Basins of attraction for cascading maps

We study a finite uni-directional array of "cascading" or "threshold coupled" chaotic maps. Such systems have been proposed for use in nonlinear computing and have been applied to classification problems in bioinformatics. We describe some of the attractors for such systems and prove general results about their basins of attraction. In particular, we show that the basins of attraction have infinitely many path components. We show that these components always accumulate at the corners of the domain of the system. For all threshold parameters above a certain value, we show that they accumulate at a Cantor set in the interior of the domain. For certain ranges of the threshold, we prove that the system has many attractors.Comment: 15 pages, 9 figures. To appear in International Journal of Bifurcations and Chao

Clustering in Cell Cycle Dynamics with General Response/Signaling Feedback

Motivated by experimental and theoretical work on autonomous oscillations in yeast, we analyze ordinary differential equations models of large populations of cells with cell-cycle dependent feedback. We assume a particular type of feedback that we call Responsive/Signaling (RS), but do not specify a functional form of the feedback. We study the dynamics and emergent behaviour of solutions, particularly temporal clustering and stability of clustered solutions. We establish the existence of certain periodic clustered solutions as well as "uniform" solutions and add to the evidence that cell-cycle dependent feedback robustly leads to cell-cycle clustering. We highlight the fundamental differences in dynamics between systems with negative and positive feedback. For positive feedback systems the most important mechanism seems to be the stability of individual isolated clusters. On the other hand we find that in negative feedback systems, clusters must interact with each other to reinforce coherence. We conclude from various details of the mathematical analysis that negative feedback is most consistent with observations in yeast experiments.Comment: To appear in J. Theoretical Biology 292 (2012), 103-11

Polygonal approximation for flows

Ph.D.Konstantin Michaiko

Signed distance functions: A new tool in binary classification, ArXiv preprint: CS.LG/0511105

Abstract From a geometric perspective most nonlinear binary classification algorithms, including stateof the art versions of Support Vector Machine (SVM) and Radial Basis Function Network (RBFN) classifiers, and are based on the idea of reconstructing indicator functions. We pro-pose instead to use reconstruction of the signed distance function (SDF) as a basis for binary classification. We discuss properties of the signed distance function that can be exploited inclassification algorithms. We develop simple versions of such classifiers and test them on several linear and nonlinear problems. On linear tests accuracy of the new algorithm exceeds that ofstandard SVM methods, with an average of 50 % fewer misclassifications. Performance of the new methods also matches or exceeds that of standard methods on several nonlinear problemsincluding classification of benchmark diagnostic micro-array data sets.---------------------------------- Machine Learning, Microarray Data----------------------------------1 Introduction Binary classification is a basic problem in machine learning with applications in many fields. Not only does binary classification have many potential direct applications, it is also the basis for many multi-category classification methods. Of particular interest are the applications in biology and medicine. The availability of micro-array and proteomic data sets that contain thousands or even tens of thousands of measurements have particularly made it important to develop good classification algorithms, since reliable use of these data could presumably revolutionize diagnostic medicine. Several binary classification algorithms have been developed and studied intensely over the past few years, most notable among these are the support vector machine (SVM) methods using radial basis functions and other functions as kernels. SVM methods have been shown to perform reasonably well in classifying micro-array data, demonstrating that the extraction of useful information from these large data sets is feasible.