Abstract
We propose a method for generating surrogate data that preserves all the properties of ordinal patterns up to a certain length, such as the numbers of allowed/forbidden ordinal patterns and transition likelihoods from ordinal patterns into others. The null hypothesis is that the details of the underlying dynamics do not matter beyond the refinements of ordinal patterns finer than a predefined length. The proposed surrogate data help construct a test of determinism that is free from the common linearity assumption for a null-hypothesis.
Highlights
Judging whether the underlying dynamics are deterministic or stochastic based on a given time series is an old problem and the first step for modelling such a time series
There have been a number of researches in the existing literature discussing how to characterize determinism and/or stochasticity: The best known approaches could be the ones using the parallelness of neighboring orbits [3,11] and the optimal neighborhood size for local linear predictions [12]
We focus on inequality relations among consecutive measurements s(t), s(t + 1), . . . , s(t + L − 1) over time period between t and t + L − 1
Summary
Judging whether the underlying dynamics are deterministic or stochastic based on a given time series is an old problem and the first step for modelling such a time series. For the test of determinism-stochasticity, we use the properties of permutations [6,7,8], which are inequality relations among consecutive measurements: If the underlying dynamics is deterministic and verifies some assumptions (see Section 3 for details), the number of appearing permutations increases exponentially when the length of permutations is prolonged. Currently this approach has a problem—we need a long time series of length 1,000,000 to classify stationary time series appropriately [4]. We focus on inequality relations among consecutive measurements s(t), s(t + 1), . . . , s(t + L − 1) over time period between t and t + L − 1.
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