Bivariate Elliptical Copulae

EllipticalCopulae {fCopulae}

R Documentation

Bivariate Elliptical Copulae

Description

A collection and description of functions to investigate bivariate elliptical copulae.

Elliptical Copulae Functions:

rellipticalCopula	Generates elliptical copula variates,
pellipticalCopula	computes elliptical copula probability,
dellipticalCopula	computes elliptical copula density,
rellipticalSlider	displays interactive plots of variates,
pellipticalSlider	displays interactive plots of probability,
dellipticalSlider	displays interactive plots of density.

Usage

    
rellipticalCopula(n, rho = 0.75, param = NULL, type = c("norm", "cauchy", 
    "t"))
pellipticalCopula(u = 0.5, v = u, rho = 0.75, param = NULL, 
    type = ellipticalList(), output = c("vector", "list"), border = TRUE)
dellipticalCopula(u = 0.5, v = u, rho = 0.75, param = NULL, 
    type = ellipticalList(), output = c("vector", "list"), border = TRUE)

rellipticalSlider(B = 100)
pellipticalSlider(type = c("persp", "contour"), B = 20)
dellipticalSlider(type = c("persp", "contour"), B = 20)

Arguments

B	[*Slider] - the maximum slider menu value when the boundary value is infinite. By default this is set to 10.
border	[pellipticalCopula][dellipticalCopula] - a logical flag. If the argument u is an integer, say N, greater than one than all points on a square grid [(0:N)/N]^2 are computed. If border is FALSE than the border points are removed from the returned value, by default this is not the case.
n	[rellipticalCopula][ellipticalCopulaSim] - the number of random deviates to be generated, an integer value.
output	[pellipticalCopula][dellipticalCopula] - a character string specifying how the output should be formatted. By default a vector of the same length as u and v is returned. If specified as "list" then u and v are expected to span a two-dimensional grid as outputted by the function grid2d and the function returns a list with elements $x, y, and z which can be directly used for example by 2D plotting functions. For the grid version, when u is specified as an integer greater than one, always the output in form of a list will be returned.
rho	[*ellipticalCopula] - is the numeric value setting the correlation strength, ranging between minus one and one.
param	[*ellipticalCopula][gfunc] - additional distributional parameters: for the Sudent-t distribution this is "nu", for the Kotz distribution this is "r", and for the Exponential Power distribution these are "r" and "s". If the argument param=NULL then default values are taken. These are for the Student-t param=c(nu=4)), for the Kotz distribution param=c(r=1)), and for the exponential power distribution param=c(r=1,s=1). Note, that the Kotz and exponential power copulae are independent of r, and that r only enters the generator, the density, the probability and the quantile functions.
type	[*ellipticalCopula][gfunc] - the type of the elliptical copula. A character string selected from: "norm", "cauchy", "t", "logistic", "laplace", "kotz", or "epower". [*ellipticalSlider] - a character string which indicates what kind of plot should be displayed, either a perspective plot if type="persp", the default value, or a contour plot if type="contour".
u, v	[*ellipticalCopula] - two numeric values or vectors of the same length at which the copula will be computed. If u is a list then the the $x and $y elements will be used as u and v. If u is a two column matrix then the first column will be used as u and the the second as v. If u is an integer value greater than one, say N, than the values for all points on the [(0:N)/N]^2 grid spanning the unit square will be returned.
...	[ellipticalCopulaFit] - arguments passed to the optimization function nlminb.

Value

Copula Functions:


The functions [rpd]ellipticalCopula return a numeric vector of random variates, probabilities, or densities for the specified copula computed at grid coordinates u|v.

The functions [rpd]ellipticalSlider display an interactive graph of an perspective copula plot either for random variates, probabilities or densities. Alternatively, an image underlayed contour plot can be shown.

Copula Dependence Measures:


The functions ellipticalTau and ellipticalRho return a numericc value for Kendall's Tau and Spearman's Rho.

Copula Tail Coefficient:


The function ellipticalTailCoeff returns the coefficient of tail dependence for a specified copula. The function ellipticalTailPlot displays a whole plot for the upper or alternatively for the lower tail dependence as a function of u for a set of nine rho values.

