[seqfan] Re: Chewing an old bone

Benoît Jubin benoit.jubin at gmail.com
Tue Oct 25 10:52:41 CEST 2011

David W, I agree that qlog is an injective function, whereas absolute
value is not (say, on Z), so you are right for the information
content.  But I think that qlog is not always better than log(|.|+1)
when it comes to visualization... and visualization is actually the
purpose of plots.

Charles's example above is the basic example, and conversely imagine
also this one: let a(n) be the sequence defined by:
a(n) = exp(3*n) if n even
a(n) = -exp(pi*n) if n odd
Then from a qlog plot one might conjecture that a(n) ~ (-1)^n*exp(c.n)
for some c, but on the log(|.|+1) plot one will see the small
discrepancy between even and odd arguments.  There are probably some
"real life" examples of such sequences already in the OEIS.

I think that these are the simplest examples, and that there are many
others where, roughly speaking, the log(|.|+1) plot makes it easier to
see some patterns or discrepancies in the scattering of dots than the
qlog plot, in particular regarding asymptotic behavior.  Actually, the
logarithmic scale is not the issue, here: sometimes, it's just more
useful to plot |a(n)| than a(n) -- but I am not suggesting to add this
one: I'm happy with the current graphs.

Finally, I still agree that the qlog plot is a very nice addition --
and I said: "addition"!


On Mon, Oct 24, 2011 at 8:31 PM, David Wilson <davidwwilson at comcast.net> wrote:
> On 10/24/2011 8:21 PM, David Applegate wrote:
>>> I really hope that the qlog plot will remain *supplementary* to the
>>> log plot (for large signed sequences), and will not replace it.  Each
>>> of these plots can be the more useful one depending on applications.
>> Do you have an example of when plotting log(|a(n)|+1) or
>> log(a(n)+1) is more informative than plotting qlog(a(n))?
>> David Applegate
> - For positive n, a qlog(n) plot retains all of the visual information of a
> log(|n|+1) plot.
> - qlog(n) is odd and monotonically increasing, hence order and sign
> preserving. In a
> qlog(n) plot, positive values lie above the x-axis, negative below.
> Increasing sequences
> trend upward, decreasing downward. None of this is true of the log(|n|+1)
> plot.
> - For all positive n, qlog(n) is closer to the holy grail, log(n), than is
> log(|n|+1).
> - Admittedly, the qlog(n) scale is not a standard plotting scale, but then,
> neither is
> the log(|n|+1) scale.  Apart from the OEIS, what other source uses the
> log(|n|+1) scale?
> Is it a builtin in any graphing package?
> For these reasons, the qlog(n) scale is the equal or superior of the
> log(|n|+1) in every
> respect. To answer David Applegate's question, there is no information in a
> log(|n|+1)
> plot that is not better represented in a qlog(n) plot. Indeed, this was my
> motivation for
> constructing the qlog(n) scale in the first place: to address the deficits
> of the log(|n|+1)
> scale in regard to plotting negative values, while retaining its virtues
> with regard to
> plotting large ranges (and by sheer luck, qlog(n) happens to be closer to
> log(n)).
> The new scale was intended to supplant the old, not to supplement it. If we
> adopt
> the qlog(n) scale, the log(|a(n)|+1) scale becomes superfluous and can be
> safely retired.
> There would be no technical reason to keep it, and its weakness with regard
> to plotting
> negative values would argue to retire it.
> In the end I suppose it is NJAS's decision. I hope he reads this mail.
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