Erik Kropat

Modeling, inference and optimization of regulatory networks based on time series data

In this survey paper, we present advances achieved during the last years in the development and use of OR, in particular, optimization methods in the new gene-environment and eco-finance networks, based on usually finite data series, with an emphasis on uncertainty in them and in the interactions of the model items. Indeed, our networks represent models in the form of time-continuous and time-discrete dynamics, whose unknown parameters we estimate under constraints on complexity and regularization by various kinds of optimization techniques, ranging from linear, mixed-integer, spline, semi-infinite and robust optimization to conic, e.g., semi-definite programming. We present different kinds of uncertainties and a new time-discretization technique, address aspects of data preprocessing and of stability, related aspects from game theory and financial mathematics, we work out structural frontiers and discuss chances for future research and OR application in our real world.

A system dynamics model for intentional transmission of HIV/AIDS using cross impact analysis

The system dynamics approach is a holistic way of solving problems in real-time scenarios. This is a powerful methodology and computer simulation modeling technique for framing, understanding, and discussing complex issues and problems. System dynamics modeling and simulation is often the background of a systemic thinking approach and has become a management and organizational development paradigm. This paper proposes a system dynamics approach for modeling the phenomenon of intentional transmission of HIV/AIDS by non-disclosure. The model is built using the Cross Impact Analysis (CIA) method of relating entities and attributes relevant to the risky conduct of HIV+ individuals in any given community. The proposed model uses a hypothetical cross impact matrix that relates pairs of attributes. The factors that impact non-disclosure are identified by simulating the model through dynamic difference equations. After the simulation results are reviewed, two policies are introduced and tested in order to observe the progress in the system state.

A survey on OR and mathematical methods applied on gene-environment networks

In this paper, we survey the recent advances and mathematical foundations of gene-environment networks. We explain their interdisciplinary implications with special regard to human and life sciences as well as financial sciences. Special attention is paid to applications in Operational Research and environmental protection. Originally developed in the context of modeling and prediction of gene-expression patterns, gene-environment networks have proved to provide a conceptual framework for the modeling of dynamical systems with respect to errors and uncertainty as well as the influence of certain environmental items. Given the noise-prone measurement data we extract nonlinear differential equations to describe and analyze the interactions and regulating effects between the data items of interest and the environmental items. In particular, these equations reflect data uncertainty by the use of interval arithmetics and comprise unknown parameters resulting in a wide variety of the model. For an identification of these parameters Chebychev approximation and generalized semi-infinite optimization are applied. In addition, the time-discrete counterparts of the nonlinear equations are introduced and their parametrical stability is investigated by a combinatorial algorithm which detects the region of parameter stability. Finally, we analyze the topological landscape of the gene-environment networks in terms of structural stability. We conclude with an application of our analysis and introduce the eco-finance networks.

A review on data mining and continuous optimization applications in computational biology and medicine.

An emerging research area in computational biology and biotechnology is devoted to mathematical modeling and prediction of gene-expression patterns; it nowadays requests mathematics to deeply understand its foundations. This article surveys data mining and machine learning methods for an analysis of complex systems in computational biology. It mathematically deepens recent advances in modeling and prediction by rigorously introducing the environment and aspects of errors and uncertainty into the genetic context within the framework of matrix and interval arithmetics. Given the data from DNA microarray experiments and environmental measurements, we extract nonlinear ordinary differential equations which contain parameters that are to be determined. This is done by a generalized Chebychev approximation and generalized semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. In addition, we analyze the topological landscape of gene-environment networks in terms of structural stability. As a second strategy, we will review recent model selection and kernel learning methods for binary classification which can be used to classify microarray data for cancerous cells or for discrimination of other kind of diseases. This review is practically motivated and theoretically elaborated; it is devoted to a contribution to better health care, progress in medicine, a better education, and more healthy living conditions.

On the classical Maki–Thompson rumour model in continuous time

In this paper, the Maki–Thompson model is slightly refined in continuous time, and a new general solution is obtained for each dynamics of spreading of a rumour. It is derived an equation for the size of a stochastic rumour process in terms of transitions. We give new lower and upper bounds for the proportion of total ignorants who never learned a rumour and the proportion of total stiflers who either forget the rumour or cease to spread the rumour when the rumour process stops, under general initial conditions. Simulation results are presented for the analytical solutions. The model and these numerical results are capable to explain the behaviour of the dynamics of any other dynamical system having interactions similar to the ones in the stochastic rumour process and requiring numerical interpretations to understand the real phenomena better. The numerical process in the differential equations of the model is investigated by using error-estimates. The estimated error is calculated by the Runge–Kutta method and found either negligible or zero for a relatively small size of the population. This pioneering paper introduces a new mathematical method into Operations research, motivated by various areas of scientific, social and daily life, it presents numerical computations, discusses structural frontiers and invites the interested readers to future research.

