Package 'PUcopula'

losses from natural perils data set

Description

This is the 19-dimensional data set presented in Neumann et al. (2019), Tab. 2 and Tab. 3.

Details

The data contains insurance losses from a non-life portfolio of natural perils in 19 areas in central Europe over a time period of 20 years. The monetary unit is 1 million €.

References

Neumann, A., Bodnar, T., Pfeifer, D., & Dickhaus, T. (2019). Multivariate multiple test procedures based on nonparametric copula estimation. Biometrical Journal, 61(1), 40-61.

---

Create a PUCopula object

Description

Constructs an object of class PUCopula. This is the main user-facing function for creating partition of unity copulas.

Usage

PUCopula(
  dim = 0,
  family = c("binom", "nbinom", "poisson", "sample", "gamma", "beta", "power"),
  pars.a = c(10, 10),
  patch = c("rook", "uFrechet", "lFrechet", "varwc", "Bernstein", "Gauss", "sample"),
  patchpar = list(NULL),
  data,
  numericCDF = FALSE
)

Arguments

dim	Integer. Dimension of the copula. If 0, it is inferred from data.
family	Character vector specifying the copula family. Possible values include "binom", "nbinom", "poisson", "sample", "gamma", "beta", "power".
pars.a	Numeric vector of parameters controlling the family.
patch	Character. Patchwork type, e.g. "rook", "uFrechet", "lFrechet", "Bernstein", "Gauss", "sample", "varwc".
patchpar	List. Parameters for the patchwork (e.g., correlation for Gaussian copula).
data	Matrix. Input data used to construct the copula.
numericCDF	Logical. Whether to use numerical CDF approximations.

Value

An object of class PUCopula.

Examples

data(stormflood)

x <- PUCopula(
  family = "gamma",
  pars.a = c(40, 43),
  patch = "lFrechet",
  data = stormflood
)

plot(x@ranks)

---

S4 class representing a partition of unity copula

Description

The PUCopula class implements partition-of-unity copulas as developed in Pfeifer et al. (2016, 2017, 2019). It provides a framework for constructing copulas using general partitions of unity, including both discrete and continuous cases.

Value

An object of class PUCopula.

Slots

dim

Numeric scalar. Dimension of the copula.

family

Character vector. Specifies the family of the PU copula. (and therefore the densities fdenss)

par.factor

Numeric vector. Scaling parameter for ranks.

pars.a

Numeric vector. Parameters controlling the distribution of $\varphi$....

patchpar

List. Parameters for the copula driver, e.g. rho in case of Gauss copula driver, K,m in case of Bernstein,...

p

Matrix. Internal parameter representation.

phi

Function. Base density $\varphi(u, s)$.... (obsolete due to phis)

psy

Function. Secondary density function.... (obsolete due to phis)

phis

List of functions. Collection of component densities.... $\varphi_k(s,u)$ for $k=1,dots,d\in N$. Either continuous case: A list of functions which represent Lebesgue densities of distributions over R with a parameter u in (0,1), i.e.

$\varphi_k(s,u)\geq 0 \text{ and } \int_{-\infty}^\infty \varphi_k(s,u)\, ds = 1 \text{ for } u \in (0,1),$

Or (discrete case): A list of functions which represent discrete probabilities over Z+ with a parameter u in (0,1).

continuous

Logical vector. Indicates whether components are continuous....

alph

Function. Integral of $\varphi$....

beta

Function. Integral of $\psi$....

alphs

List of functions. Marginal densities.... Integral over the elements of phis w.r.t. u, i.e.

$\alpha_k(s):=\int_0^1 \varphi_k(s,u)\, du \in (0,\infty)$

alphsCDF

List of functions. CDFs corresponding to densities alphs, i.e.

