Covariance matrix of linear model

Question

I want to write the expression for covariance matrix of following linear model

$$x= As+ v$$ Where $A\in R^{N\times p}, s\in R^{p\times 1}$, $v$ consists of MVN noise only with mean $Bz$ and covariance $R=\sigma^2I$, $v\sim \mathcal{N}[Bz, \sigma^2I ]$: $B\in R^{N\times t}, \phi\in R^{t\times 1}, t<N-p$. Now the matrices $A$ and $B$ are linearly independent, i.e $A^TB \neq 0 $ and each matrix is full rank.

Although, I know how to write the covariance matrix expression for following signal $x$ which obeys linear subspace model

$$x=As+n$$ Where $s$ is independent of noise,then covariance $C=AE[ss^T]A^T+\sigma^2I$ for Gaussian noise of zero mean. What if I write the covariance matrix expression for first model as $C=AE[ss^T]A^T+ BE[zz^T]B^T+\sigma^2I$, is it the right way ?

Please suggest as I can not find a reference.

Appreciate your suggestions!

Answer 1

Your statement for the distribution of $v$ should be $v|z\sim\mathcal{N}[Bz,\sigma^2I]$. That is, you specified conditional distribution of $v$ on $z$. If you condition on $z$, the first model is as easy as the second model to describe. The conditional variance of $x$ given $z$ is $$Var(x|z)=A\mathbb{E}[ss^T|z]A^T + \sigma^2I$$ Since you want the unconditional variance, you can apply the familiar formula there. $$Var(x)=E[Var(x|z)]+Var(E[x|z]) = A\mathbb{E}[ss^T]A^T + \sigma^2I + Var(A\mathbb{E}[s|z]+Bz)$$ Similar to what you got, right? Only if you assume that $s$ and $z$ are independent, or more directly $E[s|z]$ is constant (which is a weaker condition), then your formula is right.

[EDIT] You can't do much about the term $Var(A\mathbb{E}[s|z]+Bz)$. You can try to expand as $$AVar(\mathbb{E}[s|z])A^T + ACov(\mathbb{E}[s|z],z)+ Cov(z,\mathbb{E}[s|z])A^T + B\mathbb{E}[zz^T]B^T$$ which has the term you came up with. This also shows that a sufficient condition for your solution being right is that $\mathbb{E}[s|z]=C$, where $C$ is a constant.