Talk:Interpolation

Crikey! This page is strangely marked-up.

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The section on cosine interpolation includes the following text

This is a slight improvement over linear interpolation, the resulting function is continuously differentiable, but the differentiable is predictable since the interpolation still has only linear accuracy.

What does it mean that the differentiable is predictable? A reference for this method of interpolation would also be appreciated. -- Jitse Niesen 22:39, 23 May 2004 (UTC)

I removed the section on cosine interpolation. Searching for it on MathSciNet gives no hits, while I got hundreds for polynomial interpolation and spline interpolation. So it seems that cosine interpolation is not that important to be mentioned here. -- Jitse Niesen 22:13, 5 Jun 2004 (UTC)

There is such a thing as cosine interpolation, but in terms of its accuracy it is pretty much the same as linear interpolation but is more expensive to compute. Basically, it tries to avoid some of the artifacts created by linear interpolation by lending the function the continuous differentiability property of the cosine curve. My opinion is that the computational costliness involved is why you had difficulty finding it, cubic interpolation has a similar cost but is far more effective, so cosine interpolation has become a mathematical oddity with no real use. I believe what I have said has quite an overlap with the article. Apart from some diagrams and the equation to generate the function, I do not see what else could go into an article. However, I do think on principle even if this topic represents a mathematical evolutionary dead end there should be an article, for the clarification of a term that I assume from your research was difficult to find information on. Chadernook 23:30, 15 January 2007 (UTC)

Too difficult to understand

I basically make game modifications using some old software, and interpolation is one of the tools we use to make things move around seemlessly. Now, I can describe how it looks in action, but when I came here to understand what it is, I am being a little blown away but I am trying to understand. So can someone make this a little easier to understand in the header? A mathematical method of "predicting" points from a limited number of points? Kind of confusing. Use simpler values than resorting to a decimal system for examples. — Preceding unsigned comment added by 98.98.130.166 (talk) 23:44, 16 November 2011 (UTC)

I cant find the topic "Spatial interpolation"

I was looking for information and techniques for Spatial interpolation and was surprised to see wikipedia, not having this topic.

Can anyone look into this?

Thanks.

See the page on multivariate interpolation. --Berland 18:31, 23 June 2007 (UTC)

Fix the table

Could someone who understands tables in html fix the table in the Example section? The columns are so close together that if 2 appears in the left column and −3 in the right, then it looks as if "2 − 3" appears there. Michael Hardy 00:37, 14 Feb 2005 (UTC)

It should be better now. -- Jitse Niesen 17:21, 1 Mar 2005 (UTC)

Remark (moved from the article page)

But there are some cases where a polynom interpolate in end point. (anoninmous comment, moved from the article page by Oleg Alexandrov 19:44, 5 Jun 2005 (UTC))

Polynomial interpolation --> "infinitely differentiable"?

The section on polynomial interpolation states:

"the interpolant is a polynomial and thus infinitely differentiable. So, we see that polynomial interpolation solves all the problems of linear interpolation."

This is referring to the remark that a linear interpolation cannot be differentiated at its endpoints. However, surely this is also the case for generalised polynomial interpolation?

Oli Filth 09:57, 15 October 2006 (UTC)

I may be wrong but at or beyond the endpoints of the interpolation I think it is still infinitely differentiable, but using the function here makes the process technically extrapolation? Chadernook 23:30, 15 January 2007 (UTC)

Lagrange polynomial

I find this page (and linked pages too, such as Linear interpolation, polynomial interpolation) to be too focussed on the practicalities of interpolation using computers, and too little on the mathematical theory of the subject. At the very least the foundations of the subject (Newton, Lagrange, Hermite, Bessel) should be covered, and references made to and analogies pointed out with the related subject of numerical integration.

