generalized extreme value (GEV) distribution using (trimmed) L-moments (see TLMoments and parameters) from a vector or matrix of observation. Based on these parameters, a p-quantile of the GEV will be calculated for the jth station.

Usage

RegioGEV(x, p, j = 1, leftrim = 0, rightrim = 0, na.rm = TRUE, ...)

Arguments

Details

The optimal weights will be calculated as described in "Kinsvater, Fried and Lilienthal (2015): Regional extreme value index estimation and a test of tail homogeneity, Environmetrics, DOI: 10.1002/env.2376, Section 3.2". If it's not possible to calculate optimal weights (negative eigenvaules of an estimated covarinace matrix), simple weights will be calculated: w_j=\frac{n_j}{sum_{j=1}^d n_j}

Value

List of

• quant quantile calculation from an estimated GEV with a regional shape-parameter.

• param estimated parameter vector from a GEV (using L-moments or trimmed L-moments).

• w optimal or simple weighting (just returned if x is a matrix).

Examples

library("evd")
# sample observations of 75 years at one station:
x <- rgev(75) # x is a vector
RegioGEV(x=x, p=0.95)

x2 <- c(NA, NA, x[1:60], NA, x[61:75]) # vector of observations with missing values
RegioGEV(x=x2, p=0.95) # NAs will be removed

# sample observations of 100 years at 4 stations:
set.seed(1053)
x <- matrix(rgev(400, 2, 1, 0.3), ncol=4)
RegioGEV(x=x, p=0.9, j=3, leftrim=0, rightrim=0) # optimal weighting
RegioGEV(x=x, p=0.9, j=3, leftrim=0, rightrim=1) # optimal weighting

# With missing values:
x2 <- x
x2[c(54, 89, 300)] <- NA
RegioGEV(x=x2, p=0.9, j=3, leftrim=0, rightrim=0)

# sample again observations of 100 years at 4 stations:
set.seed(958)
x <- matrix(rgev(400, 2, 1, 0.3), ncol=4)
RegioGEV(x=x, p=0.9, j=3, leftrim=0, rightrim=0) # simple weighting

Seasonal regional (or local) parameter and quantile estimation

Description

Calculates regional (or local) parameters of a two-component GEV distribution (product of two GEVs) using (trimmed) L-moments (see TLMoments and parameters) from two vectors or two matrices of observation, e.g. winter and summer observations from one or more than one station. Based on these two parameter vectors, a p-quantile of the two-component GEV will be calculated for the jth station.

Usage

RegioGEVSeas(x1, x2, p, j = 1, leftrim = 0, rightrim = 0, na.rm = TRUE,
  ...)

Arguments

Value

List of

• quant quantile calculation from an estimated two-component GEV with a regional (or local) shape-parameters.

• param1 estimated parameter vector from season 1 from a GEV (using L-moments or trimmed L-moments).

• param2 estimated parameter vector from season 2 from a GEV (using L-moments or trimmed L-moments).

Examples

library("evd")
# Seasonal observations of 80 years at one station:
x1 <- rgev(80, 2, 1, 0.2) # observations from season 1
x2 <- rgev(80, 3, 1, 0.3) # observations from season 2
RegioGEVSeas(x1=x1, x2=x2, p=0.95)

# Missing values in both seasons in the same year(s):
x1a <- c(NA, x1, NA)
x2a <- c(NA, x2, NA)
RegioGEVSeas(x1a, x2a, p=0.99, leftrim=0, rightrim=0, na.rm=TRUE)

# Missing values in both seasons in different year(s):
x1b <- x1
x1b[c(4,19)] <- NA
x2b <- x2
x2b[77] <- NA
RegioGEVSeas(x1b, x2b, p=0.99, leftrim=0, rightrim=0, na.rm=TRUE)

