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<math>Cov[\tilde{x}_n,\tilde{x}_m] = E\left[\left(\sum_{i=1}^na_{ni}x_i\right)\left(\sum_{j=1}^m a_{mj}x_j\right)\right] = \sum_{i=1}^n\sum_{j=1}^m a_{ni}a_{mj}E[x_ix_j] \Rightarrow</math> | <math>Cov[\tilde{x}_n,\tilde{x}_m] = E\left[\left(\sum_{i=1}^na_{ni}x_i\right)\left(\sum_{j=1}^m a_{mj}x_j\right)\right] = \sum_{i=1}^n\sum_{j=1}^m a_{ni}a_{mj}E[x_ix_j] \Rightarrow</math> | ||
| − | <math>Cov(\tilde{x}_n,\tilde{x}_m)</math> = <math>\sum_{i=1}^{\min(m,n)}a_{ni}a_{mi}</math>, since <math>x_i</math>'s are independent, <math>E[x_i] = 0</math> and <math>Var[x_i] = 1</math> | + | <math>Cov(\tilde{x}_n,\tilde{x}_m)</math> = <math>\sum_{i=1}^{\min(m,n)}a_{ni}a_{mi}</math>, since <math>x_i</math>'s are independent, <math>E[x_i] = 0</math> and <math>Var[x_i] = 1</math>. |
| − | + | We are now left with the problem of defining <math>a_{ni}</math>'s so that the form of <math>Cov(\tilde{x}_n,\tilde{x}_m)</math> follows the form of <math>\Sigma_{nm}</math>: i.e. | |
| − | + | <math>\Sigma_{nm}</math> = <math>Cov(\tilde{x}_n,\tilde{x}_m)</math> = <math>\sum_{i=1}^{\min(m,n)}a_{ni}a_{mi} | |
| + | </math> | ||
| − | + | <math> </math>|sigrelation|, where <math>A</math> is lower triangular, and <math>\Sigma</math> is positive definite. Therefore, <math>A</math> follows the form of what is called the Cholesky decomposition of <math>\Sigma</math>. | |
| − | + | ||
| − | |sigrelation|, where | + | |
Thus, to summarize, to generate samples |xdist| from samples |x| drawn from |normdist|, simply let |xxtildrel|, where |A| is the Cholesky decomposition of |sig|. | Thus, to summarize, to generate samples |xdist| from samples |x| drawn from |normdist|, simply let |xxtildrel|, where |A| is the Cholesky decomposition of |sig|. | ||
Revision as of 15:52, 20 March 2008
{ Summary: To generate "colored" samples $ \tilde{x}\in\mathbb{R}^n \sim \mathcal{N}(\mu,\Sigma) $ from "white" samples $ x $ drawn from $ \mathcal{N}(\vec{0},I_n) $, simply let $ \tilde{x} = Ax + \mu $, where $ A $ is the Cholesky decomposition of $ \Sigma $, i.e. $ \Sigma = AA^T $}
Consider generating samples $ \tilde{x}\in\mathbb{R}^n \sim \mathcal{N}(\mu,\Sigma) $. Many platforms (e.g. Matlab) have a random number generator to generate iid samples from (white) Gaussian distribution. If we seek to "color" the noise with an arbitrary covariance matrix $ \Sigma $, we must produce a "coloring matrix" $ A $. Let us consider generating a colored sample $ \tilde{x} = [\tilde{x}_1,\tilde{x}_2,\ldots,\tilde{x}_n]^T $ from $ x = [x_1,x_2,\ldots,x_n]^T $, where $ x_1, x_2, \ldots, x_n $ are iid samples drawn from $ \mathcal{N}(0,1) $. (Note: Matlab has a function, mvnrnd.m, to sample from $ \mathcal{N}(\mu,\Sigma) $, but I discuss here the theory behind it). Relate $ \tilde{x} $ to $ x $ as follows:
$ \tilde{x}_1 = a_{11} x_1, $
$ \tilde{x}_2 = a_{21} x_1 + a_{22} x_2 $,
...
$ \tilde{x}_n = \sum_{i=1}^n a_{ni}x_i $.
