As part of a larger problem, I would like to divide some elements of a vector $ \vec y $ and combine those with the original vector elements like so:
$$ \vec y = \begin{pmatrix} x_1 \\ x_2 \\ y_1 \\ y_2 \end{pmatrix} $$
$$ \vec u = \begin{pmatrix} x_1 \\ x_2 \\ y_1 \\ y_2 \\ x_1/y_1 \\ x_2/y_2 \end{pmatrix} := \begin{pmatrix} x_1 \\ x_2 \\ y_1 \\ y_2 \\ r_1 \\ r_2 \end{pmatrix}$$
In particular, I would like to propagate uncertainties from $ \vec y $ to $ \vec u $. Normally I would approximate the transformation from $ \vec y $ to $ \vec u $ through the Jacobian matrix:
$$ \vec u \approx \mathbf{J} \vec y $$
$$ \mathbf{J} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1/y_1 & 0 & -r_1/y_1 & 0 \\ 0 & 1/y_2 & 0 & -r_2/y_2 \end{pmatrix} $$
And then the covariance in $\vec y$, $\mathbf{\Sigma}_y$, would be propagated into $\vec u$:
$$ \mathbf{\Sigma}_u \approx \mathbf{J} \mathbf{\Sigma}_y \mathbf{J}^T $$
Where I would expect significant correlations between $r_i$ and $x_i, y_i$.
However, as you may notice, actually calculating $ \mathbf{J} \vec y $ will always result in $0$ for the last two elements.
Where am I going wrong? I don't see any obvious errors in my derivation, so I think one of my assumptions must be incorrect. The most logical option being that the Jacobian matrix approximation simply does not hold for this operation. If this is true, what should I do instead?
Best Answer
You never apply the Jacobian to $\vec{y}$, you apply it to a (first-order) change in $\vec{y}$.
In a covariance calculation, you take the expectation of $\Delta r \Delta r^T$, where $\Delta r=r-E(r)$.
Let $\Delta y= y-E(y)$, i.e. $y=E(y)+\Delta y$. Then the first-order approximation is $r(y)\approx r(E(y))+J\Delta y$. Taking expectations on both sides we see that in this approximation $E(r)=r(E(y))$ and so $r\approx E(r)+J\Delta y$, i.e. $\Delta r \approx J \Delta y$. Then, under those assumptions,
$$Cov(r,r)=E(\Delta r \Delta r^T)\approx E(J \Delta y \Delta y^T J^T)=J E(\Delta y \Delta y^T )J^T=JCov(y,y)J^T,$$
as wanted.
There is no problem.