[R] Maximum likelihood estimate of bivariate vonmises-weibulldistribution

[Chaouch, Aziz]

 Hi Dimitrios,

1) you propose to compare copula models using the same Kendall's tau. If
I understand correctly, I should use the Kendall's tau between wind
direction and wind speed and then compute the different copula models
using that Kendall's tau, right? However as Wind direction is a circular
variable modelled with a Von Mises distribution (or a mixture of),
Kendall's tau should be inefficient at measuring a circular-linear rank
correlation. I'm aware that Mardia (1976) has proposed a circular-linear
correlation coefficient (based on Pearson's one) but I'm not sure about
the existence of a circular-linear version of Kendall's tau.

2) Anyway I'm probably better using the fitMvdc function in package
copula to estimate the copula model parameter by MLE (doing this for
every copula model and see which one better fits my data). However do
you have recommandations on how to choose good starting values for the
parameter in the fitMvdc function? In the help of the fitMvdc function
(see below), the starting values for a gumbel copula are c(1,1,2)
although the gumbel copula has only one parameter (not three). So what
does this vector c(1,1,2) means?

gmb <- gumbelCopula(3, dim=2)
myMvd <- mvdc(gmb, c("exp","exp"), list(list(rate=2),list(rate=4)))
x <- rmvdc(myMvd, 1000)
fit <- fitMvdc(x, myMvd, c(1,1,2))

3) How did you do to choose the copula parameter that is associated to a
specific Kendall's tau in your example? Would such a parameter (based on
a measured Kendall's tau on my variables providing that such a measure
for circular-linear relationships exists) be a good starting value as a
parameter for each copula model??

Thanks a lot!

Aziz


PS: the email address of the maintainer of copula package seems to be
not working

-----Original Message-----
From: Dimitrios Rizopoulos [mailto:Dimitris.Rizopoulos at med.kuleuven.be] 
Sent: May 12, 2006 4:35 PM
To: Chaouch, Aziz
Subject: RE: [R] Maximum likelihood estimate of bivariate
vonmises-weibulldistribution

look at the following code:

library(copula)
par(mfrow = c(2, 2))
x <- mvdc(normalCopula(sin(0.5 * pi /2)), c("norm", "norm"),
list(list(mean = 0, sd = 1), list(mean = 0, sd = 1))) contour(x, dmvdc,
xlim = c(-2.7, 2.7), ylim = c(-2.7, 2.7))

x <- mvdc(frankCopula(5.736276), c("norm", "norm"), list(list(mean = 0,
sd = 1), list(mean = 0, sd = 1))) contour(x, dmvdc, xlim = c(-2.7, 2.7),
ylim = c(-2.7, 2.7))

x <- mvdc(gumbelCopula(2), c("norm", "norm"), list(list(mean = 0, sd =
1), list(mean = 0, sd = 1))) contour(x, dmvdc, xlim = c(-2.7, 2.7), ylim
= c(-2.7, 2.7))

x <- mvdc(claytonCopula(2), c("norm", "norm"), list(list(mean = 0, sd =
1), list(mean = 0, sd = 1))) contour(x, dmvdc, xlim = c(-2.7, 2.7), ylim
= c(-2.7, 2.7))


the values of the association parameter I've chosen in each copula 
correspond to Kendall's tau 0.5; assuming also standard normal 
marginal distributions look at the different shapes you get!

If possible try something similar for you case (i.e., using von Mises 
and Weibull marginals) and check if the association shape for a 
specific copula is more appropriate for your application. If this is 
not possible fit models assumig different copulas and check which one 
provides a better fit to your data.

I hope it helps.

Best,
Dimitris



Quoting "Chaouch, Aziz" <achaouch at NRCan.gc.ca>:

