Gaussian Bayesian Network EKF

• GBN-EKF is a non-linear state estimation approach that integrates Gaussian Bayesian Networks with the Extended Kalman Filter to address stiffness and ill-conditioned measurements.
• Its methodology replaces global matrix inversions with local scalar operations, significantly improving numerical stability for stiff dynamical models.
• Comparative analysis shows that GBN-EKF outperforms classical EKF, UKF, and CKF by reducing RMSE and effectively managing near-singular measurement covariances.

The Gaussian Bayesian Network-based Extended Kalman Filter (GBN-EKF) is a non-linear state estimation methodology for continuous–discrete stochastic systems characterized by stiffness and ill-conditioned measurements. GBN-EKF advances the Extended Kalman Filter (EKF) through Gaussian Bayesian Network (GBN) formalism, enabling robust recursive state estimation without matrix inversion during measurement update steps. This yields enhanced stability for stiff dynamical models and singular or near-singular measurement covariances. The approach is presented and analyzed in the context of systems where traditional Cubature (CKF) and Unscented (UKF) Kalman Filters are numerically destabilized, particularly under ill-conditioned measurement scenarios (Behera et al., 4 Nov 2025).

1. Problem Statement and System Formulation

The systems of interest satisfy: $dx(t) = f(t, x(t))\,dt + G(t)\,dw(t),\quad t > 0$

$z_k = h(x_k) + v_k,\quad k = 1,2,\ldots$

where:

• $x \in \mathbb{R}^n$ is the state,
• $w$ is Brownian motion with covariance $Q(t)$,
• $z_k \in \mathbb{R}^m$ are measurements at discrete times $t_k$,
• $v_k \sim \mathcal{N}(0, R_k)$ is Gaussian measurement noise.

Stiffness corresponds to the Jacobian $\partial f/\partial x$ having large positive eigenvalues, yielding rapid mode dynamics. Ill-conditioned measurements manifest when $H_k P_{k|k-1} H_k^T + R_k$ approaches singularity, compromising the numerical stability of classical filtering updates. The goal is to estimate state trajectories $x(t)$ using EKF principles restructured as a GBN, explicitly eliminating matrix inversion steps.

2. Probabilistic Structure and Gaussian Bayesian Network Representation

The prediction and correction steps are framed through joint Gaussian distributions: $$[x_k, z_k]^T \sim \mathcal{N}\left(\mu_{\text{aug}}, \Sigma_{\text{aug}}\right)$$ with: $$\mu_{\text{aug}} = \begin{bmatrix} x_{k|k-1} \ h(x_{k|k-1}) \end{bmatrix},\quad \Sigma_{\text{aug}} = \begin{bmatrix} P_{k|k-1} & P_{k|k-1} H_k^T \ H_k P_{k|k-1} & H_k P_{k|k-1} H_k^T + R_k \end{bmatrix}$$ The joint covariance is decomposed via a recursive regression structure: $$x_j = \sum_{i<j} B_{ij} x_i + \varepsilon_j, \quad \varepsilon_j \sim \mathcal{N}(0, v_j)$$ where parameters $(B,V)$ are obtained through: $$B_{kj} = [P_{1:(j-1), 1:(j-1)}]^{-1} \Sigma_{1:(j-1),j},\quad V_j = \Sigma_{jj} - \Sigma_{j,1:(j-1)} B_{1:(j-1), j}$$

Conditioning on measurements ($z_k$) proceeds via arc reversals and local updates, never requiring the inversion of $\Sigma_{\text{aug}}$ or any of its constituent blocks.

3. Filter Recursion and Update Dynamics

3.1 Time Update (Prediction)

State mean and covariance are propagated by the standard matrix differential equations (MDEs): $$\dot{\hat{x}}(t) = f(t, \hat{x}(t)),\quad \hat{x}(t_{k-1}) = \hat{x}_{k-1|k-1}$$

$$\dot{P}(t) = J(t,\hat{x}(t)) P(t) + P(t) J(t,\hat{x}(t))^T + G(t) Q(t) G(t)^T$$

where $J = \partial f/\partial x$ at $\hat{x}(t)$. Integration over $[t_{k-1}, t_k]$ yields the predicted state and covariance.

3.2 Linearization and Measurement Model

At update step $k$, the measurement function is linearized: $$H_k = \left. \frac{\partial h}{\partial x} \right|_{x_{k|k-1}}, \quad h_{k|k-1} = h(x_{k|k-1})$$

3.3 EKF and GBN-EKF Update Contrast

Step	Conventional EKF	GBN-EKF Approach
Update Formula	Involves $S_k^{-1}$ (matrix inversion)	Uses arc reversal on $(B,V)$
Numerical Stability	Sensitive to ill-conditioning	Robust, no inversion
Computation Type	Matrix operations	Scalar multiplies, adds/divides

In EKF:

• $S_k = H_k P_{k|k-1} H_k^T + R_k$
• $K_k = P_{k|k-1} H_k^T S_k^{-1}$
• Update: $$\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k(z_k - h_{k|k-1})$$

In GBN-EKF, the update proceeds in $(B,V)$ space:

Augmentation:

$$\mu_{\text{aug}} = [\hat{x}_{k|k-1}; h_{k|k-1}],\quad B_{\text{aug}} = \begin{bmatrix} B_x & H_k^T \ 0 & 0 \end{bmatrix},\quad V_{\text{aug}} = \operatorname{diag}(V_x, R_k)$$

Conditioning:

Sequential arc reversals for each measurement dimension, for parent $i \to$ child $j$: $$B'_{ji} = \frac{1}{B_{ij}},\quad V'_j = V_j + \frac{V_i}{B_{ij}^2}$$ After full conditioning, recover posterior mean and covariance: $$P = B'^{-T} \operatorname{diag}(V') B'^{-1},\quad \hat{x} = \mu$$ All updates are locally scalar, entirely free of global matrix inversions.

