kerdiest (version 1.2)

mrp: 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. The plug-in method of Polansky and Baker (2000) is used by default.

lambda

The mean activity rate.

Value

• Returns a list containing:
• Estimated_values
  Vector containing the estimated function.
• grid
  The used grid.
• bw
  Value of the bandwidth.

Details

The mean return period is usually calculated assuming that the occurrence process of the events follows a Poisson one. 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).

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 non-parametric techniques for area-characteristic seismic hazard parameters. Geophysical Journal International 180, pp. 339--346. Quintela-del-Rio, A. (2011) On bandwidth selection for nonparametric estimation in flood frequency analysis. Hydrological Processes 25, pp. 671--678. Quintela-del-Rio, A. and Estevez-Perez, G. (2012) Nonparametric Kernel Distribution Function Estimation with kerdiest: An R Package for Bandwidth Choice and Applications, Journal of Statistical Software 50(8), pp. 1-21. URL http://www.jstatsoft.org/v50/i08/.

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)
# we consider the data with magnitude greater than 3
mg<-nwip$magnitude[nwip$magnitude>3.0]
x1<-nwip$year
x2<-nwip$month
x3<-nwip$day
ys<-paste(x1,x2,x3)
earthquake_date<-as.character(ys)
y1s<-as.date(earthquake_date, order = "ymd")
# we compute the total number of years
y2s<-as.POSIXct(y1s)
z<-years(y2s)
n.years<-length(levels(z))
# the mean rate of earthquakes per year
lambda<-length(mg)/n.years
  # we estimate the mean return period (in years) between earthquakes of 
# the same magnitude
est2<-mrp(vec_data=mg, lambda=lambda)
plot(est2$grid, est2$Estimated_values, type="l", 
xlab="magnitude", ylab="Mean return period (years)")
##  Working with hydrological data: annual peak instantaneous flow of the
# Salt River near Roosevelt, AZ, USA, for 1924 - 2006.
data(saltriver)
peak<-saltriver$peakflow
year<-saltriver$year
plot(year,peak, type="l",ylab="Annual peak flow")
 #  mean return period for the Saltriver data 
rp<-mrp(type_kernel="n", vec_data=peak, lambda=1)
 plot(rp$grid, rp$Estimated_values, type="l",
xlab="Flow level", ylab="Years ", main="Mean return period")