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### 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/
```
Associated Topics:
College Statistics

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