Where to Measure: Epistemic Uncertainty-Based Sensor Placement with ConvCNPs

TL;DR

This work tackles optimal sensor placement for spatio-temporal processes by leveraging ConvCNPs and addressing the conflation of epistemic and aleatoric uncertainty in predictive variance. It extends ConvCNPs with a Mixture Density Network head to separately estimate epistemic uncertainty, and introduces an acquisition function based on the expected reduction in epistemic uncertainty to guide sensor deployment. Empirical results on Baltic Sea SST data show that epistemic-driven placement reduces RMSE and NLL more effectively than methods based on total uncertainty, highlighting the importance of uncertainty disentanglement. The work outlines limitations and future directions, including the choice of mixture components and evaluations over multiple days.

Abstract

Accurate sensor placement is critical for modeling spatio-temporal systems such as environmental and climate processes. Neural Processes (NPs), particularly Convolutional Conditional Neural Processes (ConvCNPs), provide scalable probabilistic models with uncertainty estimates, making them well-suited for data-driven sensor placement. However, existing approaches rely on total predictive uncertainty, which conflates epistemic and aleatoric components, that may lead to suboptimal sensor selection in ambiguous regions. To address this, we propose expected reduction in epistemic uncertainty as a new acquisition function for sensor placement. To enable this, we extend ConvCNPs with a Mixture Density Networks (MDNs) output head for epistemic uncertainty estimation. Preliminary results suggest that epistemic uncertainty driven sensor placement more effectively reduces model error than approaches based on overall uncertainty.

Figures (4)

• Figure 1: Model test error RMSE (left) and negative log-likelihood (right) as a function of the number of placed sensors. Acquisition values are computed for the test date of 1 January 2022, and the resulting sensor placements are evaluated on the same day. The random placement results are averaged over three different seeds.
• Figure 2: Top row: Sample training tasks for the two synthetic data distributions. Bottom row: Corresponding uncertainty estimates produced by the predictive model. (a) Training tasks for the noisy scenario; (b) Training tasks for the multiple-function scenario; (c) Aleatoric uncertainty estimated by the model $p_\theta(y \mathop{\mathrm{|}}\nolimits x, \mathcal{C})$ trained on noisy data; (d) Epistemic uncertainty estimated by the model $p_\theta(y \mathop{\mathrm{|}}\nolimits x, \mathcal{C})$ trained on multiple functions. In the bottom row, ground-truth target values $y$ are shown as blue dots, and the predictive mean $\hat{\mu}$ for target locations $x$ is depicted as a green line for a single test task.
• Figure 3: Greedy sensor placement using baseline acquisition function $\alpha_{\Delta_\mathrm{Var}}$ for 10 sensors in the Western Baltic Sea region. The leftmost panel displays the true Sea Surface Temperature$y$ on January 1, 2022 while the rightmost panel shows acquisition values. The displayed quantities $\hat{\mu}$, $|y-\hat{\mu}|$, and ${\hat{\sigma}_\mathrm{Var}}^2$ represent the predicted mean, absolute error, and total predictive variance, respectively.
• Figure 4: Greedy sensor placement using proposed acquisition function $\alpha_{\Delta_\mathrm{Ep}}$ for 10 sensors in the Western Baltic Sea region. The leftmost panel displays the true Sea Surface Temperature$y$ on January 1, 2022 while the rightmost panel shows acquisition values. The displayed quantities $\hat{\mu}$, $|y-\hat{\mu}|$, and ${\hat{\sigma}_\mathrm{Var}}^2$ represent the predicted mean, absolute error, and total predictive variance, respectively.