Re: graphical analysis menagerie (was: Need Ideas)

[John Denker <jsd@MONMOUTH.COM>]

At 02:09 PM 12/28/99 -0500, Robert Carlson wrote:
> I have a new course starting next semester and need some ideas.  The course
> will concentrate on graphical analysis with some additional miscellaneous
> topics.  In general, the course will cover graphical techniques to find the
> relationship of a set of data pairs (not concentrating on theory)

> Possible Topics:
>
> Linear:
> Resistance as a function of length
> Constant velocity
> Z versus atomic number and n+p versus atomic number

1) Hmmmm.  The graph of Z versus atomic number should definitely be
linear.  Very, very linear. :-)

> Quadratic:
>
> Log-log:
>
> Semi-log:
>
> Sinusoidal:
>
> Miscellaneous:

2) Here's a contribution for the "miscellaneous" category.  If you take
ordinary Allen-Bradley carbon resistors and cool them down, the resistance
depends on temperature.  This makes them useful as secondary thermometers
in the range of roughly 77 Kelvin to maybe 10 milliKelvin.  There is a
large range where the resistance is
      exponential in the square root of temperature
to remarkably high precision.  No kidding.

Analyzing data of this form is an interesting challenge because
  a) It isn't exactly the first functional form that occurs to the typical
student.
  b) It is a clear exception to the rule-of-thumb that says "everything" can
be made to look linear by using log-log or semi-log paper.
  c) Even after you plot it on semi-log paper, you are left with something
(sqrt) that can't be well fitted by a polynomial (unless you transpose the
axes, which most people don't immediately do because they're stuck on the
idea that T is the *independent* variable).

If you are interested in this as a physics problem (as opposed to a
data analysis problem "not concentrating on theory") you can start by
reading up on Richardson's equation (although I'm not 100% convinced that
the resistor physics is the same as the thermionic physics that Sir Owen
Willans Richardson studied).

--------

3) Here's another example for you:  Consider the magnitude and phase of the
transfer function (i.e. output voltage over input voltage) of a simple
electrical circuit such as an RC or RCRC or RLC network.

There is (shall we say) a certain amount of real-world significance to such
functions.

There are about ten good reasons to plot these on log-log paper.

This is interesting because the result is not a straight line.  The
asymptotes are straight (which is one of the reasons for doing it this way)
but there's more to the story than the asymptotes.

This is an important contrast to the original list, which contained an
unbalanced overabundance of relationships which resulted in straight lines
when plotted a certain way.

================================

4) This brings us to a deeper issue.  I'm not at all convinced that it is
good science or good pedagogy to collect physical phenomenal and classify
them using terms like "log-log" or "semi-log".  I'm even skeptical that
such a list is a good way to organize a course on graphical analysis.

A)  Linear....

As the saying goes,
        "to first order everything is linear"
so, for instance, every quadratic relationship will have a range where it is
well approximated by a straight line.

[BTW, for huge extra credit, give an example from physics of a situation
where the given saying does not apply -- where there is *no* first-order
approximation for interesting and fundamental reasons.]

Conversely, with a few exceptions (mostly trivial, like plotting Z versus
atomic number) if you've got something that is linear over a certain range,
if you push it harder it becomes nonlinear.

So I'm not sure it is helpful to classify physical phenomena as "linear"
versus "nonlinear".  It is usually better to speak of a linear regime and a
nonlinear regime.

B)  Power Law....

Any power law looks straight on log-log paper.  This even includes the
first-order power law i.e. the "proportionality" relationship.  If you have
something that adheres to this humblest of relationships over a very wide
range, you might well want to plot it on log-log paper.

On the other hand, there are lots of power laws which, although you *could*
plot them log-log paper, you probably *should* not.

