Predictability of Dow Jones Index via Chaotic Symbolic Dynamics

We define alignment scores, the Hurst exponent and root mean square variation and use them along with Shannon entropy to analyze the Dow Jones index for the years 1985-2010. It is seen that the dynamical behavior of the US stock market is characterized by the temporal variations of the Hurst exponents, the Shannon entropy, the scores and the root mean square variation. We conclude that these measures can be used as precursors to describe the health of the US market. Key word:Schaotic time series Symbolic dynamic analysis Dow Jones index Hurst exponent Shannon entropy INTRODUCTION periodic orbits of a dynamical system can often be Equity markets exhibit a natural daily fluctuation probabilities of occurrence for different symbol sequences sometimes marked by large surges in both price and constitute the symbol sequence statistics. In this study, volume of trades. An investor is often interested in the symbol sequence statistics are used as the target for determining when to buy and when to sell an individual detecting interesting features in the series. stock. Most securities follow the trend of the major The concept of entropy and the tools derived from indices such as the DJIA or the NASDAQ which provide chaotic symbolic dynamics are used together in an information about the overall market trend. However, the attempt to detect and monitor the chaos. All complexity level of an index by itself does not provide sufficient measures based on static entropy (Shannon entropy, information for an investor to make a decision. The other pattern entropy, renormalized entropy) quantify statistical side of the coin is information about the volume of shares order in the time series. At the core of these measures is trading hands. Exogenous and endogenous impacts or the probability distribution of the objects defined from the political and economic impacts can have a considerable data. Entropy serves as an indicator for the complexity of effect on the dynamics on financial markets. the underlying processes that gave rise to the variability This article attempts to detect and monitor in the time series data. Entropy, especially Kolmogorovthe chaos and to assess their utility in flagging Sinai entropy, characterizes the degree of randomization crisis events. Different complexity measures quantify of chaotic orbits. different kinds of patterns of the given time series. The aim of this article is to examine the variability in The Hurst Exponent provides a measure which represents a chaotic time series using symbolic dynamic analysis. For the long-range correlation properties of the data and can this purpose, sequences of symbols are examined using be viewed as a self-similarity parameter characteristic of information entropies, alignment scores and fluctuations fractals. of alignments. The scores obtained for each sequence as We represent the original time series as sequence of in Yamano et al. abnormalities occurring in the series. symbols of a finite alphabet defined by the turning points Scores alone however do not provide sufficient of a map. Finding the symbolic dynamics of a time series information about abnormalities occurring in the series. is a mathematical-statistical technique aimed at the Hence, in order to compare pairs of sequences, we detection of the underlying topological and metrical perform a detrended fluctuation analysis (DFA) and structures in that time series [1]. Information about the compute the Shannon entropy for each sequence. captured in terms of these sequences of symbols. The