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This paper explores the conceptual background to ﬁnancial time series analysis and ﬁnancial signal processing in terms of the Efﬁcient Market Hypothesis. By revisiting the principal conventional approaches to market analysis and the reasoning associated with them, we develop a Fractal Market Hypothesis that is based on the application of non-stationary fractional dynamics using an operator of the type
∂2 / ∂x2 − σq(t) * ∂ q(t)/ ∂tq(t)
where σ−1 is the fractional diffusivity and q is the Fourier dimension which, for the topology considered, (i.e. the one-dimensional case) is related to the Fractal Dimension 1 < DF < 2 by q = 1 − DF + 3/2.
We consider an approach that is based on the signal q(t) and its interpretation, including its use as a macroeconomic volatility index. In practice, this is based on the application of a moving window data processor that utilises Orthogonal Linear Regression to compute q from the power spectrum of the windowed data. This is applied to FTSE close-of-day data between 1980 and 2007 which reveals plausible correlations between the behaviour of this market over the period considered and the amplitude ﬂuctuations of q(t) in terms of a macroeconomic model that is compounded in the operator above.
Blackledge, J.: Application of the Fractal Market Hypothesis for Modelling Macroeconomic Time Series. ISAST Transactions on Electronics and Signal Processing. ISSN 1797-2329, issue: No. 1, Vol. 2, pages: 89-110, 2008.