We can rewrite this in matrix form as $ \tilde{x} = Ax $, where matrix $ A $ is lower triangular. We have, then, that
$ E[\tilde{x}_n] = \sum_{i=1}^n a_{ni}E[x_i] = 0 $, and
$ Cov[\tilde{x}_n,\tilde{x}_m] = E\left[\left(\sum_{i=1}^na_{ni}x_i\right)\left(\sum_{j=1}^m a_{mj}x_j\right)\right] = \sum_{i=1}^n\sum_{j=1}^m a_{ni}a_{mj}E[x_ix_j] \Rightarrow $
$ Cov(\tilde{x}_n,\tilde{x}_m) $ = $ \sum_{i=1}^{\min(m,n)}a_{ni}a_{mi} $, since $ x_i $'s are independent, $ E[x_i] = 0 $ and $ Var[x_i] = 1 $.
We are now left with the problem of defining $ a_{ni} $'s so that the form of $ Cov(\tilde{x}_n,\tilde{x}_m) $ follows the form of $ \Sigma_{nm} $: i.e.
$ \Sigma_{nm} $ = $ Cov(\tilde{x}_n,\tilde{x}_m) $ = $ \sum_{i=1}^{\min(m,n)}a_{ni}a_{mi} $
|sigrelation|, where $ A $ is lower triangular, and $ \Sigma $ is positive definite. Therefore, $ A $ follows the form of what is called the Cholesky decomposition of $ \Sigma $.
Thus, to summarize, to generate samples |xdist| from samples |x| drawn from |normdist|, simply let |xxtildrel|, where |A| is the Cholesky decomposition of |sig|.
.. |aat| image:: tex
- alt: tex: \Sigma = AA^T
.. |xdist| image:: tex
- alt: tex: \tilde{x}\in\mathbb{R}^n \sim \mathcal{N}(\mu,\Sigma)
.. |sig| image:: tex
- alt: tex: \Sigma
.. |signm| image:: tex
- alt: tex: \Sigma_{nm}
.. |A| image:: tex
- alt: tex: A
.. |xtildfull| image:: tex
- alt: tex: \tilde{x} = [\tilde{x}_1,\tilde{x}_2,\ldots,\tilde{x}_n]^T
.. |xfull| image:: tex
- alt: tex: x = [x_1,x_2,\ldots,x_n]^T
.. |xiid| image:: tex
- alt: tex: x_1, x_2, \ldots, x_n
.. |1Dnormdist| image:: tex
- alt: tex: \mathcal{N}(0,1)
.. |normdist| image:: tex
- alt: tex: \mathcal{N}(\vec{0},I_n)
.. |normdistarb| image:: tex
- alt: tex: \mathcal{N}(\mu,\Sigma)
.. |x| image:: tex
- alt: tex: x
.. |xtild| image:: tex
- alt: tex: \tilde{x}
.. |xxtild1| image:: tex
- alt: tex: \tilde{x}_1 = a_{11} x_1,
.. |xxtild2| image:: tex
- alt: tex: \tilde{x}_2 = a_{21} x_1 + a_{22} x_2,\ldots
.. |xxtildn| image:: tex
- alt: tex: \tilde{x}_n = \sum_{i=1}^n a_{ni}x_i
.. |xxtildMat| image:: tex
- alt: tex: \tilde{x} = Ax
.. |xxtildrel| image:: tex
- alt: tex: \tilde{x} = Ax + \mu
.. |E(n)| image:: tex
- alt: tex: E[\tilde{x}_n] = \sum_{i=1}^n a_{ni}E[x_i] = 0
.. |Cov(n,m)def| image:: tex
- alt: tex: Cov[\tilde{x}_n,\tilde{x}_m] = E\left[\left(\sum_{i=1}^na_{ni}x_i\right)\left(\sum_{j=1}^m a_{mj}x_j\right)\right] = \sum_{i=1}^n\sum_{j=1}^m a_{ni}a_{mj}E[x_ix_j] \Rightarrow
.. |Cov(n,m)deffinal| image:: tex
- alt: tex: \sum_{i=1}^{\min(m,n)}a_{ni}a_{mi}
.. |Cov(n,m)| image:: tex
- alt: tex: Cov(\tilde{x}_n,\tilde{x}_m)
.. |xi| image:: tex
- alt: tex: x_i
.. |xivar| image:: tex
- alt: tex: Var[x_i] = 1
.. |ximean| image:: tex
- alt: tex: E[x_i] = 0
.. |ani| image:: tex
- alt: tex: a_{ni}
.. |sigrelation| image:: tex
- alt: tex: \Rightarrow \Sigma = AA^T