# Seasonal observations of 100 years at 4 stations:
x1 <- matrix(rgev(400, 2, 1, 0.3), ncol=4)
x2 <- matrix(rgev(400, 4, 1, 0.2), ncol=4)
# estimate quantile for station 1 and 2 (consider the same shape-parameters):
RegioGEVSeas(x1, x2, p=0.99, j=1, leftrim=0, rightrim=0)
RegioGEVSeas(x1, x2, p=0.99, j=2, leftrim=0, rightrim=0)

# With missing values:
x3 <- x1
x4 <- x2
x3[c(54, 89, 300)] <- NA
RegioGEVSeas(x3, x4, p=0.99, j=1, leftrim=0, rightrim=0)

Regional EVI estimator

Description

Estimation of the positive extreme value index (EVI) based on multiple local Hill estimators. We assume heavy-tail homogeneity, i.e., all local EVI's are the same.

Usage

RegioHill(x, k, k.qu = 20, type = "evopt", alpha = 0.05, ci = "nonlog")

Arguments

Value

List of

• est a weighted average of local Hill estimates.

• Sigma an estimate of the corresponding variance matrix.

• CI a confidence interval.

Examples

library("evd")
x1 <- rgev(150, loc = 2, scale = 1, shape=0.4)
hill(x1, k=20)
x2 <- rgev(100, loc = 2.5, scale = 1, shape=0.4)
x2 <- c(x2, rep(NA, 50))
x <- cbind(x1, x2)
k <- c(40, 30)
RegioHill(x, k)

Quantile estimation: Weissman's extrapolation

Description

Estimation of the p-quantile based on multiple local Hill estimators and Weissman's extrapolation formula. We assume heavy-tail homogeneity, i.e., all local EVI's are the same.

Usage

RegioWeissman(x, j = 1, p, k, k.qu = 20, type = "evopt", alpha = 0.05)

Arguments

Value

List of

• est Point estimate of p-quantile of column j

• CI the corresponding alpha-confidence interval

• EVI estimate of the extreme value index

• k tail sample size

• p a probability

• u.kn (n-k)-th largest observation, where n is the sample length at station j after removing missing values

• description a short description.

Examples

library("evd")
# sample observations of 75 years at one station:
x <- rgev(75, 0, 1, 0) # x is a vector
RegioWeissman(x=x, p=0.95)

x2 <- c(NA, NA, x[1:60], NA, x[61:75]) # vector of observations with missing values
RegioWeissman(x=x2, p=0.95) # NAs will be removed

# sample observations of 100 years at 4 stations:
set.seed(1053)
x <- matrix(rgev(400, 2, 1, 0.3), ncol=4)
RegioWeissman(x=x, p=0.9, j=3)

# With missing values:
x2 <- x
x2[sample(100, 12),2] <- NA
RegioWeissman(x=x2, p=0.9, j=3)

Quantile estimation: Weissman's extrapolation for seasonal data

Description

Estimation of the p-quantile based on multiple local Hill estimators and Weissman's extrapolation formula with seasonality. We assume heavy-tail homogeneity within each season, i.e., all local EVI's are the same.

Usage

RegioWeissmanSeas(x, j = 1, p, ...)

Arguments

Value

Point estimate of seasonal p-quantile of column j and a short description.

Examples

library("evd")
# Local & seasonal (observations of 80 years at one station):
x1 <- rgev(80, 2, 1, 0.2) # observations from season 1
x2 <- rgev(80, 3, 1, 0.3) # observations from season 2
x <- list(x1, x2)
RegioWeissmanSeas(x=x, j=1, p=0.99)

x1 <- matrix(rgev(400, 3, 1, 0.1), ncol=4) # observations of season 1 at 4 stations
x2 <- matrix(rgev(400, 2, 1, 0.3), ncol=4) # observations of season 2 at 4 stations
x <- list(x1, x2)
RegioWeissmanSeas(x=x, j=1, p=0.99)

Heavy-tail ANOVA

Description

A test of heavy-tail homogeneity, that is, equality of the positive extreme value index for all d columns of x.

Usage

TailAnova(x, k, k.qu = 20, type = "evopt", cf = TRUE)

Arguments

Value

Test statistic and p-value.