> Hi Dimitris,
> 
> I'm not sure to understand your suggestion. How would you build that
> contour plot for a particular copula and on what is computed the
> Kendall's tau? 
> 
> Thanks,
> 
> Aziz
> 
> -----Original Message-----
> From: Dimitris Rizopoulos
> [mailto:dimitris.rizopoulos at med.kuleuven.be] 
> Sent: May 12, 2006 9:57 AM
> To: Chaouch, Aziz; hydinghua at gmail.com
> Cc: r-help at stat.math.ethz.ch
> Subject: Re: [R] Maximum likelihood estimate of bivariate
> vonmises-weibulldistribution
> 
> the choice of the copula is, in fact, a model selection problem. 
> First, you could have a look at the contour plots of different
> copulas
> (preferably for the same value of Kendall's tau), and decide if some
> of
> them assume a more appropriate association structure for your
> application, compared to the others. Afterwards, you may fit various
> copula functions, check the fit on the data, compute AIC (since
> these
> are typically not nested models), etc.
> 
> regarding the Von Mises distribution and if could be used in mvdc(),
> that I don't know. It'd be better to contact the copula package
> maintainer and ask.
> 
> I hope it helps.
> 
> Best,
> Dimitirs
> 
> ----
> Dimitris Rizopoulos
> Ph.D. Student
> Biostatistical Centre
> School of Public Health
> Catholic University of Leuven
> 
> Address: Kapucijnenvoer 35, Leuven, Belgium
> Tel: +32/(0)16/336899
> Fax: +32/(0)16/337015
> Web: http://www.med.kuleuven.be/biostat/
>      http://www.student.kuleuven.be/~m0390867/dimitris.htm
> 
> 
> ----- Original Message -----
> From: "Chaouch, Aziz" <achaouch at NRCan.gc.ca>
> To: "Dimitris Rizopoulos" <dimitris.rizopoulos at med.kuleuven.be>;
> <hydinghua at gmail.com>
> Cc: <r-help at stat.math.ethz.ch>
> Sent: Friday, May 12, 2006 3:13 PM
> Subject: RE: [R] Maximum likelihood estimate of bivariate
> vonmises-weibulldistribution
> 
> 
> Thanks a lot! I wasn't aware of that copula package and it could well
> be
> appropriate to use it for my application. However if I read the
> copula
> help correctly, I still need to know what kind of copula to use to
> link
> the distribution of wind speeds and directions. Is there a way to
> determine this in R?
> 
> Moreover can I use the Von Mises distribution from the circular or
> CircStats package into the mvdc function of the copula package or
> does
> the mvdc function only recognize distributions available "natively"
> within R?
> 
> Thanks again to all, your help is highly appreciated for a newbie
> like
> me!
> 
> Regards,
> 
> Aziz
> 
> -----Original Message-----
> From: Dimitris Rizopoulos
> [mailto:dimitris.rizopoulos at med.kuleuven.be]
> Sent: May 12, 2006 3:01 AM
> To: Philip He; Chaouch, Aziz
> Cc: r-help at stat.math.ethz.ch
> Subject: Re: [R] Maximum likelihood estimate of bivariate
> vonmises-weibulldistribution
> 
> 
> ----- Original Message -----
> From: "Philip He" <hydinghua at gmail.com>
> To: "Chaouch, Aziz" <achaouch at nrcan.gc.ca>
> Cc: <r-help at stat.math.ethz.ch>
> Sent: Thursday, May 11, 2006 11:21 PM
> Subject: Re: [R] Maximum likelihood estimate of bivariate
> vonmises-weibulldistribution
> 
> 
> > On 5/11/06, Chaouch, Aziz <achaouch at nrcan.gc.ca> wrote:
> >>
> >> Hi,
> >>
> >> I'm dealing with wind data and I'd like to model their
> distribution 
> >> in order to simulate data to fill-in missing values. Wind
> direction 
> >> are typically following a vonmises distribution and wind speeds 
> >> follow a weibull distribution. I'd like to build a joint
> distribution
> 
> >> of directions and speeds as a VonMises-Weibull bivariate 
> >> distribution.
> >
> >
> > In order to built a bivariate distribution from two marginal 
> > distributions (wind direction, wind speed) , more information is 
> > needed to specify the relation between these two marginal 
> > distributions.For example, a conditional distribution may help.
> >
> 
> 
> An alternative in such cases (i.e., when marginals are available but
> the
> joint is difficult to postulate) is to use copulas, which can
> construct
> multivariate distributions from univariate marginals. If this is
> appropriate for this application, the "copula" package might be of
> help.
> 
> Best,
> Dimitris
> 
> ---
> Dimitris Rizopoulos
> Ph.D. Student
> Biostatistical Centre
> School of Public Health
> Catholic University of Leuven
> 
> Address: Kapucijnenvoer 35, Leuven, Belgium
> Tel: +32/(0)16/336899
> Fax: +32/(0)16/337015
> Web: http://www.med.kuleuven.be/biostat/
>      http://www.student.kuleuven.be/~m0390867/dimitris.htm
> 
> 
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> 
> 
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