## 35 Reputation

12 years, 145 days

## MaplePrimes Activity

### These are replies submitted by mah00

SO even if k is purely symbolik, maple cannot perform the integral?

SO even if k is purely symbolik, maple cannot perform the integral?

## subs...

The subs version is the most adequate to my application.

## subs...

The subs version is the most adequate to my application.

## Exactly...

Thanks!

I think it is exactly what I want.

## Exactly...

Thanks!

I think it is exactly what I want.

## example...

Here is an example:

U=Vector[column]([[x],[y]]);

A:=Matrix(3,3,(i,j)->1);

f(U):=exp(U'*A*U);

Then my question is how to calculate int(f(U),x=-infinity..infinity,y=-infinity..infinity);

But in 2D it is easy and I just wrote it.

My question is, is there a more clever way to write this in high diemension?

I hope it is clear enough.

## Actually I have a function f of U=Vector...

Actually I have a function f of U=Vector[column]([[x],[y],[z]]) and I want to calculate the inetgral of f over U, something like:

int(f(U),U=-infinity..infinity);

In my case, U is a 12 dimensional vector.

## In this example, the covariance matrix i...

In this example, the covariance matrix is given and it is not computed using the random vector.

Here is how he did it:

CorrMat := Matrix([seq([seq(`if`(i = j, 1, rho), j = 1 .. N)], i = 1 .. N)]);

C1 := simplify(S.Transpose(S));

CoVar := Zip(`*`, C1, CorrMat);

Here is my code, may be it is more clear this way:

T:=20;

d:=1;

`> xi0:=Vector([[0],[0],[0],[0]]):> Mxi:=Vector([[0],[0],[0],[0]]):> Rxi:=sqrt(d)*IdentityMatrix(4):> xi1:=BrownianMotion(xi0,Mxi,Rxi):> xi:=SamplePath(xi1(t),t=0..T,timesteps =T/d):`
`> eta0:=Vector([[0],[0]]):> Meta:=Vector([[0],[0]]):> Reta:=sqrt(d)*IdentityMatrix(2):> eta1:=BrownianMotion(eta0,Meta,Reta):> eta:=SamplePath(eta1(t),t=0..T,timesteps = (T/d),replications=4):`
` `

zeta:=k->Vector[column]([[xi[1,1,k+2*d]-xi[1,1,k+d]],[xi[1,2,k+2*d]-xi[1,2,k+d]],[xi[1,3,k+2*d]-xi[1,3,k+d]],[xi[1,4,k+2*d]-xi[1,4,k+d]],[eta[1,1,k+2*d]-eta[1,1,k+d]],[eta[1,2,k+2*d]-eta[1,2,k+d]],[eta[2,1,k+2*d]-eta[2,1,k+d]],[eta[2,2,k+2*d]-eta[2,2,k+d]],[eta[3,1,k+2*d]-eta[3,1,k+d]],[eta[3,2,k+2*d]-eta[3,2,k+d]],[eta[4,1,k+2*d]-eta[4,1,k+d]],[eta[4,2,k+2*d]-eta[4,2,k+d]]]):

sigma1:=k->Matrix(12,12,(i,j)->2*nu*(Statistics[ExpectedValue](zeta(k)[i]*zeta(k)[j])-Statistics[ExpectedValue](zeta(k)[i])*Statistics[ExpectedValue](zeta(k)[j])));

Now when you compute Determinant(sigma1(k)), it equal to 0 for any k!

## In this example, the covariance matrix i...

In this example, the covariance matrix is given and it is not computed using the random vector.

Here is how he did it:

CorrMat := Matrix([seq([seq(`if`(i = j, 1, rho), j = 1 .. N)], i = 1 .. N)]);

C1 := simplify(S.Transpose(S));

CoVar := Zip(`*`, C1, CorrMat);

Here is my code, may be it is more clear this way:

T:=20;

d:=1;

`> xi0:=Vector([[0],[0],[0],[0]]):> Mxi:=Vector([[0],[0],[0],[0]]):> Rxi:=sqrt(d)*IdentityMatrix(4):> xi1:=BrownianMotion(xi0,Mxi,Rxi):> xi:=SamplePath(xi1(t),t=0..T,timesteps =T/d):`
`> eta0:=Vector([[0],[0]]):> Meta:=Vector([[0],[0]]):> Reta:=sqrt(d)*IdentityMatrix(2):> eta1:=BrownianMotion(eta0,Meta,Reta):> eta:=SamplePath(eta1(t),t=0..T,timesteps = (T/d),replications=4):`
` `

zeta:=k->Vector[column]([[xi[1,1,k+2*d]-xi[1,1,k+d]],[xi[1,2,k+2*d]-xi[1,2,k+d]],[xi[1,3,k+2*d]-xi[1,3,k+d]],[xi[1,4,k+2*d]-xi[1,4,k+d]],[eta[1,1,k+2*d]-eta[1,1,k+d]],[eta[1,2,k+2*d]-eta[1,2,k+d]],[eta[2,1,k+2*d]-eta[2,1,k+d]],[eta[2,2,k+2*d]-eta[2,2,k+d]],[eta[3,1,k+2*d]-eta[3,1,k+d]],[eta[3,2,k+2*d]-eta[3,2,k+d]],[eta[4,1,k+2*d]-eta[4,1,k+d]],[eta[4,2,k+2*d]-eta[4,2,k+d]]]):

sigma1:=k->Matrix(12,12,(i,j)->2*nu*(Statistics[ExpectedValue](zeta(k)[i]*zeta(k)[j])-Statistics[ExpectedValue](zeta(k)[i])*Statistics[ExpectedValue](zeta(k)[j])));

Now when you compute Determinant(sigma1(k)), it equal to 0 for any k!

## Thanks, I will do....

Thanks, I will do.

## Thanks, I will do....

Thanks, I will do.

## Great explanation. Thanks.   My p...

Great explanation. Thanks.

My problem is actually the following:

I want to create a Gaussian PDF so I need to calculate Determinant(sigma) with sigma the covariance matrix of a gaussian variable.

If we call this variable alpha, then sigma_ij=ExpectedValue(alpha_i*alpha_j)-ExpectedValue(alpha_i)*ExpectedValue(alpha_j)

and this is zero most of the time! So the covariance matrix is singular and the determinant is zero. And this is because ExpectedValue(alpha_i) is different from zero (at least that's what I think).

Do you think that the problem is elsewhere?

## Great explanation. Thanks.   My p...

Great explanation. Thanks.

My problem is actually the following:

I want to create a Gaussian PDF so I need to calculate Determinant(sigma) with sigma the covariance matrix of a gaussian variable.

If we call this variable alpha, then sigma_ij=ExpectedValue(alpha_i*alpha_j)-ExpectedValue(alpha_i)*ExpectedValue(alpha_j)

and this is zero most of the time! So the covariance matrix is singular and the determinant is zero. And this is because ExpectedValue(alpha_i) is different from zero (at least that's what I think).

Do you think that the problem is elsewhere?

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