Robust Wrapped Gaussian Process Inference for Noisy Angular Data

Andrew Cooper

Virginia Tech Department of Statistics

Los Alamos National Laboratory

What is this talk about?

• Storing and maintaining nuclear assets is vital for national security.

• Radio-Frequency IDentification (RFID) technology allows researchers to track hundreds of assets simultaneously.

• Noisy signal information makes localizing assets based on RFID information difficult.

• We propose a novel statistical model for RFID localization purposes that is robust to signal noise.

Talk Roadmap

1. Introduce the nuclear materials laboratory setting and our RFID localization strategy with phase angle.

2. Overview experiments performed in an RFID laboratory for studying phase-frequency behavior.

3. Explain our novel wrapped Gaussian process framework for robust phase-frequency modeling.

4. Apply our statistical modeling framework for the purposes of ranging assets in nuclear materials laboratories.

Part 1: RFID technology

• Technology researched in early 1970’s at Los Alamos National Laboratory.

• There has since been widespread adoption of RFID.

• Signals from antennae communicate with “passive” RFID tags attached to objects of interest.

https://wallester.com

https://www.rfidcard.com

Tracking nuclear materials with RFID

• Passive RFID tags are placed on containers and tracked with fixed antennae.

• RFID tells you whether an asset is in the laboratory.

• What researchers really want to know is where that asset is.
  • Knowing the approximate location of tags would significantly save time searching for materials of interest.

Localizing RFID tags

• RFID does not directly share location information, but we can infer location based on available communication data.

• Previous attempts to localize RFID tags leverage the received signal strength indicator (RSSI).

  • RSSI has been shown to be an unreliable metric for localization in indoor environments (Chandrasekaran et al. 2009).

• Multipath propagation causes signals to inferfere with each other and create noise.

https://www.atlasrfidstore.com

RFID phase angle

• Communication between an antenna and a RFID tag elicits a phase angle.

• Phase will change when we change either…

• the signal’s frequency,

• or the distance between the tag and the antenna.

• There is a know relationship between frequency \(f\), phase angle \(\phi\), and distance \(d\):

\[\begin{align*} d &= \dfrac{c}{4\pi}\dfrac{\partial \phi}{\partial f}. \end{align*}\]

Part 2: RFID experiment setup

• Phase-frequency relationship of RFID signals studied in a “mock” lab environment.

• Static antennae aligned with two pathways in lab.

• Cart with RFID-tagged containers moved up and down pathways.

RFID experiment data

• We employed a random design of \(22\) distances to collect phase angles.

• Antennae cycled through \(50\) frequencies between \(902.75\)-\(927.25\) MHz for two minutes.

Part 3: Modeling phase-frequency

• Gaussian processes (GPs) assume a priori \(\phi\mid f \sim \mathcal{N}(\alpha + \beta f, \tau^2\Sigma(f))\).
  • Allows for flexible, non-linear modeling.

• An ordinary GP struggles to capture phase-frequency behavior.

Modeling an angular response

• Angular responses \(\phi\in[0,2\pi)^n\) lie in circular space.

Wrapped modeling approach

• We can think of angular observations as being a “wrapped” version of some real-valued response \(Z\).
  • What if we could “unwrap” phase angle?

• To do this, we must know the corresponding wrapping numbers \(K\in\mathbb{Z}^n\):

\[\begin{align*} Z &= \phi + 2\pi K. \end{align*}\]

Wrapped Gaussian processes

• Wrapped distributional modeling assumes a latent random variable in Euclidean space is wrapped around the unit circle (Mardia and Jupp 2000).

• Suppose \(Z\mid f \sim N_n(\alpha + \beta f,\tau^2\Sigma(f))\) is a latent GP.

• \(\phi = Z \hspace{2mm}\text{mod}\hspace{2mm} 2\pi\) represents a wrapped GP (Jona-Lasinio et al. 2012).

