Covariance matrix

The covariance matrix \Sigma of n random variables X_1,\cdots, X_n is an n\times n matrix that \Sigma_{i, j}=Cov(X_j, X_j). The correlation matrix is the covariance matrix of standardized random variables X_1/\sigma(X_1), \cdots, X_n /\sigma(X_n) .

For any random variable Y=\sum_{i=1}^{n} a_i X_i, we have Var(Y)=a^T \Sigma a, where a=(a_1, \cdots, a_n)^T.

So the covariance matrix is positive-semidefinite, since the variance is non-negative.

Given correlation matrix, we can generate the random variables X_1, \cdots, X_n  using Cholesky decomposition \Sigma=LL^*. Generate a vector of uncorrelated samples u, then we have X=Lu.

Another way is to use eigenvalue method. \Sigma=A\Lambda A^*.

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About cgao

Cambridge, MA
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