pFailure: Probability of Falling into a Failure Region

In ExtremalDep: Extremal Dependence Models

Probability of Falling into a Failure Region

Description

Compute the empirical probability of falling into two types of failure regions, based on exceedances simulated from a fitted extreme-value model.

Usage

pFailure(n, beta, u1, u2, mar1, mar2, type, plot, xlab, ylab, nlevels = 10)

Arguments

n	Integer. Number of points generated to compute the empirical probability.
beta	Numeric vector. Bernstein polynomial coefficients.
u1, u2	Numeric vectors of positive thresholds.
mar1, mar2	Numeric vectors. Marginal GEV parameters.
type	Character. Type of failure region: "or" - at least one exceedance, "and" - both exceedances, "both" - compute both probabilities.
plot	Logical. If TRUE, display contour plots of the probabilities.
xlab, ylab	Character strings for axis labels in plots.
nlevels	Integer. Number of contour levels for plots.

Details

The two failure regions are:

A_u = \{ (v_1, v_2): v_1 > u_1 \ \textrm{or}\ v_2 > u_2 \}

and

B_u = \{ (v_1, v_2): v_1 > u_1 \ \textrm{and}\ v_2 > u_2 \}

for (v_1, v_2) \in \mathbb{R}_+^2, with thresholds u_1,u_2 > 0.

Exceedance samples are generated following Algorithm 3 of Marcon et al. (2017). The empirical estimates are

\hat{P}_{A_u} = \frac{1}{n}\sum_{i=1}^n I(y_{i1} > u_1^* \ \textrm{or}\ y_{i2} > u_2^*)

and

\hat{P}_{B_u} = \frac{1}{n}\sum_{i=1}^n I(y_{i1} > u_1^* \ \textrm{and}\ y_{i2} > u_2^*)

.

Value

A list with elements AND and/or OR, depending on type. Each element is a matrix of size length(u1) x length(u2).

Author(s)

Simone Padoan [e-mail] (https://faculty.unibocconi.it/simonepadoan/) \ Boris Beranger [e-mail] (https://www.borisberanger.com)

References

Marcon, G., Naveau, P. and Padoan, S.A. (2017). A semi-parametric stochastic generator for bivariate extreme events. Stat, 6, 184–201.

See Also

dExtDep, rExtDep, fExtDep, fExtDep.np

Examples

# Example: Wind speed and gust data
data(WindSpeedGust)
years <- format(ParcayMeslay$time, format = "%Y")
attach(ParcayMeslay[years %in% 2004:2013, ])

WS_th <- quantile(WS, .9)
DP_th <- quantile(DP, .9)

pars.WS <- evd::fpot(WS, WS_th, model = "pp")$estimate
pars.DP <- evd::fpot(DP, DP_th, model = "pp")$estimate

data_uf <- trans2UFrechet(cbind(WS, DP), type = "Empirical")
rdata <- rowSums(data_uf)
r0 <- quantile(rdata, probs = .90)
extdata <- data_uf[rdata >= r0, ]

SP_mle <- fExtDep.np(x = extdata, method = "Frequentist", k0 = 10, type = "maxima")


pF <- pFailure(
  n = 50000, beta = SP_mle$Ahat$beta,
  u1 = seq(19, 28, length = 200), mar1 = pars.WS,
  u2 = seq(40, 60, length = 200), mar2 = pars.DP,
  type = "both", plot = TRUE,
  xlab = "Daily-maximum Wind Speed (m/s)",
  ylab = "Differential of Pressure (mbar)",
  nlevels = 15
)

ExtremalDep documentation built on Aug. 21, 2025, 5:57 p.m.