Entropy: The Markov Ordering Approach

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the {\em Markov order}). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally "most random" distributions.Comment: 50 pages, 4 figures, Postprint version. More detailed discussion of the various entropy additivity properties and separation of variables for independent subsystems in MaxEnt problem is added in Section 4.2. Bibliography is extende

Principal manifolds and graphs in practice: from molecular biology to dynamical systems

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen's self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.Comment: 12 pages, 9 figure

Fractional norms and quasinorms do not help to overcome the curse of dimensionality

The curse of dimensionality causes the well-known and widely discussed problems for machine learning methods. There is a hypothesis that using of the Manhattan distance and even fractional quasinorms lp (for p less than 1) can help to overcome the curse of dimensionality in classification problems. In this study, we systematically test this hypothesis. We confirm that fractional quasinorms have a greater relative contrast or coefficient of variation than the Euclidean norm l2, but we also demonstrate that the distance concentration shows qualitatively the same behaviour for all tested norms and quasinorms and the difference between them decays as dimension tends to infinity. Estimation of classification quality for kNN based on different norms and quasinorms shows that a greater relative contrast does not mean better classifier performance and the worst performance for different databases was shown by different norms (quasinorms). A systematic comparison shows that the difference of the performance of kNN based on lp for p=2, 1, and 0.5 is statistically insignificant

Singularities of transient processes in dynamics and beyond

This note is a brief review of the analysis of long transients in dynamical systems. The problem of long transients arose in many disciplines, from physical and chemical kinetic to biology and even social sciences. Detailed analysis of singularities of various `relaxation times' associated long transients with bifurcations of $\omega$-limit sets, homoclinic structures (intersections of $\alpha$- and $\omega$-limit sets) and other peculiarities of dynamics. This review was stimulated by the analysis of anomalously long transients in ecology published recently by A. Morozov and S. Petrovskii with co-authors

Visualization of Data by Method of Elastic Maps and Its Applications in Genomics, Economics and Sociology

Technology of data visualization and data modeling is suggested. The basic of the technology is original idea of elastic net and methods of its construction and application. A short review of relevant methods has been made. The methods proposed are illustrated by applying them to the real economical, sociological and biological datasets and to some model data distributions. The basic of the technology is original idea of elastic net - regular point approximation of some manifold that is put into the multidimensional space and has in a certain sense minimal energy. This manifold is an analogue of principal surface and serves as non-linear screen on what multidimensional data are projected. Remarkable feature of the technology is its ability to work with and to fill gaps in data tables. Gaps are unknown or unreliable values of some features. It gives a possibility to predict plausibly values of unknown features by values of other ones. So it provides technology of constructing different prognosis systems and non-linear regressions. The technology can be used by specialists in different fields. There are several examples of applying the method presented in the end of this paper

Transition states and entangled mass action law

The classical approaches to the derivation of the (generalized) Mass Action Law (MAL) assume that the intermediate transition state (i) has short life time and (ii) is in partial equilibrium with the initial reagents of the elementary reaction. The partial equilibrium assumption (ii) means that the reverse decomposition of the intermediates is much faster than its transition through other channels to the products. In this work we demonstrate how avoiding this partial equilibrium assumption modifies the reaction rates. This kinetic revision of transition state theory results in an effective `entanglement' of reaction rates, which become linear combinations of different MAL expressions.Comment: Significantly extended version with more explanation, illustrations, and reference