Abstract
Relatively few methods have been proposed, that could be used in the fitting of curves to data without external criterion, i.e, data with no clear distinction between objective and explanatory variables. Hastie and Stuetzle (1989) have proposed Principal Curves as an algorithm to fit curves to such data. The algorithm repeats alternately a Projection Step and an Expectation Step until a convergence condition is satisfied. But in the Projection Step, we must search the nearest neighbouring points on the curve from each of N data points, thus computational complexity of order N^2 is necessary in the straightforward search. In this paper, we propose an effective algorithm for the Projection Step, which untilizes a recursive binary-tree search. We evaluate the computational complexities of the straiforward and the refined algorithm for some special cases, and show efficiency of our algorithm on some examples.
