Entropy versus extropy in permutation-based complexity measures: A comparative study of GWPE and a new weighted extension, GWPExt

Permutation entropy and its weighted variants are standard tools for quantifying time series complexity. Extropy, the dual measure to Shannon entropy, has recently been combined with the same ordinal-pattern framework to produce Permutation Extropy. We complete this construction by introducing Weighted Permutation Extropy and its continuous generalization GWPExt, and prove four supporting mathematical results (an invariance property, a concavity lemma, a closed-form maximum-divergence result, and an asymptotic expansion), each checked independently by symbolic and numerical computation. We then compare the entropy- and extropy-based families on two tasks. First, tracking a bifurcation diagram: both families track the diagram’s overall shape comparably, though extropy-based measures show roughly half the resolution within the chaotic region itself. Second, separating chaos from noise: across six chaos/noise pairs and 40 realizations each, the entropy-based measure is more discriminative at standard settings in every realization tested, while the extropy-based measure only wins in a narrow, probabilistic region at strongly negative q . Four real series (lynx populations, two solar-activity records, and a stock index) mostly confirm this pattern, with one longer, strongly periodic series showing additional structure we could not fully explain. Together these results give a precise, statistically supported answer to when the extropy-based generalization helps, when it does not, and concrete recommendations for choosing between the two.