ARTICLE | doi:10.20944/preprints202008.0072.v2
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: Autoreflectivity; Identity; Isoreflectivity; Kabirian coefficient; Similarity; Symmetry
Online: 13 September 2021 (13:27:39 CEST)
The key concepts in symmetry detection and similarity, identity measures are automorphism and isomorphism respectively. Therefore, methods for symmetry detection and similarity, identity measures should be functionally bijective, inverse, and invariance under a set of mathematical operations. Nevertheless, few or no existing method is functional for these properties. In this paper, a new methodological paradigm, called optinalysis, is presented for symmetry detections, similarity, and identity measures between isoreflective or autoreflective pair of mathematical structures. The paradigm of optinalysis is the re-mapping of isoreflective or autoreflective pairs with an optical scale. Optinalysis is characterized as invariant under a set of transformations and its isoreflective polymorphism behaves on polynomial and non-polynomial models.
ARTICLE | doi:10.20944/preprints201911.0268.v2
Subject: Mathematics & Computer Science, Numerical Analysis & Optimization Keywords: isomorphic optinalysis; dispersion; statistical mirrors; estimators; statistical properties
Online: 13 September 2021 (13:26:35 CEST)
The statistical properties of a good estimator include robustness, unbiasedness, efficiency, and consistency. However, the commonly used estimators of dispersion have lack or are weak in one or more of these properties. In this paper, I proposed statistical mirroring as a good alternative estimator of dispersion around defined location estimates or points. In the main part of the paper, attention is restricted to Gaussian distribution and only estimators of dispersion around the mean that functionalize with all the observations of a dataset were considered at this time. The different estimators were compared with the proposed estimators in terms of alternativeness, scale and sample size robustness, outlier biasedness, and efficiency. Monte Carlo simulation was used to generate artificial datasets for application. The proposed estimators (of statistical meanic mirroring) turn out to be suitable alternative estimators of dispersion that is less biased (more resistant) to contaminations, robust to scale and sample size, and more efficient to a random distribution of variable than the standard deviation, variance, and coefficient of variation. However, statistical meanic mirroring is not suitable with a mean (of a normal distribution) close to zero, and on a scale below ratio level.
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