Let $\mathbf{x}$ be a random $n\times1$ real vector, $\mathbf{x}\in\Bbb{R}^n$, which is distributed normally with mean $\bar{\mathbf{x}}$ and covariance matrix $\Sigma_x\in\Bbb{R}^{n\times n}$, i.e. $\mathbf{x}\sim\mathcal{N}(\bar{\mathbf{x}},\Sigma_x)$.

I would like to generate a new random vector, $\mathbf{y}\in\Bbb{R}^m$, with $m<n$, for which the following requirements should be meet:

  1. $\mathbf{y}\sim\mathcal{N}(\bar{\mathbf{y}},\Sigma_y)$
  2. The covariance matrix $\Sigma_y$ should follow from $\Sigma_x$ via a PCA (Principal Component Analysis) process.

The above conditions arise from the fact that I need to apply PCA on $\Sigma_x$ such that only a fraction of the energy is preserved (say $90\%$), and the projection matrix, say $P$, transforms the original random vector $\mathbf{x}$ into $\mathbf{y}=P\mathbf{x}$.

I am looking for a discussion here, if you like to share your thoughts on this. Thanks a lot in advance!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.