2
$\begingroup$

I have two datasets, 'A' and 'B', comprising N measurements of one quantity, that I would like to compare to the results of a simulation, let's call this last dataset 'S'. This comparison got me thinking and I have the following questions (which are quite general I guess).

Is a dataset completely determined by its standardized moments? That is to say, given a dataset I can always calculate its zeroth order moment, the first, and so forth. If I were able to calculate them all, I'll have a function of the discrete variable 'q', the order of the standardized moment. Is this function unique? That is to say, is the following statement a mathematical truth: "if two datasets have the same standardized moments, the datasets are the same" ?

Of course, I'm looking for an educated answer on this but also for some references where I could read more about this subject.

Thanks in advance for all your help.

$\endgroup$
  • $\begingroup$ What do you mean by "standardized moments"? $\endgroup$ – Yuval Filmus Jun 10 '15 at 21:58
  • $\begingroup$ If you are interested in the case where there is an infinite number of moments (i.e. a continuous density), then you can take a look at en.wikipedia.org/wiki/Hamburger_moment_problem. Either way, that page will lead in the direction of the general "problem of moments". $\endgroup$ – muaddib Jun 10 '15 at 22:17
1
$\begingroup$

The moments define the dataset up to a permutation. Let the dataset be $x_1,\ldots,x_n$. You are given the values of $m_k = \frac{1}{n} \sum_{i=1}^n x_i^k$ for all $k$ (in fact, $k=1,\ldots,n$ suffice). Using the Newton identities, you can compute the elementary symmetric functions $\sigma_k = \sum_{\substack{S \subseteq [n] \\ |S|=k}} \prod_{i \in S} x_i$ (for example, $\sigma_1 = x_1+\cdots+x_n$ and $\sigma_2 = x_1x_2+\cdots+x_{n-1}x_n$, with $\binom{n}{2}$ terms for all possible pairs of indices). This allows you to form the polynomial $$ P(x) = x^n - \sigma_1 x^{n-1} + \sigma_2 x^{n-2} - \cdots \pm \sigma_n = (x-x_1) \cdots (x-x_n), $$ whose roots are $x_1,\ldots,x_n$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks for your answer Yuval, it's pretty clear to me now. By standardized moments I meant en.wikipedia.org/wiki/Standardized_moment. Can you tell me a little more about those elementary symmetric functions that you defined? I want to make sure I understand the limits of that summation in \sigma_k. Thanks again. $\endgroup$ – Guest Jun 10 '15 at 23:03
  • $\begingroup$ Never mind, I found the answer to the question in my last comment. I'm sorry for asking that, i'm a chemistry undergraduate with an interest in maths but I didn't know about elementary symmetric functions. I'm marking your answer as 'the' answer. Thanks again for all your help. $\endgroup$ – Guest Jun 10 '15 at 23:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.