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Conformalized Kalman Filters for State Estimation with Trustworthy Confidence Regions

Olga Weisman, Nir Shlezinger, Bracha Laufer-Goldshtein

arXiv:2609.27506v1eess.SP

TL;DR

Covariance-derived Kalman-filter confidence regions can be unreliable under nonlinearities, non-Gaussian disturbances, and model mismatch. The paper conformalizes regions built from recursively estimated posterior moments, and reports prescribed coverage with informative regions across diverse dynamic systems.

  • Problem

    Gaussian covariance-based confidence regions can be unreliable under nonlinearities, non-Gaussian disturbances, and model mismatch, despite accurate point estimates.

    Source: §I · p. 1

  • Method

    The framework uses Kalman-type filters’ time-varying posterior moments to construct and conformally calibrate Gaussian, quantile-based, or learned Gaussian-mixture confidence regions.

    Source: Contributions · p. 2, C. Discussion · p. 8

  • Results

    Across diverse dynamic systems, the methods attain prescribed coverage while producing informative regions, with the relative merits of the constructions varying across posterior characteristics.

    Source: B. Scalar-State Results · pp. 9–10, C. Multivariate-State Results · p. 10, §V · p. 11

  • Takeaways & Limitations

    The constructions combine model-based filtering with conformal calibration and offer complementary tradeoffs between computational simplicity and region flexibility.

    Source: §V · p. 11

    The coverage guarantees rely on exchangeability between calibration and test trajectories, which may be violated when operating conditions change.

    Source: C. Discussion · p. 8

Abstract

from arXiv · show

Kalman-type filters are widely used for tracking dynamic systems, yet the confidence regions commonly derived from their estimated covariances can become unreliable under nonlinearities, non-Gaussian disturbances, and model mismatch. In this work, we develop a conformal prediction (CP) framework for equipping Kalman-type filters with statistically reliable confidence regions. Unlike CP applied to black-box estimators, our approach exploits the recursively estimated first- and second-order moments of the state posterior, which capture the time-varying uncertainty induced by the underlying dynamics. Based on these statistical features, we propose three complementary constructions. The first conformally calibrates Gaussian confidence regions induced by the filter moments. The second employs quantile regression to map the estimated moments into the boundaries of adaptive convex regions, which are subsequently calibrated. The third learns a Gaussian-mixture representation of the posterior and conformalizes the resulting density-based regions, enabling the characterization of multimodal and non-convex uncertainty sets. We establish finite- sample coverage guarantees for both sample-wise confidence, which controls miscoverage at individual time instances, and trajectory- wise confidence, which jointly covers the state sequence over a prescribed horizon. Numerical experiments across diverse linear and nonlinear dynamic systems demonstrate that the proposed methods attain the prescribed coverage while producing tight and informative confidence regions, and highlight the relative merits of the three constructions under different posterior characteristics.

I. INTRODUCTION

The paper addresses unreliable covariance-based confidence regions in Kalman-type tracking by using conformal prediction with the filter’s time-varying posterior moments. It proposes Gaussian, convex quantile-based, and mixture-based regions, with coverage guarantees and evaluations across dynamic systems.

