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.