I was thinking about Lagrange too! Could someone mention it in the right place? --Adoniscik (talk) 09:20, 17 February 2008 (UTC)

Piecewise constant interpolation

I made a graph similar to the existing ones (same dataset) to illustrate nearest neighbor interpolation. Should it be included alongside linear and spline-interpolation in this article? --Berland 18:54, 22 June 2007 (UTC)

That's a good idea. However, the vertical lines at x = 1/2 etc. where the interpolant is discontinuous are officially not part of the graph. I'd prefer it if you could get rid of them, or replace them by dashed lines, or something like that. -- Jitse Niesen (talk) 11:07, 23 June 2007 (UTC)

I totally agree on the vertical lines, I'll see what I can do with gnuplot. All these graphs should possibly also be converted to SVG. --Berland 14:46, 23 June 2007 (UTC)

Here is my go at it--Cronholm¹⁴⁴ 15:58, 23 June 2007 (UTC)

Umm... Berland, you kinda steamrolled my work. Look at the history of you svg upload--Cronholm¹⁴⁴ 16:08, 23 June 2007 (UTC)

Yes, I just noticed, and now I see I have broken the style you applied to all of them. However, I liked my style better, have a look at my gnuplot code. --Berland 16:10, 23 June 2007 (UTC)

I saw the code. If you export them all as svg's, then there can be some preservation of the style. Unfortunately all of my work will go to waste :( --Cronholm¹⁴⁴ 16:14, 23 June 2007 (UTC)

Jitse made the original png's, maybe he has an opinion. Anyway, it is a pity your images don't come with (gnuplot?) source code so that they are reproducible. --Berland 16:50, 23 June 2007 (UTC)

Nope, they are hand-drawn in inkscape, I use gnuplot for plotting curves, I have never used it for data interpolation. I agree about waiting for Jitse. It would be nice if you could tweak his code to export as an svg for comparison. I would do it myself, but I don't have quite the handle on the program that you seem to have(I only started using 5 days ago). Anyway, I am not attached to those images and I certainly can't compete with a graphing program. :)--Cronholm¹⁴⁴ 18:54, 23 June 2007 (UTC)

I ended up replacing your figures with svg made in Gnuplot. I eventually figured I had to use Gnuplot 4.2 for smooth lines in SVG. Also, I added a stub section on piecewise constant interpolation, please feel free to elaborate. --Berland 21:30, 25 June 2007 (UTC)

Spline interpolation before Polynomial interpolation

I moved the section on spline interpolation atop of polynomial interpolation, but an IP-editor reverted it. I think it is more natural that splines come first. Polynomial interpolation generalizes constant, linear and spline interpolation, but if often generates one function passing through all points, which is a different procedure than the other ones. Also, its usability, due to Runge's phenomenon, is much less, and I therefore clearly prioritize spline interpolation. --Berland 20:50, 26 June 2007 (UTC)

Microsphere interpolation?

Do we want a section on "Microsphere interpolation"? Is it original research? See http://en.wikipedia.org/wiki/Talk:Microsphere_projection. Andrew Moylan 14:53, 3 September 2007 (UTC)

I removed the section in this page. I think it violates WP:NOR, at least enough to warrant this removal. --Berland 16:53, 3 September 2007 (UTC)

Simplest form of interpolation?

On this page, the "simplest form" of interpolation is two things: midpoint and piecewise constant. Maybe this should be altered. --Axel 129.125.178.61 14:45, 9 October 2007 (UTC)

Spline interpolation

"In this cas we get f(2.5)=0.5972"

Hmmm?

Maxima - function cspline - f(2.5)=.5962098076923081

Ex.