$A_k(s) := \int_{-\infty}^s \alpha_k(w) \, dw \text{ for } s \in R$

opstart

List. Starting values for optimization....

cdflim

List. Limits for numerically stable CDF evaluation....

cdfdom

List. Effective domain of the CDFs....

alphsquantf

List. Fast quantile approximations....

cdfappx

List. Approximate CDF functions....

alphsobjective

List. Objective functions for quantiles....

alphsquant

List. Quantile functions....

fdens

Function. Density function.... (obsolete due to fdenss)

gdens

Function. Secondary density.... (obsolete due to fdenss)

fdenss

List of functions. Contains densities obtained from normalizing the functions $\varphi(s,u)$ (from slot phis) w.r.t. $u \in 0,1$, i.e.

$f_k(s,u) := \frac{\varphi_k(s,u)}{\alpha_k(s)}, \, u \in (0,1) \text{ for } s \in R$

cpatch

Function. Copula driver density....

PUdens

Function. Density of the (continuous) PU copula defined by

$c(\mathbf{u}) := \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty p(s_1,\cdots,s_d) \prod_{k=1}^d f_k(s_k, u_k) \, ds_1 \cdots ds_d, \mathbf{u} = (u_1,\cdots,u_d) \in (0,1)^d$

where $p(s_1,\cdots,s_d)$ denotes the density of an arbitrary $d$-dimensional random vector $\mathbf{S}=S_1,\cdots,S_d)$ over $R^d$ with marginal densities $\alpha(cdot)$ for $S_k$.

fq

Function. Auxiliary function.

gq

Function. Auxiliary function.

patch

Character. Patchwork type.

dPWork

Function. Patchwork density....

data

Matrix. Input data.

ranks

Matrix. Rank-transformed data.

relRanks

Matrix. Relative ranks.

rpatch

Function. Patchwork sampler - generates random samples from the specified patchwork.

rand

Function. Random number generator to simulate from PU copula.

References

Pfeifer, D., Tsatedem, H. A., Mändle, A., and Girschig, C. (2016). New copulas based on general partitions-of-unity and their applications to risk management. Dependence Modeling, 4(1). doi:10.1515/demo-2016-0006

Pfeifer, D., Mändle, A., and Ragulina, O. (2017). New copulas based on general partitions-of-unity and their applications to risk management (part II). Dependence Modeling, 5(1), 246–255. doi:10.1515/demo-2017-0014

Pfeifer, D., Mändle, A., Ragulina, O., and Girschig, C. (2019). New copulas based on general partitions-of-unity (part III) — the continuous case. Dependence Modeling, 7(1), 181–201. doi:10.1515/demo-2019-0009

Examples

# Use dataset stormflood
data(stormflood)

# example: choose Gamma copula model
x <- PUCopula(family="gamma", pars.a=c(40, 43), patch="lFrechet",data=stormflood)

# The following plots show the common ranks and several copula drivers;
#   these plots do not depend on the above chosen family of the PUcopula.
# 2*3 plots in one
par(mfrow=c(2,3))
# plot the empirical rank vectors
plot(x@ranks, sub="emp. rank vectors", xlab="", ylab="")
# plot the lower Fréchet driver, i.e. phi=-1
plot(x@rpatch(2000,"lFrechet"), sub="lower Fréchet driver, phi=-1", xlab="", ylab="")
# plot for phi=-0.8
plot(
  x@rpatch(2000,"Bernstein",list(m=20,K=20)),
  sub="Bernstein copula, m=20, K=20",
  xlab="", ylab=""
)
# plot for phi=0 (rook copula)
plot(x@rpatch(2000,"rook"), sub="rook copula, phi=0", xlab="", ylab="")
# plot for phi=0.9
plot(x@rpatch(2000,"Gauss",0.9), sub="phi=0.9", xlab="", ylab="")
# plot for phi=1 (upper Fréchet driver)
plot(x@rpatch(2000,"uFrechet"), sub="upper Fréchet driver, phi=1", xlab="", ylab="")
# single plot window
par(mfrow=c(1,1))

