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Radio Map Construction with Post-Hoc Location Calibration under Quasi-Static Positioning Errors: Joint Estimation, Performance Bounds, and GNSS-Based Evaluation

Koki Kanzaki, Katsuya Suto, Koya Sato

arXiv:2609.11142v1eess.SPcs.NI

TL;DR

Temporally correlated GNSS errors can shift an entire sensing trajectory and leave radio measurements spatially misregistered. The paper jointly estimates trajectory offsets and radio-propagation parameters from RSS, using that spatial structure for post-hoc calibration. The method reduces the MSE gap to ideal GPR and outperforms a KF–RTS baseline on the GNSS-derived error models considered.

  • Problem

    Urban GNSS errors can persist across measurements and coherently displace a trajectory, causing spatial misregistration that more samples along that trajectory do not average out.

    Source: §I · pp. 1–2

  • Method

    The framework jointly estimates sensor-specific quasi-static offsets and radio-propagation parameters from RSS, exploiting path-loss and shadowing structure to calibrate locations for radio-map construction.

    Source: §I · p. 2, §VIII · p. 13

  • Results

    The proposed method's MSE gap was approximately 3.26 dB^2 versus approximately 10 dB^2 for position-error-agnostic and NIGP methods, and it outperformed KF + RTS on the GNSS-derived error models considered.

    Source: §VIII · p. 13

  • Takeaways & Limitations

    RSS measurements can provide spatial information for post-hoc calibration as well as observations for radio-map reconstruction.

    Source: §VIII · p. 13

    The method assumes a constant positioning offset for each sensor and does not explicitly model remaining time-varying positioning errors.

    Source: §VIII · p. 13

Abstract

from arXiv · show

Radio maps enable environment-aware wireless and Internet-of-Things applications and can be constructed from location-tagged received signal strength (RSS) measurements collected by mobile devices. In urban environments, temporally correlated GNSS errors can shift an entire sensing trajectory, causing systematic spatial misregistration that is not mitigated by collecting more measurements. This paper presents a radio-map construction framework that uses the radio measurements themselves to calibrate erroneous location tags after data collection. The dominant positioning error is modeled as a sensor-specific quasi-static offset, which is jointly estimated with radio-propagation parameters in a Gaussian process regression (GPR) framework by exploiting complementary spatial information from distance-dependent path loss and spatially correlated shadowing. We establish lower and upper bounds on the conditional Bayes risk and show that, under a translation-invariant trajectory model, trajectory information alone cannot identify the quasi-static offset, thereby motivating the use of RSS-derived spatial information for calibration. Numerical evaluations across propagation conditions show that the proposed method reduces the mean squared error (MSE) gap from ideal GPR to approximately $3.26\mathrm{dB}^2$, compared with about $10\mathrm{dB}^2$ for position-error-agnostic and noisy-input GPR baselines. Evaluation using positioning-error models derived from smartphone GNSS measurements shows that the proposed method outperforms a KF--RTS trajectory-smoothing baseline despite unmodeled time-varying positioning errors, remaining within approximately $5\mathrm{dB}^2$ of ideal GPR at the median MSE. These results demonstrate that RSS measurements can serve not only as observations for radio-map reconstruction but also as spatial cues for post-hoc calibration of imperfectly geotagged sensing data.

I. INTRODUCTION

Urban GNSS errors can coherently displace measurements along a trajectory, so additional samples do not remove the resulting spatial misregistration. The paper uses RSS spatial structure to jointly calibrate those locations and construct a radio map.

