Derives a single or ensemble of bivariate design events

Calculates the isoline and relative probability of events on the isoline, where the data contains events from two populations. Outputs the single "most-likely" design event or an ensemble of possible design events obtained by sampling along the isoline according to these relative probabilities. The design event under the assumption of full dependence is also computed. Isoline is derived by calculating annual exceedance probabilities from both copula models on a user-specified grid rather by overlaying the partial isolines from the two copula models as in Design_Event_2D.

Usage

Design_Event_2D_Multi_Pop(
  Data,
  Data_Con1,
  Data_Con2,
  Data_Con3,
  Data_Con4,
  u1,
  u2,
  u3,
  u4,
  Thres1 = NA,
  Thres2 = NA,
  Thres3 = NA,
  Thres4 = NA,
  N_Both_1,
  N_Both_2,
  Copula_Family1,
  Copula_Family2,
  Copula_Family3,
  Copula_Family4,
  Marginal_Dist1,
  Marginal_Dist2,
  Marginal_Dist3,
  Marginal_Dist4,
  Marginal_Dist1_Par = NA,
  Marginal_Dist2_Par = NA,
  Marginal_Dist3_Par = NA,
  Marginal_Dist4_Par = NA,
  Con1 = "Rainfall",
  Con2 = "OsWL",
  Con3 = "Rainfall",
  Con4 = "OsWL",
  GPD1 = NA,
  GPD2 = NA,
  GPD3 = NA,
  GPD4 = NA,
  Rate_Con1 = NA,
  Rate_Con2 = NA,
  Rate_Con3 = NA,
  Rate_Con4 = NA,
  Tab1 = NULL,
  Tab2 = NULL,
  Tab3 = NULL,
  Tab4 = NULL,
  mu = 365.25,
  GPD_Bayes = FALSE,
  Decimal_Place = 2,
  Grid_x_min = NA,
  Grid_x_max = NA,
  Grid_y_min = NA,
  Grid_y_max = NA,
  Grid_x_interval = NA,
  Grid_y_interval = NA,
  RP,
  x_lab = "Rainfall (mm)",
  y_lab = "O-sWL (mNGVD 29)",
  x_lim_min = NA,
  x_lim_max = NA,
  y_lim_min = NA,
  y_lim_max = NA,
  Isoline_Probs = "Sample",
  N = 10^6,
  N_Ensemble = 0,
  Sim_Max = 10,
  Plot_Quantile_Isoline = FALSE
)

