Consider a real-valued polynomial function of one variable. Knowing the "rules" function dictates we can plot the graph of that function with a given accuracy.
Question: is the reversed process possible?
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You mean, given a graph, can you get the function? – gt6989b Jul 09 '13 at 16:03
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Yes, that is exactly what I meant. – Constantine Jul 09 '13 at 16:04
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well, you kind of have to specify what you're willing to admit as a "function", otherwise you'd just say $f(x)$ is the unique $x$ mapping to the appropriate point. if you restrict what class of functions you're considering the graph of you can get a "better" answer i guess, like if you restrict to polynomials – citedcorpse Jul 09 '13 at 16:13
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Changed the question appropriately. – Constantine Jul 09 '13 at 16:15
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Given the "rule-based" part of your comment, if you demand perfect accuracy, the answer is no. Just take any curve that goes through $(0,y)$ at an indefinable real number. If you accept some error in both directions, you can use an interpolating polynomial of high enough degree.
Ross Millikan
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Is this what you're looking for?
Jonathan Dewein
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As far as I understand, no. The question can also be posed like this: let's assume I've drawn an arbitrary graph in Paint. Does three exist a symbolic, "rule-like" representation of the function this graph presumably represents? – Constantine Jul 09 '13 at 16:31
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I assume that, beside the graph, you have a table with some values $(x,f(x))$. In some cases, say equispaced $x$ points, building a finite difference table should allow to guess the degree of the polynomial and, subsequently, to compute the coefficients by interpolation. If you know that the data is produced by a polynomial, then iterative regressions (or interpolations) with different polynomial specifications must ultimately estimate the right coefficients giving null error up to roundoff.
More generally, symbolic regression may help, see http://www.ams.org/notices/201306/rnoti-p713.pdf for an interesting description of a software that can guess the symbolic form of your function. You can specify the search space to include only "+" and "*" (sum and product), effectively reducing the search to polynomials.
The software can be downloaded at http://formulize.nutonian.com/ and a serious treatment appeared on Science, http://www.sciencemag.org/content/324/5923/81.abstract
user1783444
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