A definitive test of discrete scale (relativity, numerology)

I have decided to take two hours away from dead island and do something
equally productive with my time: test the numerology of Robert
Oldershaw.

Exhibit "A" is the VizieR catalog "J/A+A/352/555/table1", sourced from
the Hipparcos catalog [1]. I am sure Robert is working on this very
thing, as he has had more than a decade to find this publication, and
about 2 weeks to find this catalog after I have first given it to him. I
have only given myself two hours because dead island is fun but I only
need a break rather than a sabattical.

The quality of the mass data is rather impressive. Statistics:

12,090 stars to 10% or better.
3028 stars to 5% or better.
185 stars are known 1% or better.
30 with zero error :P
641 stars known to 0.02 M_sun or better.

Procedu

* Yoink the data off VizieR.
* Export to ascii with the only meaningful/relevant parameters being
mass and the uncertainty of the measurement.
* Grep out invalid entires.

I, personally, find zero error unlikely.

Maybe they are good, maybe not. Don't feel like making big effort for
0.17% of the catalog.

* Take the mass of each star, and subtract off the nearest integer
multiple of 0.145. A computational step, not a physics step.

What survives is the absolute difference from the predicted
quantization, and the error in the measurement.

I tried plotting out the 1% data with gnuplot, and even that small block
of quality data is visually _meaningless_. Overlapping error bars make
it impossible to get anything of value out of the data that way.

So might as well go all the way and use all the stars where the masses
are determined to 5% or better. Five percent or so, is about as large as
you can go because the average mass of a star in this catalog is 1.3
M_sun and I'd like the results to be useful to at least one standard
deviation.

Results of DOES_STUFF.PL [2]:

3028 stars with masses determined to 5% or better:
37 exactly as predicted
1938 within 1 standard deviations
630 off by 1 to 2 standard deviations
215 off by 2 to 3 standard deviations
115 off by 3 to 4 standard deviations
40 off by 4 to 5 standard deviations
53 off by more than 5+ standard deviations
Average standard deviation per star: 1.09
Average mass of star: 1.43 solar masses

Seeing that about half the data is wrong by a standard deviation or more
isn't all that telling given that the binning being tested for is only
twice the average error of the sample. Let's go deeper.

On a related note, Robert has frequently complained that rand("he can't
find", "there doesn't exist", "he hasn't bothered to look for") a
dataset that has masses determined to 1% or better. Negative twelve
years after making that complaint, science has finally come through for
him!

Re-run with $cutoff = 0.01:

185 stars with masses determined to 1% or better:
3 exactly as predicted
42 within 1 standard deviations
36 off by 1 to 2 standard deviations
28 off by 2 to 3 standard deviations
23 off by 3 to 4 standard deviations
16 off by 4 to 5 standard deviations
37 off by more than 5+ standard deviations
Average standard deviation per star: 2.97
Average mass of star: 1.35 solar masses

Now all the subtleties of averaging a sample of data aside, I think it
is pretty safe to say Robert's numerology is wrong when 75% of a
precisely determined sample of relevant data is more than one standard
deviation different from the predicted binning. Also when the _average_
standard deviation per sample is 3. Sure, there could be a corner case
or twenty in there that could be jacking up the average by a large
amount but I argue that tearing apart the data is an exercise for the
reader at this point.

None of this includes other data sets such as the various smaller sets
such as ones that include eclipsing binaries, which are determined to
fractions of a percent, or the Sun which has already been determined to
not agree with Robert's numerology by 100 standard deviations and
change.

The catalog seems to cut out at about 0.88 solar masses on the low side,
so small mass stars aren't really present here. There are other catalogs
of low mass stars that Robert can ignore, so it isn't really an issue.

Note that only 37 out of 3,028 stars are binned exactly as predicted. I
doubt I will hear a sarcastic rendition of my mathematical point of the
gauranteed existence of some matches, with such a small number.

