📊 statistics

Regularised Iterative Generalised Least Squares with Optimal Selection of the Hyper-Parameter for Identifying Nonlinear Phenomenological Models

This paper introduces a regularised iterative generalised least squares method with an automated, information-theoretic approach to optimally select the ridge regression hyper-parameter, enabling the reliable identification of nonlinear phenomenological models with confounded parameters in the presence of heteroscedastic and serially correlated data.

Original authors: Mark Cary, Charles Bokor

Published 2026-08-20

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Mark Cary, Charles Bokor

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Batteries are the silent workhorses of modern life, powering everything from smartphones to electric vehicles. Yet, like any machine, they age. Over time, they lose their ability to hold a charge and deliver power, a process known as degradation. To keep these systems safe and efficient, engineers need to know exactly how much life is left in a battery, a metric called the state of health. To predict this, scientists build mathematical models that mimic the physical chemistry inside the battery. These models are not just guesses; they are based on real physical laws, describing how heat and time wear down the materials. However, these models come with a catch: they contain many hidden numbers, or parameters, that must be figured out by looking at experimental data. Often, the way these numbers are arranged in the equations makes them impossible to separate from one another. It is as if two different ingredients in a recipe are so perfectly mixed that tasting the final dish cannot tell you how much of each was used. When this happens, the model becomes unreliable, and the predictions fail.

This is the specific puzzle tackled by researchers M. Cary and Charles Bokor. They focused on a common type of battery model where the parameters are multiplied together, a structure that frequently causes this confusion. In their work, they developed a new way to solve these tangled equations without throwing away the physical meaning of the model. Instead of trying to force the data to fit a simplified version of the model, which would lose important details about how the battery ages, they introduced a mathematical "tether." This tether gently pulls the estimated numbers toward a reasonable range, preventing them from wandering into impossible values. They call this technique ridge regression. The brilliance of their approach lies in how they choose the strength of this tether. Too weak, and the numbers remain confused; too strong, and the model becomes too rigid to reflect reality. The researchers created an automated system that constantly adjusts this strength, finding the perfect balance for every step of the calculation.

The team tested their method using computer simulations that mimicked real battery data, including data that was noisy or behaved unpredictably. They found that their new system could quickly and accurately untangle the confused parameters, even when the data was difficult. The method works by iterating, or repeating, a cycle of calculation where it refines the model and then immediately re-evaluates the best setting for the tether. This cycle converges very rapidly, meaning it finds the solution in very few steps. The researchers demonstrated that this process works well for models that account for complex real-world issues, such as data where the error changes over time or where errors are linked to one another. By using a specific measure of how well the model explains the data, their system automatically selects the optimal setting without human intervention.

The result is a robust tool for identifying the true characteristics of battery aging models. The researchers showed that their method preserves the complex, physics-based structure of the model while making the math solvable. This is crucial because simplifying the model to make the math easier often strips away the very details scientists need to understand specific aging mechanisms. Their simulations confirmed that the approach is effective, providing reliable estimates where previous methods might have failed or produced wildly inaccurate results. The work offers a practical path forward for engineers who need to monitor battery health, ensuring that the models used to predict when a battery should be replaced are both physically meaningful and mathematically sound.

Technical Summary: Regularised Iterative Generalised Least Squares with Optimal Selection of the Hyper-Parameter

Problem Statement
The paper addresses the challenge of identifying parameters in nonlinear phenomenological models, specifically within the context of State of Health (SoH) estimation for Lithium-ion batteries. These models often rely on quasi-physical motivations, utilizing multiplicative structures (e.g., power laws combined with Arrhenius-like terms) that frequently result in fully or partially confounded parameters. Such confounding leads to poor numerical conditioning, making parameters difficult or impossible to estimate reliably. While traditional approaches to identifiability often involve simplifying the model structure or re-parameterizing to reduce the number of estimable parameters, this paper argues that such simplifications may conflict with the underlying physical thinking required to rank the significance of specific ageing mechanisms. The core problem is to preserve the desired model formulation while improving numerical conditioning to obtain reliable parameter estimates, even in the presence of heteroscedasticity and serial correlation in the data.

