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Re: T Copula Calibration

• To: mathgroup at smc.vnet.net
• Subject: [mg78643] Re: T Copula Calibration
• From: dwstrickler at tx.rr.com
• Date: Fri, 6 Jul 2007 03:18:37 -0400 (EDT)
• References: <200707030944.FAA18997@smc.vnet.net><f6fpep$84j$1@smc.vnet.net>

On Jul 4, 4:32 am, Carl Woll <c... at wolfram.com> wrote:
> dwstrick... at tx.rr.com wrote:
> > All,
>
> > In the interest of full disclosure and genuine humility, I don't code
> > in Mathematica so much as I wrestle it into submission from time to
> > time. Accordingly, I've come up against an optimization problem in
> > which Mathematica flatly refuses to see things my way. In a nutshell,
> > I have an n x 3 table of interdependent log-delta price data (to the
> > extent it matters, I have a corresponding table of alpha-Stable
> > standardized log-delta price data), and I'm trying to construct a
> > simulation algorithm that will effectuate a Student T copula
> > relationship between the three variables. (FWIW, I've managed to
> > create Clayton and Gumbel copula RNGs for similar data; they work
> > fine).
>
> > The problem is the Student T copula density function, and in
> > particular, the Student T DOF parameter, v. I've tried NMinimize,
> > FindMinimum, and FindRoot, and I simply cannot get Mathematica to return a
> > numerical value for v. In fact, the problem is broader than that - I
> > can't get Mathematica to optimize ANY function that contains a Sum or
> > Product
> > term - but I digress.
>
> > In terms of the density function, here's an example of what I've
> > tried:
>
> > tCopulaPDF[y__,v_,R_]:=Module[{dims,ret},dims=Length[Transpose[y]];ret=((Gamma
> > [(v+dims)/2]*(Gamma[v/2])^dims-1)/(Gamma[(v+1)/
> > 2]^dims)*(Det[R])^1/2)*NProduct[(1+
> > (y[[i]]^2/v))^v+1/2,{i,1,dims}]*(1+y.R^-1*y/v)^-v+dims/2;Return[ret]];
>
>                                                           ^^^^^^
> It's unnecessary to use Return here. Module automatically returns the
> last expression.
>
> Without creating data to test your code, I suspect that the problem is
> that NMinimize attempts to evaluate tCopulaPDF with a symbolic v. If so,
> the simple workaround is to use
>
> tCopulaPDF[y_, v_?NumberQ, R_] := ...
>
> If the above doesn't solve your problem, make sure that
>
> tCopulaPDF[data, 1, corrmat]
>
> evaluates to a number.
>
> Another possible issue is the use of R^-1. Did you mean to use
> Inverse[R] here instead?
>
> Carl Woll
> Wolfram Research
>
>
>
> > where y is the n x 3 table, v is the DOF parameter, and R is the
> > correlation matrix for y. Gamma[] is Mathematica's built-in Euler Gamma
> > function.
>
> > Then I try to optimize with something like:
>
> > NMinimize[{tCopulaPDF[data,v,corrmat],v>0},v]
>
> > No luck. Mathematica gives me the standard NAN warning, and then
> > returns the
> > function unevaluated.
>
> > Thinking the source of the problem might be the NProduct term, I also
> > tried expressing the density function a different way:
>
> > tCopulaPDF[y__,v_,R_]:=Module[{dims,ret,cols,a,b,c},dims=Length[Transpose[y]];cols=y/.
> > {a_,b_,c_}:>Apply[Times,(1+a^2/v)^v+1/2]+Apply[Times,(b^2/v)^v
> > +1/2]+Apply[Times,(c^2/v)^v
> > +1/2];ret=((Gamma[(v+dims)/2]*(Gamma[v/2])^dims-1)/(Gamma[(v+1)/
> > 2]^dims)*(Det[R])^1/2)
> > *cols*(1+y.R^-1*y/v)^-v+dims/2;Return[ret]];
>
> > Same result. Any suggestions? I apologize in advance if the solution
> > is blindingly obvious to everyone but me.

Carl,

Thanks for your suggestions.  I tried the NumberQ and Inverse[R]
fixes, but these still didn't allow me to optimize.  In accordance
with your suggestion, I did however make sure that the tCopulaPDF
function itself was working.  When I input:

dims = Length[Transpose[xdata]]; tCopulaPDF =
   (((Gamma[(v + dims)/2]*Gamma[v/2]^
        (dims - 1))/(Gamma[(v + 1)/2]^dims*
       Det[R]^(1/2)))*Product[(1 + y^2/v)^
       ((v + 1)/2), {i, 1, dims}])/
    (1 + y . R^(-1)*(y/v))^((v + dims)/2);
  test = tCopulaPDF /. {y -> xdata, v -> 2,
     R -> xcorrmat}

I get a clean, n * 3 matrix of real numbers, albeit with values that
are 3-5 times too large.  If I input:

test2 = tCopulaPDF /. {y -> xdata, R -> xcorrmat}

the result simplifies nicely around the unspecified v parameter.
Trial-and-error confirms that increasing the DOF parameter (v)
decreases the values that are returned, but there's still the matter
of calibration.  When I try:

FindMinimum[test, {v, 5}], or
NMinimize[{test, v > 5}, v]

I get the familiar error message:

NMinimize::nnum: The function value {<<1>>} is not a number at {v} =
{5.063707273968251`}.  More . . .

To the extent it helps, here's a 6 * 3 sample from the standardized
data set:

xtrunc = {{0.19067, 0.269248, 0.248315}, {0.857068,
     0.371992, 0.943073}, {0.714966, 0.908612, 0.889339}, {0.255969,
     0.791845, 0.790311}, {0.693816, 0.988873, 0.560541}, {0.247043,
     0.384309, 0.196687}};

And here's the correlation matrix:

xcorrmat = {{1, .608018033570726, .759864730700654}, {.
608018033570726, 1, .5916085946121}, {.759864730700654, .
5916085946121, 1}};

Any ideas?  BTW, thanks again for your help.

• Follow-Ups:
  • Re: Re: T Copula Calibration
    • From: Carl Woll <carlw@wolfram.com>

• References:
  • T Copula Calibration
    • From: dwstrickler@tx.rr.com