4. Algorithmic Implementation

The following steps summarize the recursive application:

Input: x̂₀|₀, P₀|₀ for k = 1, ..., K do 1) Time-update: integrate prediction MDEs → x̂ₖ|ₖ₋₁, Pₖ|ₖ₋₁ 2) Linearize: compute Hₖ, hₖ|ₖ₋₁ 3) Convert Σₖ|ₖ₋₁ to (B, V) via forward recursions 4) Form augmented (B_aug, V_aug) with Hₖ, Rₖ 5) For each measurement dimension j: a) Reverse arc(s) to condition x on zₖ using updates above 6) Recover (x̂ₖ|ₖ, Pₖ|ₖ) from updated (B', V') end for

Each arithmetic operation is limited to scalar addition, multiplication, and division by regression coefficients, ensuring robustness even for highly ill-conditioned measurement matrices.

5. Numerical Stability and Conditioning Advantages

Matrix inversion in the conventional EKF update step amplifies rounding errors when the innovation covariance $S_k$ is nearly singular. GBN-EKF circumvents this issue by decomposing global updates into $O(n^2)$ local conditioning steps:

• Each local operation involves inverting a scalar regression coefficient $B_{ij}$ and adding variances.
• These distributed updates propagate measurement information gently, preserving positive semi-definiteness of the posterior covariance.
• Even when $R_k$ is nearly singular, propagation of conditioning is stabilized by the architecture of the GBN.

This mechanism demonstrably avoids catastrophic numerical errors and breakdowns associated with stiff, ill-conditioned measurement updates.

6. Comparative Performance Analysis

6.1 Root Mean Squared Error (RMSE) Metric

Average RMSE (ARMSE) over $L$ Monte Carlo runs and $K$ steps: $$\text{ARMSE} = \sqrt{\frac{1}{LK} \sum_{\ell=1}^L \sum_{k=1}^K \|x^{ref,\ell}(t_k) - \hat{x}^{\ell}_{k|k}\|_2^2 }$$

6.2 Empirical Evaluation Summary

• Dahlquist-type SDE ($\mu = -10^4$):
  • Linear regime ($j = 1$): all filters tie.
  • Nonlinear regime ($j = 3$): EKF/GBN $\approx$ UKF/CKF at small $\delta$; CD-UKF/CKF slightly better until large stiffness.
• Van der Pol Oscillator ($\mu = 10^4$), well-conditioned:
  • CD-EKF and CD-GBN-EKF ARMSE $\approx 0.02$ at $\delta \leq 0.4$.
  • UKF/CKF fail for $\delta \geq 0.6$.
• Van der Pol, ill-conditioned $H$ ($\sigma = 10^{-6}$):
  • CD-GBN-EKF ARMSE $\approx 0.05$
  • CD-EKF ARMSE $\approx 0.12$
  • Classical EKF ARMSE increases by factor $\approx 2.5$
  • GBN-EKF remains stable.

6.3 Stability Analysis

• Covariance propagation ($\dot{P} = J P + P J^T$) in stiff regimes induces rapid eigenvalue growth for EKF.
• UKF/CKF incorporate higher-order terms $\partial^2 f\, P$, which are stabilizing when $\partial f/\partial x < 0$ but exacerbate instability for stiff ($\partial f/\partial x > 0$) systems.

7. Graphical Model Architecture and Information Flow

State evolution and measurement updates are represented as a directed acyclic graph (DAG):

xₖ₋₁ → xₖ → h(xₖ) → zₖ ↑ ↑ wₖ vₖ

• $x_{k-1} \to x_k$: process model (dynamics)
• $x_k \to z_k$: measurement arc, linearized by $H_k$
• $w_k \to x_k$: process noise injection
• $v_k \to z_k$: measurement noise injection

Arc reversals during evidence absorption convert $z_k$ to a root node with observed value, distributing the evidence to $x_k$ and recursively updating $(B,V)$. Each measurement dimension undergoes this process independently.

8. Summary and Context

GBN-EKF preserves EKF's first-order accuracy in stiff dynamical regimes while eliminating vulnerable matrix inversion steps during measurement updates. Its core innovation is the use of distributed, scalar local conditioning steps within a Gaussian Bayesian Network architecture, resulting in dramatically improved numerical stability and robust state estimation under ill-conditioned measurement models. Comparative evaluations establish its advantage over classical EKF, UKF, and CKF in scenarios susceptible to numerical breakdown, particularly with near-singular innovation covariances and stiff system dynamics (Behera et al., 4 Nov 2025).