Jacob Roll

Identification of piecewise affine systems via mixed-integer programming

This paper addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes and Wiener piecewise affine autoregressive exogenous models, in which the regressor space is partitioned into polyhedra with affine submodels for each polyhedron. 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 the estimation data only seldom switches between the different submodels, we also suggest a way of trading off between optimality and complexity by using a change detection approach.

Nonlinear system identification via direct weight optimization

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.

Systems biology: model based evaluation and comparison of potential explanations for given biological data

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.

Identification of hybrid systems via mixed-integer programming

Addresses the problem of identification of hybrid dynamical systems, by focusing the attention on hinging hyperplanes and Wiener piecewise affine 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.

Model-Based Hypothesis Testing of Key Mechanisms in Initial Phase of Insulin Signaling

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.

A non-asymptotic approach to local modelling

Local models and methods construct function estimates or predictions from observations in a local neighborhood of the point of interest. The bandwidth, i.e., how large the local neighborhood should be, is often determined based on asymptotic analysis. In the paper, an alternative, non-asymptotic approach that minimizes a uniform upper bound on the mean square error for a linear estimate is proposed. It is shown, for the scalar case, that the solution is obtained from a quadratic program, and that it maintains many of the key features of the asymptotic approaches. Moreover, examples show that the proposed approach in some cases is superior to an asymptotically based local linear estimator.

Piecewise linear solution paths with application to direct weight optimization

Recently, pathfollowing algorithms for parametric optimization problems with piecewise linear solution paths have been developed within the field of regularized regression. This paper presents a generalization of these algorithms to a wider class of problems. It is shown that the approach can be applied to the nonparametric system identification method, Direct Weight Optimization (DWO), and be used to enhance the computational efficiency of this method. The most important design parameter in the DWO method is a parameter (@l) controlling the bias-variance trade-off, and the use of parametric optimization with piecewise linear solution paths means that the DWO estimates can be efficiently computed for all values of @l simultaneously. This allows for designing computationally attractive adaptive bandwidth selection algorithms. One such algorithm for DWO is proposed and demonstrated in two examples.

Recent techniques for the identification of piecewise affine and hybrid systems

The problem of piecewise affine identification is addressed by studying four recently proposed techniques for the identification of PWARX/HHARX models, namely a Bayesian procedure, a bounded-error procedure, a clustering-based procedure and a mixed-integer programming procedure. The four techniques are compared on suitably defined one-dimensional examples, which help to highlight the features of the different approaches with respect to classification, noise and tuning parameters. The procedures are also tested on the experimental identification of the electronic component placement process in pick-and-place machines.

The use of nonnegative garrote for order selection of ARX models

Order selection of linear regression models has been thoroughly researched in the statistical community for some time. Different shrinkage methods have been proposed, such as the Ridge and Lasso regression methods. Especially the Lasso regression has won fame because of its ability to set less important parameters exactly to zero. However, these methods do not take dynamical systems into account, where the regressors are ordered via the time lag. To this end, a modified variant of the nonnegative garrote method will be analyzed.

On the Input-Output Representation of Piecewise Affine State Space Models

This paper addresses the conversion of discrete-time piecewise affine (PWA) state space models into input-output form. Necessary and sufficient conditions for the existence of equivalent input-output representations of a given PWA state space model are derived. Connections to the observability properties of PWA models are investigated. Under a technical assumption, it is shown that every finite-time observable PWA model admits an equivalent input-output representation. When an equivalent input-output model exists, a constructive procedure is presented to derive its equations. Several examples illustrate the proposed results.