A New Formulation for Total Least Square Error Method in d-dimensional Space with Mapping to a Parametric Line

TL;DR

This paper introduces a novel Total Least Square Error method tailored for fitting a line in d-dimensional space, addressing limitations of traditional TLSE formulations and applicable to various physical science problems.

Contribution

A new TLSE formulation for fitting lines in d-dimensional space, extending beyond traditional methods and applicable to physical sciences.

Findings

Applicable to general d-dimensional data fitting

Provides a different formulation from existing TLSE methods

Suitable for physical science applications

Abstract

There are many practical applications based on the Least Square Error (LSE) or Total Least Square Error (TLSE) methods. Usually the standard least square error is used due to its simplicity, but it is not an optimal solution, as it does not optimize distance, but square of a distance. The TLSE method, respecting the orthogonality of a distance measurement, is computed in d-dimensional space, i.e. for points given in E2 a line p in E2, resp. for points given in E3 a plane in rho in E3, fitting the TLSE criteria are found. However, some tasks in physical sciences lead to a slightly different problem. In this paper, a new TSLE method is introduced for solving a problem when data are given in E3 and a line p in E3 is to be found fitting the TLSE criterion. The presented approach is applicable for a general -dimensional case, i.e. when points are given in E^d a line E^d is to be found. This formulation is different from the TLSE formulation.