This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes (HHARX) and Wiener piecewise affine (W-PWARX) autoregressive exogenous models. In particular, we provide algorithms based on mixed-integer linear or quadratic programming which are guaranteed to converge to a global optimum. For the special case where switches occur only seldom in the estimation data, we also suggest a way of trading off between optimality and complexity by using a change detection approach.

Systems biology and its usage of mathematical modeling to analyse biological data is rapidly becoming an established approach to biology. A crucial advantage of this approach is that more information can be extracted from observations of intricate dynamics, which allows nontrivial complex explanations to be evaluated and compared. In this minireview we explain this process, and review some of the most central available analysis tools. The focus is on the evaluation and comparison of given explanations for a given set of experimental data and prior knowledge. Three types of methods are discussed: (a) for evaluation of whether a given model is sufficiently able to describe the given data to be nonrejectable; (b) for evaluation of whether a slightly superior model is significantly better; and (c) for a general evaluation and comparison of the biologically interesting features in a model. The most central methods are reviewed, both in terms of underlying assumptions, including references to more advanced literature for the theoretically oriented reader, and in terms of practical guidelines and examples, for the practically oriented reader. Many of the methods are based upon analysis tools from statistics and engineering, and we emphasize that the systems biology focus on acceptable explanations puts these methods in a nonstandard setting. We highlight some associated future improvements that will be essential for future developments of model based data analysis in biology.

A general framework for estimating nonlinear functions and systems is described and analyzed in this paper. Identification of a system is seen as estimation of a predictor function. The considered predictor function estimate at a particular point is defined to be affine in the observed outputs, and the estimate is defined by the weights in this expression. For each given point, the maximal mean-square error (or an upper bound) of the function estimate over a class of possible true functions is minimized with respect to the weights, which is a convex optimization problem. This gives different types of algorithms depending on the chosen function class. It is shown how the classical linear least squares is obtained as a special case and how unknown-but-bounded disturbances can be handled.Most of the paper deals with the method applied to locally smooth predictor functions. It is shown how this leads to local estimators with a finite bandwidth, meaning that only observations in a neighborhood of the target point will be used in the estimate. The size of this neighborhood (the bandwidth) is automatically computed and reflects the noise level in the data and the smoothness priors.The approach is applied to a number of dynamical systems to illustrate its potential. Keywords 2005-09-13Abstract A general framework for estimating nonlinear functions and systems is described and analyzed in this paper. Identification of a system is seen as estimation of a predictor function. The considered predictor function estimate at a particular point is defined to be affine in the observed outputs, and the estimate is defined by the weights in this expression. For each given point, the maximal mean-square error (or an upper bound) of the function estimate over a class of possible true functions is minimized with respect to the weights, which is a convex optimization problem. This gives different types of algorithms depending on the chosen function class. It is shown how the classical linear least squares is obtained as a special case and how unknown-but-bounded disturbances can be handled.Most of the paper deals with the method applied to locally smooth predictor functions. It is shown how this leads to local estimators with a finite bandwidth, meaning that only observations in a neighborhood of the target point will be used in the estimate. The size of this neighborhood (the bandwidth) is automatically computed and reflects the noise level in the data and the smoothness priors.The approach is applied to a number of dynamical systems to illustrate its potential.

Type 2 diabetes is characterized by insulin resistance of target organs, which is due to impaired insulin signal transduction. The skeleton of signaling mediators that provide for normal insulin action has been established. However, the detailed kinetics, and their mechanistic generation, remain incompletely understood. We measured time-courses in primary human adipocytes for the short-term phosphorylation dynamics of the insulin receptor (IR) and the IR substrate-1 in response to a step increase in insulin concentration. Both proteins exhibited a rapid transient overshoot in tyrosine phosphorylation, reaching maximum within 1 min, followed by an intermediate steady-state level after approximately 10 min. We used model-based hypothesis testing to evaluate three mechanistic explanations for this behavior: (A) phosphorylation and dephosphorylation of IR at the plasma membrane only; (B) the additional possibility for IR endocytosis; (C) the alternative additional possibility of feedback signals to IR from downstream intermediates. We concluded that (A) is not a satisfactory explanation; that (B) may serve as an explanation only if both internalization, dephosphorylation, and subsequent recycling are permitted; and that (C) is acceptable. These mechanistic insights cannot be obtained by mere inspection of the datasets, and they are rejections and thus stronger and more final conclusions than ordinary model predictions.

This paper addresses the conversion of discrete-time single input-single output piecewise affine (PWA) models from state space to input-output form. Necessary and sufficient conditions are given for a PWA state space model to admit equivalent input-output representations. When an equivalent input-output model exists, a constructive procedure is presented to derive both its parameters and the partition of the regressors domain. It is shown that the number of modes and the number of parameters may grow considerably when converting a PWA state space model into an equivalent input-output representation. Numerical examples highlight the role of the derived necessary and sufficient conditions for input-output realization of PWA state space models