We propose a novel class of mixed fluctuations with different orientations and fractal scaling features as a model for anisotropic two-dimensional (2D) trajectories hypothesized to appear in complex systems. Furthermore, we develop the oriented fractal scaling component analysis (OFSCA) to decompose such mixed fluctuations into the original orientation components. In the OFSCA, the original orientations are detected based on the principle that the original angles are orthogonal to the angles with the minimum and maximum scaling exponents of the mixed fluctuations. In our approach, the angle-dependent scaling properties are estimated using the Savitzky–Golay-filter-based detrended moving-average analysis (DMA), which has a higher detrending order than the conventional moving-average-filter-based DMA. To illustrate the OFSCA, we demonstrate that the numerically generated time-series of mixed fractional Gaussian noise (fGn) processes with non-orthogonal orientations and different scaling exponents is successfully decomposed into the original fGn components. We demonstrate the existence of oriented components in the 2D trajectories by applying OFSCA to real-world time-series, such as human postural fluctuations during standing and seismic ground acceleration during the great 2011 Tohoku-oki earthquake.

A b s t r a c t A method to analyze self-affinities is introduced and applied to the large scale fold geometries of Quaternary and Tertiary sediments or geographical topographies in the inner belt of the Northeast Honshu Arc, Japan. Based on this analysis, their geometries are self-affine and can be differently scaled in different directions. We recognize a crossover from local to global altitude (vertical) variation of the geometries of folds and topographies. The characteristic length for the crossover of topographies (landforms) is about 25 km and is related to the half wavelength of the crustal buckling folds or possible maximum magnitude of inland earthquakes in the Northeast Honshu Arc. Moreover, self-affinity of the folds and topographies can be connected with the b-value in GutenbergRichter's law. We obtain two average Hurst exponents obtained from the self-affinities of folds in the Northeast Honshu Arc. This indicates that there are two possible seismic modes for the smaller and larger ranges in the focal regions in the Northeast Honshu Arc.

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