log y 'axis' of histogram

Hadley,

I have counts ranging over 4-6 orders of magnitude with peaks
occurring at various 'magic' values.  Using a log scale for the
y-axis enables the smaller peaks, which would otherwise
be almost invisible bumps along the x-axis, to be seen

That doesn't justify the use of a _histogram_  - and regardless of

The usage highlights meaningful characteristics of the data.
What better justification for any method of analysis and display is
there?

what distributional display you use, logging the counts imposes some
pretty heavy restrictions on the shape of the distribution (e.g. that
it must not drop to zero).

Does there have to be a recognized statistical distribution to use R?
In my case I am using R for all of the analysis and graphics in a
new book.  This means that sometimes I have to deal with data sets
that are more or less a jumble of numbers with patterns in a few
places.  For instance, the numeric value of integer constants
appearing as one operand of the binary bitwise-AND operator (see
figure 1224.1 of www.knosof.co.uk/cbook/usefigtab.pdf, raw data
at: www.knosof.co.uk/cbook/bandcons.hist.gz)

qplot(band, binwidth=8, geom="histogram") + scale_y_log()
does a good job of highlighting the peaks.

It may be useful for your purposes, but that doesn't necessarily make
it a meaningful graphic.

Doesn't being useful for my purpose make it meaningful, at least for me
and I hope my readers?

Hadley is correct about the problem of where to end the bars when trying 
to draw a log-histogram: basically you have to decide to cut them off 
somewhere. He is also right that a log-histogram is perhaps not a great 
graphic to use. However, they are used and indeed there is one in the 
Fieller, Flenley, Olbricht paper (published in Applied Statistics, now 
JRSS C) for example. I haven't searched for others, but certainly when I 
wrote a log-histogram routine it wasn't because I thought of doing such 
a plot all on my own.

A number of authors, including Barndorff-Nielsen in at least some of his 
papers (I haven't gone back and checked all his older work) just plot 
the midpoints of the tops of the log-histogram. (That is an option in 
logHist). Another approach is to fit an empirical density to the data 
and plot the log-density. That matches the advice often seen in this 
forum that plotting empirical density functions is preferable to drawing 
histograms. My feeling is that either of these two approaches is 
probably preferable to using log-histograms for the reasons Hadley 
enunciated. When plotting data plus a fitted curve, the midpoints 
approach does have the advantage of distinguishing data and theoretical 
curve more clearly.

Overall the idea of a plot with a logged y-axis is definitely a good one 
and its use is endemic in literature concerned with heavy-tailed 
distributions, particularly finance. The advantage is the clarity 
offered regarding tail behaviour, where for example exponential tails in 
the density correspond to straight lines in the logged y-axis plot.

Hope this helps.

David Scott

_________________________________________________________________
David Scott	Department of Statistics
		The University of Auckland, PB 92019
		Auckland 1142,    NEW ZEALAND
Phone: +64 9 923 5055, or +64 9 373 7599 ext 85055
Email:	d.scott at auckland.ac.nz,  Fax: +64 9 373 7018

Director of Consulting, Department of Statistics