Extrapolation versus Interpolation

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Extrapolation and interpolation are basically the acts of continuing a curve beyond its data set and drawing a similar curve between two sets of the same data where there is a gap, respectively. Both continue or utilize an assumed pattern of the Cartesian curves they are used upon. However, an interpolated curve is finite whereas an extrapolated curve can be infinite. The larger the gap of curvature interpolated, the more chance for inaccuracy, but since it is finite there can only be so much inaccuracy to be had by interpolation. Yet for extrapolation the extrapolated curvature can be extended infinitely allowing for infinite inaccuracy eventually.

So, which pattern matching method has more inherent accuracy and which has more inherent inaccuracy?

well, just so this one has a response of some kind [which it deserves], i can only speak of the limited area in which i use interpolation and extrapolation, which is in the field of audio restoration- lots of my old tape recordings have drop-outs [sudden magnetic nonlinearities] of variable length, most of which are in the range of a quarter to a half-second or so, which is in the range where you interpolate by extrapolating the audio material before and after the drop-out- backfilling and front-filling, combined with subtlely stretching each gap-end towards the middle, combined with selective cutting copying and pasting of similar-sounding material from other parts of the recording, to fill the gap.

pure extrapolation comes in when i have a truncation/amputation of material at the end and/or beginning of the soundfile, and in order to restore the beginning material i find that using ordinary extrapolation of existing material at the amputation site often imparts a pre-echo, so i end up having to subtly time-stretch and/or borrow appropriate material from elsewhere, or reconstruct plausible replacement material from various points in the soundfile, for insertion at the beginning. it works better at the end, where i can use a combo of copy/paste/reverse/fade-out with reverberant masking to produce a proper semblance of an untruncated ending.

pure interpolation comes in handy with very short interruptions in the signal, such as a very brief click/spike on the upslope/downslope of a rising/falling waveform/wavelet, in which the more powerful [longer-duration gap event] extrapolation technique [in the absense of musical masking] can introduce audible artifacts.

after saying all this, i must say that in my little arena of expertise, neither technique is the be-all-end-all, they each have their own areas of appropriateness, and each of them can sound awful if not used judiciously and with restraint.

Learning curves topic

Interesting response by Auntie

Keet, can you give a concrete example in everyday use, like Auntie, in which accuracy was the most important, and the end result was the most desired?

It depends what area you are working in.

In real analysis, a two dimensional curve can perfectly define any given, finite data set provided that certain basic criteria are in place (no more than one value for any given x). In that context, it is possible to create a function that will perfectly replicate the observed data and produce a reliable result for any real value of x.

However, in practical applications one must rely on a practice of "Best Fit", and suppose the nature of the function that we are looking to impose on the data set. As soon as you start to account for error in your observed data, the reliability of curve fitting diminishes. The stronger your data, then, the stronger your interpretive exercises.

Interpolation, notionally, is stronger, because the data for which are are seeking to find solutions lies within the range of data that you have observed. But this intuitive strength presupposes that your best-fit curve will be increasingly less accurate the farther from your sample data that you go.

When one is looking at statistical samples, then that intuitive strength is a valid assumption. When one is looking at observed data from, say, experimental practices, then the validity of extrapolation might be more appropriately tied to the scale of the experiement, rather than the departure from the range of the sample set. (For example, observations about gravity cannot be extrapolated to the quantum level).

Poles apart topic

Thanks, VG. The wisdom is knowing differences and contexts.

I read the post yesterday and last night I dreamt about the answer so I going to post it even if it does not add too much to the discussion. Interpolation was more useful to me. In my disertation I used spline interpolation to "fill the holes" of a World Bank´s database. I guess it took me two or three days to finish the interpolation of the dataset using microsoft excell and a slow computer (the database contained 18 variables for 180 countries and 60 years). Then I ran the panel least squares fixed effects estimator for the estimations that I wanted. I used interpolation to generate the data in the first place because when I first tried to estimate the equation I recieved the message "insufficient data" from the econometric software. I didn´t want to discard the whole research so I decided that I needed to "create" some more data and used spline interpolation to "fill the holes" of the database (mainly years when the research was not done). One of my friends who was doing his masters in economics helped me with the disseration and I asked him if besides the interpolation I could perform extrapolation of the dataset for the missing years generating more data and only then estimate. He said to me that I shoudn´t do that and that it would cause a significant bias when I estimated the equation. He said that I only could do that (extrapolation) if I used Bayesian or non-parametric methods. I couldnt find any software which would do that and dind´t have time to learn that either. So for me interpolation was more useful and more accurate!

Interesting post.

my version of interpolation/extrapolation was far easier than for the numerate ones on this thread, the software i use did all the math for me, i just had to arrange things so that they sounded passable.