I. INTRODUCTION

  • Gaussian covariance-based regions are justified for linear Gaussian models but may be miscalibrated or excessively conservative under nonlinearities, non-Gaussian disturbances, or model mismatch.Accurate point estimates do not ensure reliable uncertainty regions.Source: §I · p. 1
  • Existing multivariate CP methods generally treat the predictor as generic regression, while sequential methods often rely on revealed past targets or do not use filter uncertainty to shape regions.Bayesian CP work cited in the paper addresses different settings rather than recursive estimation of unobserved dynamic states.Source: §I · pp. 1–2
  • The framework uses recursively estimated posterior moments to construct calibrated confidence regions tailored to Kalman-type tracking, rather than treating the filter as a black box.The moments retain time-varying uncertainty information propagated by the filter.Source: Contributions · p. 2
  • CGKF calibrates Gaussian regions, CQKF learns convex boundaries from filter moments, and CDKF calibrates density-based regions from a learned Gaussian mixture.CQKF accommodates asymmetric and non-Gaussian but primarily unimodal posteriors; CDKF can represent multimodal posteriors and potentially non-convex regions.Source: Contributions · p. 2
  • The framework establishes finite-sample guarantees for sample-wise coverage at individual times and trajectory-wise coverage across a trajectory.Trajectory-wise coverage controls the probability that any state along the trajectory falls outside its region.Source: Contributions · p. 2
  • The experiments report prescribed coverage with tight, informative regions across diverse dynamic systems, comparing the relative merits of the three constructions under different posterior characteristics.The paper also motivates trustworthy regions because downstream detection, control, planning, and decision-making depend on uncertainty as well as state estimates.Source: §I · p. 1, Contributions · p. 2

A. State-Space Models

  • State-space models describe the evolution of a latent state and its associated observation at each time step.The paper denotes the state transition and observation functions by f_t and h_t, respectively.Source: A. State-Space Models · p. 2

B. KF-Type Tracking of Dynamic Systems

Kalman-type filters recursively estimate state moments through prediction and update steps. The paper seeks compact confidence regions with either per-time or joint trajectory coverage under nonlinear, non-Gaussian, and potentially mismatched dynamics.

B. KF-Type Tracking of Dynamic Systems

  • Kalman-type filters recursively propagate estimates of the state’s first- and second-order moments using current and past observations.The predict step estimates state and observation moments; the update step balances predicted moments with the current observation using a Kalman gain.Source: B. KF-Type Tracking of Dynamic Systems · p. 3
  • The classical KF is MSE-optimal when the model is linear and Gaussian, while EKF and UKF extend Kalman filtering to nonlinear settings.The UKF uses a deterministic sigma-point transform instead of the EKF’s local linearization.Source: B. KF-Type Tracking of Dynamic Systems · p. 3

C. Problem Formulation

  • The target setting permits nonlinear state and observation functions, non-Gaussian process or observation noise, and mismatch between the estimator’s model and the true dynamics.The mechanism is also intended to work with a broad family of Kalman-type estimators.Source: C. Problem Formulation · p. 3
  • The problem formulation assumes access to multiple trajectories generated by the underlying dynamical system.These trajectories are used in addressing the stated nonlinear, estimator-compatibility, and model-mismatch requirements.Source: C. Problem Formulation · p. 3
  • The objective is to construct confidence regions that are as tight as possible while providing either sample-wise or trajectory-wise coverage.Sample-wise coverage applies independently at each time; trajectory-wise coverage jointly covers the full trajectory with probability at least 1−α.Source: C. Problem Formulation · p. 3

D. Preliminaries: Conformal Prediction

Conformal prediction constructs prediction regions with coverage guarantees using calibration data. The reviewed multivariate quantile-regression approach learns directional limits and calibrates a convex region.

D. Preliminaries: Conformal Prediction

  • Split conformal prediction uses a separate calibration set to construct prediction regions with coverage at least 1−α.The stated guarantee is finite-sample and distribution-free under exchangeability.Source: D. Preliminaries: Conformal Prediction · p. 3

1) Quantile Regression:

  • Directional quantile regression learns interval limits for projections of the multivariate output along selected directions.A dedicated neural network maps an input and direction vector to a directional quantile limit, then calibration scores measure violations across directions.Source: 1) Quantile Regression · p. 3
  • Calibration uses the empirical quantile of nonconformity scores to produce a convex prediction region; in the scalar case with directions ±1, DQR specializes to CQR.The figure contrasts generic DCP and DQR regions with Gaussian and rectangular special cases for m = 2.Source: 1) Quantile Regression · p. 4, Figure 1 · p. 4

2) Distributional CP:

The section treats filter moments as dynamically adapted features rather than a complete posterior model, then combines them with conformal methods to form calibrated confidence regions. Its methods vary in region geometry, flexibility, and data requirements.