Here's a concrete example:  I once had some gas (monatomic hydrogen --
whee!!!) that was decaying with time according to a hyperbolic law:

       density = 1 over (constant + t)

Using a really sensitive pressure gauge, I could watch the decay.  Now in
favorable cases, the data was a really nice straight line if you plotted (1
over density) versus time.  But plotting it that way and fitting a straight
line to it would have been an incredibly dumb thing to do.  That's because
there was additive noise on the density signal.  So that leads to an
"equation" of the form

       1 over (density plus noise) ?=? constant + t

and in cases where the signal-to-noise ratio was approaching
unity, you could wind up taking the reciprocal of something that was
practically zero, and making an *infinitely* large mistake.

The correct solution was to write the equation the right way around,

       (density plus noise) = 1 over (constant + t)

and curve-fit to it using nonlinear techniques, and *then* plot the
residuals.

The choice of how to plot the function y(x) depends partly on the ideal
functional form of y(x), partly on how many orders of magnitude the data
covers, and partly on what sort of *nonidealities* you are looking
for.  Suppose there was (a) some random noise or (b) some interesting
hitherto-undiscovered systematic deviation from the expected power law --
how would this show up in your chosen plotting scheme?

Just grabbing some transformation that (in ideal cases) produces a straight
line is *not* good physics and nor good graphical analysis.

=============

To restate point (4) in more positive terms:  A better way of organizing a
tour of physics-functions might be something like this:

Monomial power laws:  constant, proportional, second-order, et cetera.

Polynomials:  constant, linear, quadratic, et cetera.

Orthogonal polynomials:  Hermite, Legendre, Tschebyshev, et cetera.

Rational functions (such as the RLC circuits).

Fractional or mixed power laws (e.g. Kepler's 1-2-3 law, renormalized
critical exponents).

Exp() and its friends:  exp(), log(), sinh(), cosh(), sin(), cos(), et cetera.

Elliptic integrals and elliptic functions (e.g. pendulum with
non-infinitesimal amplitude).

Commonly-used eigenfunctions and Green functions (e.g. Bessel functions,
spherical harmonics, et cetera).

........ and then you still have to think.  For example, if you have what
appears to be a pure second-order power law, you should ask yourself
whether it is really a second-order polynomial with a first-order term that
is hiding somewhere.  Did the first-order term vanish by symmetry, or did
you just overlook it?  Why?  Graphical analysis won't answer such questions
for you.

==========

Just to come at point (4) from yet another angle:  there are several quite
distinct tasks:
  a) recognizing what function you've got,
  b) curve fitting, data reduction, and error analysis, and
  c) deciding how to present the data and/or residuals (log-log paper, pie
chart, or whatever) in a way that helps with (a) and/or (b).

There are big fat classic books on each of these subjects.

=======================================

5) In general, I am greatly troubled by the statement:

> the course will cover graphical techniques to find the
> relationship of a set of data pairs (not concentrating on theory)

Such an approach (munging the data with minimal theoretical guidance) is a
bad idea in typical cases -- and in the general case it is PROVABLY
IMPOSSIBLE to find a relationship without *crucially* depending on
theoretical guidance to narrow the search-space.   If you don't believe me,
try analyzing the very simple data at
  http://www.monmouth.com/~jsd/physics/sin-data.gif

I'll even give you a big hint:  the data is well represented by a sine
wave;  all you need to do is tell me the amplitude, frequency, and
phase.  I'll bet you will quickly come up with a solution that looks a lot like
  http://www.monmouth.com/~jsd/physics/sin-fit.gif

which is a good fit, with only a small amount of residual noise.  OK, are
you with me so far?  Is everybody happy?  Well, then, please take a look at
  http://www.monmouth.com/~jsd/physics/sin-otherfit.gif

and tell me if you think this is or is not a better fit.

The point is that given the data by itself, without *crucial* additional
guidance as to what the data should look like and what the noise should
look like, graphical analysis provably has no chance of saying what is the
"correct" description of the data.  If you want to see the full proof, look
in Vladimir Vapnik's books.

----

The role of theory in graphical analysis is like the role of water in
biochemistry -- you're not going to get very far without it.  It is so
important that among experts it might even go without saying -- but when
talking to beginners you'd better cover it in depth or there will be hell
to pay later.