binteg(px,py,z,prm,pout,model="log",prob="AEP",nz=100,ninc=1000)This package is illustrated for the example of coastal flood levels (the response variable) forced by extreme rainfall (x) and extreme storm tides (y). For this problem, the low bounds of the marginal distributions, i.e. no rainfall (x) and a suitable tide level (such as mean sea level or the lowest astronomical tide) are denoted as ARI=0/AEP=1. "binteg" estimates the flood risk allows for joint dependence between the extreme values of two forcing variables (rainfall and storm tides). Here the logistic and negative logistic model are used to model dependence due to their simplicity and great ability (see Zheng et al. 2014a). The results from "binteg" also include estimates for the cases of complete dependence (alpha=0) and independence (alpha=1) to enable a comparison with the observed dependence strength.
An issue in the use of the proposed integral method is when the response variable table (e.g.flood level table) does not encapsulate the entirety of the boundary function. This can be seen from the example of this package whenever a flood contour intersects the upper or right-hand bound (as with contours 1.8m and higher that intersect the upper bound of the figure). In order to address this issue, return probabilities of two alternate cases are computed. Taking the h=2 m in (use "plot2" to generate the figure) for illustration, the first case is to project the red contour line (h=2 m) horizontally back to the y axis at the storm tide intersection ARI=500. This caps the largest storm tide at ARI=500, which will result in an overestimate of the return probability. The second case is to extend the red contour line (h=2 m) vertically to infinity, i.e., assuming the largest flood level produced by the storm tides is h=2 m, even if the storm tide ARI tends to infinity . For this case, the return probability is underestimated. Using this approach, the two cases which give over and under estimates are respectively shown different flood levels (see the figure provided by "plot2"). For the given ARIs/AEPs, if the differences of the flood levels from the two cases are insignificant, the provided response variable table is sufficient for estimation. Otherwise a more extended table with larger marginal ARIs/AEPs is required.
Zheng, F., Westra S. Sisson S. and Leonard M. (2014a). Modelling the dependence between extreme rainfall and storm surge to estimate coastal flood risk, Water Resources Research, under review.
Zheng, F., Leonard M. and Westra S. (2014b). An efficient bivariate integration method for joint probability analysis of flood risk, Water Resources Research, under review.
library(jointPm)
data(flood)
px=flood$px;py=flood$py;z=flood$flood_table;prm=flood$prm;pout=flood$pout
binteg(px,py,z,prm,pout,model="log",prob="ARI",nz=100,ninc=1000)
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