The most representative shape of a family of curves

摘要

Suppose that we are given a family of curves, either continuous or discrete, and we wish to determine a curve that is most representative of the family. Using least squares theory, we find that in both cases we seek the dominant eigenvalue and eigenvector of a symmetric matrix. Numerical approaches to the eigenvalue problem are then given. One involves the standard inverse power method, and the other involves a new differential equation approach. Faddeev's method for obtaining the characteristic polynomial is also employed. Two illustrative examples, one for a family of continuous curves and one for a family of discrete curves, are presented.