Minimising interpolation error between two data sets

Question

In the top of the diagrams below we can see some value (y-axis) changing over time (x-axis).

As this happens we are sampling the value at different and unpredictable times, also we are alternating the sampling between two data sets, indicated by red and blue.

When computing the value at any time, we expect that both red and blue data sets will return similar values. However as shown in the three smaller boxes this is not the case. Viewed over time the values from each data set (red and blue) will appear to diverge and then converge about the original value.

Initially I used linear interpolation to obtain a value, next I tried using Catmull-Rom interpolation. The former results in a values come close together and then drift apart between each data point; the latter results in values which remain closer, but where the average error is greater.

Can anyone suggest another strategy or interpolation method which will provide greater smoothing (perhaps by using a greater number of sample points from each data set)?

Answer 1

Try B-splines: Catmull-Rom interpolates (goes through the data points), B-spline does smoothing.
For example, for uniformly-spaced data (not your case)

Bspline(t) = (data(t-1) + 4*data(t) + data(t+1)) / 6

Of course the interpolated red / blue curves depend on the spacing of the red / blue data points, so cannot match perfectly.

Answer 2

I believe what you ask is a question that does not have a straight answer without further knowledge on the underlying sampled process. By its nature, the value of the function between samples can be merely anything, so I think there is no way to assure the convergence of the interpolations of two sample arrays.

That said, if you have a prior knowledge of the underlying process, then you can choose among several interpolation methods to minimize the errors. For example, if you measure the drag force as a function of the wing velocity, you know the relation is square (a*V^2). Then you can choose polynomial fitting of the 2nd order and have pretty good match between the interpolations of the two serieses.

Answer 3

I'd like to quote Introduction to Catmull-Rom Splines to suggest not using Catmull-Rom for this interpolation task.

One of the features of the Catmull-Rom spline is that the specified curve will pass through all of the control points - this is not true of all types of splines.

By definition your red interpolated curve will pass through all red data points and your blue interpolated curve will pass through all blue points. Therefore you won't get a best fit for both data sets.

You might change your boundary conditions and use data points from both data sets for a piecewise approximation as shown in these slides.

Answer 4

I agree with ysap that this question cannot be answered as you may be expecting. There may be better interpolation methods, depending on your model dynamics - as with ysap, I recommend methods that utilize the underlying dynamics, if known.

Regarding the red/blue samples, I think you have made a good observation about sampled and interpolated data sets and I would challenge your original expectation that:

When computing the value at any time, we expect that both red and blue data sets will return similar values.

I do not expect this. If you assume that you cannot perfectly interpolate - and particularly if the interpolation error is large compared to the errors in samples - then you are certain to have a continuous error function that exhibits largest errors longest (time) from your sample points. Therefore two data sets that have differing sample points should exhibit the behaviour you see because points that are far (in time) from red sample points may be near (in time) to blue sample points and vice versa - if staggered as your points are, this is sure to be true. Thus I would expect what you show, that:

Viewed over time the values from each data set (red and blue) will appear to diverge and then converge about the original value.

(If you do not have information about underlying dynamics (except frequency content), then Giacomo's points on sampling are key - however, you need not interpolate if looking at info below Nyquist.)

Answer 5

When sampling the original continuous function, the sampling frequency should comply to the Nyquist-Shannon sampling theorem, otherwise the sampling process introduces an error (also known as aliasing). The error, being different in the two datasets, results in a different value when you interpolate.

Therefore, you need to know the highest frequency B of the original function and then collect samples with a frequency at least 2B. If your function has very high frequencies and you cannot sample that fast, you should at least try to filter them away before sampling.