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

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

Detrended cross-correlations between returns, volatility, trading activity, and volume traded for the stock market companies

We consider a few quantities that characterize trading on a stock market in a fixed time interval: logarithmic returns, volatility, trading activity (i.e., the number of transactions), and volume traded. We search for the power-law cross-correlations among these quantities aggregated over different time units from 1 min to 10 min. Our study is based on empirical data from the American stock market consisting of tick-by-tick recordings of 31 stocks listed in Dow Jones Industrial Average during the years 2008-2011. Since all the considered quantities except the returns show strong daily patterns related to the variable trading activity in different parts of a day, which are the best evident in the autocorrelation function, we remove these patterns by detrending before we proceed further with our study. We apply the multifractal detrended cross-correlation analysis with sign preserving (MFCCA) and show that the strongest power-law cross-correlations exist between trading activity and volume traded, while the weakest ones exist (or even do not exist) between the returns and the remaining quantities. We also show that the strongest cross-correlations are carried by those parts of the signals that are characterized by large and medium variance. Our observation that the most convincing power-law cross-correlations occur between trading activity and volume traded reveals the existence of strong fractal-like coupling between these quantities

Criticality Characteristics of Current Oil Price Dynamics

Methodology that recently lead us to predict to an amazing accuracy the date (July 11, 2008) of reverse of the oil price up trend is briefly summarized and some further aspects of the related oil price dynamics elaborated. This methodology is based on the concept of discrete scale invariance whose finance-prediction-oriented variant involves such elements as log-periodic self-similarity, the universal preferred scaling factor lambda=2, and allows a phenomenon of the "super-bubble". From this perspective the present (as of August 22, 2008) violent - but still log-periodically decelerating - decrease of the oil prices is associated with the decay of such a "super- bubble" that has started developing about one year ago on top of the longer-term oil price increasing phase (normal bubble) whose ultimate termination is evaluated to occur in around mid 2010.Comment: to appear in Acta Physica Polonica