[SOLVED] Quantitative risk management using copulas, fmsn65/masm33 computer assignment 3
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1. Present in detail the characterization of bivariate extreme value distributions with unit
Fréchet margins.
2. What are the properties of the dependence function?
3. Write the formula for extreme value copula and give at least three examples of parametric
bivariate extreme value copulas. How can one check that a given copula is an extreme
value copula?4. Explain in detail how one can construct a bivariate extreme value distribution with arbitrary univariate extreme value margins?
5. Derive the formula for Pickand’s estimator of the dependence function.Getting started with the computational part:
If you plan to carry out the assignment on your own computer, you need to download and install
R from http://www.r-project.org.If you want to carry out the assignment in the computer room MH:230, you just need to start
one of the PCs first. Then choose the latest version of R from the Start menu.In this assignment you need to use evd, SimCop, copula and mgpd packages. It is strongly recommended that you download the corresponding PDF files of the manuals from cran.r-project.org to
use as a reference in the assignment. Type library(evd), library(SimCop), library(copula)
and library(mgpd) to load the packages to your R-session.These packages are installed in all
computers in the computer room MH:230. If you use your own laptop you need to install the
packages yourself; see the help page for install.packages for more information.In all exercises below you will plot several datasets. Note that you can use mfrow to collect any
number of the plots in one single figure.1. In the package evd nine parametric bivariate extreme value models have been implemented.
Study each family and specify which parameter(s) stand for strength of dependence in each
family. If the model is asymmetric specify also which parameter stands for asymmetry. Note
that the parametrization of some families in evd package (as you can see in the help page
for bvevd in the evd manual which you can download from cran.r-project.org) is different
from the course literature and lecture notes (see e.g. the logistic model).2. Choose one symmetric and one asymmetric family and generate 200 observations from
each model using the package evd. In order to check your answer to the previous part, in
each case simulate for a couple of different values of parameters in the model and make a
scatter plot of the data. Does the dependence parameter have the expected effect on the
dependence of simulated data?For each pair of the simulated sample calculate the empirical values of Pearson’s correlation
ρ, Kendall’s τ and Spearman’s ρS.3. In the package SimCop nine parametric bivariate and multivariate copulas have been implemented. Download the reference manual SimCop.pdf and study each family and specify
which parameter(s) stand for strength of dependence in each family. If the model is asymmetric specify also which parameter stands for asymmetry.4. Generate 200 observations from the bivariate asymmetric logistic extreme value copula for
different combinations of the parameters and plot the results. For each plot comment on
the effect of changing the copula parameters on the dependence structure of the model.Check the help documents for GenerateRV.CopApprox and NewBEVAsyLogisticCopula
to see examples of how these functions can be used to generate random observations
from copulas. You do not need to use Metropolis-Hastings algorithm (MH) argument) in
your simulations. See also the help document for plot.CopApprox. Note that by default
the plot function for the class CopApprox creates interactive plots but by setting the
type=öriginal” you can create the traditional 2-dimensional plots.This part of the assignment is concerned with the annual maximum sea-level in Fremantle and
Portpirie in Australia. The datasets ”fremantle.R” and ”portpirie.R” can be downloaded from
the following locations
• The page Datasets in R under the Pages in the homepage of the course in Canvas,
• http://www.maths.lth.se/matstat/kurser/fmsn15masm23/datasetsR.html.Note that these datasets should be loaded to R using source command.
1. Find out for which years the maximum sea level is available in each location.2. Create a data frame which contains data for those years for which maximum sea level is
available in both locations. One way of doing this is to use the function merge in R. See
the help file of the function for details.3. Create a scatter plot of maximum sea level in both locations.
4. Choose at least two parametric models (one symmetric and one asymmetric) and fit the
models to the dataset. In both cases use full maximum likelihood (FML) and inference for
margins (IFM) methods to estimates the parameters. Compare your results.5. Choose at least one non-parametric model and fit it to the dataset. Use both parametric
and empirical transformation of margins to estimate the dependence function (see the
argument epmar to abvnonpar). Plot both estimates and compare them.
