# Estimating the dimension of a data set

A colleague in applied statistics sent me this:

"I was wondering if you know any way to find out the true dimension of the domain of a function. For example, a circle is a one dimensional function in a two dimensional space. If I do not know how to draw, is there a statistic that I can compute that tells me that it is a one dimensional object in a two dimensional space? I have to do this in high dimensional situations so cannot draw pictures. Any help will be greatly appreciated."

The notion of dimension here is obviously ill-defined. I mean, I could run a curve through any finite collection of points in high dimensional space, and say that my data is one-dimensional. But, depending on the configuration, there may be an easier or more efficient way to describe the data as a higher dimensional set.

Such issues must have been considered in the statistics literature, but I'm not familiar with it. Any pointers or suggestions? Thanks!

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See

Levina, E. and Bickel, P. (2004) “Maximum Likelihood Estimation of Intrinsic Dimension.” Advances in Neural Information Processing Systems 17

http://books.nips.cc/papers/files/nips17/NIPS2004_0094.pdf

Their idea is that if the data are sampled from a smooth density in $R^m$ embedded in $R^p$ with $m < p$, then locally the number of data points in a small ball of radius $t$ behaves roughly like a poisson process. The rate of the process is related to the volume of the ball which in turn is related to the intrinsic dimension.

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+1 Nice find! The paper also has a brief discussion of the PCA approach (as well as some other methods). –  whuber Jan 6 '11 at 23:08
Thanks very much, I think that this is the closest to what my colleague was looking for. –  Byron Schmuland Jan 8 '11 at 19:17