kerdiest (version 1.2)

ef: Exceedance function estimation

Description

We compute the exceedance probability, that is, the probability that a specified value c (a magnitude of a seismic event, a flow level... ) will be exceeded in D time units.

Usage

ef(type_kernel = "n", vec_data, c,
 bw = PBbw(type_kernel = "n", vec_data, 2),
 Dmin = 0, Dmax = 15, size_grid = 50, 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...)

c

The concrete level in which we want to compute the exceedance probability.

bw

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

Dmin

Minimum value for D time units (years, days... ). Default is Dmin=0.

Dmax

Maximum value for D time units (years, days... ). Default is Dmax=15.

size_grid

Length of a grid in which we compute the exceedance function. The size is 50 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 exceedance function is usually calculated assuming that the occurrence process of events follows a Poisson one. In this case, the exceedance function, that is, the probability of an specific value c is calculated as $$R(c,D) = 1- exp(-\lambda D(1-F_h(c)).$$ See, for example, Orlecka-Sikora (2008) or Quintela del Rio (2010) for earthquake data applications.

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. 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$day
ys<-paste(x1,x2,x3)
earthquake_date<-as.character(ys)
y1s<-as.date(earthquake_date, order = "ymd")
# we compute the total number of years
y2s<-as.POSIXct(y1s)
z<-years(y2s)
n.years<-length(levels(z))
# the mean rate of earthquakes per year
lambda<-length(mg)/n.years
# we estimate the exceedance probability for a value of the
# the magnitude = 4
est<-ef(vec_data=mg, m_c=4, lambda=lambda)
plot(est$grid, est$Estimated_values, type="l", 
xlab="years", ylab="Probability of Exceedance")