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Variable Dimension IMM and GPB2 Filters for Tracking in Multiple Model Systems

Roberto Pérez-Pérez, Ángel F. García-Fernández

arXiv:2609.31416v1eess.SP

TL;DR

Multiple-model filtering must handle modes whose state vectors have different dimensions, which makes standard IMM mixing's sums of differently sized means and covariances undefined. The paper develops a Bayesian formulation and KLD-based VD-IMM and VD-GPB2 filters. These filters outperform standard IMM algorithms adapted to unequal dimensions, especially at low process noise, with higher computational burden.

  • Problem

    Multiple-model filtering with unequal state dimensions is challenging because standard IMM mixing requires sums of means and covariances of different sizes, which are not defined mathematical operations.

    Source: §I · p. 1

  • Method

    The paper formulates variable-dimension filtering in a Bayesian setting and derives VD-IMM and VD-GPB2 filters that use KLD minimisation to retain a Gaussian state density for each mode.

    Source: §I · p. 2, §VIII · p. 13

  • Results

    The resulting filters outperform standard IMM algorithms adapted for different state dimensions, especially at low process noise, at the expense of higher computational burden.

    Source: §VIII · p. 13

  • Takeaways & Limitations

    The proposed filters handle heterogeneous motion models of different dimensionality without external modifications or tuning parameters.

    Source: §I · p. 2

    The standard GPB1 filter's single Gaussian representation across modes is not feasible in principled variable-dimension multiple-model filtering.

    Source: §V · p. 7

Abstract

from arXiv · show

This paper presents mathematically principled filters for multiple-model systems with states of different dimensionality, specifically the variable-dimension interacting multiple model (VD-IMM) filter and the variable-dimension generalised pseudo-Bayesian filter of order 2 (VD-GPB2). To do so, we first provide a Bayesian modelling of a variable dimensional dynamic system, and its measurements. Then, for variable dimensional linear Gaussian dynamic and measurement models, the VD-IMM filter is derived by assuming a posterior that has a Gaussian density for each mode, and then performing a Kullback-Leibler Divergence (KLD) minimisation after each prediction step to keep the Gaussian density form for each mode. Subsequently, the Bayesian update step is performed, keeping a Gaussian density for each mode. The VD-GPB2 filter considers the same model as the VD-IMM filter and also assumes that the posterior for each mode is Gaussian. In contrast, the VD-GPB2 filter propagates a Gaussian mixture for each mode in the prediction step. In the update step, the VD-GPB2 filter performs a KLD minimisation to have a Gaussian density for each mode. Simulation results show the benefits of the VD-IMM and VD-GPB2 filters compared to previous alternatives.

I. INTRODUCTION

The paper addresses multiple-model filtering when motion modes have different state dimensions, a setting where standard IMM mixing requires undefined sums of differently sized means and covariances. It formulates the problem in a variable-dimension Bayesian framework and derives principled filters using KLD minimisation.

I. INTRODUCTION

  • Unequal state dimensions arise because motion parameters may be relevant in some modes but not others.The filtering objective is to compute the state density given all past observations when the state includes both mode and kinematic state.Source: §I · p. 1
  • Earlier VD-MM approaches adapted fixed-dimensional filters, while unequal-dimension IMM mixing may require augmentation or reduced-subspace methods with additional bias or tuning issues.Zero-padding can bias additional components, and reduced-subspace interpolation introduces tuning parameters.Source: §I · pp. 1–2
  • The proposed VD-MM filters avoid sums of differently sized means and covariances, external tuning parameters, and modifications for heterogeneous motion models.The paper develops a Bayesian formulation and derives VD-IMM and VD-GPB2 filters using KLD minimisation.Source: §I · p. 2

A. Models

  • The variable-dimension state space is a disjoint union of mode-specific spaces, and the state transition density depends on both the previous and current modes.That dependence accounts for transitions between modes with different state dimensions.Source: A. Models · p. 2

B. Variable Dimension Multiple Mode Bayesian Filtering

  • The paper derives a Bayesian prediction and update recursion, then seeks computationally efficient Gaussian approximations for variable-dimension linear-Gaussian models.Its VD-IMM applies KLD minimisation at prediction, while VD-GPB2 applies it at update.Source: §I · p. 2, B. Variable Dimension Multiple Mode… · p. 3

III. VARIABLE DIMENSION IMM FILTER

The VD-IMM uses linear-Gaussian transition and measurement models sized for mode changes. It approximates each predicted mode-conditional Gaussian mixture with a Gaussian by KLD minimisation, then applies a Bayesian update that preserves the per-mode Gaussian form.