I figure taking 2 hours, wasting half of it re-learning gnuplot and
realizing that there's no meaningful way to visualize this data, and
spending the other half doing something useful with it, is a better
investment of my time than repeatedly saying "HEY! HERE'S THE DATA! STOP
CRYING AND BE THE SCIENTIST YOU CLAIM TO BE."

I am going back to dead island so I can light undead minorities and
Australians on fire. [3]

----

[1] : "Fundamental parameters of nearby stars from the comparison with
evolutionary calculations: masses, radii and effective temperatures.",
Allende Prieto C., Lambert D.L., Astron. Astrophys. 352, 555 (1999)

[2] :

#!/usr/bin/perl

use warnings;
use strict;
use Math::Round ':all';
open (DATA, "stars2.txt");
my $cutoff = 0.01;
my $percent = 100 * $cutoff; my $zero = 0; my $one = 0;
my $two = 0; my $three = 0; my $four = 0; my $five = 0;
my $morethanfive = 0; my $totaldeviations; my $count; my $totalmass = 0;

while (my $line = DATA) {
my @data = split(' ', $line);
my $mass = $data[2]; my $mass_error = $data[3];
my $multiple = round( nearest(0.145, $mass) / 0.145 );
my $residual = $mass - ($multiple * 0.145);
my $truncated_residual = substr($residual, 0, 5);
my $deviations = substr( abs($residual / $mass_error), 0, 4);
if ($mass_error / $mass le $cutoff) {
$totaldeviations += $deviations;
$count++;
$totalmass += $mass;
if ($deviations eq 0) { $zero++; next; };
if ($deviations le 1 && $deviations ne 0) { $one++;
next; };
if ($deviations le 2 && $deviations gt 1) { $two++;
next; };
if ($deviations le 3 && $deviations gt 2) { $three++;
next; };
if ($deviations le 4 && $deviations gt 3) { $four++;
next; };
if ($deviations le 5 && $deviations gt 4) { $five++;
next; };
if ($deviations gt 5) { $morethanfive++; next; };
}
}
my $average_deviations = substr($totaldeviations / $count, 0, 4);
my $average_mass = substr($totalmass / $count, 0, 4);
print "$count stars with masses determined to $percent% or better:
$zero exactly as predicted
$one within 1 standard deviations
$two off by 1 to 2 standard deviations
$three off by 2 to 3 standard deviations
$four off by 3 to 4 standard deviations
$five off by 4 to 5 standard deviations
$morethanfive off by more than 5+ standard deviations
Average standard deviation per star: $average_deviations
Average mass of star: $average_mass solar masses
";

[3] : Fire is the cleanser.

In article ,
eric gisse wrote:
I have decided to take two hours away from dead island and do something
equally productive with my time: test the numerology of Robert
Oldershaw.

To help people doing this sort of work: there's a standard way of
saying whether a given number of standard deviations away from a model
is significant, the chi^2 test. Eric's code almost does this already,
but below[1] is a slightly modified version which computes the chi^2
statistic, which is simply the sum of the squares of the deviations
over the errors. The higher chi^2 is, the worse the model agrees with
the data. The advantage of doing that is that the probability of
obtaining a value of chi^2 as extreme, or more extreme than the one
that you actually see under the 'null hypothesis' that the data are
actually consistent with the model can be computed: if this
probability p is low, then we say that the model is ruled out at the
(1-p) confidence level. Various online calculators or the Perl module
Statistics:istributions can be used to do this[2]. Thus it's possible
to show, for example, that taking the stars with 5% or better errors
on masses, the model is ruled out at the 99.9999999999% confidence
level. Similar tests could be done with other databases.