Examples

library("evd")
set.seed(6754)
x1 <- rgev(150, loc = 2, scale = 1, shape=0.4)
x2 <- rgev(150, loc = 2.5, scale = 1, shape=0.1) # H_0 violated because of different shapes
x <- cbind(x1, x2)
TailAnova(x)

x1 <- rgev(150, loc = 2, scale = 1, shape=0.3)
x2 <- rgev(150, loc = 2.5, scale = 1, shape=0.3) # H_0 not violated because of same shapes
x <- cbind(x1, x2)
TailAnova(x)

Estimated confidence intervals using TL

Description

Estimated regional (or local) (1-alpha)-confidence intervals for an estimated quantile by using annual TL(0,1)-moments (TL).

Usage

confInt_TL(x, p, j = 1, alpha = 0.05)

Arguments

Value

List of

• ci confidence interval.

• quant estimated quantile from a GEV using trimmed L-moments (leftrim=0, rightrim=1).

Examples

library("evd")
# Seasonal observations of 80 years at one station:
x1 <- rgev(80, 2, 1, 0.2) # observations from season 1
x2 <- rgev(80, 3, 1, 0.3) # observations from season 2
x <- seas2ann(x1, x2) # calculaes annual maxima of the two seasons
confInt_TL(x=x, p=0.95, alpha=0.05)

# Seasonal observations of 100 years at 4 stations:
x1 <- matrix(rgev(400, 2, 1, 0.3), ncol=4) # observations from season 1
x2 <- matrix(rgev(400, 2, 1, 0.2), ncol=4) # observations from season 2
x <- seas2ann(x1, x2) # calculaes annual maxima of the two seasons
confInt_TL(x=x, j=2, p=0.95, alpha=0.05)

Estimated confidence intervals using W

Description

Estimated regional (or local) (1-alpha)-confidence intervals for an estimated quantile by using the annual Weissman estimator (W).

Usage

confInt_W(x, p, j = 1, alpha = 0.05, ...)

Arguments

Value

List of

• ci confidence interval.

• quant estimated quantile using Weissman's estimator.

Examples

library("evd")
# Seasonal observations of 80 years at one station:
x1 <- rgev(80, 2, 1, 0.2) # observations from season 1
x2 <- rgev(80, 3, 1, 0.3) # observations from season 2
x <- seas2ann(x1, x2) # calculaes annual maxima of the two seasons
confInt_W(x=x, p=0.95, alpha=0.05)

# Seasonal observations of 100 years at 4 stations:
x1 <- matrix(rgev(400, 2, 1, 0.3), ncol=4) # observations from season 1
x2 <- matrix(rgev(400, 2, 1, 0.2), ncol=4) # observations from season 2
x <- seas2ann(x1, x2) # calculaes annual maxima of the two seasons
confInt_W(x=x, j=2, p=0.95, alpha=0.05)

Estimated confidence intervals using sTL

Description

Estimated regional (or local) (1-alpha)-confidence intervals for an estimated quantile by using seasonal TL(0,1)-moments (sTL).

Usage

confInt_sTL(x1, x2, p, j = 1, alpha = 0.05)

Arguments

Value

List of

• ci confidence interval.

• quant estimated quantile from a two-component GEV using trimmed L-moments (leftrim=0, rightrim=1).

• var variance of the estimated quantile.

Examples

library("evd")
# Seasonal observations of 80 years at one station:
x1 <- rgev(80, 2, 1, 0.2) # observations from season 1
x2 <- rgev(80, 3, 1, 0.3) # observations from season 2
confInt_sTL(x1=x1, x2=x2, p=0.95, alpha=0.05)

# Seasonal observations of 100 years at 4 stations:
x1 <- matrix(rgev(400, 2, 1, 0.3), ncol=4)
x2 <- matrix(rgev(400, 4, 1, 0.2), ncol=4)
confInt_sTL(x1=x1, x2=x2, j=2, p=0.95, alpha=0.05)

Two-component generalized extreme value distribution (GEV)

Description

Density, distribution function, quantile function and random generation for a two-component GEV distribution (product of two GEVs).

