Time Series Analysis for a 1/tβ Memory Function and Comparison with the Lyapunov Exponent using Volatility Scaling
Paddy Walsh and Jonathan Blackledge
Being able to provide accurate forecasts on the trending behaviour of time series is important in a range of applications involving the real-time evolution of signals, most notably in financial time series analysis, but control engineering in general. This paper reports on the use of an indicator that is based on a Memory Function of the form ∼ 1/tβ , β > 0, and, in terms of a comparative analysis, the Lyapunov Exponent λ coupled with an approach whereby both parameters (i.e. λ and β − 1) are scaled according to the corresponding Volatility σ of the time series. A ‘back-testing’ procedure is used to evaluate and compare the performance of the indices (β − 1)/σ and λ/σ for forecasting and quantifying trends over a range of time scales. However, in either case, a critical solution for providing high accuracy forecasts is the filtering operation used to identify the position in time at which a trend occurs subject to a time delay factor that is inherent in the filtering strategy used. The paper explores this strategy and presents some example results that provide a quantitative measure of the accuracy obtained.
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