Copula Generator Function:


The function gfunc computes the generator function for the specified copula, by default the normal copula. If the argument x is missing, then the normalization constand lambda will be returned, otherwise if x is specified the values for the function g(x) will be freturned. The selected type of copula is added to the output as an attribute named "control". The function gfuncSlider allows to display interactively the generator function, the marginal density, the marginal probability, and the contours of the the bivariate density.

Copula Simulation and Parameter Fitting:


The function ellipticalCopulaSim returns a numeric two-column matrix with randomly generated variates for the specified copula.

The function ellipticalCopulaFit returns a fit to empirical data for the specified copula. The returned object is a list with elements from the function nlminb.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## [rp]ellipticalCopula -
   # Default Normal Copula:
   rellipticalCopula(10)
   pellipticalCopula(10)

## [rp]ellipticalCopula -   
   # Student-t Copula Probability and Density:
   u = grid2d(x = (0:25)/25)
   pellipticalCopula(u, rho = 0.75, param = 4, 
     type = "t", output = "list")
   d = dellipticalCopula(u, rho = 0.75, param = 4, 
     type = "t", output = "list")   
   persp(d, theta = -40, phi = 30, col = "steelblue")
   
## ellipticalTau -
## ellipticalRho -
   # Dependence Measures:
   ellipticalTau(rho = -0.5)
   ellipticalRho(rho = 0.75, type = "logistic", subdivisions = 100)
   
## ellipticalTailCoeff -
   # Student-t Tail Coefficient:
   ellipticalTailCoeff(rho = 0.25, param = 3, type = "t")

## gfunc -
   # Generator Function:
   plot(gfunc(x = 0:10), main = "Generator Function")
   
## ellipticalCopulaSim -
## ellipticalCopulaSim -
   # Simualtion and Parameter Fitting:
   rv = ellipticalCopulaSim(n = 100, rho = 0.75)
   ellipticalCopulaFit(rv)

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R Corner: A Toy Copula1 ERM Model in R

R Corner: A Toy Copula¹ ERM Model in R²

by Steve Craighead

Editor's note: R Corner is a series by Steve Craighead introducing readers to the "R" language used for statistics and modeling of data. See footnote 2 at the end of the article for information on installing and learning more about R.
A simple ERM model entails setting up various risk sub-models and creating a dependency relationship between these risks. After this is accomplished all one needs to do is to simulate for a given number of trials (say 1,000) and aggregate the dependent risk values. From these aggregated results one can then either determine VaR or conditional tail expectation (CTE) at a specific percentile. Below, we will construct our ERM toy model, which will demonstrate how to link statistical sub-models with copula dependency models using the R copula package.
The R copula package models the Frank, Gumbel and Clayton copulas within the Archimedean family, as well as, the multivariate normal and Student t copulas.
The Archimedean copulas are limited to dimensions less than seven since the resultant multivariate probability density function (PDF) is not available due to difficulty in symbolically differentiating the associated multivariate cumulative distribution function (CDF).
In brief, the steps that the reader needs to follow are:

1. Specify the copula that will model the dependence.
2. Specify the multivariate distribution using the copula defined in step A by using the mvdc() function. You must also define the marginal distributions associated with each risk. You will also specify the marginal parameters which will be discussed below.
3. Supply the marginal data associated with each risk.
4. Fit the model.
5. Examine the results.

