Re: Axis rescaling

[James R. V. Z. <jr...@co...>]

I've just uploaded a patch that implements probability axes:

1757226  	  Provide "probability" scaling and axes    	   2007-07-19 21:31  	 5  	nobody  	vanzandt

Rationale:

Probability axes simplify the presentation of certain kinds of data.

Recall that if Y varies as the exponential of X, then data values are
best shown in a log plot - i.e. where you plot the logarithm of Y
against X.  More generally, if there are very small and very large Y
values, but all are positive, then it is convenient to use a log Y
axis.  E.g. if you plot Y values of .002, .02, and 200 along a linear
axis, it would be hard to tell the first two apart.  However, it would
be easy with a log axis.

Suppose you have a collection of n data values that you think are
normally distributed.  Find the "empirical distribution function": a
cumulative probability distribution function that concentrates
probability 1/n at each of the n numbers in the sample.  I.e. sort the
values and label them X(1) through X(n).  Then let Y(k)=k/n.  If you
plot Y vs. X, you get something like a sigmoid curve, which starts
near the line Y=0, rises quickly near the mean of the X values, then
gradually approaches the line Y=1.  Probability axes are the way of
transforming Y values such that, if the X values were really normally
distributed, then the transformed data points fall approximately along
a straight line.

More generally, if there are both very small Y values and values very
near 1, but none outside the range 0 to 1 (e.g. if Ys represent
probabilities), then probability axes can make the data easier to see.

Summary of changes: Most of the added code is in the new files
"transform.c" and the corresponding header file "transform.h".

The "probability" transformation (inverse of normal CDF) is the only
one implemented here.  However, the logic for labeling the axis is
quite general.  The goal is to select a set of tics and minitics such
that:
 - tic labels are "simple"
 - tic labels do not overlap
 - the distances between tics are roughly equal
 - the numeric value corresponding to any minitic is unambiguous
 - the distances between minitics are roughly equal, and small enough
   that the numeric value of any point can be estimated by eye.

The current code for log axes uses the functional forms for both the
data transform log(x) and its inverse 10^x.  This patch for
probability axes does the same.  However the part that calculates tic
and minitic locations uses the function form only for the forward
transformation -- inverse values are found numerically.  My hope is
that this more general implementation can be easily applied to other
transform functions, including user-supplied ones without a convenient
inverse.

axis_array[axis].log is now an enum, with values SCALE_LINEAR,
SCALE_LOG, and SCALE_PROBABILITY, and AXIS_LOG_VALUE and
AXIS_DE_LOG_VALUE are functions instead of macros.

TODO: 
 - Rename axis_array[axis].log to axis_array[axis].scale
 - Convert names AXIS_LOG_VALUE and AXIS_DE_LOG_VALUE to lower case.
 - Implement several other kinds of axes, including "weibull plots".
 - Implement a user-supplied scaling function.
 - Implement scaling only of axis labels, without scaling the data.
 - New command syntax [axis] "set scale linear|log|probability|f(x) x|y|..."


Ethan Merritt wrote:
> > > 
> > > My request was simply that you leave the existing log scale code
> > > in place until a general mechanism was ready to replace it.
> > 
> > What do you mean by general?  Able to apply mappings other than
> > log scale?
>
> Exactly.  I gave a bunch of examples earlier.  The most common
> request (and hardest to work around) is marking off the X-axis as
> 1/x.  For instance, we crystallographers have to deal with
> inconsistent usage all the time; some quantities are given in terms
> of energy while others are given in terms of wavelength.  There is
> an inverse relationship:
>	   Energy = k / Wavelength
> where k is an appropriate constant.  It is a pain to try to persuade
> gnuplot to mark off both Energy and Wavelength on axes of the same plot.
>
> I would like to be able to do something like:
>
>	   set axis x1  x                 # will use for Energy
>	   set axis x2  k/x               # will use for Wavelength
>	   set axis y1  log(y)            # log scale value to be plotted
>
> Even better if there is a way to automatically lock x1 to x2, so that
> they are forced to span the same range.
>
> -- 
> Ethan A Merritt

Ethan wants to rescale one or both axes without changing the data.  I
would like to think that this patch is a first step toward that
capability.  The patch rescales the data as well as the axes.
At least, it could easily handle the tic/minitic placement.

(By the way, I do have CVS write access, but I figured this would be
too big a change to apply to the head.  I am not comfortable enough
with my CVS skills to start a new branch, let alone to try and merge
later on.)

              - Jim Van Zandt