Noisy Data Known to Estimate a Function

Date: 05/15/2003 at 20:56:37
From: Kharem Espinoza
Subject: Finding the maximum value of a function with noise

How do we find the maximum value of a function with a high level of 
noise?

Date: 05/16/2003 at 18:51:46
From: Doctor Douglas
Subject: Re: Finding the maximum value of a function with noise

Hi Kharem,

Thanks for writing to the Math Forum.

Do you mean that you have a set of data (which are noisy) that are 
known to estimate some function and you are trying to find the maximum 
of the function?  I say that the data are "known to estimate a 
function" so that you might not want to simply take the maximum of the 
data set, which may be affected by the noise.

Here are some ways you could proceed:

  1. you could just take the maximum of the data and live with
     the noise.

  2. you could "fit" the data in order to obtain a mathematical 
     function (that might be differentiable, or otherwise easy 
     to extract the maximum), given what you know about the function
     that describes the data.

  3. you could interpolate the data with a polynomial, choosing
     the lowest order consistent with the number of points.  Then
     you could find the maximum from the polynomial.

I hope this helps you get started. There are some delicate issues
that arise in fitting and interpolation, and these are compounded
by the problem of noise.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/