An R Package for Univariate and Bivariate Peaks Over Threshold ...

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4 A Concrete Statistical Analysis of Peaks Over a ThresholdIn this section, we provide a full and detailed analysis of peaks over a threshold for the river Ardières atBeaujeu. Figure 10 depicts instantaneous flood discharges - blue line.As this is a time series, we must selects independent events above a threshold. First, we fix a relativelylow threshold to “extract” more events. Thus, some of them are not extreme but regular events. This isnecessary to select a reasonable threshold for the asymptotic approximation by a GPD - see section 2.> summary(ardieres)timeobsMin. :1970 Min. : 0.0221st Qu.:1981 1st Qu.: 0.236Median :1991 Median : 0.542Mean :1989 Mean : 1.0243rd Qu.:1997 3rd Qu.: 1.230Max. :2004 Max. :44.200NA's : 1.000> events0 par(mfrow = c(2, 2))> mrlplot(events0[, "obs"])> abline(v = 6, col = "green")> diplot(events0)> abline(v = 6, col = "green")> tcplot(events0[, "obs"], which = 1)> abline(v = 6, col = "green")> tcplot(events0[, "obs"], which = 2)> abline(v = 6, col = "green")From Figure 11, a threshold value of 6m 3 /s should be reasonable. The Mean residual life plot - top leftpanel- indicates that a threshold around 10m 3 /s should be adequate. However, the selected thresholdmust be low enough to have enough events above it to reduce variance while not too low as it increasethe bias 3 .Thus, we can now “re-extract” events above the threshold 6m 3 /s, obtaining object events1. This isnecessary as sometimes events1 is not equal to observations of events0 greater than 6m 3 /s. We cannow define the mean number of events per year “npy”. Note that an estimation of the extremal index isavailable.> events1 npy print(npy)[1] 1.707897> attributes(events1)$exi[1] 0.1247265Let’s fit the GPD.> mle par(mfrow = c(2, 2))> plot(mle, npy = npy)24
timeobsMin. :1970 Min. : 0.0221st Qu.:1981 1st Qu.: 0.236Median :1991 Median : 0.542Mean :1989 Mean : 1.0243rd Qu.:1997 3rd Qu.: 1.230Max. :2004 Max. :44.200NA's : 1.000Mean Residual Life PlotDispersion Index PlotMean Excess0 5 10 15 20Dispersion Index0.6 1.0 1.4 1.85 10 150 10 20 30 40ThresholdThresholdModified Scale−100 −60 −20 20●●●●●●●●●●●● ●●●● ●●●● ● ● ●●●Shape−1 1 2 3 4 5●●●●●●●●●●●● ● ●●● ●● ●● ● ● ●● ●5 10 155 10 15ThresholdThresholdFigure 11: Threshold selection for river Ardières at Beaujeu.25
Page 3 and 4: z ≤ 0, Coles et al. (1999) use th
Page 5 and 6: 3.2 Threshold SelectionThe location
Page 7 and 8: Mean Residual Life PlotMean Excess0
Page 9 and 10: ●●●L−Moments Plotτ 40.0 0.
Page 11 and 12: 3.3 Fitting the GPD3.3.1 The univar
Page 13 and 14: Estimator: MLEDeviance: -173.0829AI
Page 15 and 16: Pickands' Dependence FunctionA (x)0
Page 17 and 18: 3.4 Confidence IntervalsOnce a stat
Page 19 and 20: conf.inf conf.sup7.220334 10.219962
Page 21 and 22: ˆ which = 4 for a return level plo
Page 23: obs0 10 20 30 401970 1975 1980 1985
Page 27 and 28: The result of function fitgpd gives
Page 29 and 30: ADependence Models for Bivariate Ex
Page 31: Liang Peng and A.H. Welsh. Robust e

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