Integral equivalent to fitting a curve to a sum of functions

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Discussion Overview

The discussion revolves around the search for a mathematical technique that allows for a continuous equivalent of fitting a curve to a sum of functions, specifically using variable kernels in an integral form. Participants explore concepts related to transforms, convolutions, and Gaussian functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks a transform akin to the Fourier transform but applicable to an arbitrary function with a variable kernel, suggesting an integral form involving Gaussians.
  • Another participant questions the clarity of the problem, specifically regarding the status of the function t(y) and whether it is given or needs to be determined.
  • A later reply confirms that both t(y) and f(y) are functions to be found, indicating a need for clarity in the formulation of the problem.
  • One participant suggests that the Weierstrass transform may be relevant, noting its general invertibility and the implication that there may not be enough information to uniquely determine t(y).
  • A reference to heteroscedastic Gaussian Processes is provided as a potential avenue for exploration, although its relevance to the original question is uncertain.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the formulation of the problem and the information available, particularly about the function t(y). There is no consensus on how to proceed or whether the Weierstrass transform adequately addresses the inquiry.

Contextual Notes

The discussion highlights limitations in the information provided, particularly concerning the functions t(y) and f(y), and the implications of the Weierstrass transform's invertibility on the ability to solve the problem.

admixtus
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Hello,

I am searching for some kind of transform if it is possible, similar to a Fourier transform, but for an arbitrary function.

Sort of an inverse convolution but with a kernel that varies in each point.

Or, like I say in the title of this topic a sort of continuous equivalent of fitting a curve to a sum of functions.

For example if I want to use Gaussians, I want to reproduce a function F(x)

As:

F(x) = \int \frac{f(y)}{\sqrt{4\pi t(y)}}e^{-\frac{(x-y)^2}{4 t(y)}} dy

Notice how t is a function of y.
This is easy for a finite sum of Gaussians with linear regression, but I'm searching for a continuous equivalent.

The closest thing that I found for Gausses is a Weierstrass transform. But the 'standard deviation' of the gausses doesn't vary in each point.

There are a ton of subjects that come close (linear regression, inverse convolution, Weierstrass transform,..) but they either are discrete or lack the variability of the convoluting kernel.

Does someone know a mathematical technique that can do this? Or know in what direction I have to look? Thanks!
 
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I'm not quite clear on what is given. Obviously F is given, and you want to find f, but how about t? Is t(y) a given function?
 
haruspex said:
I'm not quite clear on what is given. Obviously F is given, and you want to find f, but how about t? Is t(y) a given function?

Yes, t(y) and f(y) are functions that I want fo find, yes. Maybe I should have written it explicitly like that instead of implying it by saying the the kernel was variable.
 
admixtus said:
Yes, t(y) and f(y) are functions that I want fo find, yes. Maybe I should have written it explicitly like that instead of implying it by saying the the kernel was variable.
From my reading of the subject (totally new to me until I saw your post) the Weierstrass transform is exactly that, a transform, so is, generally speaking, invertible. This means there is not enough information to find t. Your mission would make more sense if t(y) were given. Am I missing something?

Not sure if this is what you are after, but look at the discussion of heteroscedastic Gaussian Processes at https://www.cs.cmu.edu/~andrewgw/andrewgwthesis.pdf
 
Last edited:

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