I. INTRODUCTION

  • Temporally correlated GNSS errors can coherently displace a sensing trajectory, and collecting more measurements does not average out the shared offset.Such persistent errors can misregister measurements in the reconstructed radio map.Source: §I · pp. 1–2
  • Pointwise uncertain-input methods address uncertainty in individual locations, whereas a shared trajectory bias coherently shifts measurements and leaves trajectory-level misregistration.Trajectory smoothing can reduce time-varying errors while leaving residual absolute spatial misregistration.Source: §I · p. 2
  • The framework jointly estimates sensor-specific positioning offsets and radio-propagation parameters from RSS, then constructs the map using calibrated measurement locations.It exploits distance-dependent path loss and spatially correlated shadowing as complementary spatial information.Source: §I · p. 2
  • Under a translation-invariant trajectory law, the reported trajectory alone does not update the prior distribution of a trajectory-wise constant offset.The paper also derives lower and upper bounds on conditional Bayes risk and bounds an estimator's excess risk without evaluating the Bayes-optimal estimator.Source: §I · p. 2

II. SYSTEM MODEL

  • Mobile devices report RSS measurements with observed coordinates to a server, which spatially interpolates RSS at unobserved locations while estimating propagation parameters.The system model considers received power from a single base station in a two-dimensional target area.Source: §II · p. 3

III. RADIO MAP CONSTRUCTION WITHOUT ACCOUNTING FOR POSITION ERRORS

The position-error-agnostic baseline treats reported coordinates as the true Gaussian-process inputs. It estimates propagation parameters from the RSS observations and uses the resulting posterior mean to form the radio map.

III. RADIO MAP CONSTRUCTION WITHOUT ACCOUNTING FOR POSITION ERRORS

  • The baseline treats reported measurement locations as exact GP inputs, ignoring positioning errors.It collects the model parameters and estimates them by maximizing the log marginal likelihood.Source: §III · p. 4
  • Its observation model uses a mean vector and covariance matrix evaluated at the reported locations to represent the RSS data.The parameters are estimated with a gradient-based method; the implementation uses Adam.Source: §III · p. 4
  • The baseline predicts RSS at query locations with the posterior mean, and those predictions over the query set form the radio map.The procedure outputs the map and estimated model parameters.Source: §III · pp. 4–5

IV. RADIO MAP CONSTRUCTION AND JOINT ESTIMATION OF PROPAGATION PARAMETERS AND POSITION ERRORS

The proposed construction method replaces exact-location inputs with jointly estimated sensor-specific offsets. Those offsets alter both the distance-dependent RSS mean and the spatial covariance, which the method exploits for estimation.

IV. RADIO MAP CONSTRUCTION AND JOINT ESTIMATION OF PROPAGATION PARAMETERS AND POSITION ERRORS

  • The method relaxes the assumption that reported measurement locations are exact GP inputs by treating sensor-specific positioning offsets as unknowns.The offsets are jointly estimated with propagation parameters.Source: §IV · p. 5
  • Position offsets shift the GP inputs and therefore affect both the distance-dependent mean and the spatial covariance of the RSS observations.Source: §IV · p. 5
  • The proposed method exploits the RSS mean and covariance structures to jointly estimate model parameters and sensor-specific positioning errors.Source: §IV · p. 5

A. Joint Estimation of Propagation Parameters and Position Errors

The joint estimation uses RSS mean and covariance structure to constrain positioning offsets, with a prior regularizing weakly constrained directions. The resulting conditional MAP/maximum-likelihood optimization is generally nonconvex and uses plug-in estimates.

A. Joint Estimation of Propagation Parameters and Position Errors

  • The RSS mean constrains absolute placement through distances to the known transmitter, while covariance primarily constrains relative offsets between sensors.The covariance gradient vanishes for pairs from the same sensor; limited trajectory or inter-sensor direction diversity can make different offsets yield similar RSS means and covariances.Source: A. Joint Estimation of Propagation… · p. 5
  • A prior on positioning errors anchors each sensor-specific offset around its prior mean and regularizes directions weakly constrained by the RSS likelihood.The formulation jointly applies MAP estimation to positioning errors and maximum-likelihood estimation to propagation parameters, conditional on reported locations.Source: A. Joint Estimation of Propagation… · p. 6
  • The objective is generally nonconvex, so the numerical optimizer is not guaranteed to find its global maximum; the predictor uses plug-in estimates rather than marginalizing posterior uncertainty.The implementation obtains a numerical solution using Adam.Source: A. Joint Estimation of Propagation… · p. 6

B. Radio Map Construction

The construction procedure uses estimated sensor positioning errors to form corrected measurement locations, then uses those locations and estimated propagation parameters to predict RSS across query points.