[1]
#!/usr/bin/perl

use warnings;
use strict;
use Math::Round ':all';
use Statistics:istributions;

open (DATA, "stars2.txt");
my $chi2=0;
my $count=0;
my $cutoff=0.05;
my $confidence=1e-12;

while (my $line = DATA) {
# $line just contains mass and error
my @data = split('\s+', $line);
my $mass = $data[0]; my $mass_error = $data[1];
if ($mass_error/$mass=$cutoff) {
my $multiple = round( nearest(0.145, $mass) / 0.145 );
my $residual = $mass - ($multiple * 0.145);
my $deviation = $residual / $mass_error;
$chi2+=$deviation**2;
$count++;
}
}

print("Chi^2 is $chi2 for $count stars.\n");
printf("Probability under null hypothesis: %g\n",Statistics:istributions::chisqrprob($count ,$chi2));
printf("(chi^2 required to rule out null hypothesis at (1-%g)\n confidence level is %g)\n",$confidence,Statistics:istributions::chis qrdistr($count,$confidence));

[2] I haven't used this module before and can only say that the
numbers it produces look entirely reasonable.


--
Martin Hardcastle
School of Physics, Astronomy and Mathematics, University of Hertfordshire, UK
Please replace the xxx.xxx.xxx in the header with herts.ac.uk to mail me

The problem is that all those stars could have invisible companions.

That is why I looked at Alpha centauri system, since I thought that in
such a close star that problem would be solved...

As I see it, this prediction is impossible to verify really. As far
as the data that eric shows, the answer is negative: there is no quanta
of matter when star systems are measured.

Is that the final truth?

We will know when we know for sure the exact characteristics of
thousands of systems. (And I mean "exact" i.e. when we can rule out any
unseen companions, we know the masses of all the planets, etc)

On Sep 18, 5:43*am, Martin Hardcastle
wrote:

print("Chi^2 is $chi2 for $count stars.\n");
printf("Probability under null hypothesis: %g\n",Statistics:istributions::chisqrprob($count ,$chi2));
printf("(chi^2 required to rule out null hypothesis at (1-%g)\n * * confidence level is %g)\n",$confidence,Statistics:istributions::chis qrdistr($count,$confidenc*e));

[2] I haven't used this module before and can only say that the
numbers it produces look entirely reasonable.
--------------------------------------------------------------------------------------------------------------

I am currently looking at the paper "Accurate masses and radii of
normal stars: Modern results and applications" by Torres, Andersen and
Gimenez.

This catalog is available at VizieR, and published Astronomy &
Astrophysics Review, vol. 18, 2010.

It is also available at arxiv.org (search Torres, 2009, astro-ph)

This sample is heavily weighted toward "massive" stars, but it might
yield some interesting results in the 1 to 4 solar mass range for
total system masses.

An independent analysis of the data would be most welcome.

RLO
Fractal Cosmology

Le 18/09/11 18:53, jacob navia a écrit :
The problem is that all those stars could have invisible companions.


And the problem could be also that they have LOST visible companions.
A couple of stars can be destroyed in gravitational interactions when
the couple is born, both of them becoming lone stars and leaving
no trace of their common origin...

See:
http://www.sciencedaily.com/releases...0915083715.htm
or the scientific article:
http://arxiv.org/abs/1109.2896

I think that proving or disproving Robert's hypothesis could be VERY
difficult. The only way to know if he is right would be to weight star
systems when they are just born...

Martin Hardcastle wrote in
:

In article ,
eric gisse wrote:
I have decided to take two hours away from dead island and do
something equally productive with my time: test the numerology of
Robert Oldershaw.

To help people doing this sort of work: there's a standard way of
saying whether a given number of standard deviations away from a model
is significant, the chi^2 test. Eric's code almost does this already,
but below[1] is a slightly modified version which computes the chi^2
statistic, which is simply the sum of the squares of the deviations
over the errors.

[...stuff...]

Irritating perl bug: The value of $residual is going to be negative a
significant portion of the time, but squaring the negative quantity
triggers some sort of overflow/insanity. Might need to do some
reporting/squashing after this.

