<p>The inverse problem of modeling biochemical processes mathematically from measured time course data falls into the category of system identification and parameter estimation. Analyzing the time course data would provide valuable insights into the model structure and dynamics of the biochemical system. Based on the types of biochemical reactions, such as metabolic networks and genetic networks, several modeling frameworks have been proposed, developed and proved effective, including the Michaelis-Menten equation, the Biochemical System Theory (BST), etc. One bottleneck in analyzing the obtained data is the estimation of parameter values within the system model.</p> <p>As most models for molecular biological systems are nonlinear with respect to both parameters and system state variables, estimation of parameters in these models from experimental measurement data is thus a nonlinear estimation problem. In principle, all algorithms for nonlinear optimization can be used to deal with this problem, for example, the Gauss-Newton iteration method and its variants. However, these methods do not take the special structures of biological system models into account. When the number of parameters to be determined increases, it will be challenging and computationally expensive to apply these conventional methods.</p> <p>In this research, several methods are proposed for estimating parameters in two classes of widely used biological system models: the S-system model and the linear fractional model (LFM), by utilizing their structure specialties. For the S-system, two estimation methods are designed. 1) Based on the two-term structure (production and degradation) of the model, an alternating iterative least squares method is proposed. 2) A separation nonlinear least squares method is proposed to deal with the partially linear structure of the model. For the LFM, two estimation methods are provided. 1) The separation nonlinear least squares method can also be adopted to treat the partially linear structure of the LFM, and moreover a modified iterative version is included. 2) A special strategy using the separation principle and the weighted least squares method is implemented to turn the cost function into a quadratic form and thus the estimates for parameters can be analytically solved. Simulation results have demonstrated the effectiveness of the proposed methods, which have shown better performance in terms of estimation accuracy and computation time, compared with those conventional nonlinear estimation methods.</p>

<p>The inverse problem of modeling biochemical processes mathematically from measured time course data falls into the category of system identification and parameter estimation. Analyzing the time course data would provide valuable insights into the model structure and dynamics of the biochemical system. Based on the types of biochemical reactions, such as metabolic networks and genetic networks, several modeling frameworks have been proposed, developed and proved effective, including the Michaelis-Menten equation, the Biochemical System Theory (BST), etc. One bottleneck in analyzing the obtained data is the estimation of parameter values within the system model.</p> <p>As most models for molecular biological systems are nonlinear with respect to both parameters and system state variables, estimation of parameters in these models from experimental measurement data is thus a nonlinear estimation problem. In principle, all algorithms for nonlinear optimization can be used to deal with this problem, for example, the Gauss-Newton iteration method and its variants. However, these methods do not take the special structures of biological system models into account. When the number of parameters to be determined increases, it will be challenging and computationally expensive to apply these conventional methods.</p> <p>In this research, several methods are proposed for estimating parameters in two classes of widely used biological system models: the S-system model and the linear fractional model (LFM), by utilizing their structure specialties. For the S-system, two estimation methods are designed. 1) Based on the two-term structure (production and degradation) of the model, an alternating iterative least squares method is proposed. 2) A separation nonlinear least squares method is proposed to deal with the partially linear structure of the model. For the LFM, two estimation methods are provided. 1) The separation nonlinear least squares method can also be adopted to treat the partially linear structure of the LFM, and moreover a modified iterative version is included. 2) A special strategy using the separation principle and the weighted least squares method is implemented to turn the cost function into a quadratic form and thus the estimates for parameters can be analytically solved. Simulation results have demonstrated the effectiveness of the proposed methods, which have shown better performance in terms of estimation accuracy and computation time, compared with those conventional nonlinear estimation methods.</p>

Les modèles linéaires non entiers ont prouvé leur efficacité dans la modélisation de la diffusion thermique pour de faibles variations de température. Cependant, pour de grandes variations de température, les paramètres thermiques dépendent de la température. Par conséquent, la diffusion thermique est régie par un modèle non linéaire non entier. Dans cette thèse, une classe de modèles non linéaires non entiers, basée sur les séries de Volterra non entières, est proposée. Les paramètres non linéaires, tels que les s^n-pôles et l’ordre commensurable, sont estimés au même titre que les coefficients linéaires. Ensuite, le comportement thermique d’un échantillon de fer ARMCO est modélisé pour de grandes variations de température. / Linear fractional differentiation models have proven their efficacy in modeling thermaldiffusive phenomena for small temperature variations. However, for large temperature variations,the thermal parameters are no longer constant but vary along with the temperatureitself. Consequently, the thermal system could be modeled by non linear fractional differentialmodels. Volterra series are first extended to fractional derivatives. Volterra seriesare then used for modeling a non linear thermal system, constituted of an ARMCO iron sample,for large temperature variations.