Possibility of subset selection for state estimation problem

[VersLerclec]

Hi @sleepymalc,

Thank you for sharing the code.
I found this work is really interesting, and I was wondering the possibility of finding most influential subset in a state estimation problem.

Specifically, the problem I want to solve is to find outlier measurements which affect the state estimates the most. In a state estimation problem, we often want to solve the following equation:

$$ y = h(x) + \epsilon \approx Hx + \epsilon $$

where $y$ is measurement vector, $H$ is partial derivatives of measurement to states, and $\epsilon$ is measurement noise. Normally, $H$ is known and the estimated parameters is $x$ vector. The problem I want to solve is to know how the outlier measurements in $y$ vector would affect the quality of state estimation.

In this paper, you mentioned that the MISS is designed for excluding k points and k<<n. However, in my case, normally n is smaller than 30, but the number of outlier can be 6 or 7. I would like to know if the influence analysis still applicable in a state estimation problem to find a most influential subset?

Thanks!

[Mirnegg]

Hi, thanks for the great question and for sharing the details.

A couple of thoughts based on your setting ($n < 30$, $k=7$):

• Brute-force (i.e., enumerating all subsets) may be feasible here ($C_{30}^7 \approx 2.0 \times 10^6$). Whether this is practical depends on the cost of re-solving the estimator for each subset.
• In the paper we focus on $k \ll n$ mainly because it is the regime where first-order approximations are theoretically justified (linearization is accurate). That said, in your $n < 30$, $k=7$ setup, influence-based greedy heuristics can still be a reasonable and practical approach in practice. If the estimator remains well-conditioned, we would expect vanilla influence-based greedy to perform reasonably well. In addition, our adaptive greedy variant is designed to better handle cases where interactions between removed points matter, and could plausibly work even better here.
• One important nuance is that most influential measurements are not always identical to outliers. Highly influential measurements can also arise from high leverage, even if they are clean. If your goal is specifically outlier detection, robust estimation and residual-based tests are also worth considering, while MISS serves as a complementary diagnostic.

Let us know if you have any further questions.