Kalman-Bucy Filtering with Randomized Sensing: Fundamental Limits and Sensor Network Design for Field Estimation

The gist The authors derive a closed-form upper bound on the expected estimation covariance for continuous-time Kalman–Bucy filtering under randomized sensing, establishing fundamental performance limits for spatiotemporal field estimation.

Theoretical Framework and Bounds

The work develops a general continuous-time framework to analyze the Kalman filter under randomly lost measurements where both the measurement matrix and noise covariance evolve as random processes. Within this setting, they derive a closed-form upper bound on the expected estimation covariance for continuous-time Kalman filtering. This framework is applied to spatiotemporal field estimation, where the field is modeled as a Gaussian process observed by randomly located, noisy sensors. Using clarity, introduced in their earlier work as a rescaled form of the differential entropy of a random variable, they establish a grid-independent lower bound on the spatially averaged expected clarity. This result exposes fundamental performance limits through a composite sensing parameter that jointly captures the effects of the number of sensors, noise level, and measurement frequency.

Continuous-Time Upper Bound Derivation

The analysis proceeds by deriving structural properties of the Riccati operator to bound the expected covariance. Theorem 1 establishes a continuous-time upper bound on the expected estimation covariance for Kalman–Bucy filtering under randomized sensing, which admits a closed-form steady-state expression. The proof utilizes Jensen’s inequality together with the concavity of RG established in Lemma 1. In the canonical case where system dynamics and process excitation are isotropic, a closed-form characterization of the steady-state covariance is provided by Theorem 2.

Application to Spatiotemporal Field Estimation

The framework is applied to spatiotemporal field estimation, modeling the field as a Gaussian process with separable kernels for temporal and spatial correlations. The continuous-time upper-bound results are directly applied to the spatiotemporal Gaussian process Kalman filter (STGPKF). This leads to a discrete-time linear measurement model where the sensing matrix H(r(tk)) and measurement noise covariance V(r(tk)) are determined by the instantaneous sensor locations. The STGPKF is then defined using the Kalman covariance recursion in Eq. (45)–(46).

Estimation Performance and Clarity

To evaluate estimation performance, clarity [9] is employed, which normalizes the differential entropy of a continuous random variable onto the range [0, 1]. The spatially averaged clarity at time tk is defined as q¯k:= Ng-1 Σ Ng i=1 (1 + Πii,k). Theorem 3 establishes a lower bound on the averaged expected clarity, q¯EΠ, based on the upper bound covariance.

Fundamental Limits and Design Guidelines

The analysis characterizes the fundamental limits of estimation performance by deriving a closed-form expression for the averaged information matrix G bar. The sensing parameter θ:= Nr / (σ 2m ∆t) quantifies the sensing strength, jointly determined by the number of sensors, their measurement noise level, and the measurement rate. Theorem 4 shows that as grid spacing δ → 0, the steady-state lower bound of the averaged expected clarity q¯∆Π∞ converges to a finite limit that depends solely on the sensing parameter θ. This implies that beyond a certain system size, adding more measurement points no longer improves estimation performance. The agent-number design problem is formulated as min Nr s.t. q¯∆Π∞ ≥ qtarget.

Conclusion

The paper develops a continuous-time framework for quantifying the fundamental performance limits of multiagent spatiotemporal estimation under stochastic sensing. The framework yields a grid-independent performance law governed by a parameter combining agent number, measurement noise, and measurement rate. Simulation results confirm that the continuous-time bound accurately predicts empirical estimation performance and provides a reliable tool for pre-deployment sensor number design without costly simulations. Future research will consider incorporating the dynamics and motion constraints of mobile sensors into the estimation framework and address non-uniform location sampling arising from heterogeneous sensor distributions.

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Kalman–Bucy Filtering with Randomized Sensing: Fundamental Limits and Sensor Network Design for Field Estimation Xinyi Wang, Devansh R. Agrawal, and Dimitra Panagou Abstract—Stability analysis of the Kalman filter under randomly lost measurements has been widely studied. We revisit this problem in a general continuous-time framework, where both the measurement matrix and noise covariance evolve as random processes, capturing variability in sensing locations. Within this setting, we derive a closed-form upper bound on the expected estimation covariance for continuous-time Kalman filtering. We then apply this framework to spatiotemporal field estimation, where the field is modeled as a Gaussian process observed by randomly located, noisy sensors. Using clarity, introduced in our earlier work as a rescaled form of the differential entropy of a random variable, we establish a grid-independent lower bound on the spatially averaged expected clarity. This result exposes fundamental performance limits through a composite sensing parameter that jointly captures the effects of the number of sensors, noise level, and measurement frequency. Simulations confirm that the proposed bound is tight for the discretetime Kalman filter, approaching it as the measurement rate decreases, while avoiding the recursive computations required in the discrete-time formulation. Most importantly, the derived limits provide principled and efficient guidelines for sensor network design problem prior to deployment. Index Terms—Kalman filter, Spatiotemporal field estimation, Multi-agent sensor network