A. Dynamic and Measurement Models

  • The VD-IMM uses linear-Gaussian transition and measurement models whose parameters account for changes in state dimension.The transition model is conditioned on the current and previous modes, and the measurement model is linear Gaussian.Source: A. Dynamic and Measurement Models · p. 3

B. Prediction

  • At prediction, the VD-IMM minimises KLD on the variable-dimension state space to approximate each mode's Gaussian-mixture state density with a Gaussian.The true predicted density is a Gaussian mixture for each mode; the filter retains a Gaussian density conditioned on each mode.Source: B. Prediction · p. 3, Figure 1 · p. 4

C. Update

  • Given a Gaussian state density conditioned on each mode, the VD-IMM Bayesian update produces an updated density of the same Gaussian form.The update is closed-form under the stated assumption.Source: C. Update · p. 4

D. Mode and State Estimation

The VD-IMM estimates the most probable mode and uses that mode's posterior mean as the state estimate. The VD-GPB2 uses the VD-IMM dynamic and measurement models and propagates a Gaussian mixture for each mode during prediction.

D. Mode and State Estimation

  • The VD-IMM selects the mode with the highest posterior probability and estimates the state using the posterior mean conditioned on that mode.The algorithm then uses this estimate as its input for the next prediction step.Source: D. Mode and State Estimation · p. 4

A. Dynamic and Measurement Models

B. Prediction

  • With a Gaussian posterior state density for each mode, the VD-GPB2 prediction produces a Gaussian mixture state density for each mode.The predicted mode density is given by the Bayesian mode prediction.Source: B. Prediction · p. 5

C. Update

VD-GPB2 approximates each mode’s updated Gaussian mixture by a Gaussian through KLD minimisation. The discussion contrasts principled variable-dimension modeling with fixed-dimension methods that require additional handling of mode-dependent state dimensions.

C. Update

  • The VD-GPB2 update applies KLD minimisation on space X to convert each mode’s Bayesian-updated Gaussian mixture into a Gaussian state density.The predicted state density and the Bayesian-updated density are both Gaussian mixtures for each mode.Source: C. Update · p. 5

D. Mode and State Estimation

  • VD-GPB2 uses the same mode and state estimation as VD-IMM, with its filtering recursion also specified in Algorithm 2.The figure shows the updated Gaussian density becoming the input to the next prediction.Source: D. Mode and State Estimation · p. 5, Figure 2 · p. 6

V. DISCUSSION: FIXED VERSUS VARIABLE DIMENSION FILTERS

  • VD filters require transition matrices and covariances to depend on both current and previous modes, which also accommodates changes in state representation such as Cartesian-to-polar coordinates.Using only current-mode matrices can simplify computations, but does not provide principled variable-dimension filtering.Source: §V · pp. 6–7
  • Standard IMM-GPB2 transition models depending only on the current mode omit transitions between different state dimensions, whereas the VD filters handle them with KLD minimisation on the variable-dimensional state.The two-mode example uses 1-D and 2-D states; its standard transition model is incomplete and requires external procedures to address dimensionality inconsistencies.Source: §V · p. 6, Figure 3 · p. 6
  • For mode changes, VD-filter dynamics must account for the states’ definitions, physics, and geometry; equal-dimensional modes with different representations also require previous-mode dependence.When the mode does not change, the dynamic model can be selected similarly to a fixed-dimensional system using relevant physical properties.Source: §V · p. 7

VI. EXTENSION TO NONLINEAR MODELS

The paper extends VD-IMM and VD-GPB2 to nonlinear models using EKF approximations. The extensions apply the linear-model filters after first-order Taylor approximations of the nonlinear dynamics and measurements.

