The Ordinal Power Spectrum: A graph spectral representation of ordinal pattern distributions

Ordinal pattern distributions are the fundamental objects underlying permutation entropy, statistical complexity, and many distribution-based ordinal descriptors. These approaches are usually summarized by scalar quantities, which quantify global uncertainty or global departure from uniformity but do not specify how the non-uniform component of the ordinal distribution is arranged over the permutation space. In this work, we introduce the Ordinal Power Spectrum (OPS) as a graph-spectral representation of Bandt–Pompe ordinal distributions. For a selected permutation graph G , the ordinal distribution p : S m → [ 0,1 ] is treated as a graph signal, and the centered distribution p − u is decomposed through the spectral projectors of the normalized graph Laplacian. The resulting eigenspace powers P G ( ν ; p ) = ‖ Π ν ( p − u ) ‖ 2 2 define a graph-indexed spectrum of ordinal non-uniformity. In the present manuscript, G is the adjacent-transposition Cayley graph, whose graph distance coincides with Kendall’s inversion distance. We show that the López–Ruiz–Mancini–Calbet disequilibrium is exactly the total positive-frequency OPS power, D LMC ( p ) = ∑ ν > 0 P G ( ν ; p ) , whereas the graph Dirichlet energy is the first frequency-weighted spectral moment. This identity is specific to the Euclidean geometry of D LMC ; the OPS is therefore presented as a graph-spectral decomposition of Euclidean ordinal disequilibrium, not as a general decomposition of all information divergences or statistical-complexity measures. Analytic examples show that ordinal distributions with identical entropy and identical Euclidean disequilibrium can allocate their non-uniform power differently across graph frequencies. Numerical experiments with the logistic map and AR(1) processes, together with a real physiological waveform proof of concept based on ventilator signals, illustrate that the OPS can reveal graph-frequency organization of ordinal disequilibrium beyond scalar summaries. We also discuss stability as a downstream transform of Bandt–Pompe distributions, computational scaling in the practical range m = 3 –6, and limitations related to graph choice, finite-sample estimation, and non-Euclidean probability geometries.