MATLAB: Extrapolation!!

curve fittingextrapolation problemfitting

Hi everyone! I've got a big problem here…

I have some x and y values:

x=[14.57261875 14.72416687 14.85546972 14.96644958 15.09755813 15.23869218 15.33941339 15.4501601 15.58088007 15.67126492 15.75150771 15.79153196 15.8315562 15.89159258 ]

y=[9159.164387 9388.584929 9602.741855 9832.286745 10123.07262 10459.80477 10735.35812 11056.85775 11500.90152 11853.1149 12205.35936 12419.79607 12634.23278 12955.88784]

I use cftool to make a polynomial fit (3rd order). As you see the lower limit of x values is ~14.5. I want to extrapolate to x=9 but it's necessery to put some conditions for that. To explain if i use the fittedmodel(from cftooll) the required values is not what i want because ,since the equation is a third degree, the curvature changes.

If i know that curvature changes about x=7 how can i require from matlab not change the tilt-curvature?

Excuse me for my english…

Thanks a lot

Best Answer

Shiver. You have data that lies between 14.5 and 15.9, and you wish to extrapolate down to x = 9???????????? Have you learned too little from the works of Mark Twain?

“In the space of one hundred and seventy six years the Lower Mississippi has shortened itself two hundred and forty-two miles. That is an average of a trifle over a mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oölitic Silurian Period, just a million years ago next November, the Lower Mississippi was upwards of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-pole. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo [Illinois] and New Orleans will have joined their streets together and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.” "Life on the Mississippi", Mark Twain, 1884

Ok. if you insist on doing this extrapolation down to an arbitrarily virtually random value at x = 9, then use a tool that can do it in style, and will have the properties you desire for that curve. My SLM tools will do exactly what you need.

slm = slmengine(x,y,'plot','on','knots', ...
   [9 14.5 15 15.5 16],'increasing','on','concaveup','on');

Find SLM on the file exchange:

http://www.mathworks.com/matlabcentral/fileexchange/24443-slm-shape-language-modeling

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MATLAB: Does Matlab interp2 (Spline) OverEstimates/buggy?!

Extrapolation using a spline is insane. Period. You get what you deserve, i.e., a virtually random result. In fact, any extrapolation tends to be dangerous, unless you use a viable model that makes physical sense. So polynomials are also useless.

You can use my own gridfit tool (found on the File Exchange) to extrapolate, as at least it tends to extrapolate as smoothly and linearly as possible. Even so, extrapolate too far and you will get what Mark Twain said...

"In the space of one hundred and seventy-six years the Lower Mississippi has shortened itself two hundred and forty-two miles. This is an average of a trifle over one mile and a third per year. Therefore, any calm person, who is not blind or idiotic, can see that in the Old Oolitic Silurian Period, just a million years ago next November, the Lower Mississippi River was upward of one million three hundred thousand miles long, and stuck out over the Gulf of Mexico like a fishing-rod. And by the same token any person can see that seven hundred and forty-two years from now the Lower Mississippi will be only a mile and three-quarters long, and Cairo and New Orleans will have joined their streets together, and be plodding comfortably along under a single mayor and a mutual board of aldermen. There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact."

Mark Twain, Life on the Mississippi (1883)

So, what happens when you try to extrapolate using splines which are essentially polynomials, carved into segments? Think of what a cubic polynomial does as it goes to infinity. It swoops up (or down) with great alacrity. (Hmm, I think I'm starting to sound like Mark Twain himself.) And the shape of that swoop will depend on subtle variations of the data along the edges. It will also depend on subtle variations in the code, on exactly how the spline was implemented. So trusting a spline extrapolant is, as I said, simply insane.

In fact, trusting any extrapolant is a dangerous thing to do. You can put more faith in models used for extrapolation that are based on intelligently chosen physical principles. But even then, extrapolation too far is a dangerous thing to do. Sorry, but it just is. Just remember what Mark said.

MATLAB: Way to extend and interpolate the matrix by using thin plate spline

It is not clear in my opinion what you are asking.

If your question is how should one simply resample the domain by interpolation, so the new array will be more finely sampled over the same domain, then just use interp2. You have a choice of methods there.

If your question is how does one explicitly extrapolate a matrix out a significant distance, and you want to use a thin plate spline?

Mark Twain answered that question better than anyone else I know of:

Life on the Mississippi (1884)

Using a spline in any form to do significant extrapolation is asking for trouble and silly results, much like the quote above. The fact is, one of the most common interpolation problems I see is when people try to use an interpolant to extrapolate. Expect random crap as a result, or at least don't be surprised at that event.

You might decide to try my gridfit or inpaint_nans tools, both found on the file exchange. Either could be used here. In the default mode, both will aim to extrapolate as linearly as possible, so they are not too poorly behaved. Both tools also have an option that attempts to extrapolate as a constant, to the extent that is possible.

Best Answer

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MATLAB: Does Matlab interp2 (Spline) OverEstimates/buggy?!

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