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I have the equation:

$x = \mathrm{V}^{\frac{1}{2}}y$

which takes a vector of random numbers $y$ and induces a correlation structure according to a matrix $\mathrm{V}$, which is a square symmetric toeplitz matrix constructed from the autocorrelation function required for $x$. $\mathrm{V}$ is positive-definite, and its square root can be easily found by the Cholesky decomposition.

Therefore, I can take a vector of random numbers and induce a required correlation structure. However, $\mathrm{V}$, can be expressed as $\mathrm{V}_1 + \mathrm{V}_2 + \mathrm{V}_3$, as there are three distinct structures in the autocorrelation function. Rather than inducing all three correlation structures at the same time, I want to do the three individually and then add them together, but:

$x = \mathrm{V}^{\frac{1}{2}}y \neq \left(\mathrm{V}_1^{\frac{1}{2}} + \mathrm{V}_2^{\frac{1}{2}} + \mathrm{V}_3^{\frac{1}{2}}\right) y$

How can this be done?

($\mathrm{V}_i$ are positive-definite, and their square roots can also be easily found by the Cholesky decomposition.)

user26857
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hydrologist
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  • Unless you want to try for a series solution (since you're introducing a square root of a sum of operators), I would highly suggest diagonalizing $V$ and then take the square root in the usual manner suggested by the spectral theorem. – Cameron Williams Aug 14 '14 at 14:40
  • Thank you @CameronWilliams. Can you elaborate a bit further please? – hydrologist Aug 14 '14 at 14:45
  • Take a look at Newton's generalized binomial series for more information. – Cameron Williams Aug 14 '14 at 14:46
  • @CameronWilliams - yes, I think I see that - but I want to to identify the influence of the three components. I can find the square root of all three, but this is not equivalent to the square root of the sum. What I want to do is to see what the impact of $\mathrm{V}_1$, $\mathrm{V}_2$ and $\mathrm{V}_3$ are individually. I can easily look at the impact of their sum, but I want to see this disaggregated into three components. – hydrologist Aug 14 '14 at 15:01

1 Answers1

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Perhaps you could write $$(V_1 + V_2 + V_3)^{1/2} y = V_1^{1/2} y + \left((V_1 + V_2)^{1/2} y - V_1^{1/2} y\right) + \left((V_1 + V_2 + V_3)^{1/2} y - (V_1 + V_2)^{1/2} y\right)$$ so that $V_1^{1/2} y$ is the effect of the first component alone, $(V_1 + V_2)^{1/2} y - V_1^{1/2} y$ is the difference the second component makes once you have the first component, and $(V_1 + V_2 + V_3)^{1/2} y - (V_1 + V_2)^{1/2} y$ is the difference the third component makes once you have the first two.

Robert Israel
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