Before we actually create the toy ERM model, we need to discuss some basics in R. The R language has many different univariate distributions available for modeling the marginals. A statistical distribution in R uses three separate functions for modeling (with the possibility of a fourth). For example, if you wanted to model a univariate normal distribution, you would use either the dnorm, pnorm, qnorm (and rnorm) functions. The dnorm function models the density, the pnorm models the distribution (CDF) and the qnorm is the quantile function, which is the inverse function of the CDF. The fourth function is rnorm, which can be used to generate random normal variates. Other example distributions functions are dt, pt, qt and rt for the Student t, and dexp, pexp, qexp and rexp for the exponential distribution. Each of these functions requires specific model parameters. The normal distribution has the model parameters mean and sd for the mean and standard deviation. The Student t distribution requires the parameter df for the degrees of freedom. The exponential distribution uses the rate parameter.
One must also load the copula package after starting R. This is done in Windows by choosing the copula package when using the Load Package option under the Packages option on the command list at the top of R.
When carrying out step B above, the mvdc() function creates a multivariate distribution object in R. This function has three major inputs. The first is the copula. The second is a list of the specific marginals and the third is a list containing lists of associated marginal parameters.
For instance, say you want to model a bivariate CDF using the Gumbel copula (with parameter of three) and the first marginal is a normal distribution with a mean of 10 and a standard deviation of two and the second marginal is an exponential with a rate of two.

1. First, you need to use the command
gmb<-gumbelCopula(3,dim=2)
to create the copula object. The symbol "<-" is used as the assignment operator. The "dim=2" assures that we are creating a bivariate CDF. You can specify up to six dimensions within the R Copula library.

2. Next you specify the bivariate distribution by using the command:
myCDF<- mvdc(gmb, margins=c("norm","exp"), paramMargins=list(list(mean=10,sd=2),list(rate=2)))
Notice the first parameter in mvdc is the gmb object defined in step A. The second parameter is a generic list of the two marginal distributions (produced using the c() function). The third parameter is much more complex, where the list() function is used three times. Here we have a list made of two separate lists. The first of these two separate lists are the parameters that model the normal marginal as discussed above. The second of these lists contains the parameter required to model the exponential marginal. The list() function is not as generic as the c() function, but the mvdc function expects the marginal parameter defined in this fashion.

3. Next you need to supply the risk data associated with each marginal. Since this is a toy, we will actually use the myCDF multivariate distribution as defined to generate the data (a bit incestuous, but OK for this demonstration). Here you will use the rmvdc() function which will generate random variates from myCDF. Use the command:
x <- rmvdc(myCDF,1000)
to generate 1000 samples from your bivariate CDF. You could also supply your own data x as in an R matrix with 1000 rows and two columns. For instance, if you were able to use identical scenarios and produce separate various risk values based on these scenarios, and collect these together into an R matrix, you could then design your copula multivariate distribution and use the fitting algorithm to fit your actual data.³

4. Next, take the data in the matrix x and determine the best parameters in myCDF by using the fitting function fitMvdc() (which uses maximum likelihood). This function requires the x matrix, the myCDF object and a generic c() list of initial guesses to the parameters. Note, the first parameter in this list will be the parameter associated with the Gumbel copula model (which we assumed as three above). The remaining values are associated with the parameters of the marginals. Suppose that you use the command:
Fitted<-fitMvdc(x, myCDF, c(3,9,1,1))
Here the three is the Gumbel parameter, the nine is the initial guess for the normal mean, one for the normal sd parameter and one for the exponential rate parameter. Here starting values were different than the original model, so that in Step E, we can see the goodness of fit. Note: you may get an warning message stating that there are so many separate warnings and that you should enter the command "warnings()" to display these. These warnings arise when the fitting algorithm moves the parameter estimates outside the acceptable values. At this time, you may disregard these.

5. Once the model fitting has completed, you can examine the results of the fit by just issuing the command "Fitted." This will display to the screen the results. For instance the following is the results obtained by the author. Your results may vary due to the random values that were generated in matrix x.
Fitted
The ML estimation is based on 1000 observations.
Margin 1 :


	Estimate	Std. Error
m1.mean	10.157765	0.05885397
m1.sd	1.984802	0.03243711


Margin 2 :


	Estimate	Std. Error
m2.rate	1.893372	0.05849316


Copula:


	Estimate	Std. Error
param	2.819898	0.08826639


The maximized loglikelihood is -1787.369
The convergence code is 0

Observe both the relative parameters and their standard error, and how close the estimates are to the original parameters originally set up.
Also, one can examine the contour plot of the bivariate density function by using the following commands:
contour(myCDF,dmvdc,xlim=c(4,15),ylim=c(0,1.7))
title("Gumbel Multivariate PDF")
These produce the following graph:
One obtains the graph of the normal marginal by using these commands:
Gumbel Bivariate PDF Graph
K <- density(x[,1]) #x[,1] gets risk 1 results
K$call <- "Normal Marginal"
plot(K)
Normal Marginal Graph
To obtain the exponential marginal, use these commands:
K <- density(x[,2]) #x[,2] obtains risk 2
K$call <- "Exponential Marginal"
plot(K)
Exponential Marginal Graph
Note how that the PDF is not symmetric, which is a characteristic of the Gumbel Copula since it is only defined in the right tail.
Also, one can examine the contour plot of the bivariate cumulative function by using the following commands:
contour(myCDF,pmvdc,xlim=c(7,15),ylim=c(0,1.7))
title("Gumbel Bivariate CDF")
This produces the following graph:
Gumbel Bivariate CDF Graph
Of course, if you add up the random samples, you will have the total dependent risks! By using these following commands, R will display the density of the total risks:
Y<- apply(x,1,"sum") #sum across columns of x
K<-density(Y)
K$call <- "Aggregate Risks"
plot(K)
Aggregate Density Graph
With just the above commands you may examine other copulas such as the Frank copula using the commands:
frank <- frankCopula(param=2,dim=2)
myCDF<-mvdc(frank,c("norm","exp"),list(list(mean=10,sd=2),list(rate=2)))
contour(myCDF,dmvdc,xlim=c(4,15),ylim=c(-0.5,1.5))
title("Frank Bivariate PDF")
You will obtain the following density contour:
Frank Bivariate Graph
Other things that you can do, is to increase the number of dimensions and use other marginals. Also, you can use the rmvdc() command as we did in step C above and simulate your dependency multivariate distribution.
Though this toy model only uses two risks, you can use up to six risks, by changing the definition of the copula as well as specifying the different sub-models. Hopefully, the examples above are explicit enough to let you do various what-ifs on your own data.

Bibliography

R Development Core Team (2006). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL: http://www.R-project.org.

Footnotes:

¹This entire article assumes that the reader is acquainted with both the concept of copulas in the modeling of dependence in ERM modeling and how to use the R language with statistical modeling. However, for further information on copulas, you may read the excellent survey article "Understanding Relationships Using Copulas" by Frees and Valdez in NAAJ Volume 2, Number 1 of 1998. If you prefer to experiment with copulas in Excel instead of R, please refer to the workbook and articles created by Sam Cox and Don Behan on http://www.behan.ws.
²The first column of R Corner was published in the October 2008 issue, and explains how to download and install the package, as well as providing a basic introduction to the language. Refer to each CompAct issue since then for additional articles in the series. The introductory article can be found on p. 24 of the October 2008 issue on the SOA Web site. The R language is a very popular open source statistical modeling environment, which you can obtain from the webpage http://cran.r-project.org. If you are interested in building the toy model in this article, you will need to download the base R Binary for the operating system of your choice, (Windows, Linux or MAC). Also, you will need to download the contributed copula package. You can do this initially when you download the base system, or you can use the package installation facility when you start R.
³The import function "read.csv" allows you to import your own numerical data, but the "read.csv" function stores the data in as a data.frame object. You must convert your data object to a matrix object by using the function "as.matrix"
Steve Craighead can be contacted at steven.craighead@towerswatson.com.

How to Install an R Package

Longhai Li, Department of Mathematics and Statistics, University of Saskatchewan

I occacionally publish R add-on packages for others to implement and test the statistical methodoglogies I discuss in my papers. R is a free software environment for statistical computing and graphics, available from http://www.r-project.org. The following is a brief instruction of installing R packages. More details can be found by typing ?INSTALL in R console.