2) Distributional CP:

  • Distributional CP constructs prediction regions as calibrated superlevel sets of a learned conditional Gaussian-mixture density.The mixture weights, means, and covariances are predicted from inputs by a dedicated neural network.Source: 2) Distributional CP · p. 4

III. CONFORMALIZED KALMAN FILTER

  • Kalman moments can inform conformal prediction while fixed-task multivariate methods miss evolving state and estimation-error distributions.The moments are informative features, not necessarily a complete probabilistic characterization of the posterior.Source: §III · p. 4
  • Three constructions use Gaussian, learned quantile, or learned distributional regions, with finite-sample sample-wise and trajectory-wise guarantees under exchangeability.They differ in learning and data requirements as well as region shape and flexibility.Source: §III · p. 4, A. Conformalized Kalman Filter Methods · pp. 4–5

1) Common Calibration and Evaluation Procedure:

  • Common calibration computes method-specific nonconformity scores from filter moments and ground-truth states, then maps scores to conformal correction terms.The filter is applied to calibration trajectories before the scores are computed.Source: 1) Common Calibration and Evaluation… · p. 5
  • Sample-wise calibration sorts scores separately at each time, whereas trajectory-wise calibration uses an aggregate score and a shared correction to control any uncovered state in the trajectory.The time-dependent sample-wise correction accounts for changes in state and estimation-error distributions along a trajectory.Source: 1) Common Calibration and Evaluation… · p. 5
  • At evaluation, the filter moments and calibrated correction determine the confidence region; a different filter implementation is permitted if the underlying state-space model is unchanged.Source: 1) Common Calibration and Evaluation… · p. 5

2) CGKF:

CGKF retains a covariance-shaped Gaussian ellipsoid but conformally adjusts its threshold using calibration data, without training an additional model.

2) CGKF:

  • CGKF uses the filter's posterior moments to form an initial Gaussian ellipsoid, whose dimensions and orientation adapt to the covariance.Under a Gaussian posterior, the ellipsoid is the corresponding highest-density region.Source: 2) CGKF · p. 5
  • Its nonconformity score is the signed deviation of the true state from the initial region boundary, distinguishing states outside from those inside.Source: 2) CGKF · p. 5
  • The calibrated threshold expands or contracts the region while its shape and orientation remain set by the propagated covariance.CGKF requires no additional model training, so all labeled trajectories can be used for calibration.Source: 2) CGKF · p. 5

3) CQKF:

CQKF maps Kalman-filter moments to directional quantiles, producing adaptive convex regions that can represent asymmetric uncertainty without imposing a parametric posterior model.

3) CQKF:

  • CQKF feeds filter moments to directional quantile regression, allowing regions to adapt to asymmetric, non-Gaussian posteriors without a parametric distributional model.Unlike CGKF, it splits available data into disjoint training and calibration sets.Source: 3) CQKF · p. 5
  • A learned model maps posterior moments to quantiles over fixed directions, whose half-spaces intersect to form a convex initial region.The training procedure uses directional quantile regression.Source: 3) CQKF · pp. 5–6
  • The conformity score is the largest violation of the learned directional constraints, calibrated using the common sample-wise or trajectory-wise procedure.Source: 3) CQKF · p. 6
  • Conformal correction relaxes or tightens the half-space constraints, preserving convexity while adapting region boundaries to state- and time-dependent filter uncertainty.Source: 3) CQKF · p. 6

4) CDKF:

CDKF learns a conditional density from Kalman-filter moments and conformally calibrates its density-based regions, supporting multimodal and potentially disconnected uncertainty sets.

