most recent 30 from http://mathoverflow.net 2013-05-23T06:38:18Z http://mathoverflow.net/feeds/question/67419 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67419/finding-number-of-independent-variables-in-statistical-dataset Szabolcs 2011-06-10T10:24:23Z 2011-06-10T15:37:19Z

<p>I would like to know if there are any standard techniques (that I don't know about) to solve the following problem.</p> <p>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. $(x, y, x^2, xy, y^2)$.</p> <p>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?</p> <p>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 <em>number</em> of independent variables for now, not the nature of relationship between them.</p> <p><strong>Note:</strong> 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.</p> <p><strong>Note 2:</strong> Please help tag the question appropriately...</p> <p><strong>EDIT</strong></p> <p>Another way to put it:</p> <p>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?</p>

http://mathoverflow.net/questions/67419/finding-number-of-independent-variables-in-statistical-dataset/67447#67447 Michael Renardy 2011-06-10T15:37:19Z 2011-06-10T15:37:19Z

<p>There should be quite an extensive literature on this type of problem. A quick Google search turned up these papers: <a href="http://www.cs.bu.edu/techreports/pdf/2011-012-intrinsic-dimension-clustering.pdf" rel="nofollow">http://www.cs.bu.edu/techreports/pdf/2011-012-intrinsic-dimension-clustering.pdf</a>, <a href="http://www.princeton.edu/~wbialek/our_papers/chigirev+bialek_04.pdf" rel="nofollow">http://www.princeton.edu/~wbialek/our_papers/chigirev+bialek_04.pdf</a></p> <p>This, and further references given there, should get you started on a more exhaustive search.</p>