Error Analysis of Approximated PCRLBs for Nonlinear Dynamics

Technical Summary: Error Analysis of Approximated PCRLBs for Nonlinear Dynamics

Problem Formulation
In the field of nonlinear filtering, assessing achievable performance is critical for algorithm design, particularly in systems like interacting multiple models (IMM) where weighted estimates are combined. While the Posterior Cramer-Rao Lower Bound (PCRLB) serves as a theoretical benchmark for filter accuracy, its recursive calculation traditionally requires knowledge of the true state trajectory. In practical online applications, the true state is unknown; only state estimates are available. Consequently, there is a need to approximate the PCRLB using these online estimates rather than true values.

The paper addresses the specific challenge of determining the accuracy and difference between two common approximation strategies:

1. Mean-based Approximation: Combining the expectation of the state estimate directly with the exact dynamic model.
2. Mean-Covariance Approximation: Utilizing both the first and second-order moments (expectation and covariance) of the state estimate, incorporating Taylor-series expansions of the dynamics.

The primary motivation is to analytically formulate the difference between these two approximations and determine which provides a closer bound to the true PCRLB.

Methodology
The authors propose a framework based on Gaussian assumptions to derive recursive approximations of the Fisher Information Matrix (FIM), from which the PCRLB is derived as the inverse.

1. Approximated Gaussian Forms (AGF):

   • Type A (Mean-Covariance): The state transition and measurement densities are approximated using a Gaussian distribution characterized by the first two moments of the state estimate. This involves Taylor-series expansions up to the second order to account for the nonlinearity of the functions $f_k$ and $h_k$. The mean and covariance are updated to include correction terms involving the Hessian matrices and the error covariance.
   • Type B (Mean-Only): The state and measurement are approximated using only the first-order moment (the state estimate) and the exact dynamic model, ignoring higher-order covariance effects in the density formulation.

2. Recursive FIM Derivation:
   Using the defined AGFs, the authors derive the components of the FIM ($D_{11}, D_{12}, D_{22}$) for both approximation types.

   • For the Mean-Covariance case, the FIM matrices are decomposed using the matrix inversion lemma. This decomposition separates the terms into a "star" component ($\Sigma^*$), which corresponds to the Mean-Only approximation, and a residual component ($\Sigma$ or $\Pi$) representing the contribution of the covariance and higher-order terms.
   • For the Mean-Only case, the FIM is derived directly from the simplified Gaussian forms, yielding a matrix denoted as $\Theta$.

3. Analytical Difference Formulation:
   By applying the matrix inversion lemma to the recursive PCRLB equations, the authors derive an explicit analytical expression for the difference between the two approximated bounds. Let $J^{-1}_{k+1}$ be the PCRLB from the Mean-Covariance model and $J^{*-1}_{k+1}$ be the PCRLB from the Mean-Only model. The difference is formulated as:
   $$\tilde{J}^{-1}_{k+1} = J^{*-1}_{k+1} - J^{-1}_{k+1} = (\Pi^{-1}\Theta + I)^{-1}\Theta^{-1}$$
   where $\Pi$ and $\Theta$ are derived from the decomposed FIM terms.

Experimental Results
The proposed methods were evaluated using a univariate nonlinear model known for its bimodal likelihood and strong nonlinearity. Two state estimators were employed to generate the necessary state estimates and covariances:

• Unscented Kalman Filter (UKF): A deterministic sampling approach accurate to the third-order term for Gaussian inputs.
• Particle Filter (PF): A Monte Carlo approach using 1000 particles and 100 simulation runs.

Key findings from the simulations include:

• Accuracy Comparison: The PCRLB derived using the Mean-Covariance model (Type A) consistently provided a tighter (lower) bound closer to the true PCRLB compared to the Mean-Only model (Type B).
• Filter Dependency: The PF generally produced more accurate approximations of the PCRLB than the UKF, attributed to the PF's ability to handle the system's strong nonlinearity and bimodality better.
• Convergence: The theoretical gap between the two approximations decreased as the accuracy of the underlying filtering method improved (e.g., by increasing the number of particles in the PF).
• Oscillations: Due to the system's acute nonlinearity, the PCRLB curves exhibited strong oscillations, yet the relative performance ranking between the approximation methods remained consistent.

Significance and Claims
The paper claims to provide a rigorous analytical framework for quantifying the error introduced when approximating the PCRLB using state estimates instead of true states. Specifically:

• It establishes that the approximation utilizing both the mean and covariance (via Taylor expansion) outperforms the simpler mean-only approximation in terms of closeness to the true bound.
• It demonstrates that the theoretical difference between these two bounds can be explicitly calculated, offering a metric to evaluate the quality of the approximation based on the filtering method's performance.
• The study confirms that improving the accuracy of the state estimator (e.g., via better filtering algorithms or increased particle counts) directly improves the accuracy of the estimated PCRLB.

The authors conclude that their approach offers a valid and significant method for assessing filtering performance in practical nonlinear systems where true states are unavailable, providing a tool to determine if imposed performance requirements are realistic.