How to measure the accuracy of a spline?

Question

If I fit a cubic spline to data points, isn't the spline function forced to go through all given points? How is it possible to measure the accuracy (as squared deviations) if all points are already intersecting with the function? Maybe I misunderstood how spline really works...

Answer 1

Spline fitting is a relatively loose term. It does not necessarily mean that the spline will exactly go through all the data points. In fact, when we use techniques like minimizing least squared error to produce a spline that only lies close to the data points, we still call this process "spline fitting". Therefore, when talking about "spline fitting", the first thing we need to understand is whether we are talking about interpolation cases or approximation cases where the former requires the spline to exactly pass through all data points and the latter does not. Of course, there are some special algorithms that will do "interproximation", which means that the spline will pass through some data point but only lie close to the others.

Fitting a spline to given set of data points is a common technique used to find an approximation for a certain function. For such cases, even when the spline passes through all data points, we still can check whether the spline is close enough to the function at other locations. A lot of research has been devoted to study issues such as how many data points to sample, how to sample and the error bound between the spline and the function,....etc.