Usage

dGEVxGEV(x, param1, param2)

pGEVxGEV(q, param1, param2)

qGEVxGEV(p, param1, param2)

rGEVxGEV(n, param1, param2)

Arguments

Details

These functions use the parametrization of the gev-functions from the package 'evd'. The distribution F of a two-component GEV is: F=F_1 \cdot F_2, where F_1 and F_2 are two distribution functions of a GEV.

Examples

# density and distribution function of a two-component GEV:
par(mfrow=c(3,1))
curve(dGEVxGEV(x, c(2,1,0.2), c(3,2,0.4)), from=0, to=20, ylab="Density", n=1000)
curve(pGEVxGEV(x, c(2,1,0.2), c(3,2,0.4)), from=0, to=20, ylab="Probability", n=1000)

# quantiles of a two-component GEV:
qGEVxGEV(p=c(0.9, 0.99), c(2,1,0.2), c(3,2,0.4))

# random numbers of a two-component GEV:
set.seed(23764)
rn <- rGEVxGEV(1000, c(2,1,0.1), c(3,2,0))
hist(rn, breaks=40, freq=FALSE, main="")
curve(dGEVxGEV(x, c(2,1,0.1), c(3,2,0)), from=0, to=20,
ylab="density", n=1000, col="red", add=TRUE)

Hill's estimator

Description

Estimation of heavy tails with Hill's estimator

Usage

hill(x, k)

Arguments

Value

Hill's estimator for each sample.

Examples

library("evd")
x1 <- rgev(100, loc = 2, scale = 1, shape=0.4)
hill(x1, k=20)
x2 <- rgev(100, loc = 2, scale = 1, shape=0.5)
hill(cbind(x1, x2), k = c(20, 25))
x2[c(4,8,39)] <- NA
hill(cbind(x1, x2), k=c(20, 25))
# if leaving out k, it will be set to floor(2*n^(2/3)/d^(1/3)) = c(34,33):
hill(cbind(x1, x2)) # is the same as:
hill(cbind(x1, x2), k=c(34,33))

Block maxima distribution

Description

Calculates quantiles of a block maxima distribution.

Usage

qBM(p, b, param)

Arguments

Details

Formular of a block maxima distribution function:

F_j(x)=F_j^{(b)}(x)=\left[2\cdot T_{1/\xi}\left(\left\{1+\xi\frac{x-\mu_j}{\sigma_j}\right\}\cdot T_{1/\xi}^{-1}\left(1-\frac{1}{2b}\right)\right)-1\right]^b,

where T_{\nu} denote the t-distribution function with \nu degrees of freedom.

Value

Quantile of a block maxima distribution.

Examples

qBM(p=c(0.75, 0.99), b=12, param=c(2,1,0.2))

Annual maxima from seasonal maxima

Description

Calculates annual maxima from seasonal maxima of two seasons.

Usage

seas2ann(x1, x2, na.rm = TRUE)

Arguments

Value

Matrix of annual observations (rows: observations, columns: stations).

Examples

set.seed(28379)
x1 <- matrix(round(rnorm(8, 20, 25)), ncol=2)
x1[2] <- NA
x2 <- matrix(round(rnorm(8, 20, 25)), ncol=2)
x2[c(2,5,6)] <- NA
x1
x2
seas2ann(x1,x2,TRUE)
seas2ann(x1,x2,FALSE)

Homogeneity test for the shape

Description

A test for assumption H_0: "shape parameter is equal for all d GEV margins" with test statistic based on (trimmed) L-moments.

Usage

xiAnova(x, leftrim = 0, rightrim = 1)

Arguments

Value

p-value of the test.

Examples

library("evd")
# sample observations of 100 years at 5 stations:
set.seed(1053)
x19 <- matrix(rgev(400, 2, 1, 0.1), ncol=4) # 4 stations with the same shape
x10 <- rgev(100, 2, 1, 0.4) # one station with a different shape
x <- cbind(x19, x10)
xiAnova(x=x, leftrim=0, rightrim=1)