4) CDKF:

  • CDKF maps filter moments to a learned conditional density that can represent multimodal posteriors and induce non-convex, possibly disconnected regions.Unlike CQKF, it is not restricted to convex confidence regions.Source: 4) CDKF · pp. 6–7
  • A neural network maps filter moments to the weights, means, and covariances of a Gaussian mixture, trained by maximizing empirical log-likelihood.Source: 4) CDKF · pp. 6–7
  • The learned density defines nested candidate regions by density threshold, with geometry able to adapt to asymmetric and multimodal posteriors.Source: 4) CDKF · p. 7
  • CDKF scores calibration states by negative log-likelihood, calibrates those scores, and uses the resulting correction to define the evaluation region.Source: 4) CDKF · p. 7

B. Theoretical Guarantees

The proposed constructions share finite-sample coverage guarantees when calibration and test trajectories are exchangeable. Learning-based guarantees are conditional on the disjoint training set and the resulting fixed parameters.

B. Theoretical Guarantees

  • All three constructions share the same validity guarantees, independent of their method-specific nonconformity scores.Their distinct data requirements and confidence-region geometries do not change those guarantees.Source: B. Theoretical Guarantees · p. 7

1) Exchangeability:

  • Exchangeability is required across trajectories, but imposes neither independence nor stationarity across time within a trajectory.For CQKF and CDKF, results are conditional on a disjoint training set and parameters fixed before calibration and test processing.Source: 1) Exchangeability · p. 7

2) Sample-Wise Calibration:

  • For exchangeable calibration and test trajectories, each proposed method achieves its prescribed sample-wise marginal coverage at a fixed time.When calibration and test scores are almost surely distinct, the finite-sample coverage exceeds the target by at most 1/(|Dcal| + 1).Source: 2) Sample-Wise Calibration · p. 7

3) Trajectory-Wise Calibration:

Trajectory-wise confidence can be obtained by tightening sample-wise thresholds or calibrating aggregated trajectory scores directly. Across the framework, region geometry and computational cost vary by construction, while the guarantees rely on exchangeability and a prescribed horizon.

3) Trajectory-Wise Calibration:

  • The sample-wise route obtains trajectory-wise confidence through a Bonferroni correction, but may be conservative because each region uses confidence level 1 − α/T.It is derived by applying the sample-wise result at the stricter miscoverage level and taking a union bound over T time instances.Source: 3) Trajectory-Wise Calibration · pp. 7–8
  • Direct trajectory-wise calibration controls the probability that any state in a test trajectory falls outside its region using one quantile of trajectory-level scores.Unlike the Bonferroni approach, it does not separately tighten confidence at every time step and has a milder calibration-data requirement.Source: 3) Trajectory-Wise Calibration · p. 8

C. Discussion

  • The framework uses evolving filter moments to shape regions without requiring them to match a Gaussian posterior, and is largely agnostic to the Kalman-type algorithm.CGKF suits roughly Gaussian posterior geometry, CQKF supports asymmetric and non-Gaussian uncertainty, and CDKF supports multimodal and non-convex regions.Source: C. Discussion · p. 8
  • CGKF has the lowest added computational and data complexity but restricted geometry; CQKF and CDKF trade additional learning and inference costs for more expressive regions.CQKF complexity and region resolution rise with the number of directions |U|; CDKF evaluation cost rises with mixture size K and state dimension.Source: C. Discussion · p. 8
  • The guarantees require exchangeability between calibration and test trajectories and apply only to the horizon represented in calibration data.Operating-condition changes may violate exchangeability, while filtering beyond that horizon requires a way to extrapolate or parameterize temporal corrections.Source: C. Discussion · p. 8

IV. NUMERICAL STUDY

The numerical study evaluates coverage, region efficiency, and computational complexity across scalar and multivariate linear and nonlinear state-space settings. Its setup compares conformalized filters with methods that use observations or simpler region constructions.

