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...

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?