VI. EXTENSION TO NONLINEAR MODELS

  • The nonlinear-model extensions use first-order Taylor approximations of the dynamic and measurement functions before applying the linear-model VD-IMM and VD-GPB2 filters.The section presents an EKF implementation for each filter.Source: §VI · p. 7

A. Variable Dimensional Nonlinear Models

  • The nonlinear transition function maps from the previous mode’s state dimension to the current mode’s dimension, with process-noise covariance sized for the current mode.The measurement function maps the current mode’s state to the measurement space, and its noise covariance is measurement-sized; the EKF assumes both functions are differentiable.Source: A. Variable Dimensional Nonlinear Models · p. 7

B. EKF VD-IMM Filter

  • The EKF VD-IMM approximates the nonlinear dynamics and measurements, then applies the VD-IMM prediction and update theorems.Its prediction uses the transition Jacobian and offset; its update uses the measurement Jacobian and offset.Source: B. EKF VD-IMM Filter · p. 7

C. EKF VD-GPB2 Filter

  • The EKF VD-GPB2 applies the nonlinear-dynamics approximation in prediction and uses mode-pair-specific approximations for its update before applying Theorem 6.The update approximation is made for each current and previous mode pair.Source: C. EKF VD-GPB2 Filter · p. 7

VII. SIMULATIONS

The simulations evaluate VD-IMM and VD-GPB2 against existing filters for unequal-size states across two scenarios. Scenario 1 uses CV and coordinated-turn models whose states differ by the turn-rate variable.

VII. SIMULATIONS

  • The evaluation compares VD-IMM and VD-GPB2 with state-of-the-art IMM filters for unequal-size states in common-variable and heterogeneous-motion scenarios.Scenario 1 has common variables across modes with extra variables in some modes; Scenario 2 represents heterogeneous motion models.Source: §VII · p. 8, A. Scenario 1 · p. 8

A. Scenario 1

  • Scenario 1 compares nearly constant velocity and coordinated-turn modes, with turn rate as an additional state in the coordinated-turn model.The selected CV, right-turn, and left-turn models differ in their offset parameter.Source: 1) Dynamic Model · p. 8

1) Dynamic Model:

  • On CV-to-turn transitions, the state dimension increases and the offset initializes angular velocity; on turn-to-CV transitions, the dimension decreases and angular-velocity influence is removed.The state vector contains position and linear velocity for CV, with angular velocity added for coordinated turn.Source: 1) Dynamic Model · p. 8

2) Mode Dynamic Model:

The simulations compare variable-dimension filters across mode, measurement, and noise settings. Across the reported conditions, VD-IMM and VD-GPB2 outperform prior alternatives, with higher computational times in the reported implementations.

3) Measurement Model:

  • The sensor measures target position in the Cartesian x and y coordinates with noisy two-dimensional measurements.Source: 3) Measurement Model · p. 9

4) Parameters:

5) Results:

  • At the CV-to-CTR transition, VD-IMM and VD-GPB2 capture the model change, while standard IMM, GPB2, and IMM-unbiased diverge and other IMM variants do not fully recover.VD-GPB2 has the best overall performance in this simulation.Source: 5) Results · p. 10, Figure 5 · p. 10
  • When process noise is high (γ > 1), filtering methods tend toward similar RMSE; under low process noise, VD-IMM and VD-GPB2 outperform previous approaches.At T = 1 s, RMSE differences increase, while the proposed filters maintain low RMSE.Source: 5) Results · pp. 10–11
  • Across Figures 6 and 7, VD-IMM and VD-GPB2 achieve lower RMSE than IMM variants and standard GPB2 across the tested process noise, measurement noise, and sampling rates.The paper reports improved performance without tuning or augmentation strategies.Source: Figure 6 · p. 10, 5) Results · p. 11, Figure 7 · p. 11
  • The proposed VD-IMM and VD-GPB2 have higher computational times than the classical IMM, IMM-unbiased, and IMM-uniform implementations.The paper attributes the added cost to variable-dimension mixing.Source: 5) Results · p. 11, Table I · p. 11

B. Scenario 2

Scenario 2 evaluates EKF implementations for a variable-dimension model pair in nonlinear settings. Its dynamic models use different coordinate representations and dimensions, with separate modes for right and left turns.

B. Scenario 2

  • Scenario 2 evaluates EKF VD-IMM and EKF VD-GPB2 against an EKF IMM with state augmentation for heterogeneous motion models.Source: B. Scenario 2 · p. 11

1) Dynamic Model:

  • The model pair combines a 2D CV model in polar coordinates with a 2D CT model in Cartesian coordinates, using state vectors of unequal dimensions.The simulations use CV-PC, CTR-CC, and CTL-CC modes.Source: 1) Dynamic Model · pp. 11–12
  • At mode changes, the nonlinear transition function accounts for kinematic and geometric considerations so the state change remains consistent.Source: 1) Dynamic Model · p. 12

2) Mode Dynamic Model:

The section reports that the proposed variable-dimension filters outperform an adapted IMM baseline, with higher running times. It concludes that the filters provide a principled Bayesian approach, with future extensions left open.

