Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures.Comment: 14 pages in Latex, Revtex + 4 figures in .ps attached (revised version, new title, several changes, to appear in CHAOS

Linguistic complexity: English vs. Polish, text vs. corpus

We analyze the rank-frequency distributions of words in selected English and Polish texts. We show that for the lemmatized (basic) word forms the scale-invariant regime breaks after about two decades, while it might be consistent for the whole range of ranks for the inflected word forms. We also find that for a corpus consisting of texts written by different authors the basic scale-invariant regime is broken more strongly than in the case of comparable corpus consisting of texts written by the same author. Similarly, for a corpus consisting of texts translated into Polish from other languages the scale-invariant regime is broken more strongly than for a comparable corpus of native Polish texts. Moreover, we find that if the words are tagged with their proper part of speech, only verbs show rank-frequency distribution that is almost scale-invariant

Sign and amplitude representation of the forex networks

We decompose the exchange rates returns of 41 currencies (incl. gold) into their sign and amplitude components. Then we group together all exchange rates with a common base currency, construct Minimal Spanning Trees for each group independently, and analyze properties of these trees. We show that both the sign and the amplitude time series have similar correlation properties as far as the core network structure is concerned. There exist however interesting peripheral differences that may open a new perspective to view the Forex dynamics.Comment: Article based on talk by S. Gworek given at FENS'08 Conference, Rzeszow, Polan

World currency exchange rate cross-correlations

World currency network constitutes one of the most complex structures that is associated with the contemporary civilization. On a way towards quantifying its characteristics we study the cross correlations in changes of the daily foreign exchange rates within the basket of 60 currencies in the period December 1998 -- May 2005. Such a dynamics turns out to predominantly involve one outstanding eigenvalue of the correlation matrix. The magnitude of this eigenvalue depends however crucially on which currency is used as a base currency for the remaining ones. Most prominent it looks from the perspective of a peripheral currency. This largest eigenvalue is seen to systematically decrease and thus the structure of correlations becomes more heterogeneous, when more significant currencies are used as reference. An extreme case in this later respect is the USD in the period considered. Besides providing further insight into subtle nature of complexity, these observations point to a formal procedure that in general can be used for practical purposes of measuring the relative currencies significance on various time horizons.Comment: 4 pages, 3 figures, LaTe

Scale free effects in world currency exchange network

A large collection of daily time series for 60 world currencies' exchange rates is considered. The correlation matrices are calculated and the corresponding Minimal Spanning Tree (MST) graphs are constructed for each of those currencies used as reference for the remaining ones. It is shown that multiplicity of the MST graphs' nodes to a good approximation develops a power like, scale free distribution with the scaling exponent similar as for several other complex systems studied so far. Furthermore, quantitative arguments in favor of the hierarchical organization of the world currency exchange network are provided by relating the structure of the above MST graphs and their scaling exponents to those that are derived from an exactly solvable hierarchical network model. A special status of the USD during the period considered can be attributed to some departures of the MST features, when this currency (or some other tied to it) is used as reference, from characteristics typical to such a hierarchical clustering of nodes towards those that correspond to the random graphs. Even though in general the basic structure of the MST is robust with respect to changing the reference currency some trace of a systematic transition from somewhat dispersed -- like the USD case -- towards more compact MST topology can be observed when correlations increase.Comment: Eur. Phys. J. B (2008) in pres

Current log-periodic view on future world market development

Applicability of the concept of financial log-periodicity is discussed and encouragingly verified for various phases of the world stock markets development in the period 2000-2010. In particular, a speculative forecasting scenario designed in the end of 2004, that properly predicted the world stock market increases in 2007, is updated by setting some more precise constraints on the time of duration of the present long-term equity market bullish phase. A termination of this phase is evaluated to occur in around November 2009. In particular, on the way towards this dead-line, a Spring-Summer 2008 increase is expected. On the precious metals market a forthcoming critical time signal is detected at the turn of March/April 2008 which marks a tendency for at least a serious correction to begin. In the present extended version some predictions for the future oil price are incorporated. In particular a serious correction on this market is expected to start in the coming days.Comment: presented by S. Drozdz at FENS2007 conference, 10 pages, 6 Figs, an extended version with the oil market included (Fig.7