0.140300*2.5^3-1.335900*2.5^2+3.24670*2.5-1.36230=0.5972

0.140250*2.5^3-1.335930*2.5^2+3.24665*2.5-1.36231=0.59616

0.140252*2.5^3-1.335928*2.5^2+3.24665*2.5-1.36231=0.59620

--ZL

diagrams

it would be better if the points used in the diagrams showed the difference between spline and polynomial —Preceding unsigned comment added by 71.167.63.79 (talk) 23:04, 12 December 2009 (UTC)

Second this. I can't see the difference between "Plot of the data with polynomial interpolation applied" and "Plot of the data with spline interpolation applied". I thought at first it was a bit of wiki-humor, just re-using the same image... Spike0xFF (talk) 21:14, 16 May 2013 (UTC)

Deleted References

I added an actual reference. And then I was faced with the prospect of what to do with the existing references. I couldn't figure out how to link any of them into the text as real references, so I deleted them, in hopes that correct references would start to accumulate from hereon in. A bit of a shame in that the fourth reference was rather quirky and interesting. Here are the references I deleted. If anyone can find a way to work them back in again, please feel free.

• David Kidner, Mark Dorey and Derek Smith (1999). What's the point? Interpolation and extrapolation with a regular grid DEM. IV International Conference on GeoComputation, Fredericksburg, VA, USA.

• Kincaid, David (2002). Numerical Analysis (3rd edition). Brooks/Cole. ISBN 0-534-38905-8. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) Chapter 6.

• Schatzman, Michelle (2002). Numerical Analysis: A Mathematical Introduction. Clarendon Press, Oxford. ISBN 0-19-850279-6. Chapters 4 and 6.

• Meijering, Erik (2002), "A chronology of interpolation: from ancient astronomy to modern signal and image processing", Proceedings of the IEEE, 90 (3): 319–342, doi:10.1109/5.993400.

Edrowland (talk) —Preceding undated comment added 05:12, 20 April 2010 (UTC).

Metrics, manifolds

I have a sense that for interpolation to make sense you need to have a metric space. Related to that, I think there is something to say about interpolation on manifolds (i.e., along geodesics). To the first point, I sometimes see "interpolation" carried out naievely on parameters of functions when conceptually one really wants to interpolate between functions. (Note this is interpolating functions not interpolating the values a function takes on between two points.) For example, suppose you have $f(x)=\sin(x)$ and $g(x)=\cos(x)$. Naievely interpolating half way between the two might give:

$h(x)={\frac {f(x)+g(x)}{2}}={\frac {\sin(x)+\cos(x)}{2}}$.

I believe this is "half way between" the two in the $L_{2}$ sense, but who is to say that the $L_{2}$ sense is appropriate? If you expect a changing phase, it would make more sense to say:

$f(x)=\sin(x)=\cos(x-\pi /2)\,$

and then interpolate as

$h_{1}(x)=\cos \left(x+{\frac {0-\pi /2}{2}}\right)=\cos(x-\pi /4)$.

While $h(x)$ is definately interpolation, $h_{1}(x)$ holds invariants (such as amplitude).

My sense is that this is related to what space/manifold consider your points are in/on (where in this case points are functions) and what norm you have. Hence, in the $L_{2}$ sense, $h(x)$ is half-way between $f(x)$ and $g(x)$ but if you consider these "points" as on the manifold of functions with an $L_{2}$ norm of π over the range [0, 2π].

A more-explicitly-topological example (that is fundamentally the same, I guess) is to interpolate betwen the matrix [1 0; 0 1] and the matrix [0 -1; 1 0]. Taking a linear combination gives the matrix 0.5 * [1 -1; 1 1] but assuming we are on SO(2) would give interpolation along a geodesic which would be sqrt(0.5) [1 1; -1 1].

I haven't seen any math texts touch on this... It seems that (1) you can't interpolate in any meaningful sense unless you have a metric and that (b) even given a metric, naieve interpolation may not be on the manifold of points make sense to your application. —Ben FrantzDale (talk) 15:07, 30 August 2010 (UTC)

This is indeed correct. I added a brief mention of the displacement interpolation problem, highly studied in transportation theory, where the interpolation is seen as moving along a geodesic in the so-called Wasserstein space. For your $\cos(x)$ and $\sin(x)$ example, that would exactly give your expected answer. Nbonneel (talk) 11:48, 25 December 2011 (UTC)