Arguments

Data: Data frame of dimension nx2 containing two co-occurring time series of length n.
Data_Con1: Data frame containing the conditional sample (declustered excesses paired with concurrent values of other variable), conditioned on the variable in the first column for population 1.
Data_Con2: Data frame containing the conditional sample (declustered excesses paired with concurrent values of other variable), conditioned on the variable in the second column for population 1.
Data_Con3: Data frame containing the conditional sample (declustered excesses paired with concurrent values of other variable), conditioned on the variable in the first column for population 2.
Data_Con4: Data frame containing the conditional sample (declustered excesses paired with concurrent values of other variable), conditioned on the variable in the second column for population 2.
u1: Numeric vector of length one specifying the threshold, expressed as a quantile, above which the variable in the first column was sampled in Data_Con1.
u2: Numeric vector of length one specifying the threshold, expressed as a quantile, above which the variable in the second column was sampled in Data_Con2.
u3: Numeric vector of length one specifying the threshold, expressed as a quantile, above which the variable in the first column was sampled in Data_Con3.
u4: Numeric vector of length one specifying the threshold, expressed as a quantile, above which the variable in the second column was sampled in Data_Con4.
Thres1: Numeric vector of length one specifying the threshold above which the variable in the first column was sampled in Data_Con1. Only one of u1 and Thres1 should be supplied. Default is NA.
Thres2: Numeric vector of length one specifying the threshold above which the variable in the second column was sampled in Data_Con2. Only one of u2 and Thres2 should be supplied. Default is NA.
Thres3: Numeric vector of length one specifying the threshold above which the variable in the first column was sampled in Data_Con3. Only one of u3 and Thres3 should be supplied. Default is NA.
Thres4: Numeric vector of length one specifying the threshold above which the variable in the second column was sampled in Data_Con4. Only one of u4 and Thres4 should be supplied. Default is NA.
N_Both_1: Numeric vector of length one specifying the number of data points in population 1 that feature in both conditional samples.
N_Both_2: Numeric vector of length one specifying the number of data points in population 1 that feature in both conditional samples.
Copula_Family1: Numeric vector of length one specifying the copula family used to model the Data_Con1 dataset.
Copula_Family2: Numeric vector of length one specifying the copula family used to model the Data_Con2 dataset.
Copula_Family3: Numeric vector of length one specifying the copula family used to model the Data_Con3 dataset.
Copula_Family4: Numeric vector of length one specifying the copula family used to model the Data_Con4 dataset.
Marginal_Dist1: Character vector of length one specifying (non-extreme) distribution used to model the marginal distribution of the non-conditioned variable in Data_Con1.
Marginal_Dist2: Character vector of length one specifying (non-extreme) distribution used to model the marginal distribution of the non-conditioned variable in Data_Con2.
Marginal_Dist3: Character vector of length one specifying (non-extreme) distribution used to model the marginal distribution of the non-conditioned variable in Data_Con3.
Marginal_Dist4: Character vector of length one specifying (non-extreme) distribution used to model the marginal distribution of the non-conditioned variable in Data_Con4.
Marginal_Dist1_Par: Character vector of length one specifying (non-extreme) parameters of Marginal_Dist1. Default is NA, specified distribution is fit within the procedure.
Marginal_Dist2_Par: Character vector of length one specifying (non-extreme) parameters of Marginal_Dist2. Default is NA, specified distribution is fit within the procedure.
Marginal_Dist3_Par: Character vector of length one specifying (non-extreme) parameters of Marginal_Dist3. Default is NA, specified distribution is fit within the procedure.
Marginal_Dist4_Par: Character vector of length one specifying (non-extreme) parameters of Marginal_Dist4. Default is NA, specified distribution is fit within the procedure.
Con1: Character vector of length one specifying the name of variable in the first column of Data_Con1.
Con2: Character vector of length one specifying the name of variable in the second column of Data_Con2.
Con3: Character vector of length one specifying the name of variable in the first column of Data_Con3.
Con4: Character vector of length one specifying the name of variable in the second column of Data_Con4.
GPD1: Output of GPD_Fit applied to variable con1 i.e., GPD fit con1. Default NULL. Only one of u1, Thres1, GPD1 and Tab1 is required.
GPD2: Output of GPD_Fit applied to variable con2 i.e., GPD fit con2. Default NULL. Only one of u2, Thres2, GPD2 and Tab2 is required.
GPD3: Output of GPD_Fit applied to variable con3 i.e., GPD fit con3. Default NULL. Only one of u3, Thres3, GPD3 and Tab3 is required.
GPD4: Output of GPD_Fit applied to variable con4 i.e., GPD fit con4. Default NULL. Only one of u4, Thres4, GPD4 and Tab4 is required.
Rate_Con1: Numeric vector of length one specifying the occurrence rate of observations in Data_Con1. Default is NA.
