How partial sensor coverage affects Gaussian process reconstructions on rings

Finite-Mask Gaussian-Process Reconstruction on a Periodic Sensor Ring: Mask-Geometry Dependence, Fourier-Mode Coupling, and Posterior Trace

Information Theory

Summary

When sensors are arranged evenly in a ring and some sensors don’t provide data, this affects how accurately a Gaussian process can reconstruct signals from the ring. The authors studied how different patterns of missing sensors influence the relationships between the frequencies in the reconstruction and the overall uncertainty. They found that even if the fraction of missing sensors is the same, different arrangements can lead to different quality in the final estimates. Their work provides formulas and examples to understand these effects in settings like microphones arranged around circular ducts.

What this means in practice

  • For acoustic system engineers: Analyze how microphone failures or dropouts around circular ducts affect signal reconstruction and uncertainty estimation.
  • For signal processing teams: Design sensor layouts and reconstruction methods accounting for specific patterns of missing sensor data on circular arrays.

Tested on simulated data.

Authors

Jun Tsuzurugi

Abstract

We analyze how an arbitrary observation mask on a finite periodic ring of equally spaced sensors affects the posterior covariance, Fourier-mode coupling, and normalized posterior trace in Gaussian-process reconstruction. Under a rotationally stationary prior and homogeneous independent measurement noise, complete observation gives independent scalar posterior formulas for the Fourier modes. For an arbitrary mask, however, the matrix $Q_M=F D_MF^\ast$ is generally non-diagonal; its off-diagonal entries are finite Fourier components of the realized mask and couple modes in the posterior precision. Consequently, masks with the same unavailable-channel fraction can have different normalized posterior traces because their geometries differ. A dimensionless 64-channel synthetic benchmark illustrates this finite-matrix effect. A circumferential array of equally spaced wall-mounted microphones at a fixed axial station of a circular fan or compressor duct provides one concrete mechanical-engineering interpretation: failed, saturated, corrupted, or dropped-out channels form the observation mask. The analysis is a finite-dimensional reference calculation under the stated rotational-stationarity and common-noise assumptions, not a performance claim for a nonuniform or unequally instrumented operating duct.
View PDFOpen arXiv