# Mean Return Period Estimation This functions computes an estimate of the time between two values of a concrete level (size of an earthquake, flow lewel, wind speed...). ```r 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). ## Returns A list containing: - **Estimated_values**: Vector containing the estimated function. - **grid**: The used grid. - **bw**: Value of the bandwidth. ## 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. ## Author(s) Graciela Estévez Pérez and Alejandro Quintela del Río ## Examples ```r ## 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") ```

mrp function