Some Generalizations of Least-Squares Algorithms

Least-squares method is often used for solving optimization problems such as line fitting of a point sequence obtained by experiments. This paper describes some extensions of the method and presents an application to some completely different geometric optimization problem. Given a sorted sequence of points (x(1),y(1)), (x(2),y(2)), ..., (x(n),y(n)), a line y = ax +b that optimally approximates the sequence can be computed by determining the constants a and b that minimizes the sum of squared distances to the line, which is given by Sigma(n)(i=1) (ax(i) + b - y(i))(2). It suffices to solve a system of linear equations derived by differentiating the sum by a and b. In this paper we extend the problem of approximating a point sequence by a 1-joint polyline. Another problem we consider is a geometric optimization problem. Suppose we are given a set of points in the plane. We want to insert a new point so that distances to existing points are as close as those distances specified as input data. If the criterion is to minimize the sum of squared errors, an application of the same idea as above combined with a notion of arrangement of lines leads to an efficient algorithm.