Support

Related to the above (but simpler and more relevant), there is no discussion of support on this page. In one-dimensional interpolation, the difference between interpolation and extrapolation is simple. However, in multivariate statistics what is "in" and what is "out" may not be clear naievely. For example, if you fit a plane to $f(x,y)$ where all your $(x,y)$ pairs fall in an ellipse close to $y=x$ and all with $(x,y)\in [0,1]$ it is easy to decide that $f(2,2)$ is extrapolation, but it may be easy to overlook the fact that $f(0.75,0.25)$ is also extrapolation. It is clear if you look at a scatter plot with no coordinate system, or if you transform the problem with PCA, but naievely the fact that $0.75\in [0,1]$ and $0.25\in [0,1]$ throws people off. With that in mind, this page should mention this subtle corner case of when what looks like interpolation is actually extrapolation. In this case, the convex hull of the $(x,y)$ points would define one version of support; something related to Mahalonobis distance would define another. —Ben FrantzDale (talk) 15:07, 30 August 2010 (UTC)

I fully agree, the idea is not clear in higher dimensions. I think interpolation is not the opposite to extrapolation, though. You can extrapolate with an interpolant! See my entry Talk:Interpolation#Doesn.27t_explicit_the_relation_to_the_given_data. Kakila (talk) 16:18, 16 February 2017 (UTC)

External links

I added the external link to my free Web e-book. It is a more standard complete treatment of piecewise interpolation for those new to the subject and aimed at graphics. I'll add references to other related Wiki pages in the future.

My e-book is both a study of the fundamentals described in simple terms as well as a reference showing many types of Polynomial Interpolation -both common types and some developed by the author. It also shows some techniques not seen elsewhere. Linear interpolation is looked at carefully and shown is as the basis for all more advanced types using only Algebra. Only after understanding how adding squared and cubed terms cause smooth curves, are the more advanced curves examined such as Bezier, Catmul-Rom, b-spline, and Hermite. More advanced mathematical concepts and notations are kept to a minimum. Some additional techniques that are suggested by the mathematics and that the author has not seen elsewhere are examined. A reference is also included with all the most common curve drawing methods and includes drawings to allow comparison with other types.

WHY: I had hoped, but failed to find a book explaining polynomial interpolation basics and thought that one must certainly exist with a collection of interpolation types. I found either purely mathematical tests or advanced graphics texts. Several years later, I started reading the original Internet Usenet "groups", comp.graphics.algorithms, and did lots of searching on the net. I saved whatever I found related to splines and curves, but did not look at it or try to understand any of it until early in 1996, I decided to look at what I had collected, and started to figure things out. I begin recording my thoughts for future reference and this is the result. The very book I wanted originally is now freely available for others.

I do not believe this has any issues with the Wiki self cite guidelines. -- Steve -- (talk) 22:46, 28 August 2011 (UTC)

lead

The current formulation of the lead seems to suggest curve fitting and regression are considered to be particular form of interpolation. However that doesn't really match the use of terms in my experience. Instead in my experience interpolation is one form of curve fitting (as described in the lead of that article) and regression is a completely different approach to a somehwat related problem, that function approximation with average error minimization without exact fit for given data points whereas the defining characteristic for interpolation is with exact fit for given data points.--Kmhkmh (talk) 05:35, 23 March 2015 (UTC)

Doesn't explicit the relation to the given data

As I see it, main point of interpolation is to generate a function that goes exactly through the given data, i.e. opposite to Smoothing. Of course one will use the obtained function to evaluate outside of the range of the given data (yaiks, Extrapolation does the same!), but the point of interpolation is that we want to recover the given data exactly. Also in higher dimensions "within the range" doesn't have an unique meaning, except maybe in convex data sets. I have seen so much confusion emerging from this jargon that I decided to write a blog entry with a new term: "intrapolation". Kakila (talk) 16:14, 16 February 2017 (UTC)