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From: "John D'Errico" <woodchips@rochester.rr.com>
Newsgroups: comp.soft-sys.matlab
Subject: Re: the mathematic relationship between two series of data
Date: Sun, 13 Jul 2008 09:32:27 +0000 (UTC)
Organization: John D'Errico (1-3LEW5R)
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"ZHANG Hong" <oceanzhhd@gmail.com> wrote in message 
<g5cf8i$eg8$1@fred.mathworks.com>...
> Hi, Mat,
> 
> Thank you very much for your help and it really works out.
> 
> I feel puzzled whether when we had got the scatter point 
> graph of two datasets, if we want to find a function to 
> feed it, it is to some extent depends on our experience. Is 
> that right or any factors can be considered?

I looked at your data. Your question is a not
uncommon one at all. What is "the" function
that represents my data?

The problem is that you need to bring along
much information that you have as the person
who generated this data, and as the person
who has a need to find a model for the data.

- Is there noise in your measurements?

- Is that noise significant, and must it be
smoothed out?

- What form of a function is acceptable to you?

- Will the resulting model be used for simple
prediction, or do you wish to then write down
that model, and study perhaps for a paper?

- What are your needs for the resulting model? 

- How accurately must the model fit your data?

- What assumptions are you willing to make
about that model?

- What knowledge do you have about the
system that generated this data? For
example, do you know it to be monotone?

- Will you try to extrapolate this curve to
some point(s)?

Some questions that are specific to the data
you listed might focus on what I saw when I
plotted it.

- It appears that the curve might have a
small break in the derivative near the middle.
Is this something that you know exists, or is
that merely noise? I've often seen artifacts
like this created when an instrument is
recalibrated in the middle of an experiment.

There are entire realms of mathematics that
try to deal with these questions, and the
issues that arise from those questions. You
may find those realms referred to by the
various names modeling, approximation,
curvefitting, and interpolation. In fact, there
are complete toolboxes from the MathWorks
that attempt to help you with these problems,
in the form of the splines toolbox, as well as
the curvefitting, optimization, neural net
toolboxes, etc. You will also find large
numbers of submission on the file exchange,
entire categories of tools, that will help you 
too.

But first, you must resolve some of the
questions I've posed above.

John