6. Suppose X and Y stand for the annual maximum sea level in Fremantle and Portpirie,
respectively. Estimate the probabilities
• P((X, Y ) > (1.7, 4.2)),
• P((X, Y ) > (1.8, 4.4)), and
• P((X, Y ) 6< (1.478, 3.850))
based on both parametric and non-parametric fits above. The values in the last probability
above are the 30%-quantiles of the sea levels in the dataset. Compare them also with the
empirical estimates of the probabilities.Remarks: For parametric models you can use pbvevd and pgev to calculate the distribution
function of bivariate and univariate extreme value distributions, respectively. There is no
such functions if the dependence function is estimated by a non-parametric method. In
this case one needs to use the following relationship as it was shown in ”Exercises 2”:
G(x, y) = G∗((1 + γ1
x−µ1
σ1
)
1
γ1
+ ,(1 + γ2
y−µ2
σ2
)
1
γ2
+ ) where G∗(x, y) = e
−(
1
x + 1
y
)A(
x
x+y
)
. The function abvnonpar can be used to find the value of dependence function in a specific point.7. With the same notation as in previous question estimate
P((X, Y ) < (1.95, 4.8)|(X, Y ) 6< (1.478, 3.850)).
8. Plot both parametric and non-parametric estimates for upper quantile curves of bivariate
extreme value distributions fitted above for p = 0.75, 0.90, 0.95 (see the help page for the
function plot.bvevd).9. The Fremantle dataset was also analyzed in the ”Computer Assignment 3” of the course on
univariate extreme values. It was noted that there seems to be a trend in Fremantle sealevel
data which might be explained by including ”Year” and ”SOI” (South Oscillation Index)
as linear trend in the location parameter. Fit the same parametric models as above but
include ”Year” and ”SOI” as covariates for Fremantle sealevel in the model (see nsloc1 and
nsloc2 arguments in the help file for fbvevd).Recall also that in order to avoid numerical
difficulties in optimization you need to normalize ”Year” to [−1, 1]. Use log-likelihood ratio
test to verify if any of these have significant effect on the fit. The functions deviance,
logLik and anova can also be used to compare nested models in R.As the package copula provides some extra functions such as goodness of fit tests for fitting
different copulas there are some advantages in using this package even for exrtreme value analysis. On the other hand there are only five families of extreme value copulas included in this
package (see the help file for evCopula)1. Fit parametric models “huslerReiss” , ”gumbel” and “galambos” implemented in the copula
package to the dataset using FML method. Compare your results with the corresponding
models in the evd package (“hr”, ”log” and “neglog” models) to make sure that the estimated
parameters are the same. What is the difference in parametrization of ”logistic” model in
these two packages?2. Use the Inference For Margins (IFM) method based on GEV margins and Canonical Maximul Likelihood (CML) method without specifying the marginal distributions and fit the
models to the dataset.3. Use the function gofEVCopula for goodness-of-fit tests for parametric bivariate extremevalue copulas which you have fitted to the dataset above.4. Use the function An.biv and find non-parametric estimators of the dependence function
according to Pickands and CFG method. Plot the estimates in a figure.5. Suppose X and Y stand for the annual maximum sea level in Fremantle and Portpirie,
respectively. Estimate the probabilities
• P((X, Y ) > (1.7, 4.2)),
• P((X, Y ) > (1.8, 4.4)), and
• P((X, Y ) 6< (1.478, 3.850))
based on both parametric and non-parametric fits above. The conditional values in the
last probability above are the 30%-quantiles of the sea levels in the dataset. Compare
them also with the empirical estimates of the probabilities. You have done this part in the
previous exercise as well using evd package. Compare your results.6. With the same notation as in previous question estimate
P((X, Y ) < (1.95, 4.8)|(X, Y ) 6< (1.478, 3.850)).One advantage of using SimCop package is that we can create a copula model based on nonparametric smoothing splines estimate of the dependence function. This can be done by using
NonparEstDepFct and SplineFitDepFct functions. Start with reading the help files for these
functions. In particular check the examples which demonstrate how the functions can be used.1. Find the Pickand’s estimator of the dependence function for the annual maximum sea
level dataset you have analyzed above. Plot the estimates for two cases corresponding to
setting convex.hull to TRUE or FALSE.
2. Find the estimate of dependence function using cubic smoothing splines. As input use
the non-parametric estimator of dependence function with the argument convex.hull =
FALSE. Plot the resulting estimate.3. Create a copula for the smoothing spline fit using the function NewBEVSplineCopula.
Find an approximation to the copula by using the function GetApprox and plot the corresponding copula.4. Suppose we want to estimate the same probabilities as in the previous exercises by using
simulated observations from the smoothing splines copulas. Generate 1000 observations
from the smoothing splines copula and estimate the following probabilities:(a) P((X, Y ) > (1.7, 4.2)),
(b) P((X, Y ) > (1.8, 4.4)),
(c) P((X, Y ) 6< (1.478, 3.850)), and
(d) P((X, Y ) < (1.95, 4.8)|(X, Y ) 6< (1.478, 3.850)).
