mrp: Mean Return Period Estimation

In kerdiest: Nonparametric Kernel Estimation of Distribution Function

Mean Return Period Estimation

Description

This functions computes an estimate of the time between two values of a concrete level (size of an earthquake, flow lewel, wind speed...).

Usage

mrp(type_kernel = "n", vec_data, y = NULL,
           bw = PBbw(type_kernel = "n", vec_data, 2), lambda)

Arguments

type_kernel	The kernel function. You can use four types: "e" Epanechnikov, "n" Normal, "b" Biweight and "t" Triweight. The normal kernel is used by default.
vec_data	The data sample (earthquake magnitudes, flow levels, wind speed...).
y	A grid or a singular value where the estimator is computed. By default, a grid of 50 values between the minimum and the maximum of the data is computed.
bw	The bandwidth parameter. By default, the plug-in method of Polansky and Baker (2000) is used.
lambda	The mean activity rate.

Details

The mean return period is usually calculated assuming that event occurrence follows a Poisson process. In this case, the mean return period of events of size c is calculated as

T(c) = \frac{1}{ \lambda (1-F_h(c))}.

In Orlecka-Sikora (2008) or Quintela-del-Rio (2010) an application to earthquake data is made. In hydrological applications, if we work with annual maxima data, the parameter of the Poisson variable is 1 (one maximum per year). The mean return period between flow levels of value c is calculated as

T(c) = \frac{1}{ 1-F_h(c)}.

See, for instance, Quintela-del-Rio (2011), for an application to data of Salt River near Roosevelt, AZ, USA (saltriver data).

Value

A list containing:

Estimated_values	Vector containing the estimated function.
grid	The used grid.
bw	Value of the bandwidth.

Author(s)

Graciela Estévez Pérez and Alejandro Quintela del Río

References

Orlecka-Sikora, B. (2008), "Resampling methods for evaluating the uncertainty of the nonparametric magnitude distribution estimation in the probabilistic seismic hazard analysis", Tectonophysics, 456, 38-51.

Quintela-del-Rio, A. (2010), "On nonparametric techniques for area- characteristic seismic hazard parameters", Geophysical Journal International, 180, 339-346.

Quintela-del-Rio, A. (2011), "On bandwidth selection for nonparametric estimation in flood frequency analysis", Hydrological Processes, 25, 671-678.

Quintela-del-Río, A. and Estévez-Pérez, G. (2012), "Nonparametric kernel distribution function estimation with kerdiest: an R package for bandwidth choice and applications", Journal of Statistical Software, 50(8), 1-21.

Examples

## Working with earthquake data. We use the catalogue of the National
## Geographic Institute (IGN) of Spain and select the data of the Northwest
## of the Iberian Peninsula.
data(nwip)
require(chron)
require(date)
## Data with magnitude greater than 3 are considered
mg <- nwip$magnitude[nwip$magnitude > 3.0]
x1 <- nwip$year
x2 <- nwip$month
x3 <- nwip$day
ys <- paste(x1, x2, x3)
earthquake_date <- as.character(ys)
y1s <- as.date(earthquake_date, order = "ymd")
## Computation of the total number of years
y2s <- as.POSIXct(y1s)
z <- years(y2s)
n.years <- length(levels(z))
## Mean rate of earthquakes per year
lambda <- length(mg)/n.years
## Estimation of the mean return period (in years) between earthquakes of
## the same magnitude
est2 <- mrp(vec_data = mg, lambda = lambda)
plot(est2$grid, est2$Estimated_values, type = "l", xlab = "Magnitude",
  ylab = "Mean return period (years)")
## Working with hydrological data: annual peak instantaneous flow of the
## Salt River near Roosevelt, AZ, USA, for 1924-2009.
data(saltriver)
peak <- saltriver$peakflow
year <- saltriver$year
plot(year, peak, type = "l", xlab = "Year", ylab = "Annual peak flow")
## Mean return period for the Saltriver data
rp <- mrp(type_kernel = "n", vec_data = peak, lambda = 1)
plot(rp$grid, rp$Estimated_values, type = "l", xlab = "Flow level",
  ylab = "Years ", main = "Mean return period")

kerdiest documentation built on June 23, 2025, 5:08 p.m.