After calculating the chi squared of the sample, I noticed I was getting
values north of 10^22. I found that to be odd.

It turns out that perl does not square a negative number correctly. So I
have to add yet another hackish/goofy computational protection in the
form of abs($residual) so the squaring doesn't go nutbar.

Now for the chi squared test, let's do that right.

Given most of my experience is with ye olde standard deviation and
associated distribution widgetry, my knowledge of the test has been
limited to "what I learned in school and forgot, and the residual notion
that a large chi squared means the result is crap".

Re-reading my statistical analysis textbook (Taylor, haven't opened it
in about 2 years) and the discussion of this, it turns out that this is
a _far_ better test of a distribution hypothesis rather than what I was
doing. I was merely generalizing my previous behavior of 'take the
latest example, and testing whether it works' which is fine for the
small scale but does not generalize.

Now, you don't want just the regular chi-squared value. That isn't
useful. What is wanted is the reduced chi-squared value, which is chi-
squared divided by the number of degrees of freedom.

For this, the hypothesis is a binning of 0.145 solar masses. This
requires figuring out the amount of bins, and can't be done beforehand
because the data set covers a wide range of masses with a wider range of
confidence.

Now this may be a point of contention, but I argue the degrees of
freedom is not the amount of stars themselves but rather the amount of
bins of 0.145 M_sun required to cover the mass range. This is actually
more generous to the testing, given that more degrees of freedom makes
it more likely that the hypothesis is true. Turns out to not matter
much, since the answer is "zero" for meaningful chunks of the data.

Knowing the probability that the resultant reduced chi squared is going
to be larger than chi squared for a certain amount of degrees of freedom
requires a computation using a variation on the gamma function. Screw
that, that is what CPAN is for. The suggested module does that!

Putting all this together, i have DOESSTUFF_mod2.pl

It turns out that the hypothesis is so wrong that it does not matter
what data quality cutoff I use.

BEEP BOOP...analyzing 17187 stars with masses determined to 100% or
better

Average standard deviation per star: 0.48
Average mass of star: 1.28 solar masses
Mass range of sample: 0.88 to 7.77 solar masses
Chi-squared of the expected binning hypothesis: 35097
Reduced chi-squared: 731.1875
The probability that the reduced chi-squared value
of 731.1875 is larger than the value of 35097
for 48 degrees of freedom is 0.

BEEP BOOP...analyzing 185 stars with masses determined to 1% or better

Average standard deviation per star: 2.97
Average mass of star: 1.35 solar masses
Mass range of sample: 1.00 to 4.63 solar masses
Chi-squared of the expected binning hypothesis: 537
Reduced chi-squared: 21.48
The probability that the reduced chi-squared value
of 21.48 is larger than the value of 537
for 25 degrees of freedom is 0.

BEEP BOOP...analyzing 10 stars with masses determined to 0.6% or better

Average standard deviation per star: 2.99
Average mass of star: 2.07 solar masses
Mass range of sample: 1.71 to 4.63 solar masses
Chi-squared of the expected binning hypothesis: 26
Reduced chi-squared: 1.3
The probability that the reduced chi-squared value
of 1.3 is larger than the value of 26
for 20 degrees of freedom is 0.16581.

The only remarkable thing here is that there is a 4.63 M_sun determined
to 0.6%!

Sorry Robert. The probability that your theory is right is competing
with the limits of floating point math.