Rate_Con2: Numeric vector of length one specifying the occurrence rate of observations in Data_Con2. Default is NA.
Rate_Con3: Numeric vector of length one specifying the occurrence rate of observations in Data_Con3. Default is NA.
Rate_Con4: Numeric vector of length one specifying the occurrence rate of observations in Data_Con4. Default is NA.
Tab1: Data frame specifying the return periods of variable con1, when conditioning on con1. First column specifies the return period and the second column gives the corresponding levels. First row must contain the return level of con1 for the inter-arrival time (1/rate) of the sample. Default is NULL.
Tab2: Data frame specifying the return periods of variable con2, when conditioning on con2. First column specifies the return period and the second column gives the corresponding levels. First row must contain the return level of con2 for the inter-arrival time (1/rate) of the sample. Default is NULL.
Tab3: Data frame specifying the return periods of variable con3, when conditioning on con3. First column specifies the return period and the second column gives the corresponding levels. First row must contain the return level of con3 for the inter-arrival time (1/rate) of the sample. Default is NULL.
Tab4: Data frame specifying the return periods of variable con4, when conditioning on con4. First column specifies the return period and the second column gives the corresponding levels. First row must contain the return level of con4 for the inter-arrival time (1/rate) of the sample. Default is NULL.
mu: Numeric vector of length one specifying the (average) occurrence frequency of events in Data. Default is 365.25, daily data.
GPD_Bayes: Logical; indicating whether to use a Bayesian approach to estimate GPD parameters. This involves applying a penalty to the likelihood to aid in the stability of the optimization procedure. Default is FALSE.
Decimal_Place: Numeric vector specifying the number of decimal places to which to specify the isoline. Default is 2
Grid_x_min: Numeric vector of length one specifying the minimum value of the variable in first column of Data contained in the grid.
Grid_x_max: Numeric vector of length one specifying the maximum value of the variable in first column of Data contained in the grid.
Grid_y_min: Numeric vector of length one specifying the minimum value of the variable in second column of Data contained in the grid.
Grid_y_max: Numeric vector of length one specifying the maximum value of the variable in second column of Data contained in the grid.
Grid_x_interval: Numeric vector of length one specifying the resolution of the grid in terms of the variable in first column of Date. Default is an interval 2 of between consecutive values.
Grid_y_interval: Numeric vector of length one specifying the resolution of the grid in terms of the variable in second column of Date. Default is an interval 0.1 of between consecutive values.
RP: Numeric vector specifying the return periods of interest.
x_lab: Character vector specifying the x-axis label.
y_lab: Character vector specifying the y-axis label.
x_lim_min: Numeric vector of length one specifying x-axis minimum. Default is NA.
x_lim_max: Numeric vector of length one specifying x-axis maximum. Default is NA.
y_lim_min: Numeric vector of length one specifying y-axis minimum. Default is NA.
y_lim_max: Numeric vector of length one specifying y-axis maximum. Default is NA.
Isoline_Probs: Character vector of length one specifying whether to calculate relative probabilities of points on the isoline from a "Sample" simulated from the fitted copula models or from the "Observations".Default is "Sample".
N: Numeric vector of length one specifying the size of the sample from the fitted joint distributions used to estimate the density along an isoline. Samples are collected from the two joint distribution with proportions consistent with the total number of extreme events conditioned on each variable. Default is 10^6
N_Ensemble: Numeric vector of length one specifying the number of possible design events sampled along the isoline of interest.
Sim_Max: Numeric vector of length one specifying the maximum value, given as a multiple of the largest observation of each variable, permitted in the sample used to estimate the (relative) probabilities along the isoline.
Plot_Quantile_Isoline: Logical; indicating whether to first plot the quantile isoline. Default is FALSE.

Value

Plot of all the observations (grey circles) as well as the declustered excesses above Thres1 (blue circles) or Thres2 (blue circles), observations may belong to both conditional samples. Also shown is the isoline associated with RP contoured according to their relative probability of occurrence on the basis of the sample from the two joint distributions, the "most likely" design event (black diamond), and design event under the assumption of full dependence (black triangle) are also shown in the plot. The function also returns a list comprising the design events assuming full dependence "FullDependence", as well as once the dependence between the variables is accounted for the "Most likley" "MostLikelyEvent" as well as an "Ensemble" of possible design events and relative probabilities of events on the isoline Contour. The quantile isolines with Quantile_Isoline_1 and Quantile_Isoline_2, and GPD thresholds with Threshold_1 and Threshold_2.

Usage

Arguments

Value

See also