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Suppose I have 2 functions, a given function f(x) and a target function g(x). Let $M_{f}^{n}$ denote the nth central moment of function f(x). In the discrete case for N samples of f(x):

$M_{f}^{n}=\tfrac{1}{N}\sum_{i=1}^{N}(f(x) - \mu)^{n}$

And $\mu$ is the mean.

In the continuous case:

$M_{f}^{n}=E[(f - E(f))^{n}]$

I would like to transform f(x) into $f_{g}^{n}(x)$ such that:

$M_{f_{g}^{n}}^{1}=M_{g}^{1}$

$M_{f_{g}^{n}}^{2}=M_{g}^{2}$

[...]

$M_{f_{g}^{n}}^{n}=M_{g}^{n}$

Upto a specified n. For n=2 case, there is a direct transformation:

$f_{g}^{2}(x) = f_1\cdot\frac{M_{g}^{2}}{M_{f_1}^{2}} + M_{g}^{2}$

Where:

$f_1(x) = f(x)-M_{f}^{1}$

I can't think of something for n=3,4. It could be the case that such a transformation might not be possible. In which case, is it possible to propose a numerical method that can find a solution upto a specified epsilon?

The only thing we know about g(x) are its n central moments.

I'm looking for a computationally efficient transformation, yet a fairly general solution.

Ideally, an affine or polynomial transformation is preferred, but not required. The target is computational efficiency.

The "transformation" for n=2 is O(1), assuming that the computation of the nth moment is O(1), and subtracting the 1st moment is O(1).

lee
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    What do you require for your "transformation", i.e. what operations are allowed? Also what you have for cases $1,2$ done looks like a Newton method. – julio_es_sui_glace Jun 15 '23 at 16:57
  • @JulesBesson no mathematical restrictions, but the "transformation" should be computationally efficient. The "transformation" I posted for n=2 is O(1), assuming that the computation of the nth moment is O(1), and subtracting the 1st moment is O(1) – lee Jun 15 '23 at 16:59
  • Ok I have a request, can you please rewrite your problem but change $f'$ for $f_2$ and $f''$ for $f_1$, these are confusing. (i'm thinking on it) – julio_es_sui_glace Jun 15 '23 at 17:02
  • @JulesBesson case 1 and 2 first changes the mean of f(x) to mean 0, then normalizes it by the ratio of the 2nd moments (this gets the 2nd moments to match), and adds back in the mean of g (this gives f the mean of g). – lee Jun 15 '23 at 17:02
  • Ok I may edit your post for it to be more clear – julio_es_sui_glace Jun 15 '23 at 17:06
  • Is you momentum defined by $M^n_f = \int |f|^n$? – julio_es_sui_glace Jun 15 '23 at 17:09
  • @JulesBesson by nth moment, I mean nth central moment. I edited it for the discrete case. – lee Jun 15 '23 at 17:13
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    Ok, so first define the nth central moment by setting $M_f^n = E((f-E(f))^n) = \int (f - \int f)^n$ or just assume $\int f = 0 $ (same up to difference), remove all of your $x$'s that cloud the vision and then add the constraint: affine transformation or polynomial transformation probably since you could always choose $f_n = g$ And don't use $\ast$ notation, it is use for convolution, use \cdot instead – julio_es_sui_glace Jun 15 '23 at 17:18
  • Is the only information you have about $g$ it's moments? – julio_es_sui_glace Jun 15 '23 at 17:23
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    @JulesBesson Yes. – lee Jun 15 '23 at 17:26
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    Ok the affine transformation probably won't work, because for $n=3$ you must satisfy a system of $3$ equations with only two parameters, we must find fitting transformation that behave well with polynomial equations of order $\leqslant n$. You might have a hint working with Legendre polynomials. – julio_es_sui_glace Jun 15 '23 at 17:31
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    Also try to work with things like generative function or characteristic functions – julio_es_sui_glace Jun 15 '23 at 17:37
  • Note that you use of the word moment is significantly different from the standard usage. See this excellent thread for the standard use : https://math.stackexchange.com/q/1002695/307944 – Carl Christian Jun 16 '23 at 13:25
  • @CarlChristian edited so that I'm talking about central moment https://en.wikipedia.org/wiki/Central_moment – lee Jun 16 '23 at 15:16

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