# plot the densities given by phis (=densities of Gamma distributions)
# plot for s=1,3,5,7,...,999 (s must be in (0,inf))
palette(rainbow(10))
xs=c(seq(0,0.01,length.out=200),seq(0.02,1,length.out=200))
# function phi for a fixed s
foo <- function(u) x@phi(u,1)
ys=foo(xs)
# plot for s=1
plot(
  x=xs,y=ys,
  col=1,
  ylim=c(0,0.4), xlim=c(0,1),
  type = "l"
) #plot(foo, col=1, ylim=c(0,0.94), xlim=c(0,1)) # ylim=c(0,1e-88)
# repeat for  s=3,5,7,...,999
for (i in 2:500) {
  foo <- function(u) x@phi(u,i*2-1)
  ys=foo(xs)
    lines(x=xs,y=ys, col=i+1) #plot(foo, add=TRUE, col=i+1)
}
# for a Gamma PUC the alphs are densities of inverse Pareto distributions...
plot(x@alphs[[1]], xlim=c(0,500))
# the fdenss are basically a normalization of the phis using alphs
# for a fixes s they are densities of exponentially transformed
# Gamma distributions
# heatplot:
us <- seq(0,1,length.out=100)
ss <- seq(1,500,length.out=100)
zs <- matrix(nrow=length(us),ncol=length(ss))
for (s in 1:length(ss)) zs[,s] <- x@fdenss[[1]](us, s)
image(x=us, y=ss, z=zs, col=terrain.colors(15))

#For the simulation from the Gamma copula, the slots alphsCDF
# and Qk/corresponnding quantile are used internally
# plot the CDF corresponding to density alphs[1]
## Not run: 
x <- PUCopula(family="gamma", pars.a=c(40, 43), patch="lFrechet",data=stormflood, numericCDF=TRUE)

warnings()
# Warning messages:
#   1: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 2: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 3: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 4: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 5: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 6: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 7: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 8: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 9: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 10: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 11: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 12: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 13: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 14: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 15: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 16: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 17: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 18: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 19: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 20: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 21: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 22: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 23: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 24: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 25: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 26: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 27: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 28: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 29: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 30: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 31: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 32: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 33: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 34: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 35: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 36: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 37: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 38: In phis[[i]](u, s) : outside range (0,inf) for s = -752.818778594017
# 39: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 40: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 41: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 42: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 43: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 44: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 45: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 46: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 47: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 48: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 49: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631
# 50: In phis[[i]](u, s) : outside range (0,inf) for s = -91.7078148698631

plot(x@alphsCDF[[1]],xlim=c(0,100))
x@alphsobjective[[1]](5,0.8)
x@alphsquant[[1]](0.8)
x@rand(1)

## End(Not run)

# Create a beta copula
x <- PUCopula(family="beta", pars.a=c(40, 43), patch="lFrechet",data=stormflood)
x@phi(0.5,0.5)
#[1] 2.210851e-11
x@alph(0.6)
#[1] 1.458522e-11
#x@fdens(0.5,0.6)
#[1] 0.0002381545
#x@gdens(0.5,0.6)
#[1] 0.0001616076
x@fdenss[[1]](0.5,0.6)
#[1] 0.0002381545
x@fdenss[[2]](0.5,0.6)
#[1] 0.0001616076
x@rand(5)
#does not work
# example 2: gamma copula
x <- PUCopula(family="gamma", pars.a=c(40, 43), patch="lFrechet",data=stormflood)
#plot alpha(s)
plot(x@alphs[[1]])
#the numerically computed alpha is quite close:
s<-10
x@pars.a[1]*s^(x@pars.a[1]-1)/((1+s)^(x@pars.a[1]+1))
#[1] 0.008034519
x@alph(s)
#[1] 0.008034519
# but of course shows some inaccuracies:
s<-2.5
x@alph(s)
#[1] 6.530307e-06
x@pars.a[1]*s^(x@pars.a[1]-1)/((1+s)^(x@pars.a[1]+1))
#[1] 6.530261e-06

---

storm and flood losses data set

Description

This is the data set used in Cottin, Pfeifer (2014). The table contains some original data from an insurance portfolio of storm and flooding losses, observed over a period of 20 years.

References

Cottin, Claudia, and Dietmar Pfeifer. "From Bernstein polynomials to Bernstein copulas." J. Appl. Funct. Anal 9.3-4 (2014): 277-288.

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