Compare the results with your answers to the corresponding probabilities in exercises 3
and 4.In this exercise the same dataset on sea level will be analyzed by using peaks over threshold method. As discussed in the course there are at least two ways of defining exceedances in higher dimensions. In the first definition a distribution is fitted to the observations
{(x, y)|(x, y) > (ux, uy)} where ux and uy are suitable thresholds for each margin. Second definition aims to fit a distribution to {(x, y)|(x, y) 6< (ux, uy)} where (ux, uy) is defined as before.These distributions will be called Type I and Type II bivariate generalized Pareto distributions
(BGPD), respectively.
Type I BGPD
The function fbvpot in package evd fits a Type I model by maximizing the censored likelihood
as given in e.g. Section 8.3.1 of Coles (2001) which is attached.1. Calculate the 30%-quantiles of the sea levels in both locations. Use these values as thresholds for the margins in the analysis below.
2. Choose at least two parametric models (one symmetric and one asymmetric) and fit a
Type I BGPD to the dataset above the thresholds.3. Plot upper quantile curves of the distributions fitted above for p = 0.75, 0.9, 0.95. (see the
help page for the function plot.bvpot). Compare these quantile curves with what you
have plotted in part 8 of exercise 2. Do they agree?4. Calculate
P((X, Y ) < (1.95, 4.8)|(X, Y ) 6< (1.478, 3.850))
based on your parametric models.Remarks: Note that you need to transform (X, Y ) to (X, e Ye) according to the theory
which has been discussed in the course (see also attached pages from Coles book). Read
also Details section in the help page for bvevd on page 14 of evd manual to check which
parametrization has been used for each model and how they can be transformed to an
arbitrary bivariate extreme value distribution with GEV margins.5. Note that if the shape parameter in GPD is negative then the distribution has finite right
end point σ/|γ|. It is important to take this into consideration when you transform the
margins to unit Frechét distribution. As an example calculate
P((X, Y ) < (2, 5)|(X, Y ) 6< (1.478, 3.850)).Type II BGPD
Type II bivariate generalized Pareto distributions can be fitted by using the package mgpd. The
main function is fbgpd. As an extra help on how to use the functions in this package you can
download and use the file ”bgpdExampleRun.R” from the following locations• The page Datasets in R under the Pages in the homepage of the course in Canvas,
• http://www.maths.lth.se/matstat/kurser/fmsn15masm23/datasetsR.html.1. Choose at least two parametric models (one symmetric and one asymmetric) and fit a
Type II BGPD to the exceedances over the same thresholds as for the Type I BGPD
above.Note that the estimation is based on a complicated likelihood function and therefore it
is strongly dependent on good start values. One way of finding suitable start values for a
model is to use estimates from a simpler model (e.g. Logistic model) as a starting point.
Further, if you get an error in R such as “invalid NA contour values”it means that the
density for some points has become not available”. In this case you should try to change
x and y vectors in the bgpdExampleRun.R.2. Calculate
P((X, Y ) < (1.95, 4.8)|(X, Y ) 6< (1.478, 3.850)).
Compare the result with your answer to question 4d in exercise 3 and question 4 in type
I BGPD.3. Plot estimates of prediction regions for p = 0.75, 0.9, 0.95 based on parametric models you
have fitted to the dataset. Compare your results with your answer to the same question
for the Type I BGPD above and comment on any differences you see.Finishing Off:
When you’ve finished, close down R by typing q(). Choose ‘Save’ when prompted as to whether
you want to retain your workspace.
154 8. Multivariate Extremes
0
‘”
“!
g
] “!
E
”
~
“‘:
“!
5 10 50 100 500 1000
Retum PeriOd (Years)
FIGURE 8.5. Comparison of return level plot for Z = min{M.,, (My- 2.5)} in
logistic model analysis of Fremantle and Port Pirie annual maximum sea-level
series with a= 0, 0.25, 0.5, 0.75 and 1, respectively. Lowest curve corresponds
to a= 1; highest to a= 0.excess model and point process model can be obtained. In this section we
give a brief description of both techniques.
