### Article

## Об энтропийных критериях отбора признаков в задачах анализа данных

The paper considers the problem of reducing the dimension of the feature space for describing objects

in data analysis problems using the example of binary classification. The article provides a detailed

overview of existing approaches to solving this problem and proposes several modifications. In which

the dimensionality reduction is considered as the problem of extracting the most relevant information

from the characteristic description of objects and is solved in terms of the Shanon's entropy. To identify

the most significant features information criteria such as crossentropy, mutual information and Kullback-

Leibler divergence are used.

Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet. An effective algorithm for calculating the entropy of an invariant Erdos measure is proposed. It is shown that, for certain values of the Bernulli parameter, the algorithm gives the Hausdorff dimension of an Erdos measure to 15 decimal places.

The paper makes a brief introduction into multiple classifier systems and describes a particular algorithm which improves classification accuracy by making a recommendation of an algorithm to an object. This recommendation is done under a hypothesis that a classifier is likely to predict the label of the object correctly if it has correctly classified its neighbors. The process of assigning a classifier to each object involves here the apparatus of Formal Concept Analysis. We explain the principle of the algorithm on a toy example and describe experiments with real-world datasets.

Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet.

This volume contains the extended version of selected talks given at the international research workshop "Coping with Complexity: Model Reduction and Data Analysis", Ambleside, UK, August 31 – September 4, 2009. The book is deliberately broad in scope and aims at promoting new ideas and methodological perspectives. The topics of the chapters range from theoretical analysis of complex and multiscale mathematical models to applications in e.g., fluid dynamics and chemical kinetics.

This book constitutes the thoroughly refereed post-conference proceedings of the 8th International Conference on Learning and Optimization, LION 8, which was held in Gainesville, FL, USA, in February 2014. The 33 contributions presented were carefully reviewed and selected for inclusion in this book. A large variety of topics are covered, such as algorithm configuration; multiobjective optimization; metaheuristics; graphs and networks; logistics and transportation; and biomedical applications.

Symbolic classifiers allow for solving classification task and provide the reason for the classifier decision. Such classifiers were studied by a large number of researchers and known under a number of names including tests, JSM-hypotheses, version spaces, emerging patterns, proper predictors of a target class, representative sets etc. Here we consider such classifiers with restriction on counter-examples and discuss them in terms of pattern structures. We show how such classifiers are related. In particular, we discuss the equivalence between good maximally redundant tests and minimal JSM-hyposethes and between minimal representations of version spaces and good irredundant tests.

In this paper, we use robust optimization models to formulate the support vector machines (SVMs) with polyhedral uncertainties of the input data points. The formulations in our models are nonlinear and we use Lagrange multipliers to give the first-order optimality conditions and reformulation methods to solve these problems. In addition, we have proposed the models for transductive SVMs with input uncertainties.

We propose extensions of the classical JSM-method andtheNa ̈ıveBayesianclassifierforthecaseoftriadicrelational data. We performed a series of experiments on various types of data (both real and synthetic) to estimate quality of classification techniques and compare them with other classification algorithms that generate hypotheses, e.g. ID3 and Random Forest. In addition to classification precision and recall we also evaluated the time performance of the proposed methods.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.