Bridging the Gap Between Deterministic and Probabilistic Approaches to State Estimation

TL;DR

This work tackles state estimation from limited, noisy measurements by bridging deterministic MNLS/DEIM methods with Bayesian MAP estimation. It derives a computable Bayes-risk difference showing the MAP estimator is always more reliable, decomposed into prior and measurement-noise components, and introduces a data-driven prior covariance derived from training snapshots. The paper also develops two sensor-placement approaches—greedy Bayesian OED and CPQR-based Q-DEIM/Q-MAP—analyzes their theoretical guarantees, and validates the framework on random harmonic functions and 2D turbulence, revealing that Bayesian placements often match or closely approximate optimal strategies. The results provide a principled, practically implementable link between deterministic and probabilistic state estimation, with implications for sensor networks and reduced-order modeling in fluid dynamics and related fields.

Abstract

We consider the problem of state estimation from limited discrete and noisy measurements. In particular, we focus on modal state estimation, which approximates the unknown state of the system within a prescribed basis. We estimate the coefficients of the modal expansion using available observational data. This is usually accomplished through two distinct frameworks. One is deterministic and estimates the expansion coefficients by solving a least-squares (LS) problem. The second is probabilistic and uses a Bayesian approach to derive a distribution for the coefficients, resulting in the maximum-a-posteriori (MAP) estimate. Here, we seek to quantify and compare the accuracy of these two approaches. To this end, we derive a computable expression for the difference in Bayes risk between the deterministic LS and the Bayesian MAP estimates. We prove that this difference is always nonnegative, indicating that the MAP estimate is always more reliable than the LS estimate. We further show that this difference comprises two nonnegative components representing measurement noise and prior uncertainty, and identify regimes where one component dominates the other in magnitude. We also derive a novel prior distribution from the sample covariance matrix of the training data, and examine the greedy Bayesian and column-pivoted QR (CPQR) sensor placement algorithms with this prior as an input. Using numerical examples, we show that the greedy Bayesian algorithm returns nearly optimal sensor locations. We show that, under certain conditions, the greedy Bayesian sensor locations are identical or nearly identical to those of CPQR when applied to a regularized modal basis.

Figures (10)

• Figure 1: Summarized workflow of the deterministic and Bayesian approaches. Arrows indicate the flow of information. Prescribed inputs are the modal basis $\Phi$, the prior covariance matrix $\Gamma_{\mathop{\mathrm{prior}}\nolimits}$, and the noise covariance matrix $\Gamma_{\mathop{\mathrm{noise}}\nolimits}$. From these inputs, CPQR and the OED criterion generate the sensor locations. DEIM and MAP then use measurements at these locations to compute $m_{\mathop{\mathrm{LS}}\nolimits}$ (the MNLS estimate of $m$) and $m_{\mathop{\mathrm{MAP}}\nolimits}$ (the MAP estimate of $m$). These approximations of the inversion parameter $m$ then generate $u_{\mathop{\mathrm{MAP}}\nolimits}:=\Phi m_{\mathop{\mathrm{MAP}}\nolimits}$ and $u_{\text{DEIM}}:=\Phi m_{\mathop{\mathrm{LS}}\nolimits}$, the MAP and DEIM approximations of the full-state $u$.
• Figure 2: Sensor placement for the harmonic functions. The algorithms place 5 sensors into 40 available locations. Brute force determines the optimal and D-optimal sensor locations, while \ref{['alg:greedy_oed']} determines the greedy D-optimal sensor locations. The optimal MAP sensor locations minimize the MAP reconstruction error with $k=5$ sensors and $n=20$ modes, while the optimal DEIM sensor locations minimize the DEIM reconstruction error with $k=n=5$. We perform CPQR on the first $k$ POD modes (i.e. the first $k$ columns of $\Phi$).
• Figure 3: The reconstructions of a single sample from the harmonic functions. The 'true data' to be reconstructed is the black curve. We use 5 modes for the DEIM reconstruction (such that the number of modes equals the number of sensors), and 20 modes for the MAP reconstruction. On this sample, D-MAP, greedy D-MAP, and Q-DEIM have relative errors of 24.68%, 31.11%, and 36.31%, respectively.
• Figure 4: The relative error of D-MAP and greedy D-MAP on the harmonic data with respect to number of modes in (a) and sensors in (b). We plot the relative error of greedy D-MAP and Q-DEIM with respect to number of modes in (c) and sensors in (d). 'Relative error' refers to average relative error across all test samples, with error bars indicating one standard deviation. The 'number of modes' refers to modes used for sensor placement and state estimation. In (b), we set the high fidelity resolution $N$ to 20 for computational tractability, while $N=40$ in (a), (c), and (d). In (d), we set the number of modes used by Q-DEIM to the number of sensors (on the horizontal axis), and the number of modes used by greedy D-MAP to 20.
• Figure 5: The components of the risk premium (i.e. $\delta_{\mathop{\mathrm{prior}}\nolimits}$ and $\delta_{\mathop{\mathrm{noise}}\nolimits}$) for the harmonic data and their respective upper bounds. We set the number of randomly-selected sensors to $k=5$.

Theorems & Definitions (21)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.1: Risk Premium
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• proof : Proof of \ref{['prop:map_ub']}