2) Mode Dynamic Model:

3) Measurement Model:

4) Parameters:

  • The stated simulation parameters are T = 1 s, ω∗ = 15 × π/180 rad/s, σ′CV = 1 m/s2, and σ′CT = 1 × π/180 rad/s1/2.Source: 4) Parameters · p. 12

5) Results:

  • The proposed EKF VD-GPB2 and EKF VD-IMM outperform the augmented EKF IMM-unbiased filter from the start; VD-GPB2 performs best, including across mode transitions.Figure 8 plots RMSE against time; VD-IMM also has lower error than the EKF IMM-unbiased filter.Source: 5) Results · pp. 12–13
  • The EKF VD-IMM and EKF VD-GPB2 implementations take longer than EKF IMM-unbiased in Scenario 2, attributed to variable-dimension mixing.Table II reports running times in milliseconds, but the supplied passage gives no values.Source: 5) Results · p. 13, Table II · p. 13

VIII. CONCLUSIONS

  • The proposed filters are derived for variable-dimension multiple-model filtering through a Bayesian formulation and KLD minimisation to obtain a Gaussian density for each mode.The conclusion reports better performance than adapted standard IMM algorithms, especially with low process noise, at the expense of higher computation.Source: §VIII · p. 13
  • The authors identify sigma-point implementations, multi-sensor fusion, and multi-target tracking as future extensions.Source: §VIII · p. 13

APPENDIX A

Appendix A formulates Bayesian filtering on the variable-dimension state space and shows how prediction and update yield mode and state-conditioned densities.

APPENDIX A

  • The variable-dimension prediction and update steps resemble fixed-dimension filtering, but their densities are defined over variable-dimensional states, with transition density depending on both current and previous modes.Source: Appendix A · p. 14

A. Integrals on the Variable Dimension Space

  • A density on space X is nonnegative and integrates to one; it can be expressed using the mode marginal and the state conditional on that mode.The conditional density is π(x|r), and π(r) is the marginal density of r.Source: A. Integrals on the Variable Dimension… · p. 14

B. Prediction

  • Prediction applies the Chapman–Kolmogorov equation on X; contributions from previous modes are summed, then marginalized to obtain the predicted mode density and state density given the mode.Source: B. Prediction · p. 14

C. Update

  • The update applies Bayes’ rule to the predicted joint density and measurement likelihood, with the denominator normalizing the posterior.Marginalizing over the state gives the posterior mode density, from which the state density conditional on mode is obtained.Source: C. Update · pp. 14–15

APPENDIX B

Appendix B presents KLD on the variable-dimension space and the minimization result used to derive the VD-IMM prediction and VD-GPB2 update. It also identifies the appendix proving the VD-IMM filtering steps.

A. KLD on space X

  • The KLD on space X decomposes into a KLD between mode probability mass functions and a mode-weighted KLD between state-conditional densities.Source: A. KLD on space X · p. 15

B. KLD Minimisation on Space X

APPENDIX C

  • Appendix C proves the VD-IMM prediction and update steps stated in Theorems 3 and 4.Source: Appendix C · p. 15

A. Prediction

The appendices prove the VD-IMM and VD-GPB2 filtering steps. VD-IMM applies KLD minimization after prediction, while VD-GPB2 carries a mixture through prediction and approximates it after update.

A. Prediction

  • For VD-IMM prediction, each mode starts with a Gaussian posterior; the predicted state density is approximated as Gaussian by KLD minimization.Source: A. Prediction · p. 16

B. Update

  • For VD-IMM update, the Gaussian predicted state density is updated using the Kalman filter, and the mode posterior is obtained from the mode update equation.Source: B. Update · p. 16

APPENDIX D

  • For VD-GPB2 prediction, a Gaussian posterior for each mode yields a predicted state density given the mode, as in the VD-IMM prediction derivation.Source: Appendix D · p. 16, A. Prediction · p. 16
  • For VD-GPB2 update, the predicted state mixture is measurement-updated and then approximated as Gaussian by moment matching; the mode posterior is also updated.Source: B. Update · p. 16
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