--------------

#!/usr/bin/perl

use warnings;
use strict;
use Math::Round ':all';
use Statistics:istributions;

open (DATA, "stars2.txt");
my $cutoff = 1; my $percent = 100 * $cutoff;
my $totaldeviations; my $count; my $totalmass = 0;
my $chisq = 0; my @stars;

while (my $line = DATA) {
my @data = split(' ', $line);
my $mass = $data[2]; my $mass_error = $data[3];
my $multiple = round( nearest(0.145, $mass) / 0.145 );
my $residual = $mass - ($multiple * 0.145);
my $truncated_residual = substr($residual, 0, 5);
my $deviations = substr( abs($residual / $mass_error), 0, 4);
if ($mass_error / $mass le $cutoff) {
$totaldeviations += $deviations;
$count++;
$totalmass += $mass;
$chisq += abs($residual / $mass_error)^2;
push (@stars, $mass);
}
}

my @sorted = sort @stars;
my $largest = $sorted[-1]; my $smallest = $sorted[0];
my $binning_largest = nearest_ceil(0.145, $largest);
my $binning_smallest = nearest_floor(0.145, $smallest);
my $degrees = ($binning_largest - $binning_smallest) / 0.145;
my $reduced_chisq = $chisq / $degrees;
my $probability = Statistics:istributions::chisqrprob ($degrees,
$chisq);
my $average_deviations = substr($totaldeviations / $count, 0, 4);
my $average_mass = substr($totalmass / $count, 0, 4);
print "BEEP BOOP...analyzing $count stars with masses determined to
$percent% or better

Average standard deviation per star: $average_deviations
Average mass of star: $average_mass solar masses
Mass range of sample: $smallest to $largest solar masses
Chi-squared of the expected binning hypothesis: $chisq
Reduced chi-squared: $reduced_chisq
The probability that the reduced chi-squared value
of $reduced_chisq is larger than the value of $chisq
for $degrees degrees of freedom is $probability.
";

"Robert L. Oldershaw" wrote in
:

On Sep 18, 5:43*am, Martin Hardcastle
wrote:

print("Chi^2 is $chi2 for $count stars.\n");
printf("Probability under null hypothesis:
%g\n",Statistics:istributions::chisqrprob($count ,$chi2));
printf("(chi^2 required to rule out null hypothesis at (1-%g)\n * *
confidence level is
%g)\n",$confidence,Statistics:istributions::chis qrdistr($count,
$conf
idenc*e));

[2] I haven't used this module before and can only say that the
numbers it produces look entirely reasonable.
----------------------------------------------------------------------
-
---------------------------------------

I am currently looking at the paper "Accurate masses and radii of
normal stars: Modern results and applications" by Torres, Andersen and
Gimenez.

This catalog is available at VizieR, and published Astronomy &
Astrophysics Review, vol. 18, 2010.

It is also available at arxiv.org (search Torres, 2009, astro-ph)

This sample is heavily weighted toward "massive" stars, but it might
yield some interesting results in the 1 to 4 solar mass range for
total system masses.

Why would you think it'd tell you anything we don't already know? ~
17,000 stars, the majority within that mass range, explicitly disproves
your theory.

Another hundred won't change anything.


An independent analysis of the data would be most welcome.

How fortunate for you that someone did your research for you. There's a
program right there for you to analyze the data with.


RLO
Fractal Cosmology

jacob navia wrote in news:mt2.0-20923-1316381120
@hydra.herts.ac.uk:

Le 18/09/11 18:53, jacob navia a écrit :
The problem is that all those stars could have invisible companions.


And the problem could be also that they have LOST visible companions.
A couple of stars can be destroyed in gravitational interactions when
the couple is born, both of them becoming lone stars and leaving
no trace of their common origin...

See:
http://www.sciencedaily.com/releases...0915083715.htm
or the scientific article:
http://arxiv.org/abs/1109.2896

I think that proving or disproving Robert's hypothesis could be VERY
difficult. The only way to know if he is right would be to weight star
systems when they are just born...


The 12k star sample have their masses determined spectroscopically.
Besides, Robert has argued that invididual stars are quantized in mass in
addition to the systems themselves.

Yes, really.

Not my theory, I don't have to justify it.

In article ,
eric gisse wrote:
Irritating perl bug: The value of $residual is going to be negative a
significant portion of the time, but squaring the negative quantity
triggers some sort of overflow/insanity. Might need to do some
reporting/squashing after this.