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Steady-state analysis of the estimation error reveals how uncertainty evolves and stabilizes over time [1]. In traditional settings, the Kalman filter (KF) provides the optimal linear minimum-variance estimator, with its steady-state covariance governed by a Riccati equation [2]–[4]. However, in modern applications such as estimating a spatiotemporal field using a multi-agent sensor network, the sensing process itself is inherently stochastic—sensor locations, measurement timing, and noise may all vary randomly [5]–[7]. This randomness makes direct steady-state analysis of the estimation error intractable. Consequently, it is more insightful to adopt a statistical viewpoint, by modeling sensing events as random processes and analyzing their collective impact on estimation performance. Within this stochastic sensing framework, we address a fundamental sensor network design problem constrained by performance requirements: How many sensors, and of what quality, are required to guarantee a prescribed level of estimation accuracy prior to deployment? This motivates two key objectives: (1) to characterize the expected estimation performance analytically, and (2) to establish the fundamental limits of estimation accuracy under stochastic sensing. In this work, we pursue these objectives through both theoretical analysis and application-driven validation. Prior work has addressed this question in the discretetime setting. In particular, Gupta [8] analyzed the stochastic sensor selection problem and derived upper bounds on the expected error covariance of the KF under randomly varying measurement availability. While insightful, these bounds are recursive in nature and therefore computationally demanding. In this work, we approximate the measurement process using a continuous-time model rather than directly analyzing the discrete-time recursion. This formulation leads to a differential inequality governing the evolution of the expected estimation covariance, from which the asymptotic behavior of the estimation system naturally follows. The continuous-time approximation offers two main advantages. First, it admits a closed-form expression for the steady-state covariance, making the analysis computationally much more efficient than the recursive discrete-time formulation. Second, in the context of multi-agent estimation of spatiotemporal Gaussian Process (GP) fields [9], where agents collect noisy measurements at locations drawn from a spatial distribution, the GP structure maps naturally to a linear statespace model, allowing the KF to produce optimal estimates. To quantify estimation uncertainty within this framework, we adopt the clarity, a rescaled form of the differential entropy of a random variable [10]. Leveraging the derived closedform upper bound and its asymptotic behavior, the proposed approach reveals that the lower bound of the averaged expected clarity is independent of the discretization grid. This property enables the identification of the fundamental limits that govern steady-state estimation performance under stochastic sensing <ref:2511.

1. Bold header: Closed-form Upper Bound for Continuous-Time Covariance Analysis. This improvement allows for efficient evaluation of estimation performance without recursive numerical computation, enabling rapid design iteration by bypassing computationally demanding discrete-time Riccati recursions when modeling stochastic sensing variability.

2. Bold header: Grid-Independent Performance Law Derivation. The framework establishes a grid-independent lower bound on the spatially averaged expected clarity, meaning the resulting performance limits are not artifacts of discretization, providing robust estimation guarantees regardless of the spatial resolution chosen for simulation or deployment planning.

3. Bold header: Optimal Sensor Resource Design via Clarity Target. The analysis enables the direct computation of sensor requirements through min Nr s.t. q¯∆Π∞ ≥ qtarget, allowing engineers to determine the minimal number of agents required to guarantee a desired clarity level prior to costly physical deployment simulations.

4. Bold header: Trade-off Characterization via Sensing Intensity Parameter. The derivation of the sensing parameter θ:= Nr / (σ2m∆t) allows for explicit trade-offs, showing that selecting a target clarity determines a required θ, enabling explicit analysis of how performance is governed by the interplay between sensor noise and measurement rate.

5. Bold header: Spatiotemporal Field Reconstruction with Uncertainty Quantification. The application to the STGPKF allows AI systems to reconstruct dynamic fields (like wind flow) while simultaneously providing clarity maps that verifies the estimation reliability, allowing for uncertainty-aware decision-making in complex spatiotemporal environments.