Parameter Inference and Estimability Analysis for Nonlinear Multi-Response Systems by Means of Profiling
Profiling , Inference , Estimability , Nonlinear regression
Determining the inference bounds for parameter estimates in multi-response models is a challenge for research and industry since potential co-dependencies among responses add to the complexity of the problem. Statistical profiling is a valuable method in such situations that provides insight into model nonlinearity and co-dependencies among parameter estimates by studying the behavior of the likelihood function. This thesis focuses on extending the application of profiling for parameter inference and estimability analysis to nonlinear multi-response models with unknown noise covariance terms. Profiling is explored based on Generalized Least Squares (GLS) and Determinant Criterion (DC). One issue is finding the distribution and the corresponding degrees of freedom of the profile likelihood function, which depends on the process of estimation of the noise covariance matrix in multi-response models. A method for estimating this matrix and performing profiling is proposed. This research proves that for multi-response models with m responses, p parameters and n experimental runs, the GLS-based profile likelihood function has a χ_1^2 or an F_(1,nm-p) distribution for cases with known or unknown noise covariance matrices respectively. Similarly, the DC-based profile likelihood is shown to have an F_(1,nm-p) distribution. The GLS-based and DC-based profiling approaches are compared theoretically and in practice, suggesting that the GLS-based method is overall more advantageous for small datasets and a combination of DC for parameter estimation and GLS for profiling generally produces reliable results when certain issues are avoided. Furthermore, application of profiling for parameter estimability analysis is explored and a profile-based exploratory analysis is proposed which studies the GLS-based profile likelihood plots and profile traces to reveal inestimability issues. This method links parameter estimability analysis to parameter inference and brings the two objectives of this thesis together. All the proposed methods are examined by means of several examples, the results of which validate the defined approaches.