درباره آنتروپی، بازارهای مالی و بازی های اقلیت

The paper builds upon an earlier statistical analysis of financial time series with Shannon information entropy, published in [L. Molgedey, W. Ebeling, Local order, entropy and predictability of financial time series, European Physical Journal B—Condensed Matter and Complex Systems 15/4 (2000) 733–737]. A novel generic procedure is proposed for making multistep-ahead predictions of time series by building a statistical model of entropy. The approach is first demonstrated on the chaotic Mackey–Glass time series and later applied to Japanese Yen/US dollar intraday currency data. The paper also reinterprets Minority Games [E. Moro, The minority game: An introductory guide, Advances in Condensed Matter and Statistical Physics (2004)] within the context of physical entropy, and uses models derived from minority game theory as a tool for measuring the entropy of a model in response to time series. This entropy conditional upon a model is subsequently used in place of information-theoretic entropy in the proposed multistep prediction algorithm.

In many areas of science and engineering a common recurring task arises: forecasting View the MathML sourceΔT steps ahead in an observed time series x1,x2,…,xNx1,x2,…,xN of some physical phenomena. Established forecasting techniques typically fall into two categories: time domain analysis and state–space modelling. Usually the process may involve finding a linear or non-linear regression function f(⋅)f(⋅) that takes as arguments past values of xtxt: equation(1) View the MathML sourcext+ΔT=f(xt,xt−1,xt−2,…,xt−p+1)+ε Turn MathJax on with εε accounting for noise and pp denoting the number of predictor variables. However, in cases where observations {xi}{xi}, also referred to as xtxt with integer time steps tt, come from financial markets for example foreign exchange currency fluctuations, the Eq. (1) fails to capture sufficiently the underlying time series generator and its often non-stationary nature. The art of predicting financial time series is notorious for being unreliable and difficult, which is evidenced by poor out-of-sample performance of models yielding apparently good in-sample fits [1]. In fact, simple Random Walk models make a good job of describing, but not forecasting values of financial time series, which is consistent with the Efficient Market Hypothesis. The inadequacy of the existing approaches has been mentioned before in for example [2] and [3], and some explanation of the perceived failures of the status quo has been contained in the works of Mandelbrot [4]. Perhaps in the absence of a compelling alternative the mainstream econometrics modelling framework (computational finance) assumes that prices of financial assets follow a simple stochastic Random Walk process [5]. However, financial time series are not generated by a Random Walk model or a linear autoregressive process. Instead they arise as a result of interactions between a large number of adaptive traders which provides a justification for applying various tools of statistical physics to computational finance [6]. Instead of order, sought after by mainstream econometricians through economic theories, a natural state for financial markets seems to be characterised by disorder, punctuated by brief periods of ordered behaviour [7]. Because of failures of conventional approaches some scientists, most notably from the statistical physics community, have turned to using entropy as a tool for analysing financial time series [3]. As another example, in Refs. [7] and [8] Molgedey and Ebeling consider the predictability of financial time series by extracting Shannon nn-gram (block) entropies from the Dow Jones Industrial Average and the German stock exchange DAX futures data. In the search for potentially viable alternatives to applying Eq. (1) directly, in subsequent sections this paper takes the entropy-based approach further by proposing a generic method for indirectly forecasting changes in time series View the MathML sourceΔT steps ahead. The approach is first demonstrated using the chaotic Mackey–Glass time series, subsequently to be followed by a similar analysis of the Japanese Yen/US Dollar intraday currency futures data. Weaknesses of the method are highlighted and a possible improvement is suggested in the form of replacing the information theoretic entropy with a physical entropy extracted from minority game theory models [9].

The study represents an imperfect attempt to utilise entropy in the hope of being able to predict financial time series. An alternative time series forecasting method has been demonstrated which relies on building a statistical model of entropy. Instead of predicting directly the underlying time series the method first extracts the corresponding entropy, subsequently performing predictions on the entropy time series. Using an information-theoretic entropy a weak trading advantage has been found in financial forecasts of foreign exchange currency futures initiated in low entropy regions, which agrees with results from other, earlier econophysics studies. Conversely, predicting time series in high entropy regions is very difficult to achieve. This follows directly from statistical physics which teaches that in a disordered state of maximum entropy complex systems lose memory of past events. Established statistical time series forecasting techniques, both linear regression and non-linear neural networks, do not take into account the physical generative aspect of financial time series. Such time series arise directly as a result of interactions between a large number of traders. As a consequence, from a physics point of view a much more attractive proposition is to try to approximate the underlying processes responsible for generating the time series in the first place. Therefore the paper attempted to replace an information-theoretic entropy with a physical entropy extracted from minority game theory models. According to literature [22] such models could provide a simplified approximation to the way real financial markets operate. One advantage of minority games is that they allow more control over the type of disorder (or complexity measure) being extracted from the time series. However, this comes at a price of having to decide how to choose the “correct” model configuration. As yet there is no principled way of dealing with this issue. During the course of experiments a number of difficulties arose. Although very good forecasting results from minority games were reported earlier in Refs. [18] and [21], other authors have also encountered a somewhat less positive experience [29]. Overall, within the context of physical entropy the study presented in this paper offers a mixed picture of standard minority games. The entropy-based approach depends crucially on accurate forecasting of entropy time series. Some configurations of minority games may result in entropy sequences that are difficult to predict. The non-stationary behaviour of entropy may also prove problematic. Optimum models used to forecast entropy need to be identified on a case-by-case basis. After improving the entropy forecasting process it has been possible to obtain positive results that are consistent with statistical physics. However, the small size of the data sample makes the results only indicative of what can be potentially achieved after perfecting the implementation. Greater consistency of results across different financial datasets needs to be obtained and the failure modes need to be investigated. The multi-step ahead forecasting method outlined in this paper offers one advantage over other models that utilise artificial stock markets and multiple agents. It removes the need for having an internal price making mechanism involved in making transitions between time steps from tt to View the MathML sourcet+ΔT. However, the often large computational cost of simulating possible future paths over long time horizons stipulates the need for using Monte Carlo sampling. Compared with conventional linear regression models, artificial neural networks have been found more adept at modelling sometimes complex entropy time series. Summing up, the paper attempted to express quantitatively a qualitative claim that “low entropy regions lead to improved predictability of financial time series”. In doing so it makes a positive contribution towards greater acceptance of econophysics by the mainstream computational finance.