Wrapped GPs for angular regression

• Most approaches leverage \(K\) to reconstruct \(Z\) (Jona-Lasinio et al. 2012)

• However, improper estimation of \(K\) can severely impact inference of \(Z\)

Our wrapped GP approach

• Consider a “de-coupled” approach to wrapped GP inference that jointly estimates \(Z\) and \(K\) (Cooper et al. 2025):

\[\begin{align*} &\phi\mid Z,K = \left(Z - 2\pi K + \epsilon\right)\mod 2\pi\\ &Z\mid f \sim \mathcal{N}_n\left(\alpha + \beta f, \tau^2\Sigma_\theta\left(f\right)\right)\\ &K\mid f \sim p(f)\\ &\epsilon_1,\dots,\epsilon_n \overset{\text{i.i.d.}}{\sim}t_{\nu}\left(0;\sigma^2\right). \end{align*}\]

• How can we infer \(P(Z,K\mid f)\)?
  • We can draw from \(P(Z\mid K, f)\) by leveraging the elliptical slice sampling algorithm (Murray et al. 2010).
  • What about \(P(K\mid Z, f)\)?
    • Jona-Lasinio et al. (2012) use a “reasonable” truncation of \(k_i\).
    • Coles (1998) employ random-walk Metropolis-Hastings.

Wrapping number behavior

• Linear structure induces monotonic wrapping behavior
  • If \(f_i\leq f_j\), then \(k_i\leq k_j\).

• We can infer wrapping numbers by estimating the wrapping \(\color{blue}{\text{locations}}\).

Wrapping number estimation

• We explore the space of wrapping numbers using three types of proposals:

1. Shifting a wrapping location.

2. “Growing” a new wrapping location.

3. “Shrinking” a wrapping location.

Part 4: Wrapped GPs for phase modeling

• We model phase at each distance with our wrapped Gaussian process approach (Cooper et al. 2025).

Phase-frequency slopes

• Shorter tag distances (red) have smaller estimated slopes compared to longer distances (yellow).

Relating phase-frequency to distance

• Wrapped GP properly captures the phase-frequency relationship \(\beta_j\) at distance \(d_j\).

• Additional hierarchical GP can capture relationship between slope and distance.

Summary and future work

• We developed a fully Bayesian frameworks for wrapped GP inference.

• Input partitioning approach improves wrapping number inference for angular setting.
  • What if wrapping behavior isn’t monotonic?
  • How to generalize to higher input dimension (\(d \geq 2\))?

• How to go from tag ranging to localization?

Acknowledgements

• This work is funded by the U.S. National Nuclear Security Agency, NA-191, under the Dynamic Material Control (DYMAC) collaboration.

• Collaborators for this project at Los Alamos National Laboratory (LANL):

  • Justin Strait, Mary Frances Dorn, Brian Weaver (Statistical Sciences)

  • Alessandro Cattaneo (Mechanical and Thermal Engineering)

  • Brendon Parsons (Safeguards Science and Technology)

References

Chandrasekaran, Gayathri, Mesut Ali Ergin, Jie Yang, et al. 2009. “Empirical Evaluation of the Limits on Localization Using Signal Strength.” 2009 6th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks, 1–9.

Coles, Stuart. 1998. “Inference for Circular Distributions and Processes.” Statistics and Computing 8: 105–13.

Cooper, Andrew, Justin Strait, Mary Frances Dorn, Robert B Gramacy, Brendon Parsons, and Alessandro Cattaneo. 2025. “Robust Wrapped Gaussian Process Inference for Noisy Angular Data.” arXiv Preprint arXiv:2512.00277.

Jona-Lasinio, Giovanna, Alan Gelfand, and Mattia Jona-Lasinio. 2012. “Spatial Analysis of Wave Direction Data Using Wrapped Gaussian Processes.” Annals of Applied Statistics.

Mardia, Kanti V, and Peter E Jupp. 2000. Directional Statistics. John Wiley & Sons.

Murray, Iain, Ryan Prescott, Adams David, and J C Mackay. 2010. Elliptical Slice Sampling.