A. Experimental Setup

  • Scalar experiments cover linear Gaussian dynamics, linear dynamics with Laplacian noise, and nonlinear dynamics with filter model mismatch.The nonlinear case generates states using a sine transition while the filter receives a linear approximation.Source: A. Experimental Setup · p. 9
  • Multivariate experiments include a nonlinear pendulum and a three-dimensional Lorenz attractor used as a chaotic tracking scenario.The Lorenz dynamics are discretized using a fifth-order Taylor expansion.Source: A. Experimental Setup · p. 9
  • Comparisons include CGKF, CQKF, and CDKF, plus Gaussian filter-posterior references and CP baselines that use observations rather than posterior moments.Multivariate comparisons also include Rec, which applies scalar CGKF per coordinate to form axis-aligned rectangles without exploiting cross-coordinate dependencies.Source: A. Experimental Setup · p. 9
  • Learning-based DQR and DCP methods use neural networks, with DQR trained by pinball loss and DCP by negative log-likelihood.DQR uses 128 directions; DCP predicts a mixture of 10 Gaussian components.Source: A. Experimental Setup · p. 9
  • Evaluation reports per-sample and per-trajectory miscoverage, with efficiency measured by scalar interval width or multivariate region volume normalized to Gaussian modeling.Scalar trials use length-100 trajectories and 200 splits; multivariate trials use length-50 trajectories and 20 random splits.Source: A. Experimental Setup · p. 9

B. Scalar-State Results

Across scalar and multivariate tests, conformalized filters restore prescribed coverage, while region compactness and inference cost vary by construction. Directional resolution also trades computation for smaller CQKF regions.

B. Scalar-State Results

  • Conformalized methods attain prescribed sample-wise or trajectory-wise reliability across scalar scenarios, unlike uncalibrated Gaussian intervals that become unreliable under model-assumption violations.Uncalibrated intervals can also vary substantially even in the matched linear Gaussian setting.Source: B. Scalar-State Results · p. 9, Table II · p. 10

C. Multivariate-State Results

  • All three constructions largely restore prescribed sample-wise and trajectory-wise coverage in multivariate scenarios despite their different region geometries.Observation-based DQR and DCP generally require larger regions than posterior-aware CQKF and CDKF; CDKF generally has the smallest learned regions, especially for the Lorenz attractor.Source: C. Multivariate-State Results · p. 10, Table III · p. 11

D. Complexity and Sensitivity

  • Inference latency increases with region flexibility: CGKF adds essentially no learned-model overhead, CQKF is substantially cheaper than CDKF, and CDKF is slowest.CDKF's mixture-density representation leads to considerably higher inference times.Source: D. Complexity and Sensitivity · pp. 10–11, Table IV · p. 11
  • Increasing CQKF's number of directional constraints refines the region boundary and reduces region size toward the ellipsoidal CGKF construction, but requires more quantile evaluations.The sensitivity experiment uses the linear Gaussian model at SNR = −15 dB.Source: D. Complexity and Sensitivity · p. 11

V. CONCLUSION

  • The conclusion presents CGKF, CQKF, and CDKF as complementary tradeoffs between computational simplicity and region flexibility, with finite-sample sample-wise and trajectory-wise guarantees.Experiments across diverse dynamical systems attain prescribed coverage with informative regions.Source: §V · p. 11

APPENDIX

The proofs use exchangeability of calibration and test scores to obtain sample-wise coverage, then extend the argument to full trajectories through maximum nonconformity scores.

A. Proof of Theorem 1

  • At a fixed time, calibration and test scores are exchangeable, so the empirical-quantile lemma yields the sample-wise coverage bound.When scores are almost surely distinct, the lemma also gives its corresponding upper bound.Source: A. Proof of Theorem 1 · p. 13

B. Proof of Corollary 1

C. Proof of Theorem 2

  • Trajectory coverage is equivalent to the maximum per-time nonconformity score falling below the calibrated threshold; exchangeability of trajectory maxima yields the coverage guarantee.Source: C. Proof of Theorem 2 · p. 13
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