Optimizing H for Accurate Outlier Detection in Weighted Linear Least Squares

[ilia1987]

I have the following problem:
I have a set of m measurements $\mathbf{\phi}$
and I estimate a set of 3 variables $\mathbf{x}$

The estimated value for $\mathbf{\phi}$ depends linearly on $\mathbf{x}$ : Hx=\Tilde{\phi}
The solution through weighted linear least squares is:
$\mathbf{x}$ = (H^TWH)^{-1}H^TW$\mathbf{\phi}$
W is diagonal matrix

Suppose there is absolutely no deviation between the estimated values of $\mathbf{\phi}$ and the measured values (perfect measurements).
Now, suppose that one measurement \phi_i deviates strongly from its nominal value. In such a case the estimated value of $\mathbf{x}$ is going to change and so will the estimated value $\mathbf{\Tilde{\phi}}$. if the actual measurements deviate by $\mathbf{\delta\phi}$, the estimated values are goind to deviate by:
$\mathbf{\Delta\Tilde{\phi}}=H (H^TWH)^{-1}H^TW$\mathbf{\delta\phi}$

I later use the difference \Tilde{\phi}-\phi to reject stray measurements.
So, a situation where one stray measurement \delta\phi_i causes a large deviation \Delta\Tilde{\phi_j} is undesirable.
I reached the conclusion that it is necessary for off diagonal elements of
H (H^TWH)^{-1}H^TW to be smaller than the diagonal elements in order for the correct measurement to be recognized as stray.

My hope is that someone on this forum can help me find a nicer, more analytical way to express that condition on H itself rather that on this monstrous expression, or at least point me in the right direction.

Thank you.

 

[ilia1987]

A small correction:

For the matrix
H (H^TWH)^{-1}H^TW - I_{m \times m}
I want the absolute value of the diagonal elements to be larger than the absolute values of the off-diagonal elements.

That's because I want:
\forall(j\neq i) \delta\phi_i > \delta\phi_j \Rightarrow r_i > r_j
where r_k = \tilde{\phi_k} - \phi_k are the residuals.