6. A nonparametric version

The term ``nonparametric'' refers to the familiar situation $y = f(x)$ or $y = f(t)$, as opposed to the parametric situation $P(t) = (f _ 1 (t), f _ 2 (t),\dots, f _ m (t))$. A method for the non-parametric version gives a parametric method, and vice-versa, as follows:

If you have a nonparametric method, just apply it separately in each coordinate to obtain a parametric version. For example, if $n=2$ and data points $(5,2),(3,7),(9,1)$ are given, find $f _ 1 (x)$ such that $f _ 1 (0) = 5$, $f _ 1 (1) = 3$, and $f _ 1 (2) = 9$, and also $f _ 2(x)$ with values $2,7,1$ at those same $x$-values. Then let $P(t) = f _ 1 (t), f _ 2 (t)$. (Notice that $t$ is now used in place of $x$.)

If you have a parametric method giving a curve $P(t)$ in R$^ m$, to get a nonparametric method just concentrate on the case $m=1$. This would give you a point moving in one dimension, i.e., a moving number $x(t)$ in R. However, you can think of the graph of $x$ against $t$ to get a better picture of the function. If you wish you can then put $y$ for $x$ and $x$ for $t$ to get a function $y = f(x)$.

The discussion so far applies to any kind of parametric versus nonparametric method, but it works in particular for uniform cubic B-splines and interpolating splines. In these cases, all functions involved are piecewise cubic. The method of Section 5 applies for interpolation, with plain numbers wherever points were mentioned. Instead of ``Bézier curves'' for the segments, we have simply Bézier functions. Instead of data points, we have simply data numbers.