[Paper Review] Sensor Selection via Randomized Sampling

Shaunak D. Bopardikar|arXiv (Cornell University)|Dec 18, 2017

Distributed Sensor Networks and Detection Algorithms17 references3 citations

TL;DR

This paper proposes a randomized sensor selection algorithm that samples sensors with replacement using tailored probability distributions to approximate the observability Gramian of a linear time-invariant system. It provides high-probability bounds on key metrics—such as trace, maximum eigenvalue, and minimum eigenvalue—ensuring non-singularity and performance guarantees, outperforming greedy methods in numerical studies while enabling efficient recursive state estimation with reduced sensor usage.

ABSTRACT

Given a linear dynamical system, we consider the problem of constructing an approximate system using only a subset of the sensors out of the total set such that the observability Gramian of the new system is approximately equal to that of the original system. Our contributions are as follows. First, we present a randomized algorithm that samples the sensors with replacement as per specified distributions. For specific metrics of the observability Gramian such as the trace or the maximum eigenvalue, we derive novel bounds on the number of samples required to yield a high probability lower bound on the metric evaluated on the approximate Gramian. Second, with a different distribution, we derive high probability bounds on other standard metrics used in sensor selection, including the minimum eigenvalue or the trace of the Gramian inverse. This distribution requires a number of samples which is larger than the one required for the trace and the maximum eigenvalue, but guarantees non-singularity of the approximate Gramian if the original system is observable with high probability. Third, we demonstrate how the randomized procedure can be used for recursive state estimation using fewer sensors than the original system and provide a high probability upper bound on the initial error covariance. We supplement the theoretical results with several insightful numerical studies and comparisons with competing greedy approaches.

Motivation & Objective

• To develop a randomized sensor selection framework that approximates the observability Gramian using fewer sensors than the full set.
• To derive high-probability bounds on standard observability metrics—trace, maximum eigenvalue, minimum eigenvalue, and trace of the Gramian inverse—under random sampling.
• To guarantee non-singularity of the approximate observability Gramian with high probability when the original system is observable.
• To enable recursive state estimation with reduced sensor count while bounding the initial and steady-state error covariance.
• To compare the proposed method empirically with greedy heuristics and demonstrate superior performance across multiple observability metrics.

Proposed method

• The method employs a randomized sampling algorithm that selects sensors with replacement according to two distinct probability distributions tailored to different performance metrics.
• For trace and maximum eigenvalue, the algorithm uses a distribution proportional to sensor contribution, with theoretical bounds on sample complexity to ensure high-probability lower bounds.
• For minimum eigenvalue and trace of the Gramian inverse, a different distribution is introduced that requires more samples but ensures non-singularity of the approximate Gramian with high probability.
• The approach leverages concentration inequalities to derive probabilistic guarantees on the approximation quality of the observability Gramian.
• The method is applied to least squares estimation and Kalman filtering, providing high-probability upper bounds on initial and steady-state error covariance matrices.
• Numerical experiments compare the method with a greedy heuristic across multiple metrics, including minimum eigenvalue, trace, and maximum eigenvalue.

Experimental results

Research questions

• RQ1 What is the minimum number of sensor samples required to achieve a high-probability lower bound on the trace or maximum eigenvalue of the observability Gramian?
• RQ2 Can a randomized sampling strategy ensure non-singularity of the approximate observability Gramian with high probability, even when the original system is only weakly observable?
• RQ3 How does the performance of the randomized algorithm compare to greedy heuristics in terms of key observability metrics like minimum eigenvalue and trace of the Gramian inverse?
• RQ4 What is the theoretical bound on the initial error covariance when using the randomized sensor selection for recursive state estimation?
• RQ5 Can the randomized sampling approach be extended to bound the steady-state error covariance in Kalman filtering with reduced sensor usage?

Key findings

• The proposed algorithm outperforms a greedy heuristic across all tested metrics—minimum eigenvalue, trace, and maximum eigenvalue—for a wide range of error tolerance ε.
• The variant allowing repeated sensor sampling consistently outperforms the greedy method, especially in minimizing the worst-case estimation error.
• The initial error covariance in recursive estimation is observed to be an order of magnitude smaller than the theoretical upper bound derived in Proposition 4.1.
• The relative error in the steady-state error covariance matrix decreases nearly linearly with ε, suggesting a predictable and scalable performance trend.
• The sample complexity for the minimum eigenvalue and Gramian inverse metrics is higher than for trace and maximum eigenvalue, but ensures non-singularity of the approximate Gramian with high probability.
• Experiments with i.i.d. standard normal C matrices yield qualitatively similar results, though sample complexity is approximately doubled compared to the baseline setup.