V. PERFORMANCE BOUND ANALYSIS

The performance-bound analysis compares estimator risk with the minimum MSE achievable from observed data. It decomposes risk into irreducible Bayes risk and estimator-dependent excess risk, while treating propagation parameters as fixed.

V. PERFORMANCE BOUND ANALYSIS

  • The analysis bounds deviation from Bayes-optimal performance using lower and upper bounds on conditional Bayes risk and the resulting excess risk.Source: §V · pp. 6–7

A. Bayes Risk and Excess Risk

  • The evaluation conditions on a fixed true propagation parameter vector and uses deterministic, positioning-error-free evaluation points.Source: A. Bayes Risk and Excess Risk · p. 7
  • Conditional Bayes risk is the minimum MSE achievable from observed data, but is generally difficult to evaluate in closed form.Source: A. Bayes Risk and Excess Risk · p. 7
  • Estimator risk decomposes into irreducible Bayes risk and estimator-dependent excess risk, which can include losses from parameter estimation and model mismatch.Source: A. Bayes Risk and Excess Risk · p. 7

B. Oracle Lower Bound

The oracle estimator has access to true measurement locations as well as the observed data, providing a lower bound on conditional Bayes risk.

  • The oracle estimator accesses true measurement locations in addition to observed data, and its risk lower-bounds Bayes risk.Source: B. Oracle Lower Bound · pp. 7–8
  • Given true locations and RSS observations, additionally conditioning on reported locations does not change the conditional distribution of the radio map.Source: B. Oracle Lower Bound · pp. 7–8
  • For fixed propagation parameters and true locations, the radio map and RSS are jointly Gaussian, enabling the oracle posterior covariance to be calculated by Gaussian conditioning.Source: B. Oracle Lower Bound · p. 8
  • The oracle risk averages the conditional MSE over the generative model with the propagation parameters fixed.Source: B. Oracle Lower Bound · p. 8

C. Working-Model Upper Bound and Its Tightness

The working-model analysis treats reported locations as fixed and omits the trajectory-generation density, yielding an upper bound on Bayes risk. It also gives conditions for this bound to be tight and supports bounds on estimator excess risk.

C. Working-Model Upper Bound and Its Tightness

D. Bounds on the Bayes Risk and Excess Risk

VI. NUMERICAL EVALUATION

Numerical evaluations compare radio-map accuracy and excess risk across propagation conditions, then test robustness using positioning-error models derived from smartphone GNSS measurements. The proposed method performs best when RSS observations retain informative spatial structure.

VI. NUMERICAL EVALUATION

  • The numerical evaluation uses 1000 trials under a reference propagation condition, varying propagation parameters and comparing the proposed method with Ideal GPR, NIGP, and position-error-agnostic methods.MSE is evaluated on a 50×50 grid over the central 150 m × 150 m region.Source: §VI · p. 9

A. Radio Map Accuracy Performance

  • Proposed GPR’s MSE is approximately 3.26 dB^2 above Ideal GPR, versus approximately 10.0 dB^2 for Without Correction and NIGP and 9.47 dB^2 for Oracle Tuned Without Correction.Oracle Tuned Without Correction uses the true radio map and is included only as a data-leaking reference.Source: A. Radio Map Accuracy Performance · pp. 9–10, Figure 4 · p. 10
  • At small η, covariance cues complement weaker distance-dependent mean information; increasing dcor adds informative inter-sensor pairs and improves all methods.The proposed method’s performance is relatively insensitive to η, while the wider correlation range provides more pairwise constraints on relative offsets.Source: A. Radio Map Accuracy Performance · p. 10, Figure 5 · p. 11
  • Increasing σf raises MSE even for Ideal GPR, while increasing σp weakens calibration cues and nearly eliminates the proposed method’s advantage at σp = 6 dB.Larger shadowing amplitude also substantially degrades methods that do not explicitly correct locations.Source: A. Radio Map Accuracy Performance · p. 10, Figure 5 · p. 11

