| Abstract: |
| We propose a unified framework for testing temporal symmetries in time series based on the
distribution of ordinal patterns. Existing ordinal-pattern approaches typically focus on specific
symmetry properties, such as time reversibility, by testing equalities between selected pattern
probabilities, often pairing each pattern with its reverse. These approaches implicitly induce a
partition of the permutation space, restricting the scope of the analysis to predefined symmetry
structures. Our framework generalizes this idea by allowing symmetry tests to be constructed
from arbitrary partitions of the ordinal-pattern space, providing a flexible and systematic way
to design tests for a wide range of temporal properties. Depending on the chosen partition, the
proposed methodology recovers classical ordinal-pattern tests as special cases while enabling the
exploration of previously unaddressed symmetry structures. We derive asymptotic results for the
proposed test statistics under broad classes of stationary processes, ensuring theoretical validity.
Extensive experiments on synthetic and real-world time series demonstrate that the proposed
tests are highly sensitive to structural temporal asymmetries, while remaining fully data-driven,
computationally efficient, and easy to implement. |
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