과도 결정 선형 시스템에서의 보통과 수직 거리 최소 제곱 해에 대한 대수 및 기하적 분석

Abstract

In this paper, we make an in-depth analysis on the ordinary least-squares solution and the orthogonal distance least-squares solution to an overdetermined linear system, having the number n of constraints greater than the number m of unknowns. Using the barycentric coordinate, algebraic and geometric analysis is made for the case of n=2 and m=3, yielding a constraint triangle, and for the case of n=3 and m=4, yielding a constraint tetrahedron. First, for an n-dimensional overdetermined system, the ordinary least squares solution that minimizes the sum of the squares of m residuals and the orthogonal distance least squares solution that minimizes the sum of the squares of m orthogonal distances are formulated. Next, for the cases of n=2 and m=3, and n=3 and m=4, the following analysis is made: The relationship between the ordinary least-squares solution and the centroid of the constraint triangle/tetrahedron is identified, the changing pattern of the ordinary least-squares solution due to the scaling of an overdetermined system is examined, and the relationship between the orthogonal distance least squares solution and the symmedian point of the constraint triangle/tetrahedron is identified. Finally, to demonstrate the applicability of the ordinary least-squares and the orthogonal distance least-squares solutions, the velocity estimation of a mobile robot using a polygonal array of optical mice is provided.