B. Deviation from the Bayes-Optimal Performance

VII. EVALUATION BASED ON MEASURED GNSS ERROR MODELS

  • GNSS-based robustness tests use AR(2) positioning-error models derived from two smartphones’ measured errors and compare the proposed method with KF–RTS smoothing.The RSS observations and true trajectories remain simulated; measured GNSS data construct the positioning-error models.Source: §VII · p. 11

A. GNSS-Derived Error Model and Evaluation Setup

The evaluation models GNSS error as a sensor-specific quasi-static offset plus a time-varying residual, then tests radio-map calibration under that mismatch. The proposed method improves on trajectory-only smoothing and other non-ideal baselines by using spatial information in RSS.

A. GNSS-Derived Error Model and Evaluation Setup

  • The evaluation combines a sensor-specific quasi-static offset with an AR(2) time-varying residual, using 20 s models matched to measured GNSS autocovariances.The fitted AR(2) models approximate the measured autocorrelation, while the time-varying residual is deliberately unmodeled by the proposed method.Source: A. GNSS-Derived Error Model and… · pp. 11–12

B. Radio Map Estimation Performance under Temporally Correlated Positioning Errors

  • At the median, the proposed method's MSE is approximately 5 dB^2 above Ideal GPR, and it outperforms KF + RTS Smoother and Without Correction under both devices' GNSS error models.KF + RTS Smoother performs close to Without Correction because the trajectory likelihood cannot distinguish a constant trajectory translation from its corresponding bias.Source: Figure 7 · p. 12, B. Radio Map Estimation Performance… · p. 12, §VIII · p. 13

VIII. CONCLUSION

  • Under the reference condition, the proposed method's MSE gap from Ideal GPR is approximately 3.26 dB^2, versus approximately 10 dB^2 for position-error-agnostic and NIGP methods.The conditional-MMSE analysis also found the excess-risk fraction approximately 24–27 percentage points lower than for the position-error-agnostic baseline.Source: §VIII · p. 13
  • The conclusion identifies road networks, walkable regions, and known anchors as possible absolute-location constraints for breaking translation symmetry.These are proposed as future additions to the calibration objective.Source: §VIII · p. 13

APPENDIX A DETAILS OF THE GNSS ERROR MODEL AND ITS 20 S

The appendix documents smartphone GNSS measurements and evaluates how well identified AR(2) models capture their temporal error correlation. The models approximate autocorrelation strength and decay, but not the full measured error distributions.

APPENDIX A DETAILS OF THE GNSS ERROR MODEL AND ITS 20 S

  • The GNSS error-model appendix describes the measurement procedure, validates AR(2) models fitted to measured errors, and explains their 20 s evaluation versions.The appendix uses these models to represent temporal correlation in the later numerical evaluation.Source: Appendix A · p. 13

A. GNSS Measurements and AR(2) Model Validation

B. Construction of the 20 s AR(2) Model

The evaluation converts measured 1 Hz AR(2) error models to approximate 20 s models by matching selected stationary autocovariances. It also specifies a KF–RTS baseline whose trajectory likelihood alone cannot resolve a constant position offset from the corresponding bias.

B. Construction of the 20 s AR(2) Model

  • The approximate 20 s AR(2) model matches stationary autocovariances at lags 0, 20, and 40 s, with its coefficients obtained from Yule–Walker equations.This is a moment-matching approximation, not an assumption that exact 20 s subsampling preserves an AR(2) process.Source: B. Construction of the 20 s AR(2) Model · p. 14
  • The KF–RTS baseline smooths the GNSS trajectory with a state-space model for position, velocity, constant bias, and AR(2) residual, then uses smoothed positions as GPR training locations.The smoother uses GNSS trajectories only; RSS enters in the subsequent GPR stage.Source: Appendix B · pp. 14–15
  • The GNSS trajectory likelihood is invariant to jointly translating a sensor's trajectory and offsetting its constant bias, so temporal smoothing alone cannot resolve that ambiguity.Only absolute-position and bias priors provide information about the translation.Source: Appendix B · p. 15
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