A linear mixed-effects model for longitudinal degradation data with several experimental levels

Schmitz, Ralph Christian; Kamps, Udo (Thesis advisor); Kateri, Maria (Thesis advisor)

Aachen : RWTH Aachen University (2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2023


In the analysis of reliability of an object, the measurement of degradation is an approach to quantify its performance. Such an approach leads to longitudinal data for which a suitable modeling framework has to be found in order to apply inferential methods for gaining insights from the data set. The obtained degradation path can be modeled by linear mixed-effects models (LMM) that were introduced in the literature as an extension of ordinary linear models by the inclusion of individual-specific random effects (RE). An exemplary field of application is the analysis of the capacity decline of lithium-ion batteries which will be considered repeatedly in the sequel. In this thesis, a specific LMM is introduced that can be fitted to data sets generated by different experimental conditions that allows for the description of different occurring aging effects. The REs are utilized for identifying trajectories of single individuals. At first, the model specification with focus on a real battery calendar aging data set at different experimental conditions described by the state of charge (SoC) is addressed. In this regard two separate parts of the model, the nonrandom fixed structure and the included REs are successively determined. Subsequently, inferential procedures in terms of prediction methods are investigated. An analytical form of point prediction methods based on the determined model is treated, using best linear unbiased predictors for REs. In this approach, the effect of different numbers of observations for specifying the REs of the predicted individual is analyzed, providing knowledge about a necessary data base for a real application of the model. Additionally, prediction intervals for the considered model are examined. Several methods are compared and evaluated. With regard to an application, the plug-in method of including estimated covariance parameters is also considered. An additional inferential focus is added by investigating the failure time distribution as direct connection to an object's reliability. Necessary constraints on the model parameters are identified and the cumulative distribution function is determined for two RE structures. The adherence to the mentioned constraints is guaranteed by adjusting an inequality constrained generalized least squares (ICGLS) estimation method and its application is analyzed in terms of bias and goodness-of-fit. Using the ICGLS correction in the parameter estimation of the model, the failure time distribution is fitted to the data set and compared to the lognormal distribution for data sets augmented by additional simulated values. In order to extend the model such that knowledge about the aging behavior at unobserved experimental conditions can be achieved, a link function is determined alongside a suitable procedure of fitting the corresponding overall model in three steps. The previously introduced inferential procedures are adjusted for this extended model. With regard to prediction methods, the ICGLS estimation is assessed additionally, since it offers a realistic physical aging behavior of objects beyond the observation range. As for the prediction methods, the failure time distribution results are applied to the fitted model considering the benefit of an enabled extrapolation to untested SoC levels. For the latter case, the importance of including predicted REs for accurate results is addressed.


  • Department of Mathematics [110000]
  • Chair of Statistics [116410]