The conceptual and theoretical backbones of symmetry/asymmetry detections or similarity/dissimilarity, and identity/unidentity study are automorphism or isomorphism respectively. However, the development of equations and methods for symmetry/asymmetry detections, similarity/dissimilarity, and identity/unidentity measures deviates from these backbones. In this article, an equation was proposed for symmetry/asymmetry detections, similarity/dissimilarity, and identity/unidentity measures, and proved that its isoreflective pairs-points are functionally bijective and inverse. The proposal, called Kabirian-based optinalysis, is based on the conceptual and theoretical frameworks of automorphism and isomorphism. The Kabirian-based optinalysis is also proven and characterized as invariant (robust) under translation (i.e., scaling and location shift), and rotation or reflection. Computing codes were written in python language for Kabirian-based optinalysis to serve as working codes for application and verification.

Optinalysis, as a method of symmetry detection, is a new algorithm that intrametrically (within elements or variables) or intermetrically (between elements or variables) computes and compares two or more univariate or multi-clustered or multivariate sequences as a mirror-like reflection of each other (optics-like manner), hence the name is driven. Optinalysis is based by the principles of reflection and moment about a symmetrical line which detects symmetry that reflects a similarity measurement. This proposed methodology was validated in comparison with Pearson method of skewness detection, and also with some algorithms for pairewise alignment and comparison of genomic sequences (Needle, Stretcher, Water, Matcher) on EMBL-EBI website. A results comparison shows that optinalysis is more advance, more sensitive, more inferential and simple alternative approach of skewness detection and pairewise sequence comparison.

Good estimators are characterized as robust, unbiased, efficient, and consistent. However, the commonly used estimators are weak or lack one or more of these properties. In this article, eight (8) estimators for statistical and geometrical estimations of symmetry/asymmetry, similarity/dissimilarity, identity/unidentity, and feature transformation were proposed following Kabirian-based optinalysis and other operations. The proposed estimators are characterized as invariant (robust) under scaling, location shift, and rotation or reflection. A computing code was written in python language for each of the proposed estimators so that peers can have working codes for application and performance evaluation.

The effect of sensitivity points (sequence order and position of every element) of sequences following comparative optinalysis under two proposed mechanisms, the paranodic and synodic skewization, was studied to develop a modeling approach to comparative optinalysis of sequences. The results show that the outcomes of comparative optinalysis (similarity measurement) in a set of paranodically skewed sequences can be modeled deterministically by suitable line regression functions. The sensitivity points (nodes) of a sequence display two important and distinct zones, the K-zones and the B-zones. Continues paranodic skewization within these zones (KB zones) operates within probability space, but at hyperskewization level and at the K-zones only, the outcomes of comparative optinalysis operate outside the probability space. Moreover, the outcomes of comparative optinalysis by synodic skewization can be modeled deterministically by some regression line functions, but a general regression function cannot be identified. At certain limit of skewization value space, following paranodic and synodic skewization, the outcomes of comparative optinalysis at the left-sided sequence form a similar pattern with the right-sided sequence.