8.3.1 Bivariate Threshold Excess Model
In Chapter 4 we derived as a class of approximations to the tail of an
arbitrary distribution function F the family
G(x) = 1 – ( { 1 + e(x; u)} -1/e, X > u. (8.18)This means there are parameters (,a and e such that, for a large enough
threshold u, F(x) ~ G(x) on x > u. Our aim now is to obtain a bivariate
version of (8.18), i.e. a family with which to approximate an arbitrary joint
distribution F(x, y) on regions of the form x > ux, y > uy, for large enough
Ux and Uy.Suppose (x1, Yl), … , (xn, Yn) are independent realizations of a random
variable (X, Y) with joint distribution function F. For suitable thresholds
Ux and uy, the marginal distributions ofF each have an approximation of
the form (8.18), with respective parameter sets ((x,ax,ex) and ((y,ay,ey)·8.3 Alternative Representations 155
The transformations
(8.19)
and
(8.20)
induce a variable (X, Y) whose distribution function F has margins that
are approximately standard Frechet for X> Uz andY> uy. By (8.5), for
large n,
_ {- }1/n F(x, y) = Fn(x, y)
~ [exp { -V(xfn,yfn)}]11n
= exp{-V(x,y)},
because of the homogeneity property of V. Finally, since F(x, y) = F(x, y),
it follows that
F(x,y) ~ G(x,y) = exp{-V(x,y)}, x > ux,Y > uy, (8.21)
with x andy defined in terms of x andy by (8.19) and (8.20). This assumes
that the thresholds Ux and uy are large enough to justify the limit (8.5) as
an approximation. We discuss this point further in Section 8.4.Inference for this model is complicated by the fact that a bivariate pair
may exceed a specified threshold in just one of its components. Let
Ro,o = (-oo,ux) x (-oo,uy),Rt,o = [ux,oo) x (-oo,uy),
Ro,t = (-oo,ux) x [uy,oo),Rt,l = [ux,oo) x [uy,oo),
so that, for example, a point (x, y) E R1,o if the x-component exceeds
the threshold ux, but the y-component is below uy. For points in R1,1,
model (8.21) applies, and the density ofF gives the appropriate likelihood
component. On the other regions, since F is not applicable, it is necessary to
censor the likelihood component. For example, suppose that (x, y) E R1,o.Then since x > Uz, but y < uy, there is information in the data concerning
the marginal x-component, but not the y-component. Hence, the likelihood
contribution for such a point is
Pr{X = x, Y $ uy} = !’. aFI
uX (x,uy}
as this is the only information in the datum concerning F. Applying similar
considerations in the other regions, we obtain the likelihood function
n
L(8; (xt,Yt), … , (xn,Yn)) =IT 1/J(B; (xi,yi)), (8.22)
i=l
156 8. Multivariate Extremes
where (} denotes the parameters of F and
‘ljJ(O; (x,y)) =
if (x,y) E R1,1,
if (x, y) E R1,o,
if (x,y) E Ro,1,
if (x, y) E Ro,o,
with each term being derived from the joint tail approximation (8.21).Maximizing the log-likelihood leads to estimates and sta~dard errors for
the parameters of F in the usual way. As with the componentwise block
maxima model, the inference can be simplified by carrying out the marginal
estimation, followed by transformations (8.19) and (8.20), as a preliminary
step. In this case, likelihood (8.22) is a function only of the dependence
parameters contained in the model for V.An alternative method, if there is a natural structure variable Z =
¢(X, Y), is to apply univariate threshold techniques to the series Zi =
¢(xi,Yi)· This approach now makes more sense, since the Zi are functions
of concurrent events. In terms of statistical efficiency, however, there are
still good reasons to prefer the multivariate model.8.3.2 Point Process Model
The point process characterization, summarized by the following theorem,
includes an interpretation of the function H in (8.6).
Theorem 8.2 Let (xl,Yl),(x2,Y2) … be a sequence of independent bivariate observations from a distribution with standard Frechet margins
that satisfies the convergence for componentwise maxima
Pr{M;,n $ x,M;,n $ y}-+ G(x,y).Let { N n} be a sequence of point processes defined by
Then,
Nn = {(n- 1xl,n-1 y1), … ,(n-1xn,n-1yn)}.
d Nn –t N
(8.23)
on regions bounded from the origin (0, 0), where N is a non-homogeneous
Poisson process on (O,oo) x (O,oo). Moreover, letting
r=x+y
the intensity function of N is
X and w=–, x+y
‘( ) _ 2dH(w) “‘r,w – 2 , r
where H is related to G through (8.5) and (8.6).
(8.24)
(8.25)
0