Not in the perl I use! (I did test the script I posted...) That would
be a very serious and basic bug if it were really present!

I think you are mistaking the operator ^ for the operator ** (see my
code). ^ in perl is not 'raise to power' but 'bitwise xor'. ** is the
exponentiation operator.

Now, you don't want just the regular chi-squared value. That isn't
useful. What is wanted is the reduced chi-squared value, which is chi-
squared divided by the number of degrees of freedom

Not so. The reduced chi^2 is a useful way of telling at first glance
whether there is something wrong with a model: a reduced chi^2 much
greater than 1 indicates a problem. However, the reduced chi^2 doesn't
tell you everything that you want to know, quantitatively -- a reduced
chi^2 of 2 for 2 degrees of freedom is very much less interesting than
a reduced chi^2 of 2 for several thousand degrees of freedom. So in
fact the standard thing to do is use the chi^2 value itself and
calculate (or look up) the critical value for a given number of
degrees of freedom. You can do the equivalent thing for a reduced
chi^2, so it's not a big deal, but chi^2/d.o.f. is what people quote
in the astrostatistics literature, so it's what I used.

For this, the hypothesis is a binning of 0.145 solar masses. This
requires figuring out the amount of bins, and can't be done beforehand
because the data set covers a wide range of masses with a wider range of
confidence.

Now this may be a point of contention, but I argue the degrees of
freedom is not the amount ouf stars themselves but rather the amount of
bins of 0.145 M_sun required to cover the mass range. This is actually
more generous to the testing, given that more degrees of freedom makes
it more likely that the hypothesis is true. Turns out to not matter
much, since the answer is "zero" for meaningful chunks of the data.

I don't think this is right. The degrees of freedom is the number of
data points, minus the number of free parameters of the model (none in
this case): see e.g.
http://en.wikipedia.org/wiki/Chi-square_statistic. So I don't think
your numbers are correct (in particular, it would seem very bizarre
if, as you suggest, including stars with larger errors caused the
model to be ruled out more stringently... in fact, if you include all
the stars with huge errors *and* calculate the chi^2 correctly, you
should find an acceptable fit, but that's only because you'd be
diluting the stars that can actually constrain the model with the many
more that can't).

However, the key point is that this test can be done, and, when it's
done with stars with accurately measured masses, it is inconsistent
with the proposed model at a very high confidence level, as I said
earlier.

Martin
--
Martin Hardcastle
School of Physics, Astronomy and Mathematics, University of Hertfordshire, UK
Please replace the xxx.xxx.xxx in the header with herts.ac.uk to mail me

In article ,
Robert L. Oldershaw wrote:
This sample is heavily weighted toward "massive" stars, but it might
yield some interesting results in the 1 to 4 solar mass range for
total system masses.

An independent analysis of the data would be most welcome.

As these are non-contact binaries and the authors say they should have
evolved independently, I did the chi^2 test as described in the
earlier posting for the individual stars with errors less than 0.145
solar masses: chi^2 of 16085 for 172 degrees of freedom, null
hypothesis ruled out at such a large confidence level that I can't
calculate it offhand, but basically such that the model can't possibly
be right.

If I add up the two components and take only the systems where the
combined error on mass is less than 0.145 solar masses, I get a chi^2
of 1154 for 82 d.o.f., again wildly inconsistent with the model.

So, again, the data are not telling you what you would like them to
tell you. It took me about ten minutes to find the data you referred
to, get them into the right format, modify and run my code, and do the
modifications needed to run it again on the sums of the masses.
Testing models, when they make quantitative predictions, is easy, and
it's a skill that any would-be-modeller ought to learn. The half-hour
or so I've spent on this today is enough for me, though.

--
Martin Hardcastle
School of Physics, Astronomy and Mathematics, University of Hertfordshire, UK
Please replace the xxx.xxx.xxx in the header with herts.ac.uk to mail me