### Article

## Predictive clustering on non-successive observations for multi-step ahead chaotic time series prediction

Predictive clustering algorithm based upon modified Wishart clustering technique is applied to predict chaotic time series. Concept of predictable and non-predictable observations is introduced in order to distinguish between reliable and unreliable predictions and, consequently, to enhance an ability to predict up to considerable number of positions ahead. Non-predictable observations are easily ascertained in the frameworks of predictive clustering, regardless used clustering technique. Clustering vectors are composed from observations according to set of patterns of non-successive positions in order to reveal characteristic observations sequences, useful for multi-step ahead predictions. The employed clustering method is featured with an ability to generate just enough clusters (submodels) to cope with inherent complexity of the series in question. The methods demonstrate good prediction quality for Lorenz system time series and satisfactory results for weather, energy market and financial time series.

In this paper, using the example of the Lorentz series, we consider several new strategies for predicting many steps ahead. The use of generalized z-vectors, composed of inconsistent observations, made it possible, within the framework of prediction approaches based on clustering, to construct for each point for which it is necessary to obtain a forecast, a sufficiently large set of possible forecast values. The analysis of these sets was carried out in two aspects: first, the determination of the possibility of obtaining a single predicted value for such a set; secondly, the construction of a single predicted value in cases where it is possible. The concept of unpredictable points made it possible to formulate a new problem of predicting many steps ahead, which assumes that the algorithm has the ability to distinguish between predictable and unpredictable points and to make a forecast in predictable ones. It was found that with increasing number of steps for which it is necessary to obtain a predicted value, the number of unpredictable points increases, while the error in the predicted points does not exceed a certain threshold value. The approaches proposed in this work to solving the problem of multistep predictionin this formulation made it possible to obtain predicted values at some points beyond the prediction horizon.

The present paper outlines a novel approach to predict popularity of topics for social network Twitter; the method is designed to identify precociously the topics able to demonstrate “explosive” growth in popularity. First of all, the predictive clustering method ascertains real (not written in hash-tags!) topics of tweets and then predicts popularity rates for the topics. The same clustering algorithm is employed both to ascertain the real topic of a message and to cluster segments of time series (in order to predict topics popularity), namely, maximum likelihood adaptive neural system based upon modelling field theory. In the course of wide-ranging simulation, typical variants of “pre-explosive” dynamics were revealed; some of them were turned out to be equal to heuristic techniques to predict topics popularity well known for PR community collaborating with the network (“crab,” “Pesavento’s butterfly,” etc.).

This article is devoted to the methodological issues of the application of artificial intelligence techniques in preventive medicine. We showed a specific example of the neural network application allows not only to diagnose cardiovascular diseases, but also on a quantitative basis to predict their emergence and development in future periods of life. This allows you to select the optimal strategy for the prevention and treatment of patients based on their individual parameters. The article concluded: recommendations for the prevention and treatment of cardiac patients should be given strictly individually, taking into account physiological peculiarities of the organism of patients. If for some patients it is useful to give up Smoking, limit the consumption of sweets, take drugs, reduce blood pressure, etc., for other patients, these recommendations may cause harm. Our intelligent system helps to identify such non-standard patients and to avoid incorrect recommendations. The prototype of the proposed system laid out in the "Projects" section on the website www.PermAi.ru.

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.