Application of computational mechanics to the analysis of natural data: an example in geomagnetism

We discuss how the ideal formalism of computational mechanics can be adapted to apply to a noninfinite series of corrupted and correlated data, that is typical of most observed natural time series. Specifically, a simple filter that removes the corruption that creates rare unphysical causal states is demonstrated, and the concept of effective soficity is introduced. We believe that computational mechanics cannot be applied to a noisy and finite data series without invoking an argument based upon effective soficity. A related distinction between noise and unresolved structure is also defined: Noise can only be eliminated by increasing the length of the time series, whereas the resolution of previously unresolved structure only requires the finite memory of the analysis to be increased. The benefits of these concepts are demonstrated in a simulated times series by (a) the effective elimination of white noise corruption from a periodic signal using the expletive filter and (b) the appearance of an effectively sofic region in the statistical complexity of a biased Poisson switch time series that is insensitive to changes in the word length (memory) used in the analysis. The new algorithm is then applied to an analysis of a real geomagnetic time series measured at Halley, Antarctica. Two principal components in the structure are detected that are interpreted as the diurnal variation due to the rotation of the Earth-based station under an electrical current pattern that is fixed with respect to the Sun-Earth axis and the random occurrence of a signature likely to be that of the magnetic substorm. In conclusion, some useful terminology for the discussion of model construction in general is introduced.