Install a new package to your computer

Method 1: Install from source

Download the add-on R package, say mypkg, and type the following command in Unix console to install it to /my/own/R-packages/:
$ R CMD INSTALL mypkg -l /my/own/R-packages/

Method 2: Install from CRAN directly

Type the following command in R console to install it to /my/own/R-packages/ directly from CRAN:
> install.packages("mypkg", lib="/my/own/R-packages/")

Load the library

Type the following command in R console to load the package
> library("mypkg", lib.loc="/my/own/R-packages/")

Instructions of using R

If you are new to R, you can learn very quickly how to use R, following for example these introductions:

• Official R manuals
• A short R tutorial with examples (PDF, 19 pages) by myself

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Back to Longhai Li's software packages

How to install an R package?

— Philippe Grosjean 2006/01/18 [last tested: 2006/02/01 on R 2.2.1]
The official documentation for such a task is included in the R Installation and Administration, a PDF manual installed with each R version. Here is a quick mini ‘HowTo’ summarizing the operations for the easiest way, that is, by using install.packages().

Linux/Unix

Look at CRAN to determine if a binary version exists for your system.
Otherwise, you still can compile packages from sources. The simplest way to download a package from CRAN and to install it is by using the install.packages() function (make sure you install also all required dependencies). For example, to download and install the package called “mypkg”:

install.packages("mypkg", dependencies = TRUE)

Of course, this only works if your computer is connected to the Internet and if all tools required to compile packages are installed (see the manual if it fails).
Another way of installing a package is in batch mode that is, without starting R you should download the package tarball (i.e. package-name.tar.gz) file you want to install and then issue as root from your usual shell (bash, csh, sh,etc.) and under the same directory in which the tarball is:

R CMD INSTALL package-name.tar.gz

Windows

Using Rgui, the default R front-end under Windows, you can install packages from the menu: Packages -> Install package(s).... The first time, R asks for a mirror. Choose the one nearest to you in the list. Then, select the package(s) you want to install in the list. If the operation fails, look at the manual and the RW-FAQ for more information.

MacOS X

You can use install.packages() or R CMD INSTALL the same way as for Linux/Unix. The Mac GUI also provides a Package Installer for convenient point-and-click installation and updating of packages from various repositories.
Unix users should be aware that the default package type for install.packages() on Mac OS X is binary unlike on other unix systems.

Do not forget to update your packages regularly from the latest CRAN version! This can be done very easily using update.packages() or the equivalent menu entries in the various R GUIs.

Frequent beginner's questions about R packages

What is a R package / a R library?

A R package is a plugin providing additional functions and/or example datasets together with online help, and sometimes “vignettes“1). A R library is a location (top directory) where one or several packages are installed.

How do I know which package to install from the (huge) list on CRAN?

If you do not already know the name of the package you want to install, browse the list of packages on CRAN (http://cran.r-project.org/web/packages/, where you can access the PDF manual of each package to learn more about the functions they provide) and on Bioconductor (http://www.bioconductor.org/docs/vignettes.html, vignettes for all bioconductor packages). Alternatively, you can look at the CRAN Task Views (http://cran.r-project.org/src/contrib/Views/) that present lists of useful packages for given applications.

I installed a package, but I still cannot access its functions

Always remember: if you want to use functions and/or datasets in an optional package, you must first load the package in your current R session. This is done with the library() function:

library(mypkg)  # From this moment on, you have acces to functions in 'mypkg'
data(mydataset) # Always remember also to load datasets with data() before using them
 
# ... Here comes your computations using 'mypkg' functions and/or 'mydataset'
 
# If you don't need 'mypkg' any more, you can unload it with:
detach("package:mypkg")

How do I automatically load an add-on package whenever R is started?

To have libraries loaded by default when R is launched for all users on a computer, add code to the “R_HOME/etc/Rprofile.site” file. For instance, the code below automatically loads the ggplot2 library (and it’s dependency libraries) when R is launched.

local({
        old <- getOption("defaultPackages")
        options(defaultPackages = c(old, "ggplot2"))
      })

See the Startup help (”?Startup”) for more information about this code. To find the R_HOME folder, use

Sys.getenv("R_home")

1) Vignettes are PDF documents with various topics like theoretical background, tutorials, application examples, etc., to explain and illustrate how to use the functions included in a packages.