A System Dynamics Model for Improving Primary Education Enrollment in a Developing Country

A System Dynamics Model for Improving Primary Education Enrollment in a Developing Country The system dynamics approach is a holistic way of solving problems in real-time scenarios. This is a powerful methodology and computer simulation modeling technique for framing, analyzing, and discussing complex issues and problems. System dynamics modeling is often the background of a systemic thinking approach and has become a management and organizational development paradigm. This paper proposes a system dynamics approach for studying the importance of infrastructure facilities on the quality of primary education system in a developing nation. The model is built using the Cross Impact Analysis (CIA) method of relating entities and attributes relevant to the primary education system in any given community. The CIA model enables us to predict the effects of infrastructural facilities on the communitys access of primary education. This may support policy makers to take more effective actions in campaigns that attempt to improve literacy.

Spline regression models for complex multi-modal regulatory networks

Complex regulatory networks often have to be further expanded and improved with regard to the unknown effects of additional parameters and factors that can emit a disturbing influence on the key variables under consideration. The concept of target-environment (TE) networks provides a holistic framework for the analysis of such parameter-dependent multi-modal systems. In this study, we consider time-discrete TE regulatory systems with spline entries. We introduce a new regression model for these particular two-modal systems that allows us to determine the unknown system parameters by applying the multivariate adaptive regression spline (MARS) technique and the newly developed conic multivariate adaptive regression spline (CMARS) method. We obtain a relaxation by means of continuous optimization, especially, conic quadratic programming (CQP) that could be conducted by interior point methods. Finally, a numerical example demonstrates the efficiency of the spline-based approach.

Dynamical Gene-Environment Networks Under Ellipsoidal Uncertainty: Set-Theoretic Regression Analysis Based on Ellipsoidal OR

We consider dynamical gene-environment networks under ellipsoidal uncertainty and discuss the corresponding set-theoretic regression models. Clustering techniques are applied for an identification of functionally related groups of genes and environmental factors. Clusters can partially overlap as single genes possibly regulate multiple groups of data items. The uncertain states of cluster elements are represented in terms of ellipsoids referring to stochastic dependencies between the multivariate data variables. The time-dependent behaviour of the system variables and clusters is determined by a regulatory system with (affine-) linear coupling rules. Explicit representations of the uncertain multivariate future states of the system are calculated by ellipsoidal calculus. Various set-theoretic regression models are introduced in order to estimate the unknown system parameters. Hereby, we extend our Ellipsoidal Operations Research previously introduced for gene-environment networks of strictly disjoint clusters to possibly overlapping clusters. We analyze the corresponding optimization problems, in particular in view of their solvability by interior point methods and semidefinite programming and we conclude with a discussion of structural frontiers and future research challenges.

Intercepting a Target with Sensor Swarms

This paper introduces a new coordination method to intercept a mobile target in urban areas with a team of sensor platforms. The task is to intercept the target before it leaves the area. The approach combines algorithmic concepts from ant colony and particle swarm optimization in order to bias the search and to spread the team in the search area. The algorithms introduced are tested in simulation experiments on grids. The success probabilities measured are relatively high for most parameter combinations, and the target is intercepted in roughly half the simulation time on average. Furthermore, the experiments reveal robust behavior with regard to the parameter setting.

Regulatory Networks under Ellipsoidal Uncertainty – Data Analysis and Prediction by Optimization Theory and Dynamical Systems

We introduce and analyze time-discrete target-environment regulatory systems (TE-systems) under ellipsoidal uncertainty. The uncertain states of clusters of target and environmental items of the regulatory system are represented in terms of ellipsoids and the interactions between the various clusters are defined by affine-linear coupling rules. The parameters of the coupling rules and the time-dependent states of clusters define the regulatory network. Explicit representations of the uncertain multivariate states of the system are determined with ellipsoidal calculus. In addition, we introduce various regression models that allow us to determine the unknown system parameters from uncertain (ellipsoidal) measurement data by applying semidefinite programming and interior point methods. Finally, we turn to rarefications of the regulatory network. We present a corresponding mixed integer regression problem and achieve a further relaxation by means of continuous optimization. We analyze the structure of the optimization problems obtained, especially, in view of their solvability, we discuss the structural frontiers and research challenges, and we conclude with an outlook.