Continued Fractions in Time Series Forecsting
Through their ergodic properties, continued fractions provide a fascinating example of a dynamical system whose properties we attempt to encode in a library of rules. For an irrational number, randomly selected from a unit interval, the probability distribution of partial quotients can be derived from the ergodic theorem, allowing one to distinguish a random time series from a time series whose elements are drawn from the probability distribution resulting from the ergodic hypothesis. Applications of ergodicity to modular transformations and chaotic cosmology are sketched. In addition to being an object of study, continued fractions are used as a tool to overcome the curse of dimensionality in rule-based forecasting. To this end, we encode the successive (possibly resealed) values of a time series, as the partial quotients of a continued fraction, resulting in a number from the unit interval. The accuracy of a ruled-based system utilizing this coding is investigated to some extent. Qualitative criteria for the applicability of the algorithm are formulated.
KeywordsForecast Error Fuzzy Controller Continue Fraction Ergodic Theorem Irrational Number
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