Model Reduction of Synchronized Lur'e Networks

In this talk, we investigate a model order reduction schemethat reduces the complexity of uncertain dynamical networks consisting of diffusively interconnected nonlinearLure subsystems. We aim to reduce the dimension ofeach subsystem and meanwhile preserve the synchronization property of the overall network. Using the upperbound of the Laplacian spectral radius, we first characterize the robust synchronization of the Lure network bya linear matrix equation (LMI), whose solutions can betreated as generalized Gramians of each subsystem, andthus the balanced truncation can be performed on the linear component of each Lure subsystem. As a result, thedimension of the each subsystem is reduced, and the dynamics of the network is simplified. It is verified that, withthe same communication topology, the resulting reducednetwork system is still robustly synchronized, and the apriori bound on the approximation error is guaranteed tocompare the behaviors of the full-order and reduced-orderLure subsyste

Inference for Differential Equation Models using Relaxation via Dynamical Systems

Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The estimation of parameters of ODE based models is essential for understanding its dynamics, but the lack of an analytical solution of the ODE makes the parameter estimation challenging. The aim of this paper is to propose a general and fast framework of statistical inference for ODE based models by relaxation of the underlying ODE system. Relaxation is achieved by a properly chosen numerical procedure, such as the Runge-Kutta, and by introducing additive Gaussian noises with small variances. Consequently, filtering methods can be applied to obtain the posterior distribution of the parameters in the Bayesian framework. The main advantage of the proposed method is computation speed. In a simulation study, the proposed method was at least 14 times faster than the other methods. Theoretical results which guarantee the convergence of the posterior of the approximated dynamical system to the posterior of true model are presented. Explicit expressions are given that relate the order and the mesh size of the Runge-Kutta procedure to the rate of convergence of the approximated posterior as a function of sample size

Modelling for understanding AND for prediction/classification - the power of neural networks in research

Two articles, Edelsbrunner and, Schneider (2013), and Nokelainen and Silander (2014) comment on Musso, Kyndt, Cascallar, and Dochy (2013). Several relevant issues are raised and some important clarifications are made in response to both commentaries. Predictive systems based on artificial neural networks continue to be the focus of current research and several advances have improved the model building and the interpretation of the resulting neural network models. What is needed is the courage and open-mindedness to actually explore new paths and rigorously apply new methodologies which can perhaps, sometimes unexpectedly, provide new conceptualisations and tools for theoretical advancement and practical applied research. This is particularly true in the fields of educational science and social sciences, where the complexity of the problems to be solved requires the exploration of proven methods and new methods, the latter usually not among the common arsenal of tools of neither practitioners nor researchers in these fields. This response will enrich the understanding of the predictive systems methodology proposed by the authors and clarify the application of the procedure, as well as give a perspective on its place among other predictive approaches

Factorial graphical lasso for dynamic networks

Dynamic networks models describe a growing number of important scientific processes, from cell biology and epidemiology to sociology and finance. There are many aspects of dynamical networks that require statistical considerations. In this paper we focus on determining network structure. Estimating dynamic networks is a difficult task since the number of components involved in the system is very large. As a result, the number of parameters to be estimated is bigger than the number of observations. However, a characteristic of many networks is that they are sparse. For example, the molecular structure of genes make interactions with other components a highly-structured and therefore sparse process. Penalized Gaussian graphical models have been used to estimate sparse networks. However, the literature has focussed on static networks, which lack specific temporal constraints. We propose a structured Gaussian dynamical graphical model, where structures can consist of specific time dynamics, known presence or absence of links and block equality constraints on the parameters. Thus, the number of parameters to be estimated is reduced and accuracy of the estimates, including the identification of the network, can be tuned up. Here, we show that the constrained optimization problem can be solved by taking advantage of an efficient solver, logdetPPA, developed in convex optimization. Moreover, model selection methods for checking the sensitivity of the inferred networks are described. Finally, synthetic and real data illustrate the proposed methodologies.Comment: 30 pp, 5 figure

Using graph-based modelling to explore changes in students’ affective states during exploratory learning tasks

This paper describes how graph-based modelling can be used to explore interactions associated with a change in students' affective state when they are working with an exploratory learning environment (ELE). We report on a user study with an ELE that is able to detect students' affective states from their interactions and speech. The data collected during the user study was modelled, visualized and queried as a graph. We were interested in exploring if there was a difference between low- and high-performing students in the kinds of interactions that occurred during a change in their affective state. Our findings provide new insights into how students are interacting with the ELE and the effects of the system's interventions on students' affective states