Re: Re: T Copula Calibration
- To: mathgroup at smc.vnet.net
- Subject: [mg78741] Re: [mg78643] Re: T Copula Calibration
- From: Carl Woll <carlw at wolfram.com>
- Date: Sun, 8 Jul 2007 06:11:02 -0400 (EDT)
- References: <200707030944.FAA18997@smc.vnet.net><f6fpep$84j$1@smc.vnet.net> <200707060718.DAA20660@smc.vnet.net>
dwstrickler at tx.rr.com wrote:
>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:
>
>
Neither FindMinimum nor NMinimize support minimizing a matrix. As I said
before, your function needs to be an expression that evaluates to a
*number* and not to a matrix when v is given a value. This is why you
are getting the not a number error message. The reason test is a matrix
is because the subexpression
(1 + y . R^(-1) * (y/v))
is a matrix. You need to change this subexpression into something that
evaluates to a number.
Carl Woll
Wolfram Research
>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.
>
>
>
- References:
- T Copula Calibration
- From: dwstrickler@tx.rr.com
- Re: T Copula Calibration
- From: dwstrickler@tx.rr.com
- T Copula Calibration