Methodology
The authors propose a Regularised Iterative Generalised Least Squares (RIGLS) framework that incorporates an automated, optimal selection of the ridge regression hyper-parameter ( $\lambda$ ) at each iteration.

Regularisation Framework: To convert the ill-posed identification problem into a well-posed one, the method introduces a ridge regression term ( $\mathcal{R}(\theta) = \theta^T\theta$ ) into the cost function. This constrains the parameters to a compact subset, trading off model fidelity against the constraint severity controlled by $\lambda$ .
Optimal Hyper-Parameter Selection: Unlike methods that rely on Generalised Cross-Validation (GCV) or fixed values, this approach derives fixed-point iteration formulae for $\lambda$ based on two information-theoretic criteria:
- Sugiura's small-sample Akaike Information Criterion ( $AIC_c$ )
- Schwarz's Bayesian Information Criterion ($BIC$)
The authors derive the necessary derivatives of the profile log-likelihood and the effective degrees of freedom with respect to $\lambda$ . These derivations allow for the construction of re-estimation formulae (Equations 19 and 20) where $\lambda$ appears on both sides, necessitating an iterative solution.
Algorithm Implementation: The proposed RIGLS algorithm operates in an iterative loop:
- It estimates model parameters ( $\theta$ ) by minimising a regularised nonlinear functional.
- At each step of the parameter estimation, the hyper-parameter $\lambda$ is re-estimated to convergence using the derived fixed-point iteration scheme.
- It estimates covariance model parameters ( $\delta$ ) to account for heteroscedasticity and serial correlation, updating the weighting matrix $W$ .
- The process repeats until the change in parameter estimates falls below a specified tolerance.
Convergence: The paper establishes a theorem guaranteeing the convergence of the fixed-point iteration for $\lambda$ , provided the derivative of the iteration function is bounded by a constant less than 1. A specific initialization strategy is proposed to select a starting $\lambda$ that satisfies these convergence conditions.

Key Contributions

Novel Re-estimation Formulae: The paper presents new analytical formulae for optimally selecting the ridge regression hyper-parameter based on $AIC_c$ and $BIC$ within a nonlinear, regularised context.
Handling of Complex Error Structures: The methodology extends beyond ordinary least squares to accommodate general covariance structures, allowing for the fitting of data with heteroscedasticity and serial correlation.
Preservation of Model Structure: By using regularisation rather than model simplification, the method retains the original phenomenological structure of the battery SoH model, enabling the estimation of parameters associated with specific physical ageing mechanisms.
Automated Optimization: The integration of the hyper-parameter selection into the iterative generalised least squares loop allows the trade-off between data fidelity and model constraints to be handled automatically during the parameter search.

Results and Validation
The efficacy of the method is demonstrated through a simulation study using the Martinez-Laserna model for battery SoH estimation. This specific model was chosen because its multiplicative structure is known to yield particularly poor identifiability.

The simulation confirms that regularisation is essential for obtaining reasonable parameter estimates in this context.
The analysis demonstrates that the fixed-point iteration for the hyper-parameter converges very rapidly, provided a suitable starting value is chosen.
The results validate that the proposed RIGLS algorithm can successfully fit both the model parameters and the covariance parameters while automatically tuning the regularisation strength.

Significance and Claims
The paper positions its contribution as a solution to the "identifiability vs. physical fidelity" dilemma in phenomenological modelling. It claims that by introducing a regularised iterative generalised least squares mechanism with optimal hyper-parameter selection, it is possible to:

Preserve the complex, multiplicative structure of physical models without resorting to simplification.
Improve the numerical conditioning of the estimation problem to handle confounded parameters.
Automatically balance the trade-off between fitting the data and satisfying the regularisation constraint at every stage of the optimisation.

The authors conclude that the procedure is effective for identifying nonlinear phenomenological models where standard approaches fail due to poor conditioning, specifically citing the application to Lithium-ion battery SoH estimation as a primary use case. The paper does not claim to solve all identifiability issues but rather provides a robust numerical framework to manage them within the context of existing physical models.

More like this