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Math Help - Construction a positive definite covariance matrix

  1. #1
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    Construction a positive definite covariance matrix

    Hi.
    I am trying to simulate two seismic traces S^1 and S^2 represented as two 1D vectors. I want the correlation between S^1 and S^2 to be a constant i.e. 0.9 and In order to simulate these two trases I am wondering how to make this into a positive definite covariance matrix.

    S^1 and S^2 is both spatially dependent, so I have introduced a gaussian correlation function and is on the form \rho(x'-x'')=exp(-abs(x'-x'')^2/d)
    let  S^j=(s_1...s_n) for j=1,2

    and I though I could create a covariance matrix B where

    \textbf{A}=\begin{bmatrix} \sigma_{s_1}^2 &\sigma_{s_1} \sigma_{s_2} \rho(x_1-x_2) &.. &.. & \sigma_{s_1} \sigma_{s_n} \rho(x_1-x_n)\\& & & &  \\& &. & &  \\& & &.  &  \\\sigma_{s_1} \sigma_{s_n} \rho(x_1-x_n)& & & & \sigma_{s_n}^2  \\\end{bmatrix}

    \textbf{C}=\begin{bmatrix} 0.9*\sigma_{s_1}^2 & & & & \\& 0.9 \sigma_{s_2}^2& & &  \\& &. & &  \\& & &. &  \\& & & &0.9\sigma_{s_n}^2  \\\end{bmatrix}



    \textbf{B}=\begin{bmatrix} A & C \\ C & A  \\\end{bmatrix}
    Why is this wrong? Why is this not positive definite?
    I also tried with

    \textbf{B}=\begin{bmatrix} A & 0.9.*A^T \\ 0.9.*A & A  \\\end{bmatrix}
    Which is indeed positive definite, but the simulation from this looks all wrong, and the second trace just got a super small variation around the mean compared to the first trace.

    I think there is something wrong with my trail of thought here, any tip is appreciated.
    Last edited by ia88; June 15th 2013 at 04:27 AM.
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  2. #2
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    Re: Construction a positive definite covariance matrix

    Hey ia88.

    For a valid co-variance matrix shouldn't the C in the second row be C^T (i.e. C transpose)?
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  3. #3
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    Re: Construction a positive definite covariance matrix

    Yes, technically, but it is symmetric so it does not matter
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  4. #4
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    Re: Construction a positive definite covariance matrix

    It will not be symmetric in general unless you use the transpose.
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