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I would like to know if there are any standard techniques (that I don't know about) to solve the following problem.

Suppose we have $n$ variables, $\mathbf{q} = (a_1, a_2, \ldots, a_n)$, but not all of them are independent. For example, the values could be determined by only a single variable, e.g. $(x, x^2, x^3)$, or only two, e.g. $(x, y, x^2, xy, y^2)$.

Now suppose we have some measurement data for $\mathbf{q}$, which might have small errors (i.e. the relationship between the variables is valid only to some error). How can we find how many of the $n$ variables are independent of each other?

If the question is not clear, please ask. I see that there are problems, e.g. in $(x, x^2, x^3)$, $x$ can determine the value of the rest of the variables, but $x^2$ cannot (if there are negative values). Nevertheless any suggestions are most welcome. I am only interested in the number of independent variables for now, not the nature of relationship between them.

Note: I know about techniques in the case when the relationship between them is linear. I am interested in the non-linear (but continues) case now.

Note 2: Please help tag the question appropriately...

EDIT

Another way to put it:

I have some points in an $n$-dimensional Euclidean space. The points lie very close to a $k$-dimensional surface. How can I estimate the value of $k$ if I know the coordinates of the points?

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There should be quite an extensive literature on this type of problem. A quick Google search turned up these papers: http://www.cs.bu.edu/techreports/pdf/2011-012-intrinsic-dimension-clustering.pdf, http://www.princeton.edu/~wbialek/our_papers/chigirev+bialek_04.pdf

This, and further references given there, should get you started on a more exhaustive search.

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  • $\begingroup$ @Michael, thank you, this is useful! I didn't react to your answer up til now because I was travelling. $\endgroup$
    – Szabolcs Horvát
    Commented Jun 13, 2011 at 14:04
  • $\begingroup$ @Michael, what keywords did you use to search for this? $\endgroup$
    – Szabolcs Horvát
    Commented Jun 13, 2011 at 18:17

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